^°l' 
 
 IN MEMORIAM 
 FLOR1AN CAJOR1 
 
 

Digitized by the Internet Archive 
 
 in 2007 with funding from 
 
 Microsoft Corporation 
 
 http://www.archive.org/details/advancedarithmetOOwentrich 
 
AN 
 
 ADVANCED ARITHMETIC 
 
 HIGH SCHOOLS, NORMAL SCHOOLS 
 AND ACADEMIES 
 
 BY 
 
 G. A. WENTWORTH, A.M. 
 
 AUTHOR OF A SERIES OF TEXT-BOOKS IN MATHEMATICS 
 
 BOSTON, U.S.A. 
 
 GINN & COMPANY, PUBLISHERS 
 
 (ZDbe &tljeu£ttm JJteaa 
 
 1898 
 
Copyright, 1898, by 
 GEORGE A. WENTWORTH 
 
 AJAj rights reserved 
 

 PREFACE 
 
 Every high school, normal school, and academy should 
 allow sufficient time for a thorough review of Arithmetic. 
 This book has been written as a text-book for that purpose, 
 and for that purpose only. It is not intended for begin- 
 ners, but assumes that pupils have previously read a more 
 elementary Arithmetic. 
 
 The shortest and surest road to a knowledge of Arith- 
 metic is by solving problems. This work is abundantly 
 supplied with welbgraded, practical problems, many taken 
 from Wentworth and Hill's High School Arithmetic, but 
 many of them are new, and of a kind to meet the require- 
 ments of the present time. These problems are designed 
 to convey a great amount of useful information, as well as 
 to furnish the very best mental training, the primary object 
 of the study. They cover to a great extent the field of 
 mercantile transactions, and so far as practicable the field 
 of science. It is not necessary for any pupil, or any class 
 even, to do all the problems. Every teacher Can select 
 such chapters and such parts of chapters as are suited to 
 the needs of his pupils. 
 
 Decimals are introduced at the beginning of the book. 
 Numbers on each side of the decimal point perform pre- 
 cisely the same office. The only difference is that numbers 
 at the left of the decimal point count ivhole units, and 
 numbers at the right count equal parts of the unit. Pupils 
 learn the notation on both sides of the decimal point as 
 
 M30GO73 
 
IV PREFACE. 
 
 easily as on one side, provided they have a clear conception 
 of the units counted. Dimes and cents are good examples 
 of tenths and hundredths of a dollar ; but decimeters, centi- 
 meters, and millimeters, marked on a meter stick, are the 
 best examples of tenths, hundredths, and thousandths of a 
 unit, in general. The Metric System is taught naturally 
 in connection with decimals, and is easily learned. Only 
 the units employed furnish any difficulty. The great 
 number of problems given under the Metric System is 
 to familiarize the learner with the units of the system, 
 to show the simplicity of the system in its application to 
 everyday problems, and at the same time to give practice 
 in operations involving decimals. This system is used in 
 the laboratories of science, and in international transac- 
 tions. Though not yet adopted by the United States in 
 the common affairs of life, it has certainly forced its 
 way to a position requiring recognition in all secondary 
 schools of the country. 
 
 The introduction of • logarithms into the High School 
 Arithmetic was warmly welcomed by progressive teachers ; 
 and the chapter on that subject in this book has been 
 written with special reference to acquiring easily the 
 practical use of a four-place table. 
 
 Every effort has been made to avoid errors in problems 
 
 and answers. The author will be very grateful to any one 
 
 who will call his attention to any mistake that may be 
 
 discovered. 
 
 G. A. WENTWORTH. 
 
 Exeter, N. H., June, 1898. 
 
CONTENTS. 
 
 CHAPTER PAGE 
 
 I. Notation and Numeration ..... 1 
 
 II. Addition and Subtraction 12 
 
 III. Multiplication 24 
 
 IV. Division 35 
 
 V. Metric Measures 58 
 
 VI. Measures and Multiples of Numbers . . 94 
 
 VII. Common Fractions 109 
 
 VIII. Compound Quantities ...... 147 
 
 IX. Problems .179 
 
 X. Metric and Common Systems .... 210 
 
 XL Ratio and Proportion 217 
 
 XII. Percentage 237 
 
 XIII. Interest and Discount 259 
 
 XIV. Stocks and Bonds, Exchange, Accounts . 288 
 XV. Powers and Roots . 306 
 
 XVI. Mensuration 317 
 
 XVII. Continued Fractions and Scales of Notation . 332 
 
 XVIII. Series 338 
 
 XIX. Common Logarithms ....... 345 
 
 XX. Applications of Logarithms .... 359 
 
 XXI. Miscellaneous Problems 372 
 
VOCABULARY. 
 
 Abstract numbers. Numbers used without reference to any par- 
 ticular unit, as 8, 10, 21. All numbers are in themselves abstract 
 whether the kind of thing numbered is mentionedor is not meat 
 
 Addition. The process of combining two or more numbers. 
 
 Agent. A person who transacts business for another. 
 
 Aliquot parts of a given number. Numbers that are contained an 
 integral number of times in the given number. 
 
 Alligation. The process of finding the value of a mixture of quanti- 
 ties of different values; or of finding the proportion of quantities of 
 different values to be used to make a mixture of a given value. 
 
 Amount. The sum of two or more numbers. In interest, the sum 
 of the principal and interest. 
 
 Ampere. The unit of quantity of electricity. The current produced 
 by a force of one volt in a circuit of a resistance of one ohm. 
 
 Analysis. The process of reasoning from the given number to one, 
 and then from one to the required number. 
 
 Annual interest. Simple interest on the principal and on each year's 
 interest from the time each interest is due until settlement. 
 
 Annuity. A sum of money to be paid at regular intervals of time. 
 
 Antecedent. The first term of a ratio. 
 
 Antilogarithm. The number corresponding to a logarithm. 
 
 Area of a surface. The number of square units in the surface. 
 
 Arithmetic. The science that treats of numbers and the methods of 
 using them. 
 
 Arithmetical progression. A series of numbers that increase or 
 decrease by a common difference. 
 
 Assets. All the property of an estate, individual, or corporation. 
 
 Average of numbers. The number that can be put for each of the 
 numbers without altering their sum. 
 
 Average of payments. The average time at which several payments 
 due at different dates may be equitably made. 
 
VOCABULARY. Vll 
 
 Bank. An establishment for the custody, loaning, and exchange of 
 
 money ; and often for the issue of money. 
 Bank discount. An allowance received by a bank for the loan of 
 
 money, paid as interest at the time of lending. 
 Bills. Written statements of goods sold, or services rendered, giving 
 
 the price and date of each item, and the parties concerned. 
 Bonds. Written contracts under seal to pay specified sums of money 
 
 at specified times, issued by national governments, states, cities, 
 
 and other corporations. 
 Breakage. In customs, an allowance of a certain per cent on liquors 
 
 in bottles ; also on glassware and china. 
 Broker. A person who buys and sells stocks and bonds for another. 
 Brokerage. The commission charged by a broker. 
 Cancellation. The striking out of a common factor from the divi- 
 dend and divisor. 
 Centrifugal force. The force that tends to make a revolving body 
 
 move in a straight line. 
 Characteristic of a logarithm. The integral part of the logarithm. 
 Check. A draft upon a bank where the maker has money deposited. 
 Cologarithm of a number. The logarithm of the reciprocal of the 
 
 number. 
 Commercial discount. A reduction from the list price of an article, 
 
 from the amount of a bill, or from the amount of a debt. 
 Commission. Compensation for the transaction of business, reckoned 
 
 at some per cent of the money employed in the transaction. 
 Common denominator. A denominator common to two or more 
 
 fractions. 
 Common factor. A factor common to two or more numbers. 
 Common fractions. Fractions expressed by two numbers, one 
 
 under the other, with a line between them. 
 Common measure of two or more numbers. A number that 
 
 exactly divides each of them. 
 Common multiple of two or more numbers. A number exactly 
 
 divisible by each of them. 
 Complex fraction. A fraction that has a fraction in one or both of 
 
 its terms. 
 Composite number. The product of two or more integral factors, 
 
 each factor being greater than unity. 
 Compound fraction. A fraction of an integer or of a fraction. 
 Compound interest. Interest not paid when due, but added to the 
 
 principal at regular intervals. 
 
Vlll VOCABULARY. 
 
 Compound quantities. Quantities expressed in two or more de- 
 nominations. 
 Concrete numbers. Numbers applied to specified things. 
 Consequent. The second term of a ratio. 
 Consignee. The person or firm to whom goods are sent. 
 Consignor. The person or firm who sends goods to another. 
 Continued fraction. A fraction whose numerator is 1, and whose 
 
 denominator is an integer plus a fraction whose numerator 
 
 is 1, and whose denominator is an integer plus a fraction, and 
 
 so on. 
 Cooperative bank. A mutual corporation for the accumulation of 
 
 a capital to be loaned to its members. 
 Corporation. An association of individuals authorized by law to 
 
 transact business as a single person. 
 Couplet The two terms of a ratio taken together. 
 Coupon. A certificate of interest attached to a bond, to be cut off 
 
 when due and presented for payment. 
 Creditor. A person or firm to whom money is due. 
 Cube root of a number. One of its three equal factors. 
 Currency. The medium of exchange I mployed in business. 
 Customs. Duties or taxes imposed by law on merchandise imported, 
 
 and sometimes on merchandise exported. 
 Debtor. A person or firm who owes money to another. 
 Decimal point. A dot placed between the number that counts whole 
 
 units and the number that counts decimal units. 
 Decimals. Fractions of which only the numerators are written, and 
 
 the denominators are ten or some power of ten. 
 Decimal system. The common system of numbers founded on their 
 
 relations to ten. 
 Demand notes. Notes that are payable on demand. 
 Denominator. The number that shows into how many equal parts 
 
 a unit is divided. 
 Difference. The number found by subtraction. 
 Discount. An allowance made for the payment of money before it 
 
 becomes due. Also, the difference between the market value and 
 
 the face value when the market value is below the face value. 
 Dividend. In division, the number to be divided. In business, the 
 
 sum paid on each share of stock from the profits of the business. 
 Division. The operation of finding the other factor, when a product 
 
 and one of its factors are given. 
 Divisor. The number by which a given dividend is to be divided. 
 
VOCABULARY. IX 
 
 Draft. A written order directing one person or firm to pay a specified 
 
 sum of money to another. 
 Drawee of a draft. The person on whom the draft is drawn. 
 Drawer of a draft. The person who signs the draft. 
 Duties. Taxes required by the government to be paid on goods im- 
 ported, exported, or put on the market for consumption. 
 Endorser of a note. A person who writes his name on the back of 
 
 the note. The endorser is responsible for the payment of the note 
 
 unless he writes above his name the words without recourse. 
 Equation. A statement that two expressions of number are equal. 
 Equivalents. Equals in value, area, or volume. 
 Even numbers. Numbers exactly divisible by 2. 
 Evolution. The process of finding the root of a number. 
 Exact interest. Simple interest reckoning 365 days to the year. 
 Example. A question to be solved. 
 Exchange. A system of paying debts, due to persons living at a 
 
 distance, by transmitting drafts instead of money. 
 Exponent. A small figure placed at the right of a number to show 
 
 how many times the number is taken as a factor. 
 Extremes. The first and last terms of a proportion. 
 Face of a note or draft. The sum of money named in it. 
 Factors. A set of numbers whose product is the given number. 
 
 In commerce, agents employed by merchants to transact business. 
 Figures. Symbols used to represent numbers in the Arabic system. 
 
 Also diagrams used to represent geometrical forms. 
 Firm. The name under which a company transacts business. 
 Foot-pound. The unit of work. The work done in raising a weight 
 
 of one pound to a height of one foot. 
 Force. That which tends to produce motion. 
 Fractions. One or more of the equal parts of a unit. 
 Fulcrum. The point or line on which a lever turns. 
 Gain. The selling price minus the cost price. 
 Geometrical progression. A series of numbers, each term of 
 
 which after the first is obtained by multiplying the preceding term 
 
 by a constant multiplier. 
 Grace. An allowance of three days, after the date a note becomes 
 
 due, within which to pay the note. 
 Gram. The unit of weight in the metric system. 
 Gravitation. The force by which all bodies attract each other. 
 Greatest common measure of two or more numbers. The 
 
 greatest number that will exactly divide each of them. 
 
X VOCABULARY. 
 
 Harmonical progression. A series of numbers, the reciprocals of 
 which form an arithmetical progression. 
 
 Holder of a note. The person who has legal possession of it. 
 
 Horse power. The power to do 33,000 foot-pounds of work a minute. 
 
 Improper fraction. A fraction whose numerator equals or exceeds 
 its denominator. 
 
 Index. A figure written at the left and above the radical sign to 
 show what root of the number under the radical sign is required. 
 A fraction written at the right of a number, of which the numer- 
 ator shows the required power of that number, and the denomi- 
 nator the required root of that power. 
 
 Instalment. A payment in part- 
 Insurance. A guarantee by an insurance company of a specified sum 
 of money to the person insured in the event of loss of property by 
 fire, by storm at sea, or by other specified disaster ; or in the event 
 of the death of the person insured, or of accident to him. 
 
 Integral numbers. Numbers that denote whole units. 
 
 Interest. Money paid for the use of money. 
 
 Involution. The process of finding a power of a number. 
 
 Latitude of a place. The distance north or south from the equator, 
 expressed in degree measures. 
 
 Leakage. In customs, an allowance of a certain per cent on liquors 
 in barrels or casks. 
 
 Least common denominator of two or more fractions. The least 
 common multiple of their denominators. 
 
 Least common multiple of two or more numbers. The least 
 number that is exactly divisible by each of them. 
 
 Lever. A rigid bar that will move freely about its fulcrum. 
 
 Liability. A debt, or obligation to pay. 
 
 Like numbers. Numbers applied to the same unit. 
 
 Line. Length without breadth or thickness. 
 
 Liter. The unit of capacity in the metric system. 
 
 Logarithm of a number. The exponent of the power to which the 
 base must be raised to obtain the given number. 
 
 Long division. The method of dividing in which the processes are 
 written in full. 
 
 Longitude of a place. The distance east or west from a standard 
 meridian, expressed in degree measures. 
 
 Loss. The cost price minus the selling price. 
 
 Mantissa of a logarithm. The decimal part of the logarithm. 
 
 Maturity of a note. The date a note legally becomes due. 
 
VOCABULARY. XI 
 
 Mean proportional. A number that is both the second and third 
 
 terms of a proportion. 
 Means. The terms of a proportion between the extremes. 
 Measures of a number. The exact divisors of the numbers. 
 Meridians. Imaginary lines drawn straight around the earth through 
 
 both poles. 
 Meter. The unit of length in the metric system. 
 Metric system. A system of weights and measures expressed in 
 
 the decimal scale. 
 Minuend. The number from which the subtrahend is taken. 
 Mixed number. A whole number and a fraction. 
 Money of a country. The legal currency of the country. 
 Multiple of a number. The product obtained by taking the given 
 
 number an integral number of times. 
 Multiplicand. The number to be multiplied by another. 
 Multiplication. The operation of taking a number of units a num- 
 ber of times. 
 Multiplier. The number by which the multiplicand is multiplied. 
 Net proceeds. The money that remains of the money received for 
 
 property after all expenses and discounts are paid. 
 Notation. A system of expressing numbers by symbols. 
 Note. A written agreement to pay, for value received, a specified 
 
 sum of money on demand or at a specified time. 
 Numbers. Expressions applied to a unit to show how many times 
 
 the unit is taken. 
 Numeration. A system of naming numbers. 
 Numerator. The number that shows how many are taken of the 
 
 equal parts into which a unit is divided. 
 Obligation. A debt, or liability to pay. 
 Odd numbers. Numbers not exactly divisible by 2. 
 Ohm. The unit of resistance to electricity. The resistance of a 
 
 column of mercury, l<imm i n cross section and 106 cm long at 0° C. 
 Order of units. The number of things in a group, as tens, hundreds. 
 Partial payment. Part payment on a note. 
 Partnership. An association of persons to carry on business. 
 Par value. Face or nominal value. 
 Pendulum. A body suspended by a straight line from a fixed point, 
 
 and moving freely about that point as a centre. 
 Percentage of a number. One or more hundredths of the number. 
 Perimeter. The length of the boundary of a plane figure. 
 Period. A group of three figures. 
 
Xll VOCABULARY. 
 
 Policy. A written contract of insurance. 
 
 Poll tax. A tax levied by the head or poll. 
 
 Power. The product of two or more equal factors. 
 
 Premium. Money paid for insurance. Also the excess of market 
 value above par value. 
 
 Present worth. The present value of a debt due at a future time. 
 
 Prime number. A number that cannot be exactly divided by any 
 number except itself and one. 
 
 Principal. Money drawing interest. 
 
 Problem. A question to be solved. 
 
 Proceeds or avails of a note or draft The amount of the note or 
 draft less the discount and exchange. 
 
 Product. The Dumber found by multiplication. 
 
 Proof. The evidence that establishes the accuracy of any result. 
 
 Proper fraction. A fraction whose numerator is less than its de- 
 nominator. 
 
 Proportion. A statement that two ratios are equal. 
 
 Protest. A notice in writing by a notary public to the endorsers of 
 a note that it has not been paid. 
 
 Pulley. A grooved wheel that turns freely within a block, fixed or 
 movable, by means of a rope that passes over the groove. 
 
 Quotient. The number found by division. 
 
 Rate per cent. Rate by the hundred. 
 
 Ratio. The relative magnitude of two numbers or of two quantities, 
 when expressed by the quotient of the first divided by the second. 
 
 Receipt. A written acknowledgment of money or goods received. 
 
 Reciprocal of a number. One divided by thit number. 
 
 Reduction. The process of changing the unit in which a quantity is 
 expressed without changing the value of the quantity. 
 
 Remainder. The number found by subtraction. 
 
 Repeating or circulating decimals. Decimals that contain a con- 
 stantly recurring figure or series of figures. 
 
 Repetend. The figure or series of figures that constantly recurs in a 
 repeating decimal. 
 
 Root of a number. One of the equal factors of the number. 
 
 Rule. The statement of a prescribed method. 
 
 Savings bank. A bank to receive deposits and pay compound inter- 
 est to the depositors. 
 
 Screw. A cylinder that has on its surface a uniform projection in 
 the form of a spiral curve, called the thread. 
 
 Security. Property used to guarantee the payment of any debt. 
 
VOCABULARY. Xlll 
 
 Share. One of a certain number of equal parts into which the capi- 
 tal of a company is divided. 
 Short division. The method of dividing in which the operations of 
 
 multiplying and subtracting are performed mentally. 
 Similar fractions. Fractions that have a common denominator. 
 Simple fractions. Fractions whose terms are integral numbers. 
 Simple quantities. Quantities expressed in a single denomination. 
 Sinking fund. The final value of sums of money put at interest at 
 
 regular intervals of time, to pay a debt due at a stated time. 
 Solid. A magnitude that has length, breadth, and thickness. 
 Solution. The process by which the answer to a question is obtained. 
 Specific gravity of a substance. The ratio of the weight of a 
 
 given volume of it to the weight of an equal volume of water. 
 Square root of a number. One of its two equal factors. 
 Stock. Capital invested in business. 
 Stock company. An association of persons under the laws of the 
 
 state for the purpose of carrying on a specified business. 
 Subtraction. The process of taking one number from another. 
 Subtrahend. The number that is taken from the minuend. 
 Sum. The number found by addition. 
 Surd. An indicated root, that cannot be exactly found. 
 Surface. A magnitude that has length and breadth. 
 Tare. In customs, an allowance for the weight of the box, cask, bag, 
 
 or other wrapping. 
 Taxes. Money required of persons and corporations for the support 
 
 of the government and for other public purposes. 
 Thermometer. An instrument for measuring temperature. 
 Time notes. Notes that are payable at a specified time. 
 Units. The standards by which we count separate objects or 
 
 measure magnitudes. 
 Velocity. Rate of motion, measured by the number of units of 
 
 space passed over in a unit of time. 
 Verify. To establish by trial the truth of any statement. 
 Volt. The unit of force of electricity. The force required to send one 
 
 ampere of electricity through a circuit of a resistance of one ohm. 
 Volume of a solid. The number of cubic units in the solid. 
 Wheel and axle. A simple machine, consisting of a wheel firmly 
 
 attached to an axle. 
 Work. The act of changing the position of a body by overcoming 
 
 resistance to the change. 
 Yard* The unit of length in the common system, 
 
XIV SHORT PROCESSES. 
 
 Short Processes. 
 
 Note. These processes should be learned as fast as they can be utilized 
 in the ordinary work of Arithmetic. The time to learn each process is left 
 to the discretion of the teacher. 
 
 1. To multiply by 25 (± of 100), 
 
 Multiply by 100 and divide the product by %. 
 
 2. To divide by 25, 
 
 Multiply by 4 and divide the product by 100. 
 
 3. To multiply by 2.5 or 2£ (\ of 10), 
 Multiply by 10 and divide the product by 4,. 
 
 4. To divide by 2.5 or 2£, 
 
 Multiply by 4 and divide the product by 10. 
 
 5. To multiply by 50 (£ of 100), 
 
 Multiply by 100 and divide the product by 2. 
 
 6. To divide by 50, 
 
 Multiply by 2 and divide the product by 100. 
 
 7. To multiply by 75 (f of 100), 
 
 Multiply by 100 and subtract from the product \ of it. 
 
 8. To divide by 75, 
 
 Divide by 100 and add to the quotient £ of it. 
 
 9. To multiply by 33£ (£ of 100), 
 Multiply by 100 and divide the product by S. 
 
 10. To divide by 33£, 
 
 Multiply by 3 and divide the product by 100. 
 
 11. To multiply by 3£ (£ of 10), 
 
 Multiply by 10 and divide the product by S. 
 
 12. To divide by 3£, 
 
 Multiply by 8 and divide the product by 10. 
 
 13. To multiply by 833$ (£ of 1000), 
 Multiply hy 1000 and divide the product by 8. 
 
SHORT PROCESSES. XV 
 
 14. To divide by 333£, 
 
 Multiply by 3 and divide the product by 1000. 
 
 15. To multiply by 16f Q of 100), 
 Multiply by 100 and divide the product by 6. 
 
 16. To divide by 16f, 
 
 Multiply by 6 and divide the product by 100. 
 
 17. To multiply by 166f (£ of 1000), 
 Multiply by 1000 and divide the product by 6. 
 
 18. To divide by 166§, 
 
 Multiply by 6 and divide the product by 1000. 
 
 19. To multiply by 66f (f of 100), 
 
 Multiply by 100 and subtract from the product £ of it. 
 
 20. To divide by 66f, 
 
 Divide by 100 and add to the quotient £ of it. 
 
 21. To multiply by 12£ (£ of 100), 
 Multiply by 100 and divide the product by 8. 
 
 22. To divide by 12£, 
 
 Multiply by 8 and divide the product by 100. 
 
 23. To multiply by 14f (i of 100), 
 Multiply by 100 and divide the product by 7. 
 
 24. To divide by 14f, 
 
 Multiply by 7 and divide the product by 100. 
 
 25. To multiply by a number that is a little less than some 
 
 multiple of 10, as 100, 1000, etc., 
 
 Multiply the multiplicand by the multiple of 10 that differs 
 little from the given multiplier. Then multiply the multipli- 
 cand by the difference between this multiple of 10 and the 
 given multiplier, and find the difference of the two products. 
 
 Thus, to multiply by 998 (1000 — 2), multiply by 1000 and then by 2, and 
 take the difference of the products. 
 
NOTICE TO TEACHERS. 
 
 Pamphlets containing the answers will be furnished 
 without charge to teachers for their classes, on appli- 
 cation to GrINN £ COMPANY', Publishers. 
 
ADVANCED ARITHMETIC. 
 
 CHAPTER I. 
 
 NOTATION AND NUMERATION. 
 
 1. Units. The standards by which we count or measure 
 are called units. 
 
 The unit may be a single thing or a definite group of things. Thus, 
 in counting the eggs in a nest the unit is an egg ; in selling eggs by 
 the dozen the unit is a dozen eggs; in selling bricks by the thousand 
 the unit is a thousand bricks; in measuring short distances the unit is 
 an inch, aioot, or a yard; in measuring long distances the unit is a 
 rod or a mile. 
 
 2. Numbers. Expressions applied to a unit to show 
 how many times the unit is taken are called numbers. 
 
 Thus, if we put an apple into an empty fruit dish, then another, 
 and then another, we shall have three apples in the dish. Here an 
 apple is the unit, and three is the number of times the unit is taken. 
 
 3. Integral Numbers. Numbers applied to whole units 
 are called whole numbers, integral numbers, or integers. 
 
 4. Figures. The following symbols, called figures, or 
 digits, are used to represent the numbers of Arithmetic : 
 
 012 3 4567 8 9 
 
 Zero One Two Three Four Five Six Seven Eight Nine 
 
 The first figure, 0, is called zero, naught, or cipher, and 
 stands for the words no number. Each of the other figures 
 stands for the number whose name is written below it. 
 
2 NOTATION AND NUMERATION. 
 
 5. Tens. The next number, ten, is expressed by writing 
 at the right of 1. Thus, ten is written 10. In this posi- 
 tion 1 signifies not one, but one group of ten ones. 
 
 Figures signifying tens are written in the second place 
 from the right. 
 
 In the same way twenty (2 tens) is expressed by 20 ; 
 thirty (3 tens) by 30 ; forty (4 tens) by 40 ; fifty (5 tens) 
 by 50 ; sixty (6 tens) by 60 ; seventy (7 tens) by 70 ; 
 eighty (8 tens) by 80 ; ninety (9 tens) by 90. 
 
 6. Tens and Ones. A number containing tens and ones 
 is expressed by writing the figure for the tens in the 
 second place from the right, and the figure for the ones in 
 the first place. 
 
 Eleven, 
 
 one ten and one, 
 
 is expressed by 11. 
 
 Twelve, 
 
 one ten and two, 
 
 is expressed by 12. 
 
 Thirteen, 
 
 one ten and three, 
 
 is expressed by 13. 
 
 Fourteen, 
 
 one ten and four, 
 
 is expressed by 14. 
 
 Fifteen, 
 
 one ten and five, 
 
 is expressed by 15. 
 
 Sixteen, 
 
 one ten and six, 
 
 is expressed by 16. 
 
 Seventeen, 
 
 one ten and seven, 
 
 is expressed by 17. 
 
 Eighteen, 
 
 one ten and eight, 
 
 is expressed by 18. 
 
 Nineteen, 
 
 one ten and nine, 
 
 is expressed by 19. 
 
 Twenty-one, 
 
 two tens and one, 
 
 is expressed by 21. 
 
 Twenty-two, 
 
 two tens and two, 
 
 is expressed by 22. 
 
 Forty-three, 
 
 four tens and three, 
 
 is expressed by 43. 
 
 Fifty-four, 
 
 five tens and four, 
 
 is expressed by 54. 
 
 Sixty-five, 
 
 six tens and five, 
 
 is expressed by 65. 
 
 And so on to ninety-nine (nine tens and nine) which is 
 by 99. 
 
 7. Hundreds. A group of 10 tens is called a hundred, 
 and figures signifying hundreds are written in the third 
 place from the right. 
 
 Thus, one hundred, two hundreds, three hundreds, etc., are ex- 
 pressed by 100, 200, 300, etc. 
 
NOTATION AND NUMERATION. 3 
 
 8. To Write Hundreds, Tens, and Ones. We write first 
 the hundreds, then the tens and ones. 
 
 Thus, two hundred seventy-six is written 276. 
 
 Express in figures the following numbers : 
 Seven hundred sixty-five. Nine hundred. 
 
 One hundred twenty-three. Five hundred eighty-one. 
 Six hundred ninety-four. Four hundred thirty. 
 
 Nine hundred forty-six. Seven hundred nine. 
 
 Two hundred twenty-nine. Seven hundred ninety. 
 
 One hundred ten. Seven hundred ninety-nine. 
 
 9. To Read Hundreds, Tens, and Ones. We read first 
 the hundreds, then the tens and ones. 
 
 Thus, the number 359 has 3 hundreds, 5 tens, and 9 ones, and is 
 read three hundred fifty-nine; the number 807 has 8 hundreds, no 
 tens, and 7 ones, and is read eight hundred seven. 
 
 Note. In reading 359, 807, or any other integral number, do not 
 introduce the word and ; that is, do not say three hundred and fifty- 
 nine, eight hundred and seven ; but simply three hundred fifty-nine, 
 eight hundred seven. 
 
 Read the following numbers, and state the number of 
 hundreds, tens, and ones in each : 
 
 507 469 101 260 206 301 808 888 
 321 694 929 300 185 340 671 999 
 
 10. Notation and Numeration. The method of writing 
 numbers is called notation, and the method of reading 
 numbers is called numeration. 
 
 11. The system of notation here explained is called the 
 Arabic system of notation. 
 
 12. Thousands. A group of 10 hundreds is called a 
 thousand, and figures signifying thousands are written in 
 the fourth place from the right. 
 
 Thus, one thousand, two thousands, three thousands, etc., are ex- 
 pressed by 1000, 2000, 3000, etc. 
 
4 NOTATION AND NUMERATION. 
 
 13. Numbers expressed by Four Figures. Numbers 
 expressed by four figures may be read as thousands, 
 hundreds, tens, and ones ; or as hundreds, tens, and ones. 
 
 The shortest way of reading numbers is the best way. 
 The best way to read 1896 is eighteen hundred ninety-six. 
 The best way to read 7005 is seven thousand five. 
 
 Read in the best way the following numbers : 
 
 1776 1924 1907 2359 5050 3627 
 7006 7076 2706 6010 5500 2036 
 
 14. Orders of Units. The ones of a number are called 
 units of the first order. The tens of a number are called 
 units of the second order. The hundreds of a number are 
 called units of the third order. The thousands of a number 
 are called units of the fourth order. 
 
 Figures in the fifth place signify ten-thousands, called 
 units of the fifth order. Figures in the sixth place signify 
 hundred-thousands, called unit* of the sixth order. 
 
 Note. The ones of a number are commonly called units, the 
 word units standing for the phrase units of the first order. Thus, 
 we say the number 459 has 4 hundreds, 5 tens, and 9 units. 
 
 15. Decimal System. Since ten units of any order are 
 equal to one unit of the next higher order, this system 
 of notation is called the decimal system; decimal being 
 derived from the Latin word decern, meaning ten. 
 
 16. Periods. When the figures of a number are five or 
 more, we separate them into groups of three figures each by 
 commas, beginning at the right. The right-hand group is 
 called the period of units ; the second group is called the 
 period of thousands ; the third group is called the period of 
 millions ; the fourth group is called the period of billions. 
 
 The unit of any period is equal to 1000 units of the next 
 lower period. 
 
NOTATION AND NUMERATION. 
 
 One million is equal to 1000 thousands, and is written 
 1,000,000; one billion is equal to 1000 millions, and is 
 written 1,000,000,000. 
 
 The left-hand period may have one, two, or three figures ; 
 every other period must have three figures. 
 
 17. To Read an Integral Number expressed in Figures. 
 
 Read the number 26217320416. 
 
 We begin at the right and point off the figures into periods of 
 three figures each. Thus, 
 
 26,217,320,416. 
 
 We begin at the left and read each period as if it stood alone, add- 
 ing the name of the period. The fourth period from the right is the 
 period of billions. Hence we read the number, 
 
 Twenty-six billion, two hundred seventeen million, three hundred 
 twenty thousand, four hundred sixteen. Therefore, 
 
 Beginning at the right, we separate the figures by commas 
 into periods of three figures each. We begin at the left and 
 read each period as if it stood alone, adding the name of each 
 period except the name of the period of units. 
 
 Note. The names of periods above billions are in order: trillions, 
 quadrillions, quintillions, sextillions, septillions, octillions, etc. 
 
 Exercise 1. 
 
 Write in periods, and read : 
 
 1. 7000. 
 
 2. 7842. 
 
 3. 5043. 
 
 4. 8375. 
 
 5. 2020. 
 
 6. 1753. 
 
 7. 18757. 
 
 8. 75764. 
 
 9. 22003. 
 10. 70856. 
 
 11. 234567. 
 
 12. 34561. 
 
 13. 123456. 
 
 14. 654089. 
 
 15. 600897. 
 
 16. 704608. 
 
 17. 350709. 
 
 18. 240682. 
 
 19. 682000. 
 
 20. 753110. 
 
 21. 703101. 
 
 22. 870890. 
 
 23. 21978564. 
 
 24. 17756423. 
 
 25. 300200100. 
 
 26. 707303202. 
 
 27. 3125476890. 
 
 28. 79501346081. 
 
 29. 3000872696. 
 
 30. 72727000000. 
 
6 NOTATION AND NUMERATION. 
 
 18. To Write an Integral Number in Figures. Write 
 in figures the number two hundred sixty-three million, six 
 hundred thirty-five thousand, two hundred one. 
 
 We consider first the periods of the number. 
 
 This number has the period of millions, the period of thousands, 
 and the period of units. 
 
 We write first the period of millions, and put a comma after it ; 
 then the period of thousands, and put a comma after it ; and then the 
 period of units. Thus, 263,635,201. 
 
 Note. Every period except the one at the left must have three 
 figures (§ 16). If any order of units of a period is lacking, we put a 
 cipher in its place ; and if any entire period is lacking we put three 
 ciphers in its place. Thus, four million, sixteen thousand, four is 
 written 4,016,004 ; sixteen million, four hundred sixteen is written 
 16,000,410. Therefore, 
 
 We begin at the left and ivrite the hundreds, tens, and 
 units of each period, putting zeros in all vacant places, and 
 putting a comma between each period and the period that 
 follows it. 
 
 Exercise 2. 
 
 Write in figures, arranged in periods: 
 
 1. Six hundred thousand, six. 
 
 2. Seven hundred thirteen thousand, three hundred 
 twenty-nine. 
 
 3. Seven thousand, eight hundred fifty-four. 
 
 4. Four million, three thousand, three hundred thirty. 
 
 5. One hundred ten million, two hundred seventy-nine. 
 
 6. Nineteen trillion, four million, three hundred nine. 
 
 7. Seven million, six hundred seventy-six thousand, 
 four hundred sixty-six. 
 
 8. Three hundred forty-seven million, six hundred fifty- 
 one thousand, seven hundred eighty-five. 
 
 9. Two hundred million, two hundred seven. 
 
 10. Four hundred billion, four hundred thousand, four. 
 
NOTATION AND NUMERATION. 7 
 
 19. The unit of measure of any kind of quantity may 
 be divided into ten smaller measures. 
 
 Thus, a dollar, as a measure of value, is divided into ten dimes, 
 or ten tenths of a dollar ; each dime into ten cents, or ten hundredths 
 of a dollar ; each cent into ten mills, or ten thousandths of a dollar. 
 
 20. The dollar sign, $, is written before the number. 
 
 21. If dollars and cents are written, a dot called the 
 decimal point is placed between the dollars and cents. 
 
 Thus, $17 is 17 dollars ; $18.20 is 18 dollars 2 dimes, or 18 dollars 
 20 cents ; $35,875 is 35 dollars 87 cents 5 mills ; and $0.08 is 8 cents. 
 
 22. Read as dollars, cents, and mills : 176.375 ; $1 63.58 ; 
 $12.50; $0,875; $1.01; $2.10; $3.08; $0.75; $0,125. 
 
 23. In reading United States money, we read the 
 number to the left of the decimal point as dollars ; the 
 number in the first two places to the right of the point as 
 cents ; and the number in the third place as mills. 
 
 24. In writing United States money, we must have the 
 cents occupy two places. J^f the number of cents is less 
 than ten, we write a cipher in the first place at the right 
 of the decimal point. 
 
 25. Parts of other measures may be expressed as tenths, 
 hundredths, etc.; just as dimes, cents, and mills are 
 respectively tenths, hundredths, and thousandths of a 
 dollar. 
 
 Thus, if we omit the dollar sign from $5,375, the expression may 
 stand for 5 yards, quarts, bushels, or any other full measures, and 
 375 thousandths of another measure. 
 
 26. Parts thus written are called Decimals. We write 
 a number to the right of the units' place just as we do to 
 the left, first marking the units' place with a decimal point 
 to its right. 
 
NOTATION AND NUMERATION. 
 
 27. In the number 
 
 9,876,543,210.123,456,789 
 
 the full point after shows that stands in the units' 
 place. The 1 to the left is 1 ten, the 1 to the right is 1 
 tenth; the 2 to the left is 2 hundreds, the 2 to the right is 
 2 hundredtJis ; the 3 to the left is 3 thousands, the 3 to the 
 right is 3 thousandths ; the 4 to the left is 4 ten-thousands, 
 the 4 to the right is 4 ten-thousandtlis ; the 5 to the left is 
 5 hundred-thousands, the 5 to the right is 5 hundred-thou- 
 sandths ; the 6 to the left is 6 millions, the 6 to the right 
 is 6 millionth* ; and so on. 
 
 Also, the 210 is the units* period ; the 543 is the thou- 
 sands', the 123 the thousandths' period, etc.; so that the 
 number may be read 9 billions, 876 millions, 543 thousands, 
 210, and 123 thousandths, 456 millionths, 789 billionths. 
 
 28. To Read a Decimal expressed in Figures. The 
 usual way of reading a decimal is : 
 
 Read the decimal precisely as if a whole number, and 
 add the fractional name of the lowest place. 
 
 Thus, 5.17 is read 5 and 17 hundredths ; 5.0017, five and 17 ten- 
 thousandths ; 6.0203107, six and 203 thousand 107 ten-millionths. 
 
 The word and is distinctly pronounced at the decimal 
 point and carefully omitted in all other places. 
 
 Thus, one hundred forty-seven means 147 ; but one hundred and 
 forty-seven thousandths means 100.047 ; and 0.147 must be read one 
 hundred forty-seven thousandths. 
 
 29. Practical computers often introduce the word 
 decimal at the place of the point, and then pronounce the 
 name of each digit in succession to the right. 
 
 Thus, 203.07051 may be read two hundred three, decimal, naught, 
 seven, naught, five, one. 
 
NOTATION AND NUMERATION. 
 
 
 Exercise 
 
 3. 
 
 Read 
 
 
 
 
 1. 
 
 6,728,642. 
 
 16. 
 
 $182,275. 
 
 2. 
 
 3.24658. 
 
 17. 
 
 $0,086. 
 
 3. 
 
 49,568.4782. 
 
 18. 
 
 $0,075. 
 
 4. 
 
 34,598,492,212. 
 
 19. 
 
 $463.87. 
 
 5. 
 
 4,002,000.02. 
 
 20. 
 
 $20,542.02. 
 
 6. 
 
 1872.17. 
 
 21. 
 
 $0.75. 
 
 7. 
 
 94.658265. 
 
 22. 
 
 428,428.428. 
 
 8. 
 
 0.0307. 
 
 23. 
 
 1542.087. 
 
 9. 
 
 100.01. 
 
 24. 
 
 642.873654. 
 
 10. 
 
 1,872,563.372. 
 
 25. 
 
 400.00004. 
 
 11. 
 
 17.008. 
 
 26. 
 
 3,543,362,338. 
 
 12. 
 
 143.00143. 
 
 27. 
 
 0.0000009. 
 
 13. 
 
 29.00081. 
 
 28. 
 
 52.02. 
 
 14. 
 
 5,262,873. 
 
 29. 
 
 56,482.56. 
 
 15. 
 
 8.7854. 
 
 30. 
 
 87,865,842.87866. 
 
 Exercise 4. 
 Write in figures : ' 
 
 1. Eighty -one thousand and three hundred forty-five 
 thousandths. 
 
 2. Thirty-seven hundred forty-one and six hundred 
 seventy-five thousandths. 
 
 3. Four hundred thirteen and eight hundredths. 
 
 4. Ninety-six and ninety-six thousandths. 
 
 5. Nine and forty-three millionths. 
 
 6. One hundred fifty-four and thirty-two ten-thou- 
 sandths. 
 
 7. Seventy-five thousandths. 
 
 8. Three tenths. 
 
 9. Forty-four million, forty-four thousand, forty-four, 
 and forty-four thousandths. 
 
10 NOTATION AND NUMERATION. 
 
 10. One hundred and forty-three millionths. 
 
 11. One hundred forty-three millionths. 
 
 12. One hundred forty and three millionths. 
 
 13. Nine hundred forty-three thousand and nine hun- 
 dred forty-three thousandths. 
 
 14. Seven hundred twenty-two ten-millionths. 
 
 15. Thirteen, decimal, naught, one, four, six, eight. 
 
 16. Four and one thousand nine ten-thousandths. 
 
 17. One hundred one and one hundred one ten-thou- 
 sandths. 
 
 18. Seventeen million, six hundred forty-nine thousand. 
 
 19. Twelve billion, twelve thousand. 
 
 20. Twelve billion and twelve thousandths. 
 
 21. Eight dollars and twelve cents. 
 
 22. One hundred twenty-seven dollars and one cent. 
 
 23. Fourteen thousand, two hundred seventy-eight dol- 
 lars and twenty-seven cents, five mills. 
 
 24. One thousand dollars and one cent, one mill. 
 
 25. Two hundred thirty-four dollars and fifty-five cents. 
 
 26. Twenty-five cents ; three cents, four mills. 
 
 27. One million, four hundred eighty-nine thousand, 
 five hundred ninety and five hundred ninety thousandths. 
 
 28. Forty-three thousand, six hundred seventy-seven 
 and four thousand six hundred-thousandths. 
 
 29. Three thousand sixty-nine and seventy-eight thou- 
 sand four hundred sixteen ten-millionths. 
 
 30. The Roman System of Notation. The Roman 
 system employs seven letters, as follows : 
 
 Letters, I V X L C D M 
 Values, 15 10 50 100 500 1000 
 
 The first nine numbers are expressed by 
 
 I II III IV V VI VII VIII IX 
 123456 7 8 9 
 
NOTATION AND NUMERATION. 11 
 
 The tens are expressed by 
 
 X XX XXX XL L LX LXX LXXX XC 
 
 10 20 30 40 50 60 70 80 90 
 
 Tens and ones are expressed by writing the expressions 
 for units at the right of the expressions for tens. Thus, 
 
 XI XII XIV XV XIX XXII LVI XCIX 
 
 11 12 14 15 19 • 22 56 99 
 
 The hundreds are expressed by 
 
 C CC CCC CD D DC DCC DCCC DCCCC 
 100 200 300 400 500 600 700 800 900 
 
 Writing M at the left of each of these expressions we 
 have 
 
 MC MCC MCCC MCD MD MDC MDCC MDCCC MDCCCC 
 
 1100 1200 1300 1400 1500 1600 1700 1800 1900 
 
 Hundreds, tens, and ones are expressed by writing the 
 hundreds, then the tens, and then the ones. 
 
 Thus, eighteen hundred ninety-six is written: MDCCC for eighteen 
 hundred, then XC for ninety, and VI for six, making MDCCCXCVI. 
 
 Exercise 5. 
 Read : 
 
 XXXVI; XL; XLVI ; LVIII ; LIX ; LXXXI; XCI ; 
 XCIII ; CIX ; CCIX ; CCXX ; CLIX ; MDCCCLXXXVI ; 
 MDCLXVI ; MDCCLXXVI ; MCDLIX ; MDLXXXIX. 
 
 Express in the Roman system : 
 
 43; 55; 81; 77; 99; 113; 128; 514; 724 ; 630 ; 1020 ; 
 1040 ; 1088 ; 1131 ; 1218 ; 1492 ; 177(5 ; 1899 ; 1319 ; 1556 ; 
 1897; 1620; 1783; 1812; 1861 ; 1872. 
 
CHAPTER II. 
 
 ADDITION AND SUBTRACTION. 
 
 31. The sign + is called plies, and means that the 
 number after it is to be counted with the number before it ; 
 that is, added to the number before it. 
 
 32. The sign = is called the sign of equality, and stands 
 for the word equals ; so that 5 + 4 = 9 is read : 5 plus 4 
 equals 9. 
 
 33. Addition. The operation of finding a number equal 
 to two or more numbers taken together is called addition .- 
 and the result is called their sum. 
 
 34. The sum of two or more numbers is the same in 
 whatever order the numbers are added. 
 
 Thus, 3 + 2 + 5 = 10, or 5 + 3 + 2 = 10. 
 
 35. Abstract Numbers. Numbers not applied to any 
 particular unit, as 7, 17, 25, are called abstract numbers. 
 
 36. Concrete Numbers. Numbers applied to a particular 
 unit, as 7 horses, 17 apples, are called concrete numbers. 
 
 37. Like Numbers. Numbers applied to the same unit. 
 expressed or understood, are called like numbers. 
 
 38. Only like numbers, and units of the same order, can 
 be added. 
 
 Exercise 6. 
 
 Count to 100 or more : 
 
 1. By 2's, beginning 0, 2, 4 ; beginning 1, 3, 5. 
 
 2. By 3's, beginning 0, 3, 6 ; beginning 1, 4, 7 ; beginning 
 2, 5, 8. 
 
ADDITION AND SUBTRACTION. 
 
 13 
 
 3. By 4's, beginning 0, 4, 8 ; beginning 1, 5, 9 ; begin- 
 ning 2, 6, 10 ; beginning 3, 7, 11. 
 
 4. By 5's, beginning 0, 5, 10 ; beginning 1, 6, 11 ; begin- 
 ning 2, 7, 12 ; beginning 3, 8, 13 ; beginning 4, 9, 14. 
 
 5. By 6's, beginning 0, 6, 12 ; beginning 1, 7, 13 ; begin- 
 ning 2, 8, 14 ; beginning 3, 9, 15 ; beginning 4, 10, 16 ; 
 beginning 5, 11, 17. 
 
 6. By 7's, beginning 0, 7, 14 ; beginning 1, 8, 15 ; begin- 
 ning 2, 9, 16 ; beginning 3, 10, 17 ; beginning 4, 11, 18 ; 
 beginning 5, 12, 19; beginning 6, 13, 20. 
 
 7. By 8's, beginning 0, 8, 16 ; beginning 1, 9, 17 ; begin- 
 ning 2, 10, 18; beginning 3, 11, 19; beginning 4, 12, 20; 
 beginning 5, 13, 21 ; beginning 6, 14, 22 ; beginning 7, 
 15,23. 
 
 8. By 9's, beginning 0, 9, 18 ; beginning 1, 10, 19 ; begin- 
 ning 2, 11, 20 ; beginning 3, 12, 21 ; beginning 4, 13, 22 ; 
 beginning 5, 14, 23 ; beginning 6, 15, 24 ; beginning 7, 16, 
 25 ; beginning 8, 17, 26. 
 
 Find the sum of 
 
 
 
 
 
 
 
 
 
 9. 10. 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 3 2 
 
 3 
 
 5 
 
 3 
 
 2 
 
 5 
 
 5 
 
 4 
 
 5 
 
 3 
 
 1 
 
 5 1 
 
 6 
 
 6 
 
 3 
 
 7 
 
 3 
 
 6 
 
 8 
 
 5 
 
 . 6 
 
 8 
 
 7 9 
 
 7 
 
 7 
 
 4 
 
 7 
 
 2 
 
 4 
 
 7 
 
 3 
 
 7 
 
 8 
 
 6 8 
 
 8 
 
 8 
 
 5 
 
 3 
 
 1 
 
 7 
 
 3 
 
 6 
 
 3 
 
 7 
 
 21. 22. 
 
 23. 
 
 24. 
 
 25. 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 
 30. 
 
 81. 
 
 32. 
 
 6 9 
 
 6 
 
 4 
 
 4 
 
 3 
 
 6 
 
 7 
 
 5 
 
 8 
 
 2 
 
 9 
 
 8 5 
 
 4 
 
 5 
 
 4 
 
 7 
 
 2 
 
 5 
 
 5 
 
 2 
 
 9 
 
 6 
 
 7 4 
 
 3 
 
 6 
 
 3 
 
 5 
 
 1 
 
 8 
 
 9 
 
 2 
 
 9 
 
 5 
 
 9 3 
 
 7 
 
 7 
 
 7 
 
 5 
 
 8 
 
 3 
 
 3 
 
 7 
 
 4 
 
 4 
 
 Note. In adding, begin at the bottom, and give results only. 
 Thus, in example 9, say 13 18, 21; do not say 6 and 7 are 13, 13 and 
 5 are 18, and 18 and 3 are 21. 
 
14 ADDITION AND SUBTRACTION. 
 
 39. Examples. 1. Find the sum of 45.32, 27.21, and 
 68.55. 
 
 Solution. Since only units of the same order can be added, we 
 write units under units, tens under tens, tenths under tenths, and 
 hundredths under hundredths, so that the decimal points stand in a 
 vertical line ; and draw a line beneath. 
 
 We first find the sum of the hundredths and write this sum, 8, 
 
 under the column of hundredths. We next find the sum of 
 
 45.32 the tenths and write the units' figure, 0, of this sum under 
 
 27.21 the column of tenths, but the tens' figure, 1, of this sum we 
 
 68.55 add with the figures of the next column, making 1 + 8 + 7 + 
 
 141.08 5 = 21. We write the units' figure, 1, of this sum under 
 
 the column of units, but the tens' figure, 2, we add with the 
 
 figures of the next column, making 2+6 + 2 + 4 = 14. We write 
 
 the entire sum of the last column on the left. We put the decimal 
 
 point in the sum directly under the column of decimal points. 
 
 2. Add 41.38, 37.89, 26.67, and 58.21. 
 
 41.38 In adding, the following wording, and no more, should be 
 
 37.89 used : one, eight, seventeen, twenty-five (emphasize five, and 
 
 26.67 write it while pronouncing it), carry two ; four, ten, eigh- 
 
 58.21 teen, twenty-one, carry two ; ten, sixteen, twenty -three, 
 
 164.15 twenty-four, carry two ; seven, nine, twelve, sixteen. 
 
 40. Hence, we have the following 
 
 Rule for Addition. Write the numbers so that units 
 of the same order shall stand in the same column. 
 
 Add the right-hand column; write the units of this sum 
 beneath, and add the tens, if any, to the next column. 
 
 So proceed with each column. Write the entire sum of the 
 last column. 
 
 If any of the numbers contain a decimal, write the decimal 
 point in the sum directly under the column of decimal points 
 in the numbers. 
 
 Proof. Add each column in reverse order. If the results 
 agree, the work may be assumed to be correct. 
 
ADDITION AND SUBTRACTION. 
 
 15 
 
 Exercise 7. 
 
 
 Find the sum of : 
 
 
 1. 
 
 231 + 764. 
 
 8. 
 
 2. 
 
 341 + 57.8. 
 
 9. 
 
 3. 
 
 430.31 + 58.61. 
 
 10. 
 
 4. 
 
 512.87 + 36.84. 
 
 11. 
 
 5. 
 
 12.78+711.56+415.86. 
 
 12. 
 
 6. 
 
 1543.1+164.7. 
 
 13. 
 
 7. 
 
 1728 + 402.56. 
 
 14. 
 
 1897.3+675.34+6897.65. 
 475.34 + 6897.65 + 1728. 
 402.56 + 164.7 + 0.5236. 
 0.7854+3.1416+2.71828. 
 2.71828+402.56 + 1897.3. 
 0.7854 + 4.12 + 30.103. 
 2.7113 + 27.53 + 341.586. 
 
 15. 230.8 + 223 + 2.63 + 373.8. 
 
 16. 32.358 + 821.9 + 23.04+73.7. 
 
 17. 202.3031 + 71.575 + 65.813. 
 
 18. 0.0078 + 7.377+653.03. 
 
 19. 653.03 + 65.303 + 6.5033. 
 
 20. 939.303 + 65.746 + 8.2794. 
 
 21. 
 
 22. 
 
 23. 
 
 2.7182818 
 
 0.4342945 
 
 1.6093295 
 
 3.1415927 
 
 0.2098882 
 
 15.4323487 
 
 0.7853982 
 
 4.81P4774 
 
 3.785 
 
 24. 
 
 25. 
 
 26. 
 
 0.4771213 
 
 1.6093295 
 
 0.6213768 
 
 0.2908882 
 
 3.2808693 
 
 3.785 
 
 4.8104774 
 
 0.3937043 
 
 0.264 
 
 2.5399772 
 
 0.5235988 
 
 15.4323487 
 
 0.3937043 
 
 0.4342945 
 
 1.7320508 
 
 27. 
 
 28. 
 
 29. 
 
 0.6213768 
 
 0.3937043 
 
 1.4142136 
 
 1.4142136 
 
 0.3047973 
 
 1.6093295 
 
 3.2808693 
 
 1.7320508 
 
 0.30103 
 
 0.3047973 
 
 2.236068 
 
 0.381966 
 
 4.8104774 
 
 0.381966 
 
 3.2808693 
 
16 
 
 ADDITION AND SUBTRACTION. 
 
 41. A simple test of the correctness of an addition is to 
 add a second time, beginning at the top instead of the bot- 
 tom of the columns, or to add two columns at once. 
 
 It is of great advantage to educate the eye to take in at 
 a glance digits enough to make 10 or more, and then these 
 sums can be added instead of the separate digits. 
 
 To illustrate, take the example in the margin. Adding from the 
 bottom, the computer says to himself (seeing 8 + 2 = 10, 
 
 61,803 and 9+ 3 = 12), 10, 12, 22; 8 ; 8, 12, 20 ; 11, 4, 15 ; 11, 
 
 43,429 10, 21. 
 
 47,712 In the two-column mode he says, 50, 32, 82; and 
 
 62,138 writes down the 82. Then he continues, 98, 52, 150, and 
 215,082 writes the 50 at the left of 82 ; then he proceeds, 11, 
 10, 21, and writes the 21. 
 
 Many bookkeepers and merchants strongly recommend 
 the addition of two columns at once, as the most expedi- 
 tious and the least liable to error. 
 
 42. Many computers begin at the bottom of the right- 
 hand column in adding, and write on a piece of waste-paper 
 the full sum of each column or double column ; then they 
 begin at the top of the left-hand column, and add each col- 
 umn or double column, also writing the full sum ; finally, 
 they add the sums obtained in the first addition, and the 
 sums obtained in the second addition, and compare the 
 results. 
 
 Thus, in the example above, 
 By Single Columns : 
 
 22 
 
 20 
 
 6 
 
 13 
 
 20 
 
 20 
 
 13 
 
 6 
 
 20 
 
 22 
 
 215082 
 
 215082 
 
 By Double Columns 
 
 82 
 
 150 
 
 215082 
 
 213 
 206 
 22 
 
 215082 
 
ADDITION AND SUBTRACTION. 17 
 
 Exercise 8. 
 
 Add the following by double columns, and test by adding 
 by single columns : 
 
 1. 2. 3. 
 
 $ 45.68 $154.31 $ 73.86 
 
 73.91 296.85 453.71 
 
 78.54 736.48 137.64 
 
 534.69 345.19 98.87 
 
 134.70 782.34 643.48 
 581.43 78.43 462.71 
 
 6. 
 
 $621.65 
 167.32 
 856.96 
 718.83 
 501.49 
 315.72 
 768.44 
 
 9. 
 
 $763.89 
 
 78.23 
 
 345.61 
 
 26.73 
 
 489.56 
 
 812.35 
 
 607.28 
 
 219.07 
 
 68.72 
 
 216.78 
 
 436.74 
 
 4. 
 
 5. 
 
 $498.50 
 
 $ 65.42 
 
 17.37 
 
 638.34 
 
 684.29 
 
 763.43 
 
 231.56 
 
 809.31 
 
 210.10 
 
 798.83 
 
 671.54 
 
 835.78 
 
 643.53 
 
 356.47 
 
 7. 
 
 8. 
 
 $791.52 
 
 $ 32.54 
 
 504.83 
 
 254.63 
 
 879.26 
 
 63.27 
 
 243.97 
 
 131.56 
 
 732.86 
 
 506.72 
 
 47.95 
 
 283.54 
 
 856.43 
 
 345.83 
 
 497.65 
 
 643.46 
 
 541.26 
 
 708.91 
 
 616.72 
 
 463.73 
 
 857.94 
 
 67.74 
 
18 
 
 ADDITION AND SUBTRACTION. 
 
 10. 
 
 11 
 
 12. 
 
 $8400.07 
 
 $1873.33 
 
 $2336.29 
 
 3212.17 
 
 6170.24 
 
 336.00 
 
 1716.41 
 
 4813.25 
 
 2456.25 
 
 1020.08 
 
 662.25 
 
 641.25 
 
 1452.44 
 
 622.64 
 
 1174.50 
 
 1829.51 
 
 692.82 
 
 326.03 
 
 1929.96 
 
 2457.75 
 
 1219.87 
 
 114.78 
 
 2126.76 
 
 226.78 
 
 89.75 
 
 5391.25 
 
 276.75 
 
 173.67 
 
 7349.86 
 
 5936.40 
 
 17.45 
 
 l 122.75 
 
 1914.78 
 
 112.44 
 
 9667.50 
 
 311.87 
 
 1098.75 
 
 6000.00 
 
 7956.00 
 
 6170.24 
 
 572.80 
 
 1919.66 
 
 13. 
 
 14. 
 
 15. 
 
 $1482.40 
 
 $ 773.72 
 
 $2406.08 
 
 2575.71 
 
 442.37 
 
 3101.24 
 
 3364.27 
 
 454.86 
 
 1452.09 
 
 689.81 
 
 358.61 
 
 3693.91 
 
 1533.61 
 
 2003.17 
 
 2054.76 
 
 735.58 
 
 179.56 
 
 1231.25 
 
 105.69 
 
 8493.75 
 
 1828.35 
 
 261.64 
 
 4179.54 
 
 1562.50 
 
 1516.56 
 
 3493.54 
 
 6937.50 
 
 2197.23 
 
 178.17 
 
 1987.57 
 
 1317.71 
 
 727.53 
 
 943.27 
 
 408.30 
 
 2889.42 
 
 2312.11 
 
 609.53 
 
 992.92 
 
 1409.28 
 
 1679.47 
 
 1183.08 
 
 2759.94 
 
ADDITION AND SUBTRACTION. 19 
 
 43. The sign — is called minus, and means that the 
 number after it is to be taken from the number before it ; 
 that is, subtracted from the number before it. 
 
 The expression 9 — 5 = 4 is read : 9 minus 5 equals 4. 
 
 44. Subtraction. The process of taking one number 
 from another is called subtraction. 
 
 45. The number taken away is called the subtrahend; 
 the number from which the subtrahend is taken, the minu- 
 end; and the number remaining, the remainder or difference. 
 
 46. The sum of the remainder and the subtrahend is 
 always equal to the minuend. Hence, 
 
 47. To test the accuracy of the work in subtraction, we 
 add the remainder and the subtrahend. The sum will be 
 equal to the minuend, if the work is correct. 
 
 48. The minuend, subtrahend, and remainder must all be 
 like numbers ; and from units of any order units of the same 
 order only can be subtracted, ones from ones, tens from tens, 
 tenths from tenths, hundredths from hundredths, etc. 
 
 Exercise 9. 
 
 1. Subtract by 2's from 20 to ; from 21 to 1. 
 
 2. Subtract by 3's from 20 to 2 ; from 21 to 0. 
 
 3. Subtract by 4's from 30 to 2 ; from 31 to 3 ; from 32 
 to ; from 33 to 1. 
 
 4. Subtract by 5's from 32 to 2 ; from 33 to 3 ; from 34 
 to 4 ; from 35 to ; from 36 to 1. 
 
 5. Subtract by 6's from 33 to 3 ; from 34 to 4 ; from 35 
 to 5 ; from 36 to ; from 37 to 1 ; from 38 to 2. 
 
 6. Subtract by 7's from 42 to ; from 43 to 1 ; from 44 
 to 2 ; from 45 to 3 ; from 46 to 4 ; from 47 to 5. 
 
 7. Subtract by 8's from 42 to 2 ; from 43 to 3 ; from 44 
 to 4 ; from 45 to 5 ; from 46 to 6 ; from 47 to 7. 
 
 8. Subtract by 9's from 55 to 1 ; from 56 to 2 ; from 57 
 to 3 ; from 59 to 5 ; from 61 to 7 ; from 62 to 8. 
 
20 ADDITION AND SUBTRACTION. 
 
 49. Examples. 1. From 359.7 take 186.3. 
 
 Solution. We write units under units, tens under tens, tenths under 
 
 tenths, and so on. Then 3 tenths from 7 tenths leaves 4 tenths, and we 
 
 write 4 under the column of tenths ; 6 units 
 
 Minuend, 359.7 from 9 units leaves 3 units, and we write 3 
 
 Subtrahend, 186.3 under the column of units ; since we cannot 
 
 Remainder, 173.4 take 8 tens from 5 tens, we change one of the 
 
 3 hundreds to 10 tens and add them to the 5 
 
 tens, making 15 tens; then 8 tens from 15 tens leaves 7 tens, and we 
 
 write 7 under the column of tens. As we have taken one of the 3 
 
 hundreds, we have only 2 hundreds remaining ; and 1 hundred from 
 
 2 hundreds leaves 1 hundred. The remainder, therefore, is 173.4. 
 
 2. From 50 take 27.65. 
 
 Solution. Since the subtrahend contains tenths and hundredths, 
 
 and the minuend has neither tenths nor hundredths, we put zeros in 
 the place of tenths and hundredths in the minuend. As 
 (4)(9)(9)(10) there are no hundredths, no tenths, and no units in the 
 5 0. minuend, 1 of the 6 tens is taken, leaving 4 tens, and 
 2 7. 6 5 changed to 10 units ; then 1 of the 10 units is taken, 
 2 2. 3 5 leaving 9 units, and changed to 10 tenths ; then one of the 
 10 tenths is taken, leaving 9 tenths, and changed to 10 
 
 hundredths. That is, 50.00 is changed to 4 tens, 9 units, 9 tenths, and 
 
 10 hundredths. Then, subtracting, we have 22.35. 
 
 50. Hence, we have the following 
 
 Rule for Subtraction. Write the subtrahend under the 
 minuend, placing units of the sams order in the same column. 
 
 Begin at the right and subtract each order of units of the 
 subtrahend from the corresponding order of the minuend. 
 Write the result beneath, step by step, and put in the decimal 
 point when reached. 
 
 If any order of the minuend has fewer units than the 
 same order of the subtrahend, increase the units of this order 
 of the minuend by 10 and subtract ; then diminish by one 
 the units of the next higher order of the minuend. 
 
 Proof. Add the remainder and subtrahend. If the sum 
 equals the minuend, the work may be assumed correct. 
 
ADDITION AND SUBTRACTION. 
 
 21 
 
 Exercise 10. 
 
 Find the remainder and prove : 
 
 1. 234-123. 11. 789-456. 
 
 2. 343-123. 12. 879-456. 
 
 3. 424-123. 13. 978-456. 
 
 4. 555-123. 14. 6378-456. 
 
 5. 676-123. 15. 6855-456. 
 
 6. 725-123. 16. 6853-456. 
 
 7. 839 — 123. 17. 7797 — 456. 
 
 8. 999-123. 18. 7006-456. 
 
 9. 1000 — 123. 19. 3542 — 456. 
 10. 5120 — 123. 20. 4000 — 456. 
 
 31. $183.45 -$76.47. 51. 
 
 32. $716.43 — $628.74. 52. 
 
 33. $647.51 -$549.64. 53. 
 
 34. $270.04— $128.31. 54. 
 
 35. $125 — $101.50. 55. 
 
 36. $247.93 — $129.47. 56. 
 
 37. $641.87 - $333.95. ' 57. 
 
 38. $56.27 — $29.89. 58. 
 
 39. 3.1415927-2.7182818. 59. 
 
 40. 0.7853982 — 0.5235988. 60. 
 
 41. 4.8104774 — 0.4342945. 61. 
 
 42. 2.5399772 — 0.3937043. 62. 
 
 43. 0.3937043-0.3047973. 63. 
 
 44. 3.2808693 — 0.3047973. 64. 
 
 45. 3.2808693-1.6093295. 65. 
 
 46. 3.785—0.6213768. 66. 
 
 47. 15.4323487 — 0.264. 67. 
 
 48. 1.7320508 — 1.4142136. 68. 
 
 49. 2.236068-1.7320508. 69. 
 
 50. 2.236068 — 0.618034. 70. 
 
 21. 974-779. 
 
 22. 368 — 249. 
 
 23. 2301 — 479. 
 
 24. 2731 — 929. 
 
 25. 708-394. 
 
 26. 1123 — 1072. 
 
 27. 891 — 773. 
 
 28. 8103 — 5621. 
 
 29. 19.001 — 3456. 
 
 30. 2180-792. 
 0.381966 - 0.30103. 
 3.1415927-0.7853982. 
 2.3561945 - 0.7853982. 
 1.5707963-0.7853982. 
 3.1415927 — 0.5235988. 
 2.6179939-0.5235988. 
 2.0943951 — 0.5235988. 
 1.5707963 - 0.5235988. 
 1.0471975-0.5235988. 
 1-0.381966. 
 1.4142136-0.618034. 
 0.618034-0.381966. 
 9,873,210 — 8,765,420. 
 8010.101 - 4187.94. 
 1,000,000 - 817,259. 
 729,434-613,488. 
 6532.18 — 1916.47. 
 1718.754 — 1389.328. 
 21,205 — 1787.563. 
 42,786.95-4278.695. 
 
22 ADDITION AND SUBTRACTION. 
 
 Exercise 11. 
 
 1. In a till are $391 in bills, $67.50 in gold, $39.75 in 
 silver, and $2.77 in copper and nickel. How much money 
 is in the till ? 
 
 2. Starting out with $315.75 in one wallet and $54.37 
 in another, I pay the grocer $127.38 ; the butcher, $64.17 ; 
 the shoemaker, $21.40 ; the landlord, $50 ; the tailor, $35. 
 What ought I to have left? 
 
 3. On a bill of $753.43, I pay $517.87. How much do 
 I still owe? If I owe $817.87, and have but $637.50, how 
 much do I lack of being able to pay ? 
 
 4. If a man was born January 1, 1812, how old was he 
 January 1, 1878? 
 
 5. America was discovered in 1492. How many years 
 after its discovery was each of the following events ? 
 
 Settlement of Florida, 1565 ; of Virginia, 1607 ; of Mas- 
 sachusetts, 1620 ; of Quebec, 1608 ; French and Indian 
 War, 1756 ; Declaration of Independence, 1776 ; Inaugura- 
 tion of Washington, 1789 ; War with England, 1812 ; Mexi- 
 can War, 1846 ; Civil War, 1861. 
 
 6. The minuend is one hundred million two hundred 
 fifty-six thousand three hundred seventy-two, and the sub- 
 trahend is nineteen million nine hundred thousand nine 
 hundred ninety-nine. Find the remainder. 
 
 7. If the minuend is 9874, and remainder 3185, what is 
 the subtrahend ? The subtrahend being 7659, and re- 
 mainder 675.68, what is the minuend ? 
 
 8. The smaller of two numbers is 7.9;">764328 ; their 
 difference is 0.00087692. What is the larger number ? 
 
 9. The larger of two numbers is 7.95764328, and their 
 difference is 7.153485. What is the smaller number? 
 
 10. If the subtrahend is 10,542, and the difference 544.2, 
 what is the minuend ? 
 
ADDITION AND SUBTRACTION. 23 
 
 11. A man pumps out of a cistern in one hour 243.75 
 gallons ; in the next hour, 227.5 gallons ; in 45 minutes 
 more, 137.75 gallons ; and the cistern is empty. How- 
 many gallons of water were in it? 
 
 12. From what number must I subtract 5 to leave 7? 
 8 to leave 9 ? From what number must I subtract 5.1736 
 to leave 8.1964? 6.231 to leave 9.6648? 74.213 to leave 
 25.787 ? 
 
 13. What must be subtracted from 1 to leave 0.5 ? to leave 
 0.53 ? to leave 0.532 ? to leave 0.5236 ? to leave 0.5235988 ? 
 
 14. I start on a journey of 3433 miles. The first day 
 I make 428 miles ; the second day, 511 miles ; the third, 
 497 miles ; the fourth, 513. How many miles of my jour- 
 ney remained for me at the close of each day ? How many 
 miles had I gone at the close of each day ? 
 
 15. Subtract 76,343 from the sum of 61,932, 51,387, 
 5193, 4674, and 8199 ; then subtract 23,657 from the re- 
 mainder. 
 
 16. Jones bought a farm and stock for $7633.90 ; sold 
 the stock for $305.75; then sold the farm for $7325. 
 How much did he lose ? ' 
 
 17. If I gave $4375 for my land, and paid for house, 
 barn, sheds, and fences, $2789.50 ; also $973.75 for horses, 
 cattle, tools, etc. ; what did my farm and stock cost ? 
 
 18. If I paid $8138.25 for land and cattle, and sold 
 part of the land for $675, and part of the cattle for $217.50, 
 what is the cost of the land and the cattle left ? 
 
 19. John has 158 cents, James has 271 cents; James 
 gives John 56 cents. Which has then more than the 
 other, and how many cents more ? 
 
 20. A cattle dealer had 228 oxen, 475 sheep, and 49 
 lambs; he sold 17 oxen, 64 sheep, and 7 lambs. How 
 many animals of each kind did he then have, and how 
 many all together ? 
 
CHAPTER III. 
 MULTIPLICATION. 
 
 51. Multiplication. The process of taking a number of 
 units a number of times is called multiplication. 
 
 52. Multiplicand. The number of units taken is called 
 the multiplicand. 
 
 53. Multiplier. The number that shows how many 
 times the multiplicand is taken is called the multiplier. 
 
 54. Product. The number found by multiplication is 
 called the product. 
 
 55. The multiplier always signifies a number of times, 
 and is, therefore, an abstract number. 
 
 56. The multiplicand and product are like numbers. 
 
 57. Factors. The numbers used in making a product 
 are called factors of the product. 
 
 58. The product of two factors is the same whichever 
 factor is taken as the multiplier. 
 
 • • • • Thus, 3 times 4 = 4 times 3. The dots in the lnar- 
 
 • • • • gin read across the page make 3 fours ; read up and 
 
 • • • • down the page they make 4 threes. 
 
 Note. The multiplicand always signifies a number of units, 
 whether the kind of units is stated or not. The only difference 
 between 15 and 15 horses is that in the first case the kind of units 
 counted is not stated, and in the second case the kind is stated. 
 
 We may interchange the multiplicand and multiplier if we refer to 
 the numbers only. Thus, in the example 3 times 4 horses, we cannot 
 say 4 horses times 3, but we may interchange the 3 and 4, and have 
 4 times 3 horses. The product in either case is 12 horses. With this 
 understanding, we may always use the smaller number as multiplier. 
 
MULTIPLICATION. 
 
 25 
 
 59. The sign of multiplication is X. When the multi- 
 plier precedes the multiplicand, the sign X is read times. 
 
 Thus, 6 X $7 = $42 is read : 6 times $7 equals $42. 
 
 When the multiplier follows the multiplicand, the sign X 
 is read multiplied by. 
 
 Thus, $7X6= $42 is read : $7 multiplied by 6 equals $42, and 
 means $7 taken 6 times equals $42. 
 
 60. The products, in all cases in which neither factor 
 exceeds twelve, should be thoroughly committed to memory. 
 They will be found in the following 
 
 Multiplication Table. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 30 
 
 33 
 
 36 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 5 
 
 10 
 
 15 
 
 20. 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 50 
 
 55 
 
 60 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 60 
 
 66 
 
 72 
 
 7 
 
 14 
 
 21 
 
 28 
 
 35 
 
 42 
 
 49 
 
 56 
 
 63 
 
 70 
 
 77 
 
 84 
 
 8 
 
 16 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 9 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 
 63 
 
 72 
 
 81 
 
 90 
 
 99 
 
 108 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 132 
 
 11 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 77 
 
 88 
 
 99 
 
 110 
 
 121 
 
 12 
 
 24 
 
 36 
 
 48 
 
 60 
 
 12 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 144 
 
 61. In this table, take the multiplicand in the upper 
 line, and the multiplier in the left-hand column; the 
 product will be found directly under the multiplicand, 
 and opposite the multiplier ; as, 12 X 7 is 84. 
 
26 MULTIPLICATION. 
 
 62. A change in the position of the decimal point of a 
 number expressed in figures affects the value of the 
 number. 
 
 Thus, if in 79.253 we move the decimal point one place to the 
 right, so that the number becomes 792.53, we increase the value of 
 each figure tenfold ; the 7 tens become 7 hundreds, the 9 units be- 
 come 9 tens, the 2 tenths become 2 units, the 6 hundredths become 6 
 tenths, and the 3 thousandths become three hundredths. The value 
 of the entire number, therefore, is increased tenfold. But multiply- 
 ing a number by 10 increases its value tenfold. Hence, moving the 
 decimal point of a number one place to the right has the same effect 
 as multiplying the number by 10. In the same way, moving the 
 decimal point two places to the right multiplies the number by 
 100, and so on. Hence, 
 
 63. To Multiply a Number by io, ioo, iooo, etc., 
 Move the decimal point in the multiplicand as many ntmeei 
 
 to the right, annexing zeros if necessary, as there are zeros in 
 the multiplier. 
 
 Again, if in 79.253 we move the decimal point one place to the 
 left, so that the number becomes 7.9253, we decrease the value of 
 each figure tenfold ; the 7 tens become 7 units, the 9 units become 9 
 tenths, the 2 tenths become 2 hundredths, the 5 hundredths become 5 
 thousandths, and the 3 thousandths become 3 ten-thousandths. The 
 value of the resulting number, therefore, is one tenth that of the 
 original number. But multiplying a number by 0.1 means to find 
 one tenth of the number. Hence, moving the decimal point of a 
 number one place to the left has the same effect as multiplying the 
 number by 0.1. In the same way, moving the decimal point two 
 places to the left has the same effect as multiplying the number by 
 0.01, and so on. Hence, 
 
 64. To Multiply a Number by o.i, o.oi, o.ooi, etc., 
 
 More the decimal point in the multiplicand as many places 
 to the left, prefixing zeros if necessary, as there are decimal 
 places in the multiplier. 
 
MtTLTtFLICATtON. 27 
 
 65. Since multiplication, as defined in § 51, is the process 
 of taking the multiplicand the number of times indicated 
 by the multiplier, the multiplier must be an integral num- 
 ber ; but the meaning of multiplication is extended to cover 
 the case in which the multiplier contains a decimal. 
 
 To multiply by a decimal is to take such a part of the 
 multiplicand as the decimal is of one. 
 
 To multiply by an integral number and a decimal is to 
 take the multiplicand as many times as is indicated by the 
 integral number, and such a part of the multiplicand as is 
 indicated by the decimal. 
 
 66. Examples. 1 . Find the product of 6 X 34.87. 
 
 Solution. We write the multiplier, 6, under the multiplicand, as 
 in the margin, and begin at the right to multiply. 6 times 7 hun- 
 dredths equals 42 hundredths, or 4 tenths and 2 hun- 
 34.87 dredths. We write the 2 hundredths in the place of 
 
 6 hundredths, and reserve the 4 tenths to add to the product 
 
 209.22 of the tenths. 6 times 8 tenths equals 48 tenths, and 48 
 tenths plus the 4 tenths reserved makes 52 tenths, or 
 5 units and 2 tenths. We write^ the 2 tenths in the place of tenths 
 and reserve the 5 units to add to the product of the units. 6 times 
 4 units equals 24 units, and 24 units plus the 5 units reserved makes 
 29 units, or 2 tens and 9 units. We write the 9 units in the place of 
 units, and reserve the 2 tens to add to the product of the tens. 6 
 times 3 tens equals 18 tens, and 18 tens plus the 2 tens reserved makes 
 20 tens. We write 20 to the left of the 9 units. 
 
 2. Find the product of 0.6 X 34.87. 
 
 Solution. The multiplier, 0.6, equals 6 X 0.1. We 
 
 34.87 therefore multiply first by 6, and this product by 0.1. 
 
 0.6 But multiplying by 0.1, simply moves the decimal point 
 
 20.922 in the product one place to the left. Hence, the product 
 has two decimal places for the decimal in the multiplicand, 
 and one more place for the decimal in the multiplier, or three 
 in all. 
 
28 
 
 MULTIPLICATION. 
 
 3. Find the product of 263 X 74.782. 
 
 Solution. The multiplier is 200 + 60 + 3. We obtain the product 
 by multiplying the multiplicand by 3, then by 60, then by 200, and 
 adding these products. 
 
 (1) 
 
 74.782 
 263 
 
 (2) 
 
 74.7*2 
 
 224.346 
 
 224346 
 
 4486.920 
 
 448692 
 
 14956.400 
 
 149664 
 
 19067.666 19667.666 
 
 3 times the multiplicand = 
 60 times the multiplicand = 
 200 times the multiplicand = 
 263 times the multiplicand = 
 Since zeros at the right of the partial products do not affect the 
 result of the addition, they may be omitted as in (2). Care must be 
 taken, however, to put the right-hand figure of each partial product 
 directly under the figure of the multiplier used in obtaining it. 
 
 4. Find the product of 2.63 X 74.782. 
 
 74.782 Solution. The multiplier, 2.63, equals 263 X 0.01. 
 
 2.63 We therefore multiply first by 263 and this product by 
 
 224346 0.01. But multiplying by 0.01 simply moves the deci- 
 
 448(592 mal point in the product two places to the left. Hence, 
 
 149564 the product has three decimal places for the decimal 
 
 19(5.67666 in the multiplicand, and two more places for the deci- 
 mal in the multiplier, or five in all. 
 
 5. Find the product of 2007 X 4587. 
 
 4587 
 
 2007 Solution. The partial products corresponding to the 
 32109 zeros in the multiplier will be zero, and therefore they 
 91 7 \ need not be written. 
 
 U2W5109 
 
 67. In each of these examples, if we interchange the 
 multiplier and multiplicand and multiply, we find that we 
 obtain the same product. 
 
 68. Hence, we have the following 
 
 Rule for Multiplication. Write the multiplier under 
 the multiplicand with their right-hand figures in a vertical 
 line, and draw a line beneath. 
 
MULTIPLICATION. 
 
 29 
 
 Begin at the right and multiply each order of units of the 
 multiplicand by each order of the multiplier. Write the units 
 of each product, and add the tens, if any, to the next product. 
 
 Place the right-hand figure of each partial product under 
 the figure of the multiplier used in obtaining it, and add the 
 partial products. 
 
 Point off from the right of the product as many figures for 
 decimals, prefixing zeros if necessary, as there are decimal 
 places in the multiplicand and multiplier together. 
 
 Proof. Interchange the multiplier and multiplicand, and 
 multiply. If the results agree, the work may be assumed to 
 be correct. 
 
 
 Exercise 12. 
 
 Find the product of : 
 
 
 
 1. 
 
 6 X 0.5235988. 
 
 19. 
 
 2.23607 X 2.236. 
 
 2. 
 
 4 X 0.7853982. 
 
 20. 
 
 0.618 X 618. 
 
 3. 
 
 3.14159265 X 5 X 5. 
 
 21. 
 
 0.618034 X 0.618035 
 
 4. 
 
 30 X 8.75. 
 
 22. 
 
 12 X 0.12936. 
 
 5. 
 
 0.07 X 6.975. 
 
 '23. 
 
 7.92801 X 0.9. 
 
 6. 
 
 700 X 7.81. 
 
 24. 
 
 58.383 X 0.39. 
 
 7. 
 
 8000 X 65.432. 
 
 25. 
 
 0.08X0.28744. 
 
 8. 
 
 300 X 7.85. 
 
 26. 
 
 0.065 X 491.205. 
 
 9. 
 
 0.009 X 10,356.78. 
 
 27. 
 
 68.325X6.25. 
 
 10. 
 
 7.37 X 0.785398. 
 
 28. 
 
 0.732 X 1.6. 
 
 11. 
 
 8.56 X 0.785398. 
 
 29. 
 
 0.438 X 1208.88. 
 
 12. 
 
 1001 X 0.785398. 
 
 30. 
 
 498 X 0.0125. 
 
 13. 
 
 0.083 X 2150.42. 
 
 31. 
 
 7 X 0.007. 
 
 14. 
 
 0.75 X 2150.42. 
 
 32. 
 
 1000 X 0.0001. 
 
 15. 
 
 0.075 X 2150.42. 
 
 33. 
 
 0.235 X 10.24. 
 
 16. 
 
 0.7071 X 1.4142136. 
 
 34. 
 
 0.00507702 X 0.0283. 
 
 17. 
 
 1.41421 X 1.4142. 
 
 35. 
 
 89.3 X 0.00752. 
 
 18. 
 
 1.732 X 1.732. 
 
 36. 
 
 74.1 X 0.0256. 
 
30 MULTIPLICATION. 
 
 69. Powers and Roots. If a product consists of equal 
 factors, it is called a power of that factor ; and one of the 
 equal factors is called a root of the product. 
 
 The power is named from the number of equal factors. 
 Thus, 
 
 25 (5 X 5) is the second poiuer, or square, of 5. 
 125 (5X5X5) is the third power, or cube, of 5. 
 625 (5X5X5X5) is the fourth power of 5. 
 
 5 is called the second root, or square root, of 25. 
 
 5 is called the third root, or cube root, of 125. 
 
 5 is called the fourth root of 625. 
 To avoid writing long rows of equal factors, a figure, 
 called the exponent, is written at the right of a number 
 to show how many times the number is taken as a factor. 
 
 Thus, 6 6 means the same as5X5X5x6x6, and is read the 
 fifth power of 6. 3 8 times 3* means 3x3x3 times 3x3x3x3 = 
 8X8X8X8X8X8X8 = 3*. Hence, 
 
 The product of two or more powers of the same number 
 may be expressed by writing the number with an exponent 
 equal to the sum of the exponents of the given powers. 
 
 Exercise 13. 
 Express the product of : 
 
 1. 7 6 X7 S ; 8 2 X8; 2 8 X 2 ; 5 4 X 5 2 . 
 
 2. 3.01 2 X 3.01 ; 0.67 s X 0.67 8 ; 0.208 X 0.208 3 . 
 
 3. 2.003 2 X 2.003 4 ; 20.03* X 20.03 ; 20.03 X 20.03 2 - 
 
 70. Casting out Nines. The product of two integral 
 factors is called a midtiple of either of its factors. 
 
 Every power of 10 is one more than some multiple of 9. 
 Thus, 10 = 9 + 1; 10 2 = 11 X 9+ 1; 108= 111 X 9 +1, etc. 
 
 Every multiple of a power of 10 by a single digit is, 
 therefore, some multiple of 9, plus that digit. 
 Thus, 500 = 55 X 9 + 5 ; 7000 = 777 X 9 + 7, etc. 
 
MULTIPLICATION. 31 
 
 But as every number consists of the sum of such multi- 
 ples of powers of 10, every number is a multiple of 9, plus 
 the sum of its own digits. 
 
 Thus, 24,573 is a multiple of 9 plus 2 + 4+5 + 7 + 3. In casting 
 out the nines from a number, the remainder, therefore, is the same 
 as that arising from casting out the nines from the sum of its digits. 
 
 In finding the remainder from casting out the nines from 
 a number, we may, of course, omit the nines, or any two 
 or three digits which we see at a glance will make 9. 
 
 Thus, to cast out the nines from 1,926,754, Ave see at once that 1, 
 2, 6, and 5, 4, make nines, and the single 7 will be the remainder. 
 So in 254,786, we reject 5, 4, and 2, 7, and add only 8, 6, from the 
 sum of which reject 9, and there is left 5. 
 
 71. Example. Multiply 761 by 147. 
 Solution. The remainder after the nines are cast out 
 
 H | Multiply. 
 
 From 761 is 5 
 
 From J47 .... 
 
 5327 From 15 . . is 6 
 3044 
 761 
 From 111867 ' is 6 
 
 The product of the two given numbers has 6 remaining, and the 
 product of the two remainders has 6 remaining after the nines are 
 cast out. Therefore, the work may be assumed to be correct. 
 
 72. Casting out Elevens. Even powers of 10 are mul- 
 tiples of 11, plus 1 ; odd powers of 10 are multiples of 11, 
 minus 1. 
 
 Thus, 10 2 , or 100 = 9 X 11 + 1 ; 10 4 , or 10,000 = 909 X 11 + 1, etc. ; 
 10 = 11 - 1 ; 10 3 , or 1000 = 91 X 11 - 1 ; 10 5 = 9091 X 11 - 1, etc. 
 
 Hence, if the elevens are cast out from a number ex- 
 pressed by two digits, the remainder equals the digit in 
 the odd place minus the digit in the even place, (the digit 
 in the odd place, if less than that in the even place, being 
 
32 MULTIPLICATION. 
 
 increased by 11) ; and, if the elevens are cast out from any 
 number, the remainder equals the sum of the digits in the 
 odd places (increased by a multiple of 11 if necessary) 
 minus the sum of the digits in the even places. 
 
 73. The proof by casting out elevens is similar to that 
 by casting out nines ; and if a process stands both tests, 
 the only possible error is a multiple both of 9 and of 11. 
 
 Multiply 67,853 by 2976, and test by casting out the 
 elevens. 
 
 2976 . . 
 
 q y Multi 
 
 407118 
 
 30 ... . 
 
 474971 
 
 
 610677 
 
 
 135706 
 
 
 201930528. . . 
 
 
 74. In applying either test to decimals, disregard the 
 decimal point. In questions involving large numbers, al- 
 ways apply one or both tests to the work. 
 
 Exercise 14. 
 
 Find the following products, and test the accuracy by 
 casting out the nines and by casting out the elevens : 
 
 1. 21.3706 X 15.243 X 1.8954. 
 
 2. 0.026891 X 5.328 X 29.74. 
 
 3. 0.0012 X 5.8281 X 0.6827. 
 
 4. 23.9875 X 12.4764 X 0.017. 
 
 5. 39.801 X 1.44 X 17.9645. 
 
 6. 0.0165 X 5.2817 X 0.8469. 
 
 7. 0.54237 X 16 X 0.00176. 
 
 8. 24.271 X 3.6485 X 15.271. 
 
 9. 13.256 X 14.125 X 30.254. 
 
MULTIPLICATION. 
 
 33 
 
 75. Contracted Multiplication of Decimals. In ordi- 
 nary calculations we seldom use a decimal smaller than 
 0.00001 of the unit. 
 
 Multiply 0.123456789 by 1.23456789. 
 
 6 
 
 49 
 
 370 
 
 2469 
 
 12345 
 
 0.123456789 
 1.23456789 
 1111111101 
 987654312 
 864197523 
 
 740740734 
 
 17283945 
 
 3827156 
 
 370367 
 
 13578 
 
 6789 
 
 0.15241578750190521 
 
 The 6 at the left of the vertical line is 
 obtained by multiplying the 1 in the mul- 
 tiplicand by 5 in the multiplier, and carry- 
 ing 1 from 5X2. The 9 below the 6 is 
 obtained by multiplying the 2 by 4, and 
 carrying 1 from 4x3. The below the 
 9 is obtained by multiplying the 3 by 3, and 
 carrying 1 from 3X4. The 9 below the 
 is obtained by multiplying the 4 by 2, and 
 carrying 1 from 2X5. The 5 below the 
 9 is obtained by multiplying the 5 by 1. 
 
 It is evident that, if five decimal places only are required 
 in the product, all the work to the right of the vertical line 
 is wasted. 
 
 To shorten the work, disregard the decimal point in 
 the multiplier, and write the multiplier under the multi- 
 plicand, so that each digit of the multiplier shall fall 
 directly under the digit of the multiplicand into which it 
 is multiplied to produce the first figure to the left of the 
 vertical line in each partial product. Thus : 
 
 0.123456789 
 987654321 
 12346 
 
 370 
 
 49 
 
 6 
 
 1 
 
 0.15241 
 
 Notice that the figures of the multiplier are re- 
 versed ; that the units' figure of the multiplier 
 falls under the last decimal of the multiplicand 
 which is to be retained, and that the decimals in 
 the result are correct. In order, however, to 
 have the required number of decimal places cor- 
 rect, it will generally be necessary to take one 
 more than the required number of decimal 
 places. Hence, 
 
34 MULTIPLICATION. 
 
 76. To Multiply Decimals by the Contracted Method, 
 
 Reverse the multiplier, and put the units? figure under the 
 last place of decimals to be retained. 
 
 Multiply each figure of the multiplier into the figure next 
 to the right above it. Do not write this result, but carry the 
 nearest ten to the next result, multiplying as usual. 
 
 Write the first figures of the partial products in a vertical 
 column. 
 
 Add the products, and point off from the sum as many 
 decimal places as were taken in the multiplicand. 
 
 If the multiplier has no units' figure, supply its place with 
 a zero. 
 
 To insure accuracy, take one decimal place more than the 
 required number. 
 
 To detect errors that may arise from displacement of the 
 decimal point, or from an erroneous arrangement of the fac- 
 tors, test the result by a rough estimate of what the product 
 should be. 
 
 Exercise 15. 
 Find to the fifth decimal the value of : 
 
 1. 0.49714987X1.75812263. 
 
 2. 0.79817987 X 0.99429975. 
 
 3. 1.09920986 X 0.24857494. 
 
 4. 0.62208861 X 0.16571662. 
 
 5. 1.75812263 X 2.05915963. 
 
 6. 0.55630251 X 0.33445375. 
 
 7. 0.75142506 X 9.98998569. 
 
 8. 0.05245506 X 0.16571662. 
 
 9. 0.33143325 X 1.79263713. 
 
 10. 0.90633287X0.6154551376. 
 
 11. 2.84657842 X 0.96695542. 
 
 12. 0.546794489X2.847697495. 
 
CHAPTER IV. 
 DIVISION. 
 
 77. Division. If the product and one factor are given, 
 the process of finding the other factor is called division. 
 
 78. Dividend. The given product, that is, the number to 
 be divided, is called the dividend. 
 
 79. Divisor. The given factor, that is, the number by 
 which the dividend is divided, is called the divisor. 
 
 80. Quotient. The required factor, that is, the number 
 found by division, is called the quotient. 
 
 81. To divide 35 apples by 7 apples is to find the number 
 of times we must take 7„ apples to obtain 35 apples. Hence, 
 
 If the divisor and dividend denote the same kind of units, 
 the quotient is an abstract number. 
 
 82. To divide 35 apples by 7 is to find the number of 
 apples in each part, when 35 apples are divided into 7 equal 
 parts. Hence, 
 
 If the divisor is an abstract number, the quotient denotes 
 units of the same kind as the dividend. 
 
 83. What is one of the parts called, if a number is 
 divided into 2 equal parts ?3?4?5?6?7?8?9? 
 
 One half is written \ ; one third, -J- ; one fourth, \ ; one 
 fifth, \-, one sixth, \\ one seventh, \; one eighth, £; one 
 ninth, \; and so on. 
 
 To divide 35 apples by 7 is to find \ of 35 apples. 
 
36 DIVISION. 
 
 84. Division is indicated by the sign of division -r-, or by 
 writing the dividend over the divisor with a line between 
 them. 
 
 Thus, 42 -j- 6 and -\ 2 have the same meaning, and each is read : 
 forty-two divided by six. 
 
 85. Remainder. If the divisor does not exactly 
 divide the dividend, the part of the dividend left from the 
 division is called the remainder. 
 
 Thus, $42 t4= $10 with remainder |2. 
 
 86. Principles of Division. The value of the quotient 
 depends upon the relative values of the dividend and 
 divisor. 
 
 Suppose we have 36 -f- 6 = 6. 
 
 If we multiply the dividend 36 by 2, what effect will this have on 
 the quotient ? 
 
 If we divide the dividend 36 by 2, what effect will this have on the 
 quotient ? 
 
 If we multiply the divisor 6 by 2, what effect will this have on the 
 quotient ? 
 
 If we divide the divisor 6 by 2, what effect will this have on the 
 quotient ? 
 
 If we multiply both the dividend and divisor by 2, what effect will 
 this have on the quotient ? 
 
 If we divide both the dividend and divisor by 2, what effect will 
 this have on the quotient ? 
 
 From the answers to these questions, we have the follow- 
 ing important principles of division : 
 
 Multiplying the dividend or dividing the divisor by a 
 number multiplies the quotient by that number. 
 
 Dividing the dividend or multiplying the divisor by a 
 number divides the quotient by that number. 
 
 Multiplying both dividend and divisor by the same number, 
 or dividing both by the same number, does not change the 
 quotient. 
 
DIVISION. 37 
 
 Short Division. 
 
 87. If the divisor is so small that the work can be per- 
 formed mentally, the process is called short division. 
 
 88. Examples. 1. Divide 976 by 3. 
 
 Solution. We write the divisor to the left of the dividend with a 
 
 curved line between them, and the quotient under the dividend, as in 
 
 the margin. 3 is contained in 9 three times ; and we write 
 
 3) 976 3 in the quotient under the 9 of the dividend. 3 is con- 
 
 325£ tained in 7 twice with remainder 1 ; and we write 2 in the 
 
 quotient in the place of tens. The remainder 1 is 1 ten 
 
 or 10 units, and 10 units + the 6 units of the dividend makes 16 units. 
 
 3 is contained in 16 five times with remainder 1. The answer is 325 
 
 with remainder 1. We may write the remainder over the divisor as a 
 
 part of the quotient; thus, 325£. 
 
 The following wording and no more should be used : 3 in 9, 3 ; in 
 7, 2 ; in 16, 5 remainder 1. 
 
 2. Divide 72.56 by 8. 
 
 Solution. 8 in 72, 9 ; in 5, ; in 56, 7. Here the 72 to be divided 
 
 by 8 is 72 units, and 72 units divided by 8 gives 9 units ; 
 
 8) 72.56 we place the decimal point in the quotient after the units' 
 
 9.07 figure 9. The next figure 5 is 5 tenths, and 5 will not 
 
 contain 8. We write in the quotient and annex the 6 
 
 hundredths to the 5 tenths, making 56 hundredths ; and 56 hundredths 
 
 divided by 8 is 7 hundredths. The quotient is 9.07. Hence, 
 
 If the divisor is an integral number, we write the decimal 
 point in the quotient as soon as we reach the decimal point in 
 the dividend. 
 
 3. Divide 72.56 by 0.08. 
 
 Solution. Since we do not change the quotient if we multiply 
 both divisor and dividend by the same number (§ 86), we 
 
 8) 7256 make the divisor an integral number by multiplying it by 
 907 100, and multiply the dividend by the same number. 
 We then divide as before. Hence, 
 
 If the divisor contains decimal places, we multiply both 
 divisor and dividend by 10, or some power of 10, so as to 
 make the divisor an integral number. 
 
38 DIVISION. 
 
 4. Divide 78.52 by 8000. 
 
 Solution. We first divide the divisor by 1000 by cutting off the 
 
 three zeros at its right ; and move the decimal 
 
 8ffff0 )O.O7852 point in the dividend three places to the left, pre- 
 
 0.009815 fixing one zero. When we reach the last figure 
 
 of the dividend, we mentally supply a zero, and 
 
 continue dividing. Hence, 
 
 If the divisor is an integral number ending in one or more 
 zeros, we cut off the zeros, and mave the decimal point in the 
 dividend as many places to the left as we cut off zeros. 
 
 89. If we add the remainder to the product of the divisor 
 and quotient, we obtain the dividend. Hence, 
 
 We find the product of the divisor and quotient, and to tl> it 
 prod mi add tJie remainder. If the result equals the dividend, 
 the work may be assumed to be correct. 
 
 Exercise 16. 
 
 Find the quotient of : 
 
 1. 126.409-^9. 15. 87,585 -^ 1200. 
 
 2. 13.31 -M0. 16. 27,485 -f- 200. 
 
 3. 13.31 -f- 11. 17. 10.01-^0.02. 
 
 4. 1.728 -T-12. 18. 0.04^0.05. 
 
 5. 37.632-^30. 19. 7.432 -f- 0.04. 
 
 6. 42,631-^20. 20. 31-^-0.005. 
 
 7. 96,464 -MOO. 21. 480-^0.012. 
 
 8. 58.775-^-600. 22. 980^-0.0007. 
 
 9. 75,230 -M00. 23. 10.98^-0.00009. 
 
 10. 8956-^80. 24. 10.98^0.09. 
 
 11. 98,254-^900. 25. 0.1098-^0.00009. 
 
 12. 82,610 -^ 7000. 26. 0.1098 -f- 0.09. 
 
 13. 83,690-!- 500. 27. 1441 -4- 0.11. 
 
 14. 96,464 -M 10. 28. 18.92 -M.l. 
 
DIVISION. 39 
 
 Long Division. 
 
 90. The process of Long Division is the same as that 
 of Short Division, except that the work is written in full, 
 and the quotient is written over the dividend, the first 
 quotient iigure being written over the right-hand figure of 
 the partial dividend used in obtaining it. 
 
 Note. The quotient may be written at the right of the dividend. 
 The advantage, however, of writing the quotient over the dividend is 
 easily seen when the quotient contains a decimal. 
 
 91. Each quotient figure may be estimated by taking for 
 a trial divisor the nearest number of tens or hundreds, 
 etc., represented by the divisor, and by taking for a trial 
 dividend the nearest number of tens or hundreds, etc., 
 represented by the partial dividend. 
 
 92. In each step of division, the product must be less than 
 the partial dividend, and the remainder less than the divisor. 
 
 93. Examples. 1., Divide 4199 by 78. 
 
 Solution. As 78 is more than 41, it is necessary to take three 
 figures of the dividend for the first partial dividend. As the nearest 
 number of tens represented by the divisor 
 &3 is 8, we take 8 for the trial divisor. As 
 
 78)4199 the nearest number of tens represented by 
 
 390 the partial dividend is 42, we take 42 for the 
 
 299 trial dividend. 8 is contained 5 times in 
 
 234 42. Hence, the first quotient figure is 5, 
 
 65 remainder, and we write 5 in the quotient over the right- 
 hand figure, 9, of the partial dividend. We 
 multiply the divisor 78 by 5 and subtract the product 390 from 419. 
 We annex the next figure 9 of the dividend to the remainder 29. 
 The nearest number of tens represented by the second partial divi- 
 dend is 30, and 8 is contained 3 times in 30. We place 3 as the 
 second figure of the quotient and multiply the divisor by 3. This 
 product subtracted from the second partial dividend leaves for a 
 remainder 65. The complete quotient may be written 53^|. 
 
40 DIVISION. 
 
 2. Divide 2791.163 by 394. 
 
 Solution. The first partial dividend is 2791. As the nearest num- 
 ber of hundreds represented by the divisor 394 is 4, we take 4 for the 
 trial divisor. As the nearest number of hundreds represented by the 
 partial dividend 2791 is 28, we take 28 
 7.084 f or the trial dividend. 4 is contained 
 
 394)2791.163 7 times in 28. We write the 7 over the 
 
 2758 1, the right-hand figure of the partial 
 
 3316 dividend. We place the decimal point 
 
 31. "i2 in the quotient directly over the decimal 
 
 1643 point in the dividend, that is, directly 
 
 1576 after the 7. We multiply the divisor 
 
 67 remainder. 394 by 7. We subtract the product 2758 
 from 2791 and have for a remainder 33, 
 to which we annex the 1 of the dividend. As 331 is less than 394, 
 the next quotient figure is 0. To 331 we annex the next figure, 6, of 
 the dividend. 4 is contained 8 times in 33. We therefore write 8 
 for the next quotient figure, and find the product of 8 X 394 to be 
 3162. The remainder obtained by subtracting 3152 from 3316 is 164, 
 to which we annex the 3 of the dividend. 4 is contained 4 times in 
 16. The product of 4 X 394 is 1576, and this subtracted from 1643 
 leaves 67 for the final remainder. The complete quotient may be 
 written 7.084^*. 
 
 17 When the quotient contains a decimal, it is not cus- 
 394^670 tomar y to write the remainder over the divisor, but 
 qo4 to continue the division by annexing zeros to the divi- 
 ^g^ dend. In this example, if we carry the division two 
 2758 decimal Peaces further, as in the margin, we have the 
 quotient 7.08417. 
 
 94. Decimals are seldom carried to more than five 
 places. If we wish to find the value of a decimal correct 
 to the nearest tenth, hundredth, thousandth, etc., we add 1 
 to the last required figure if the next figure would be five 
 or more. 
 
 Thus, the value of the answer to the last example, 7.08417, correct 
 to the nearest tenth is 7.1 ; correct to the nearest hundredth is 7.08 ; 
 correct to the nearest thousandth is 7.084 ; correct to the nearest ten- 
 thousandth is 7.0842. 
 
DIVISION. 41 
 
 95. As in Short Division, if the divisor contains deci- 
 mal places, we multiply both divisor and dividend by a 
 power of 10, so as to make the divisor an integral number. 
 
 Examples. 1. Divide 28.3696 by 1.49. 
 
 (1) (2) 
 
 19.04 19.04 
 
 149)2836.96 1.49)28.36^96 
 
 149 149 
 
 1346 1346 
 
 1341 1341 
 
 596 596 
 
 596 596 
 
 In form (1) we multiply both dividend and divisor by 100. 
 
 The same result is obtained by counting to the right from the 
 decimal point in the dividend as many places as there are decimal 
 places in the divisor, inserting a caret as shown in form (2), and 
 putting the decimal point in the quotient directly over the caret. 
 
 2. Divide 0.381876 Jay 2.63; 739.4112 by 0.1728. 
 
 0.1452 
 
 4279. 
 
 2.63)0.38 A 1876 
 
 0.1728)739.4112 A 
 
 263 
 
 6912 
 
 1188 
 
 4821 
 
 1052 
 
 3456 
 
 1367 
 
 13651 
 
 1315 
 
 12096 
 
 526 
 
 15552 
 
 526 
 
 15552 
 
 3. Divide 34.2 by 18,000. 
 
 0.0019 
 180^0) A 034. 2 Solution. Here we cut off the three zeros from 
 
 18 the divisor, and put a caret three places to the left 
 
 162 of the decimal point in the dividend ; that is, we 
 
 162 divide both the divisor and the dividend by 1000. 
 
42 DIVISION. 
 
 96. From these examples we have the following 
 
 Rule fob Long Division. Write the divisor to the left 
 of the dividend, with a curved line between them. 
 
 If the divisor contains decimal places, remove the decimal 
 point from the divisor, and move the decimal point in the 
 dividend to the right as many places as there are decimal 
 jjlaces in the divisor. 
 
 Take for the first partial dividend the fewest left-hand 
 figures thai will contain the divisor, <nnl write the quotient 
 over the right-hand figure of this partial dividend. 
 
 Multiply the divisor by tliis quotient, and place the product 
 under the partial dividend used. 
 
 Subtract this product, and to the remainder annex the next 
 figure of the dividend. 
 
 Proceed as before, and continue the process until all the 
 figures of the dividend have been used ; putting the decimal 
 jmint in the quotient as soon as the decimal point in the 
 dividend is reached. 
 
 Proof. Find the product of the divisor and quotient, and 
 to this j >r<><l net a (hi the remainder, if any. If the result equals 
 the dividend, the work mag be assumed to be correct. 
 
 Note. If the divisor is an integral number ending in one or more 
 zeros, cut off the zeros, and move the decimal point in the dividend 
 to the left as many places as there are zeros cut off, prefixing zeros 
 if necessary. 
 
 Exercise 17. 
 Find the quotient of : 
 
 1. 7553 -f- 91. 5. 35,372-^35. 9. 370,406 -r- 843. 
 
 2. 45934-73. 6. 834,561-^408. 10. 978.217 -M98. 
 
 3. 89,713 4-76. 7. 341,586 -f- 247. 11. 543,816 4-357. 
 
 4. 53,691^-88. 8. 861,3454-395. 12. 604,730 4-1289. 
 

 DtVlSlOtt. 
 
 13. 
 
 326.7 4-132. 
 
 47. 
 
 125,457.64 4- 354.2. 
 
 14. 
 
 79.72552 4- 1.121. 
 
 48. 
 
 0.75-^0.866. 
 
 15: 
 
 0.4077 -f- 9.06. 
 
 49. 
 
 96,800 4-311.13. 
 
 16. 
 
 961.9476 4-106.8. 
 
 50. 
 
 2,534,000-1-6.4037. 
 
 17. 
 
 $27,121.50 4- $387.45. 
 
 51. 
 
 6.4037 4-2,534,000. 
 
 18. 
 
 123.684792 4- 39.37. 
 
 52. 
 
 0.7407^5.504. 
 
 19. 
 
 1203.75 4-19.26. 
 
 53. 
 
 54.44 4- 1.7359. 
 
 20. 
 
 256 4-0.0016. 
 
 54. 
 
 0.3658 4-2.322. 
 
 21. 
 
 24.1802 4-3.19. 
 
 55. 
 
 $25,12 4-1.43. 
 
 22. 
 
 200-^0.3125. 
 
 56. 
 
 17-^0.036. 
 
 23. 
 
 7.704256 -i- 0.08302. 
 
 57. 
 
 $143 4-1.7892. 
 
 24. 
 
 1.6093295 4-0.479. 
 
 58. 
 
 18.7 4-121. 
 
 25. 
 
 1.6093295^-0.917. 
 
 59. 
 
 495,872.17654-1728. 
 
 26. 
 
 1.6093295^-0.017. 
 
 60. 
 
 186,517.725 4-5280. 
 
 27. 
 
 1.6093295 4-0.0087. 
 
 61. 
 
 56,287.625 4-231. 
 
 28. 
 
 3-^1.7. 
 
 62. 
 
 782,847.375 4-43,560, 
 
 29. 
 
 3-^1.73. 
 
 63. 
 
 18,520 4-272.25. 
 
 30. 
 
 34-1.732. 
 
 64. 
 
 36,581.17-^2150.42. 
 
 31. 
 
 3 4-1.7321. 
 
 65. 
 
 10,000 4- 196. 
 
 32. 
 
 1.6093295 4-5280. 
 
 66. 
 
 $219.12 -M.025. 
 
 33. 
 
 2 4- 1.4142. 
 
 67. 
 
 22,000-^5645.376. 
 
 34. 
 
 54-2.236. 
 
 68. 
 
 0.0165 4-1.331. 
 
 35. 
 
 $25,000 4-117. 
 
 69. 
 
 75,555 4-1152. 
 
 36. 
 
 162-^14.72. 
 
 70. 
 
 12.2 4- 5.5056. 
 
 37. 
 
 1.27 4-19,800. 
 
 71. 
 
 77^-10.7716. 
 
 38. 
 
 0.5632-^1.6382. 
 
 72. 
 
 66 4-7.2426. 
 
 39. 
 
 1,872,760 4- 42,360. 
 
 73. 
 
 54.55 4- 1728. 
 
 40. 
 
 1897 4-192.93. 
 
 74. 
 
 46.88 4-44.723. 
 
 41. 
 
 0.00014-0.0872. 
 
 75. 
 
 0.874 4-444. 
 
 42. 
 
 14.1658 4- 1.8246. 
 
 76. 
 
 54,728 4- 5280. 
 
 43. 
 
 $10,150.75 4- $303.77. 
 
 77. 
 
 2.034 4- 0.0018. 
 
 44. 
 
 $312.12-^24.73. 
 
 78. 
 
 0.08748 -^ 10.8. 
 
 45. 
 
 $3115.20-4176. 
 
 79. 
 
 4443.33 4- 0.0037. 
 
 46. 
 
 $2840-^5.135. 
 
 80. 
 
 0.032048 4-20.03. 
 
 48 
 
44 DIVISION. 
 
 97. The Parenthesis. If numbers are included in a 
 parenthesis ( ), the first step is to reduce these numbers to 
 a single number as the signs direct. 
 
 (9+ 7 -1)-^ 5= 15-^5. (4 X 6-9) -r 5=15-^5. 
 
 48 -r (4 X 3) = 48 4- 12. 48 -r (24 -r 4) = 48 + 6. 
 
 Note. Instead of a parenthesis we sometimes use brackets [], 
 braces { }, or a vinculum — . Thus, (8-3), [8-3], { 8 - 3 }, 
 8 — 3, all have the same meaning. 
 
 In reducing expressions containing the signs +, — , X, 
 -7-, we first perform the operations indicated by the signs 
 X and + in the order in which they stand ; then the opera- 
 tions indicated by + an( i — • 
 
 Thus, 48-r8X2- 3X2 + GX 6 -r 2 = 12 - 6 + 15 = 21. 
 
 Exercise 18. 
 Reduce to a single expression : 
 
 1. (16-11 + 2) X 5. 3. (84 + 7) + (4 + 5-6). 
 
 2. (4 X 15) + (2 X 3). 4. (44-31) X (14 -11). 
 
 5. (96+6 + 5)-(6x8 + 16). 
 
 6. (52-5X7) + (4X5) -16 + 2. 
 
 7. 52-5X7 + 4X5-16 + 2. 
 
 8. (62 + 3-15) + 10 + (6X7-30) + 3. 
 
 98. Cancellation. The work of division may often be 
 shortened by dividing out, or cancelling, equal factors from 
 the divisor and dividend. 
 
 Divide 1596 by 84. 
 
 Solution. It is readily seen that 12 is contained in 84 
 12)1596 and in 1596. Dividing each by 12, we have 7 and 133 ; 
 7 )133 and we find the quotient 19 by dividing 133 by 7 by 
 19 short division. 
 
DIVISION. 45 
 
 •• 99*. Reciprocals. If the product of two numbers is 1, 
 each of the numbers is called the reciprocal of the other. 
 
 Thus, 2 X 0.5 = 1 ; hence, 2 is the reciprocal of 0.5, and 0.5 is the 
 reciprocal of 2. Again, 0.8 X 1.25 = 1 ; hence, 0.8 is the reciprocal of 
 1.25, and 1.25 is the reciprocal of 0.8. So 4 is the reciprocal of 0.25 ; 
 3 is the reciprocal of 0.33333 ; and so on. 
 
 100. When the dividend is 1, the quotient is the recipro- 
 cal of the divisor; and when the dividend is any other 
 number than 1, the quotient is the reciprocal of the divisor 
 multiplied by that number. Hence, 
 
 101. To divide a number by a divisor gives the same result 
 as to multiply the number by the reciprocal of the divisor. 
 
 Also, to multiply a number by a multiplier gives the same 
 result as to divide the number by the reciprocal of the multi- 
 plier. 
 
 102. The processes of multiplication and division are 
 often made much simpler by using the reverse process with 
 the reciprocal of the multiplier or of the divisor. 
 
 Thus, to divide 2.71827 by 37.5, since 37.5 = 3 X 12.5, we may 
 divide 2.71827 by 3, and multiply the quotient by 0.08. 
 
 Exercise 19. 
 
 By the use of reciprocals, find the value of : 
 
 1. 8X0.25. 10. 1764 X 0.025. 
 
 2. 171-^0.25. 11. 5381-^-0.025. 
 
 3. 876X1.25. 12. 7452-^0.875. 
 
 4. 132X2.5. 13. 651X0.33333. 
 
 5. 591 -f- 2.5. 14. 456X6.66667. 
 
 6. 756-^0.125. 15. 1554X0.16667. 
 
 7. 268X25. 16. 432^-1.33333. 
 
 8. 753-^25. 17. 375 -=-16.66667. 
 
 9. 567-^625. 18. 225-^6.66667. 
 
46 DIVISION. 
 
 103. Contracted Division of Decimals. Annexing a 
 digit to a partial dividend multiplies the partial dividend 
 by 10 and adds to the product the number expressed by 
 the digit. 
 
 Cutting off a digit from the right of the divisor subtracts 
 from the divisor the number expressed by the digit and 
 divides the remainder by 10. The quotient figure, there- 
 fore, will be the same whichever we do ; and this principle 
 can be applied to shorten the labor of division in examples 
 involving long decimals. 
 
 Divide 15.4323487 by 1.4142136, to four decimal places. 
 
 10*9123 it is necessary to determine, first, 
 
 14142136)154323487. the number of significant figures 
 
 14142136 required in the quotient. 
 
 129021270 Since 1.4 is contained in 15 ten 
 
 127279224 times, it is evident that the quo- 
 
 17420460 tient will have two integral places. 
 
 14142136 The two integral figures and the 
 
 32783240 four decimal figures make six, which 
 
 28284272 will be the number of figures re- 
 
 44989680 quired in the quotient. 
 42426408 
 
 104. In general, to determine the number of significant 
 figures required in the quotient, move the decimal point 
 of the divisor to the right of the first significant figure, 
 and move the decimal point of the dividend as many 
 places, and in the same direction, as the decimal point of 
 the divisor has been moved. The number of integral places 
 in the quotient can then easily be determined. 
 
 It is best not to cut off figures from the right of the 
 divisor until the number of figures still required in the quo- 
 tient is two less than the number of digits in the divisor. 
 In multiplying the divisor by each quotient figure, multiply 
 the figure of the divisor cut off, and carry the nearest ten. 
 
DIVISION. 47 
 
 The work may be arranged as follows : 
 
 Cut off the 6. The first product is increased by 1 for the 1 X 6 
 omitted. The first remainder is in- 
 
 10-9123 creased by 1 for the 8 in the dividend. 
 
 UlfflXW) 154323487. " Cut off the 3. As the divisor is not 
 
 1414214 contained in the partial dividend, we 
 
 129021 also cut off the 1. The product by 9 
 
 127279 is increased by 1 for the 9 X 1 omitted. 
 
 1742 Cut off the 2. As 1 X 2 is less than 5, 
 
 1414 the product by 1 is not increased. Cut 
 
 328 off the 4. The product by 2 is in- 
 
 283 creased by 1 for 2 X 4 omitted. Cut off 
 
 45 the 1. The product by 3 is not in- 
 
 42 creased, for 3 X 1 is less than 5. Hence, 
 
 105. To Divide Decimals by the Contracted Method, 
 
 Determine the number of significant figures required in 
 the quotient. Begin to cut off figures from the right of the 
 divisor, when the number of figures still required in the quo- 
 tient is two less than the number of digits in the divisor. In 
 multiplying the divisor by each quotient figure, multiply the 
 figure of the divisor cut off, carrying the nearest ten. 
 
 Exercise 20. 
 
 Divide by the contracted method : 
 
 1. 11.4285285 by 3.1415927 to six decimal places. 
 
 2. 0.004239239 by 3.2783278 to five decimal places. 
 
 3. 437 by 215.253 to five decimal places. 
 
 4. 0.0053 by 72.654 to eight decimal places. 
 
 5. 6 by 0.1573 to three decimal places. 
 
 6. 0.11 by 1937.43 to eight decimal places. 
 
 7. 44.2 by 0.768547 to five decimal places. 
 
 8. 0.6587465 by 0.5475869 to five decimal places. 
 
 9. 46 by 0.00751515151 to three decimal places. 
 
48 DIVISION. 
 
 106. Equations. A statement that two expressions of 
 number have the same value is called an equation. 
 
 Every equation consists of two expressions of number 
 connected by the sign of equality, = ; the two expressions 
 are called the sides or members of the equation. 
 
 Thus, 2 X 4 = 8, and G + 4 + 5 = 18 — 3 are equations. 
 
 107. Since the two members of an equation are equal, 
 if both members are increased by, diminished by, multiplied 
 by, or divided by equal numbers, the results are equal. 
 
 108. Division of Powers of the Same Number. 
 
 SinC e f-5X5X5X5X5 =5x5 = 
 5* 5X5X5 
 
 5» 5X5X5 1 .. . 
 
 and 5' = 5X5X5x5l^5 = 5' ,theref0re ' 
 
 If the exponent of the power in the dividend is greater 
 than the exponent of the power in the divisor, 
 
 The quotient is that number with an exponent equal to the 
 exponent of the dividend minus that of the divisor. 
 
 If the exponent of the power in the divisor is greater 
 than the exponent of the power in the dividend, 
 
 The quotient is one divided by the number with an exponent 
 equal to the exponent of the divisor minus that of the dividend. 
 
 Exercise 21. 
 
 Express the value of : 
 
 1. 10 1 ; 10 2 ; 10 8 ; 10 4 ; 10 5 ; 10 6 ; 10 7 ; 10 8 . 
 
 2. 10 3 -^10 2 ; 10 8 -M0«; 10 5 -M0 8 ; 10 9 -f-10 4 . 
 
 3. 9.99 4 -f-9.99 2 ; 9.99 108 -^9.99 110 ; 9.99 16 -^9.99 18 . 
 
 4. 1.01 2 * -7-1.01"; 1.01 M -5-1.01 u ; 1.01 u -r-1.01 M . 
 
DIVISION. 49 
 
 109. Proof of Division by Casting out Nines. Divide 
 1,348,708 by 498. 
 
 If we divide 1,348,708 by 498, we have for quotient 2708 and for 
 remainder 124. 
 
 That is, 1,348,708 = 498 X 2708+124. 
 
 But, § 70, 1,348,708 = a number of nines + 4, 
 
 498 X 2708 = a number of nines + 6, 
 and 124 = a number of nines + 7. 
 
 Hence, 498 X 2708 + 124 = a number of nines + 6+7 
 
 = a number of nines + 4. 
 
 Therefore, the two members of the equation 1,348,708 = 498 X 
 2708 + 124, when divided by 9, give the same remainder 4. 
 We may arrange the work as follows : 
 
 ( 8 2708 
 
 Multiplyj g 498)1348708 4 
 
 Add 
 
 6 .... 24 996 
 
 3527 
 
 3486 
 
 4108 
 3984 
 
 _7 v 124 
 
 13 '. 
 
 In examples involving large numbers always apply this 
 test. 
 
 Exercise 22. 
 
 Find the following quotients, and test the accuracy of 
 the work by casting out the nines : 
 
 1. 157,056.41692-+ 2150.42. 
 
 2. 49,652.7896 -+5645.376. 
 
 3. 2000 -+3.1416. 
 
 4. 18.97657 -+0.7854. 
 
 5. 28,762.762 +- 14.39874. 
 
 6. 8747.119 -+6.58298. 
 
 7. 28,752.1 -+149.796. 
 
50 DIVISION. 
 
 Exercise 23. — Review. 
 Express in words : 
 
 1. 327.244. 3. 0.390012. 5. 0.0000008. 
 
 2. 80.9056. 4. 20,000.002. 6. 41.27105. 
 
 Write in figures : 
 
 7. Two hundred thirty-five and eight hundred thirty-five 
 
 thousandths. 
 
 8. Seventy-four and two hundred three thousand six mil- 
 
 lionths. 
 
 9. Twelve hundred and eight thousand three ten-mil- 
 
 lionths. 
 
 10. Five thousand sixty-four niillionths. 
 
 11. One million and four tenths. 
 
 12. Six hundivd-millionths. 
 
 Multiply and divide : 
 
 13. 789.365 by 10 ; by 100 ; by 100,000. 
 
 14. 0.004 by 100 ; by 10,000 ; by 1000. 
 
 15. 436 by 1,000,000 ; by 1000 ; by 10. 
 
 16. 0.1 by ten ; by ten millions. 
 
 Find the value of : 
 
 17. 21.3706 + 15.243 + 1.8954 + 0.026891+5.328 + 20.74. 
 
 18. 57 + 0.0057 + 6.8 + 1200 + 0.847 + 159.2 + 3. 
 
 1 9. 0.0012+ 10 + 5.8281 + 5 + 39.43 + 0.6827 + 1. 
 
 20. 23.9875 - 12.4764 ; 35.14732 - 27.62815. 
 
 21. 102.1274 - 83.072 ; 39.801 - 17.9645. 
 
 22. 30 — 5.2817; 1.7 — 0.8469. 
 
 23. 1-0.54237 ; 100 — 0.00176. 
 
 24. 24.271 — 3.6485 + 15.271 — 13.256 - 14.125. 
 
 25. 52 + 0.52 — 17.8946 — 30.254 — 0.5 + 21.12. 
 
DIVISION. 51 
 
 26. 41.289 X 0.5 ; 0.268 X 0.9 ; 0.112 X 0.2. 
 
 27. 2.435 X 4.23 ; 71.651 X 3.37 ; 0.251 X 0.04. 
 
 28. 0.0012 X 0.005 ; 2.26823 X 200 ; 5.6125 X 0.0768. 
 
 29. 0.7 X 7 X 0.07 ; 0.15625 X 23.7 X 0.00192 X 5. 
 
 30. (2.465 + 1.21) X (3.2 - 2.89). 
 
 31. (3.01) 2 ; (0.045) 2 ; (0.0081) 2 ; (5.1004) 3 ; (0.76) 3 . 
 
 32. (0.125) 2 X (0.32) 3 . 
 
 Divide : 
 
 33. 291.84 by 6 ; 0.12936 by 12 ; 7.92801 by 0.9. 
 
 34. 58.383 by 0.39 ; 0.28744 by 0.08 ; 491.205 by 0.065. 
 
 35. 68.325 by 6.25 ; 0.732 by 1.6; 1208.88 by 0.438. 
 
 36. 498 by 0.0125 ; 7 by 0.007 ; 1000 by 0.0001. 
 
 37. 0.235 by 10.24 ; 27 by 12 ; 0.00507702 by 0.0283. 
 
 38. 89.3 by 0.00752 ; 74.1 by 0.0256 ; 1 by 0.128. 
 
 39. 0.39842 by 3.7164 ; 281.5 by 13.789 ; 0.0005 by 0.0028. 
 
 40. 63.04128 by 912.85 ; 287.209 by 0.00493 ; 2000 by 
 
 0.0059. 
 
 Exercise 24. — Review. 
 Find the value of : 
 
 1. 1.4-1-2.08 + 3.895. 
 
 2. 2.8 + 2.08-1-0.28+0.028 + 0.812. 
 
 3. 1.667 + 0.4 + 0.286 + 6.08 + 0.636 + 0.931. 
 
 4. 6.125 — 0.57. 
 
 5. (4.625 + 1.146) -(1.2 + 3.571). 
 
 6. 6.913 -(2.85 — 0.937). 
 
 7. 24 — 2.4 + (5 -3.508) — 3.092. 
 
 8. 10 - (4.25 - 2.5 + 2 — 0.625 — 0.4 — 2.02) —0.295. 
 
 9. 1.5 X 0.08 X 0.5. 
 
 10. 0.1204X0.0168 X 100. 
 
 11. 0.04X3.25X0.06. 
 
 12. 36 X 0.002 X 2.05 X 0.00765. 
 
 13. 0.139 X 28 + 42 X 0.002 + 6 X 0.004 — 0.05 X 20. 
 
52 DIVISION. 
 
 14. (10 - 1.25) X 0.2 + 0.02 X 2.8 + (80.3 X 0.1 - 5.3) X 
 10 — 805.3X0.02. 
 
 15. 28.3696^-1.49. 19. 0.28744-^800. 
 
 16. 0.27^-0.00225. 20. 491.205-^650. 
 
 17. 8.8779 -f- 175.8. 21. 68.325-^6250. 
 
 18. 0.0427^92.3. 22. 0.732 -f- 16,000. 
 
 23. 1208.88 -f- 0.438. 
 
 24. 2 -r 0.01 - (0.2 -f- 0.02 + 0.8 *f- 10) + 36.48 -^ 8 
 
 — (4 -^- 0.05 — 2 + 0.6 -^ 1.25). 
 
 25. 72.2 -^- 10 - 2 -^ (0.5 -^ 1.60) + 2.125 -f- (1.75 - 0.5). 
 
 Exercise 25. — Review. 
 
 1. What number subtracted 88 times from 80,005 will 
 leave 13 as a remainder ? 
 
 2. If 7 men can build a wall in 16 days, how many men 
 will it take to build a wall three times as long in half the 
 time? 
 
 3. How many minutes are there between 25 minutes 
 past 8 in the morning and midnight? 
 
 4. If the velocity of sound is 1090 feet per second, 
 at what distance is a gun fired the report of which I hear 
 11 seconds after seeing the flash? (5280 feet make a 
 mile.) 
 
 5. How long will it take to travel 30.2375 miles at 
 the rate of 8.85 miles per hour ? 
 
 6. If the circumference of a circle is 3.1416 times the 
 diameter, find the circumference of a circle whose diameter 
 is 6.8 feet ; also, find the diameter of a circle whose cir- 
 cumference is 20 inches. 
 
 7. How much wire will be required to make a hoop 30 
 inches in diameter, allowing 2 inches for the joining ? 
 
 8. How many times would the hoop of Ex. 7 turn 
 in going half a mile ? 
 
DIVISION. 53 
 
 9. Cork, whose weight is 0.24 of the weight of water, 
 weighs 15 pounds per cubic foot. What is the weight of 
 6 cubic feet of oak, if the weight of oak is 0.934 of the 
 weight of water ? 
 
 10. From what number can 847 be subtracted 307 times 
 and leave a remainder of 49 ? 
 
 11. What is the 235th part of 141,235 ? 
 
 1 2. What will 343 barrels of flour cost at $ 6.37 a barrel ? 
 
 13. Twelve makes a dozen, and 12 dozen makes a gross. 
 How many steel pens in 28 gross ? What will a gross of 
 eggs cost at 27 cents a dozen ? 
 
 14. How much must be added to $4429 to make the sum 
 equal to 43 X $241 ? 
 
 15. What number deducted from the 26th part of 2262 
 will leave the 87th part of the same number ? 
 
 16. At the ordinary rate, 123 words a minute, how long 
 will it take a man to deliver a speech of 15 pages, if each 
 page contains 28 lines, and each line 11 words ? How long 
 would it have taken Daniel Webster to deliver the same 
 speech, whose rate was^93 words a minute ? 
 
 17. How long will it take a railway train to go from 
 New York to San Francisco, 3310 miles, at the rate of 
 1973 feet a minute ? 
 
 18. How many hours will it take to count a million, at 
 the rate of 67 a minute ? 
 
 19. If you put into a box 17 cents a day, including 
 Sundays, beginning January 1 and ending July 4, how 
 much money will there be in the box ? 
 
 20. If a man's income is $3000 a year, and his daily 
 expenses average $7.68, what does he save in a year ? 
 
 21. In a question of division the quotient was 87.83, the 
 divisor, 759. What was the dividend ? 
 
 22. What is the nearest number to 7196 that will con- 
 tain 372 without a remainder ? 
 
54 DIVISION. 
 
 23. It is 3.1416 times as far round a wheel as across it. 
 How many times will a wheel 4.5 feet across turn in 
 going 23 miles of 5280 feet each ? 
 
 24. How many gallons of 231 cubic inches are contained 
 in a cubic foot of 1728 cubic inches ? in a bushel of 2150.42 
 cubic inches ? How many cubic feet in a bushel ? How 
 many bushels in 31.5 gallons ? 
 
 25. Seven children had left to them $7186 apiece ; one 
 died, and his share was divided among the surviving six. 
 How much had each then ? 
 
 26. How long will it take 2 men to do what 1 man can 
 do in 6 days ? what 4 men can do in 3 days? what 3 men 
 can do in 4 days ? 
 
 27. Divide $1.80 among Thomas, Richard, and Henry in 
 such a way that Henry shall receive 3 cents for every 
 5 cents that Thomas gets, and Richard shall receive 2 cents 
 for every 3 cents that Henry gets. 
 
 28. Divide $87.84 between B and C so that C shall get 
 $19 as often as B gets $17. 
 
 29. Three partners received for goods: one, $371.63; 
 the second, $285.40; the third, $411.91. They paid for 
 the goods $879.34. and divided the profit equally among 
 them. How much did each receive ? 
 
 30. If there are 12 inches in a foot, how many inches 
 long is a wall 35 feet in length ? If a brick and its share 
 of mortar is 8.4 inches long, how many bricks in length 
 is the wall ? 
 
 31. If a brick and its mortar is 2.4 inches high, how many 
 bricks are required to build a wall 12 feet high, 35 feet 
 long, if the width of the wall is the width of two bricks? 
 
 32. What is the total weight of the wall of Ex. 31, if a 
 brick with its share of the mortar weighs 4.13 pounds ? 
 What is the weight after a long rain, when the weight is 
 increased to 4.27 pounds for each brick ? 
 
DIVISION. 55 
 
 33. How many pounds does each foot in length of the 
 wall of Ex. 31 weigh ? 
 
 34. If 60.98 cubic inches of brick weigh 4 pounds, how 
 many cubic inches of brick weigh 1 pound ? How many 
 pounds will a cubic foot (1728 cubic inches) weigh ? 
 
 35. If a cubic foot of water weighs 62.5 pounds, how 
 many times as heavy as water is brick ? 
 
 36. Light moves through the air at the rate of 186,500 
 miles a second. How many times can it go around the 
 earth in a second, if the distance round the earth is 
 24,897.714 miles? 
 
 37. Light moves through the air at the rate of 300,190 
 kilometers a second. How many times can it go around 
 the earth in a second, if the distance round the earth is 
 40,007.5 kilometers? 
 
 38. A minute is 60 seconds. How many miles and how 
 many kilometers can light travel through air in a minute ? 
 
 39. An hour is 60 minutes. How many miles and how 
 many kilometers can light travel in an hour ? 
 
 40. The distance round the earth, given in Ex. 37, is 
 measured on a north and south line. Around the equator 
 the distance is 40,075.45 kilometers. How many times 
 could light move round the equator in one minute ? 
 
 41. Find the reciprocal of the difference between 31.24 
 and 31.23768. 
 
 42. The Hanoverian mile is 25,400 Hanoverian feet 
 long, and each foot is 0.9542 of an English foot. Find to 
 four places of decimals the fraction that an English mile of 
 5280 English feet is of a Hanoverian mile. 
 
 43. Express in inches the length of a meter, given that 
 a meter is one ten-millionth of a quarter of the earth's cir- 
 cumference, that the circumference is 3.14159 times the 
 diameter, that the diameter of the earth is 7911.7 miles, 
 and that a mile is 5280 X 12 inches, 
 
56 
 
 DIVISION. 
 
 44. How must a number be altered that its reciprocal 
 may be doubled ? 
 
 45. What effect is produced on the sum of two numbers, 
 if the same number is added to each of them ? What 
 effect on the difference ? 
 
 46. What effect is produced on the product of two 
 numbers, if both numbers are multiplied by the same 
 number ? What effect on the quotient ? 
 
 47. What effect is produced on the remainder, if both 
 divisor and dividend are multiplied by the same number ? 
 If both are divided by the same number ? 
 
 48. In going from one planet to another, light probably 
 moves faster than in air. Suppose it moves at the rate of 
 309,800 kilometers a second, how long would it take light 
 to perforin each of the following journeys : 
 
 Moon to Earth 
 Sun to Earth 
 Sun to Mercury 
 Sun to Venus 
 Sun to Mars . 
 Sun to the Asteroids 
 Sun to Jupiter 
 Sun to Saturn 
 Sun to Uranus 
 Sun to Neptune . 
 Sun to the nearest star 
 
 375,600 kilometers. 
 . 147,250,000 
 . 56,900,000 
 . 100,400,000 " 
 
 224,100,000 " 
 
 . 400,000,000 
 
 765,400,000 " 
 
 . 1,403,000,000 
 . 2,817,000,000 " 
 
 . 4,421,000,000 
 24,000,000,000,000 
 
 49. A kilometer is about 0.6214 of a mile. How many 
 miles is each of the planets from the sun ? 
 
 50. If 11.75 tons of coal cost $82.25, what will 21.4 
 tons cost ? 
 
 51. Find the number of hours it will take a locomotive 
 running at the rate of 27 miles an hour to make the distance 
 passed over in 13.25 hours by another locomotive that has 
 a velocity of 43.5 miles an hour. 
 
DIVISION. 57 
 
 Review Questions. 
 
 What are units ? numbers ? integral numbers ? decimal numbers ? 
 abstract numbers ? concrete numbers ? like numbers ? 
 
 What is notation ? numeration ? Where is the decimal point 
 placed ? What do figures in the first place to the left of the decimal 
 point represent ? in the second place ? in the third place ? in the 
 fourth place ? in the first place to the right of the decimal point ? in 
 the second place ? in the third place ? in the fourth place ? Which 
 place do the units of a number occupy ? the tens ? the hundreds ? 
 the thousands ? the tenths ? the hundredths ? the thousandths ? At 
 what rate does the value of a figure increase from right to left ? In 
 separating a row of figures into periods where do we begin ? How 
 many figures in each period ? What is the period on the right called ? 
 the second period ? the third period ? Which period do we write 
 first ? Which period do we read first ? How do we write a decimal ? 
 How do we read a decimal ? In reading, by what word do we connect 
 the integral and decimal parts of a number ? 
 
 What is addition? What is the result called? What kind of 
 numbers only can be added ? What is the sign of addition ? Does 
 it make any difference in what order the numbers are added ? In 
 addition how do we arrange the decimal points ? How do we prove 
 that the work of addition Ts correct ? 
 
 What is subtraction ? What is the greater number called ? the 
 smaller ? the result ? What kind of numbers must the minuend, 
 subtrahend, and remainder be ? What is the sign of subtraction ? 
 In subtraction, how do we arrange the decimal points ? How do we 
 prove that the work of subtraction is correct ? 
 
 What is multiplication ? the multiplicand ? the multiplier ? the 
 product ? What kind of a number must the multiplier be ? What 
 are the factors of a product ? Does it make any difference in what 
 order the factors are multiplied ? What is the sign of multiplication ? 
 How many decimal places must the product have ? How do we prove 
 that the work of multiplication is correct ? 
 
 What is division ? the dividend ? the divisor ? the quotient ? If 
 the divisor is not an integer how can we make it an integer without 
 altering the quotient ? If the divisor is an integer where do we place 
 the decimal point in the quotient ? What is the advantage of writing 
 the quotient over the dividend in long division ? What is the sign of 
 division ? How do we prove that the work of division is correct ? 
 
CHAPTER V. 
 
 METRIC MEASURES. 
 
 110. To measure a quantity is to find the number of 
 times it contains a known quantity of the same kind, called 
 the unit of measure. 
 
 111. The metric system is a system of weights and 
 measures expressed in the decimal scale. 
 
 112. The standard meter, as defined by law, is the 
 length of a bar of very hard metal, carefully preserved at 
 Paris, accurate copies of which are furnished the govern- 
 ments of all civilized countries. 
 
 The meter was intended to be one ten-millionth of the distance 
 from the equator to the north pole, but more careful measurements 
 show that this distance is 10,001,887 meters. 
 
 113. The principal units of the metric system are : 
 
 The meter ( m ) for lengths ; 
 The square meter ( < i m ) for surfaces ; 
 The cubic meter ( ebm ) for large volumes; 
 The liter Q) (lee'ter) for smaller volumes j 
 The gram ( g ) for weights. 
 
 114. All these units are divided and multiplied deci- 
 mally, and the size of the measures thus produced is shown 
 by a prefix ; namely, deka, meaning 10 ; hekto, meaning 
 100 ; kilo, meaning 1000 ; myria, meaning 10,000 ; and deci, 
 meaning 0.1 ; centi, meaning 0.01 ; milli, meaning 0.001. 
 
 As in United States money we seldom- speak of any- 
 thing except dollars and cents, so in metric measures 
 only measures printed in black letter are in common use. 
 
METRIC MEASURES. 59 
 
 Measures of Length. 
 
 115. The principal unit of length is the meter. 
 
 Table. 
 
 10 millimeters ( mm ) = 1 centimeter ( om ). 
 
 10 centimeters ss 1 decimeter ( dm ). 
 
 10 decimeters = 1 meter ( m ). 
 
 10 meters = 1 dekameter ( dkm ). 
 
 10 dekameters = 1 hektometer ( hm ). 
 
 10 hektometers = 1 kilometer ( km ). 
 
 10 kilometers = 1 myriameter. 
 
 Note. The names of all compound units are accented H 
 on the first syllable ; thus, millimeter, kilometer. <? 
 
 116. A length given in one unit may be ex- g- 
 pressed in another unit by simply moving the ■ 
 decimal point. o 
 
 Thus, 17,856,342 mm may be written as kilo-meters by § 
 observing that milli-meters are changed to meters by 2 
 moving the point three places to the left; and these 
 meters into kilo-meters Joy carrying it three places K 
 further, making in all six places. Therefore, 
 17,856,342""!!=: i7.856342 k m. 
 
 Again, 4.876326 km may be written as centi-meters by 
 observing that kilo-meters are changed to meters by 
 moving the point three places to the right, and meters 
 to centi-meters by moving it two places further, making 
 in all Jive places. Therefore, 
 
 4.876326 km = 487,632.6cm 
 
 r, 
 
 g 
 
 117. The rule, therefore, for this conversion is : 
 
 First count the number of places needed to convert the given 
 measure into terms of the principal measure; then the number 
 needed to convert the principal into the required measure. 
 
 118. Before adding or subtracting, the quantities must 
 be written in terms of the same unit of measure. 
 
60 METRIC MEASURES. 
 
 Exercise 26. 
 
 1. Change 5427 m to kilometers ; to millimeters ; to 
 centimeters. 
 
 2. How many meters in 6853 mm ? how many centi- 
 meters ? what part of a kilometer ? 
 
 3. Write 49.7 m as centimeters ; as millimeters ; as the 
 decimal of a kilometer. 
 
 4. How many centimeters in 12.4 km ? how many milli- 
 meters ? 
 
 5. Change 1230 m to kilometers ; to centimeters. 
 
 6. Write 1230 cm as meters ; as millimeters. 
 Find in meters the value of the following : 
 
 7. 0.435 m + 852 cm + 4263 mm + O.^S 1 "" 
 
 8. 0.927 km — 6495 cm ; 4.37 cra — 42.87 mm . 
 
 9. 8 X 0.0457 km ; 3.04 X 60.93 cm ; 5.43 X 67.2 mm . 
 
 10. 38,01 9 mm 4- 0.097; 0.41 km -h 25.625. 
 
 11. At fl.87 a meter, what is the cost of 6.20 m of 
 cloth ? 
 
 12. At $0.75 a meter, what is the cost of 60™ of cloth ? 
 
 13. From a piece of cloth containing 47.60 m a tailor 
 cuts off three pieces : the first of 3.80 m , the second of 
 1.30 m , and the third of 45 cm . How many meters of the 
 cloth are left ? 
 
 14. What is the value of 60 cm of cloth at $5.20 a meter ? 
 
 15. If $6.00 is paid for a railroad ticket to travel 
 440 km , what is the fare per kilometer ? 
 
 16. If a train goes 288 km in 9 hours, how many meters 
 does it go in a minute ? (1 hour = 60 minutes.) 
 
 17. If a man walks at the rate of 6 km an hour, what 
 part of an hour will it take hini to walk 420 m ? 
 
 18. A railroad carried 412 passengers 18 km for $88,992 ; 
 at the same rate, what will it receive for carrying 350 
 passengers 35 km ? 
 
METRIC MEASURES. 61 
 
 Measures of Surface. 
 
 119. A square is a flat surface with four 
 square corners and four equal straight sides. 
 
 120. A square meter is a square one meter 
 
 Square. on a s id e> 
 
 The principal unit of surface is the square meter. 
 
 121. Suppose the square in the mar- 
 gin to represent a square meter. It is 
 divided into ten equal horizontal bands, 
 and each band is one tenth of the square 
 meter. Each band can be divided, as 
 the upper one is, into ten little squares 
 
 i i i i i i i i i : 
 
 measuring one tenth of a meter on a side. Each of these 
 squares will be 0.1 of the band, or 0.01 of the whole 
 square. The square meter, therefore, contains 10 X 10, or 
 100 square decimeters. 
 
 In like manner, a square decimeter contains 100 square 
 centimeters, and therefore a square meter contains 100 X 
 100, or 10,000 square centimeters. 
 
 In like manner, a square meter contains 1000 X 1000, or 
 1,000,000 square millimeters. 
 
 Table. 
 
 100 square millimeters (i mm ) = 1 square centimeter (<i cm ). 
 
 100 square centimeters = 1 square decimeter (<i dm ). 
 
 100 square decimeters = 1 square meter (i m ). 
 
 100 square meters = 1 square dekameter (i dkm ). 
 
 100 square dekameters = 1 square hektometer (i hm ). 
 
 100 square hektometers = 1 square kilometer (<i km ). 
 
 122. In measures of length each unit is io times as 
 large as the next smaller unit, but in measures of surface 
 each unit is ioo times as large as the next smaller unit. 
 
62 METRIC MEASURES. 
 
 123. In the measurement of land, the square meter is 
 called a centar, the square dekameter is called an ar, and 
 the square hektometer is called a hektar. 
 
 Table of Land Measures. 
 
 100 centers ( ca ) = 1 ar (•). 
 100 ars = 1 hektar (ha). 
 
 124. In surface measures, when the ar is the unit, ioo 
 ars make a hektar ; but when the square meter is the unit, 
 ioo X ioo, or 10,000 square meters make a square hekto- 
 meter or hektar. 
 
 Exercise 27. 
 
 1. Change l,854,276 qm to hektars ; to square kilome- 
 ters. 
 
 2. How many hektars in 2.7856 qkm ? 
 
 3. Write 1.7431 qm as square centimeters ; as square 
 millimeters. 
 
 4. How many square kilometers in 17,467.5 h * ? 
 
 5. How many square meters in 1.3614 qkm ? 
 
 6. How many square meters in 2.25 ha ? 
 
 7. How many square centimeters in 0.0137 qm ? 
 
 8. Write 3.57 l qcm as square millimeters. 
 
 9. If a field contains 7500 ca , how many ars does it 
 contain ? What part of a hektar ? 
 
 10. How many square meters must be added to 22,612 qm 
 to make 4 ha 62 a 17 ca ? 
 
 11. A field containing 72.4 a is sold at 15 cents a square 
 meter. What is received for the field ? 
 
 12. If C2 a 12°* of land is sold for $1366.64, what is the 
 price per square meter? 
 
 13. How many square centimeters must be taken from 
 12,473 qcm to leave l qm 14 qdm 53 qcm ? 
 
METRIC MEASURES. 
 
 63 
 
 Measures of Volume. 
 
 125. A cube is a solid bounded by six equal squares. 
 Each bounding square is called a face, and 
 the intersection of two faces is called an 
 edge. 
 
 126. A cubic meter is a cube one meter on 
 Cube - an edge. 
 The principal unit of volume is the cubic meter. 
 
 127. The cubic meter can be divided into 10 layers, 
 each a meter square and a decimeter 
 thick. Each layer will, therefore, be 
 0.1 of a cubic meter. Again, each 
 layer can be divided into 10 equal 
 parts. Each part will, therefore, be 
 0.1 of the layer, or 0.01 of the meter, 
 and will be a decimeter square and 
 a meter long. Also, each one of 
 these parts can be divided into 10 
 equal parts, each of which will be a cubic decimeter, and 
 will be 0.1 of 0.01, or 0.001 of the cubic meter. 
 
 The cubic meter, therefore, contains 1000 cubic decime- 
 ters. 
 
 In like manner, a cubic decimeter contains 1000 cubic 
 centimeters, and a cubic centimeter contains 1000 cubic 
 millimeters. 
 
 / 
 
 A 
 
 - 
 
 
 
 
 
 
 
 
 
 
 p fi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '/////// / ../. 
 
 
 
 Table. 
 
 1 1000 cubic millimeters ( cmm ) 
 1000 cubic centimeters 
 1000 cubic decimeters 
 
 = 1 cubic centimeter ( ccm ). 
 = 1 cubic decimeter ( cdm ). 
 = 1 cubic meter ( cbm ). 
 
 128. In measures of volume each unit of measure is 
 iooo times as large as the next smaller unit of measure. 
 
64 METRIC MEASURES. 
 
 129. In the measurement of wood, the cubic meter is 
 called a ster. 
 
 Ster of Wood. 
 
 Table of Wood Measures. 
 
 10 decisters ( d * 1 ) = 1 ster ( 8t ). 
 
 10 sters = 1 dekaster ( dk8t ). 
 
 Exercise 28. 
 
 1. Write 2.25 cbm as cubic centimeters. 
 
 2. Change 2,162,875 ccm to cubic meters. 
 
 3. Change 0.0175 cbm to cubic millimeters. 
 
 4. Change 46,164 ccm to cubic decimeters. 
 
 5. What is the equivalent of 0.875 dk8t in cubic meters ? 
 in cubic centimeters ? 
 
 6. How many sters are there in 14.75 dk8t of wood? 
 how many decisters ? 
 
 7. What is the cost of 28.25 dk8t of wood at $1.25 a 
 ster ? 
 
 8. Find the cost of an oak beam containing 1250 cdm at 
 $25 a cubic meter. 
 
 9. How many cubic centimeters must be added to 
 l,262,376 ccm to make 2 cbm 2 cdm 2 ccm ? 
 
 10. How many cubic millimeters must be taken from 
 22,350,000,000 cmm to leave 20 cbm 22 cdm 222 ccm ? 
 
METRIC MEASURES. 
 
 65 
 
 Measures of Capacity. 
 
 130. The principal unit of capacity is the liter. 
 
 131. In measuring liquids, grain, etc., the cubic deci- 
 meter is called a liter. 
 
 Table. 
 
 10 milliliters ( ml ) 
 10 centiliters 
 10 deciliters 
 10 liters 
 10 dekaliters 
 10 hektoliters 
 
 = 1 centiliter ( cl ). 
 = 1 deciliter (<U). 
 = 1 liter (*). 
 = 1 dekaliter (*"). 
 = 1 hektoliter ( hl ). 
 = 1 kiloliter (*i). 
 
 132. In measures of capacity each unit of measure is 
 io times as large as the next smaller unit of measure. 
 
 Exercise 29. 
 
 1. How many liters in 1.7 cbm ? in 157,854 ccm ? 
 
 2. How many cubic centimeters in 9.5 1 ? in 0.015 1 ? 
 
 3. Change 1.25 w to cubic centimeters ; to the fraction 
 of a cubic meter. 
 
 4. Change 431.88 1 to hekto- 
 liters; to the fraction of a cubic 
 meter. 
 
 5. Write 0.375 cbm as liters; as 
 cubic centimeters. 
 
 6. Write 734,159.651 ccm as 
 liters ; as hektoliters ; as cubic 
 meters. 
 
 7. How many cubic meters in 8,573,412.867 CCIn ? 
 
 8. Change 0.734578912 cbm to cubic centimeters ; to liters. 
 
 9. Change 1731. 5 1 to cubic meters ; to cubic centi- 
 meters. 
 
 Liter = Cubic Decimeter. 
 
66 METRIC MEASURES. 
 
 Measures of Weight. 
 
 133. The units of weight are the weights of units of 
 
 pure water taken at its greatest 
 
 density, that is, a little above the 
 
 freezing point. 
 cA^ The principal unit is the gram, 
 
 which is the weight of a cubic cen- 
 
 Cubic Centimeter. Gram Weight, timeter of Water. 
 
 Table. 
 
 10 milligrams ( m «) = 1 centigram (<*). 
 
 10 centigrams = 1 decigram ( d «). 
 
 10 decigrams = 1 gram (*). 
 
 10 grams = 1 dekagram ( dk «). 
 
 10 dekagrams = 1 hektogram (*>*). 
 
 10 hektograms = 1 kilogram ( k «). 
 
 1000 kilograms = 1 metric ton (*)• 
 
 Note. The kilogram is often called a kilo. 
 
 134. In measures of weight each unit of weight is io 
 times as great as the next smaller unit of weight. 
 
 135. A cubic centimeter of water weighs a gram. 
 A liter of water weighs a kilogram. 
 
 A cubic meter of water weighs a ton. 
 
 Exercise 30. 
 
 1. How many kilos in 1.73'? in 0.341 of a ton ? 
 
 2. How many kilos will a hektoliter of water weigh ? 
 
 3. Change 13,756 mg to grams ; to the fraction of a kilo. 
 
 4. What is the weight in grams of 346.1 ccm of water ? 
 
 5. Find the weight in kilograms of 0.37615 cbm of water. 
 
 6. Change 0.6778 kg to milligrams. 
 
 7. How many milligrams in the third part of 17.4* ? 
 
METRIC MEASURES. 67 
 
 Exercise 31. 
 
 1. Add 17.3 m , 87.41 m , 271 cm , 380 mm , and 1.79 m . 
 
 2. Add 15.87 m , 394.6 dm , 47.52 ra , 7538 cm , and 75.89 m . 
 
 3. Add 187 cm , 49.3 ra , 317 mm , and 6.138 m . 
 
 4. In a room the doorsill is 3 cm high; the door, 2.34 m ; 
 the finish over the door, 13.7 cm ; and the distance from the 
 finish to the ceiling is 93 cm . What is the height of the 
 room ? 
 
 5. The distance to the post office is 3.31 km ; thence to 
 the mill, 1.711 km ; thence to the store, 3.718 km ; thence 
 home, 2.543 km . How long is the circuit ? 
 
 6. The distance from Portland, Me., to Boston is 174 km ; 
 Boston to Albany, 317 km ; Albany to Buffalo, 478 kra ; Buffalo 
 to Chicago, 863 km ; Chicago to Omaha, 789 km ; Omaha to 
 Cheyenne, 830 km . How far is it from Cheyenne to Port- 
 land ? from Cheyenne to Albany ? from Boston to Chicago ? 
 from Boston to Cheyenne? 
 
 7. If I travel 789.7 km a day, how far shall I go in 7 
 days ? in 8.5 ? in 19.6? in 27.8 ? in 365 ? 
 
 8. How much will 3 m of cloth cost at $1.37 a meter ? 
 How much will 5.38 m cost at $2.63 -a meter ? 
 
 9. How much will 13.4 k s of opium be worth at $8.48 
 a kilo ? 28.79 k * at $7.96 a kilo ? 
 
 10. If one barrel of flour weighs 88.9 kg , how many 
 barrels can be filled from 444.5* of flour ? 
 
 1 1 . How many steps 80 cm long will a man take in walk- 
 ing a kilometer ? 
 
 12. At 16 cents a liter, what is the cost of 52.4 W of 
 olive oil ? 
 
 13. What is the cost of 6 dk8t 4 8t of oak wood at $1.75 
 per ster ? 
 
 14. If a pasture contains 22,408 ca , how many ars does 
 it contain ? how many hektars ? 
 
68 METRIC MEASURES. 
 
 136. A flat surface bounded by straight lines or by a 
 curved line is called a plane figure. 
 
 137. A circle is a plane figure 
 bounded by a curved line called the 
 circumfere7ice, all points of which 
 are equally distant from a point 
 within called the centre. 
 
 A straight line drawn through 
 the centre, having its ends in the 
 circumference, is called a diameter ; 
 circle - and half a diameter is called a radius. 
 
 138. If the diameter of a circle is multiplied by 8.1416, 
 the product is the length of the circumference. 
 
 15. Find the circumference of a circle l m in diameter. 
 
 16. Find to the nearest tenth of a millimeter the circum- 
 ferences of circles whose diameters are respectively 83 m ; 
 3.71 m ; 32.8 m ; 10.4 cm ; 11.8 cm ; 167.1 mm ; 39.3 mm . 
 
 17. What is the length of the earth's orbit, to the 
 nearest meter, if the diameter of the orbit is 294,481,21 7 km ? 
 
 18. What is the circumference of a carriage wheel, 1.31 m 
 in diameter ? How far will it go in turning once ? 17 times ? 
 
 19. How many times must the wheel of Ex. 18 turn in 
 going 69.429 m ? 73.513 ra ? 17.27 km ? 
 
 20. Find the reciprocal of 3.1416 to the fifth place. 
 
 139. From Ex. 20, if a circumference is multiplied by 
 0.81881, the product is the diameter. 
 
 21. How thick through is a tree whose girth is 2.97 m ? 
 
 22. What is the diameter of a wheel that turns 19.5 
 times in going 107. 25 m ? 
 
 23. What is the diameter of a rope of which the cir- 
 cumference is 20 cm ? 
 
METRIC MEASURES. 
 
 Areas. 
 
 140. A surface has two dimensions, length and breadth. 
 
 141. The unit of surface is a square, each side of which 
 is a unit of length. 
 
 142. The area of a surface is the number of square 
 units it contains. 
 
 143. The perimeter of a plane figure is the distance 
 round it. 
 
 144. A rectangle is a plane figure with four straight 
 sides and four square corners. 
 
 145. Suppose the rectangle in the margin is 3 
 and 2 cm wide. If lines are drawn as 
 represented, the surface is divided 
 into square centimeters. There are 
 2 horizontal rows of 3 qcm each ; that 
 is, in all, 2 X 3<* cm . Hence, 
 
 long 
 
 To Find the Area of a Rectangle, 
 
 Express the length and breadth of the rectangle in the same 
 linear unit ; take the product of these two numbers for its 
 area in square units of the same name as the linear unit. 
 
 The number of square units in a rectangle divided by the 
 number of linear units in one dimension gives the number of 
 linear units in the other dimension. 
 
 Exercise 32. 
 
 1. Find the area of a rectangle 17 cm by 19 cm . 
 
 2. In a rectangular township 16 km by 7 km , how many 
 hektars ? If there are in it 47.3 km of highway, averaging 
 11. 7 m wide, how much land is left for other uses ? 
 
 3. In a rectangular field 751.3 m long and 189.3 m wide 
 is a rectangular garden 31.4 m by 17.8 m . How many hektars 
 in the field ? How many, exclusive of the garden ? 
 
70 METRIC MEASURES. 
 
 4. If my garden contains 941.65 qm and my neighbor's 
 747.37 qm , what is the area in hektars of both taken to- 
 gether ? 
 
 5. If a painter can cover 8.786 qm in an hour, how many 
 square meters can he cover in 1.78 hours ? in 3.86 hours ? 
 in 4.57 hours ? 
 
 6. How many hektars in each of three rectangular 
 fields : one measuring 315.71 m by 78.91 m ; a second, 293.6 m 
 by 84.84 m ; the third, 346.8 m by 71.82 m ? How many in 
 the three together ? 
 
 7. Find the price of a rectangular field, 346.8 m by 71.82 m , 
 at $67.50 a hektar ; at $384 a hektar ; and at $2,375 a 
 square meter. 
 
 8. Find the length of a rectangle 17 cm wide that con- 
 tains 306 qcm . What length of carpet 75 cm wide is required 
 to make 27 qm ? 
 
 9. A room is 16 m long, 8 ra wide, and 8 m high ; another 
 room is 7 m long, 7 m wide, and 3 m high. How many square 
 meters of painting on the walls of both rooms, if no allow- 
 ance is made for doors and windows ? How many more 
 square meters of painting on the walls of the larger room 
 than on those of the smaller ? 
 
 146. To Find the Area of a Circle, 
 
 Multiply the square of the radius by 3.1416 ; or, multiply 
 the square of the diameter by 0.7854 (i of 8.1 416). 
 
 10. -What is the area of a circle 27 cm in diameter? of 
 a circle l m in diameter ? 
 
 1 1 . What is the area in hektars of a circular field 784 m 
 in diameter ? 
 
 12. Find the area of a circle 31 cm in diameter. 
 
 13. Find the area of a circle whose radius is 24 m . 
 
 14. If a circle has a radius of 7 cm , how many square 
 centimeters does it contaiu ? 
 
METRIC MEASURES. 71 
 
 15. In a rectangular sheet of zinc 1.76 ra long and 89 cm 
 wide are two circular openings, one of which has a radius 
 of 10.5 crn , the other a radius of 9.2 cm . What is the area of 
 the zinc left ? 
 
 16. A piece of land in the form of a circle has a radius 
 of 40 m ; in the middle of it is a pond forming a circle of 
 15 m radius. What is the total surface ? the surface of the 
 pond ? the surface of the land to cultivate ? 
 
 17. How deep is a well, if the wheel whose diameter is 
 75 cm makes 26 revolutions in raising the bucket ? 
 
 147. A sphere is a solid bounded by a curved surface, 
 all points of which are equally distant from a point within 
 called the centre. 
 
 148. A straight line drawn through the centre of a 
 sphere, having its ends in the surface, is called a diameter ; 
 half a diameter is called a radius. 
 
 149. The area of the surface of a sphere is four times 
 the area of a circle of the same diameter. Hence, 
 
 To Find the Area of the Surface of a Sphere, 
 
 Multiply the square of the diameter by 3.1^16. 
 
 18. How many square centimeters of surface on a ball 
 7 cm in diameter ? 
 
 19. How many square centimeters of surface on a ball 
 18 cm in diameter ? 
 
 20. How many square meters of surface on a hemis- 
 pherical dome 11.27 m in diameter ? 
 
 Note. A hemisphere is half a sphere. 
 
 21. What is the interior surface of a hemispherical 
 basin 12 cm in diameter ? 
 
 22. What is the interior surface of a hemispherical 
 vase 70 cm in diameter ? 
 
72 METRIC MEASURES. 
 
 Carpeting Rooms. 
 
 150. Carpeting is made of various widths and is sold 
 by the length. 
 
 In determining the number of meters required for a 
 room, we first decide whether the strips shall run length- 
 wise or across the room, and then find the number of strips 
 needed. The number of meters in a strip, including the 
 waste in matching the pattern, multiplied by the number 
 of strips will give the required number of meters. 
 
 23. How many meters of carpet 60 cm wide will be re- 
 quired for a room 6 m long and 5.4 m wide, the strips running 
 lengthwise ? how many meters would be required if the 
 carpet were 80 cm wide ? 
 
 Since the room is 640 cm wide, it will take ^-, or 9 widths of car- 
 pet CO 01 " wide ; that is, 9 X 6 m , or 54 m , will be required. If the car- 
 pet were 80 cm wide, it would take ^ 8 ^, or 7 widths. Six widths would 
 leave a surface ({O ™ wide to be covered. This surface would require 
 another strip, of which a width of 20 cm would be turned under. 
 
 24. How many meters of carpet 56 cm wide will be re- 
 quired for a room 8.32 m long and 6.6 m wide, strips running 
 lengthwise ? 
 
 25. How many meters of carpet 70 cm wide will be re- 
 quired for a room 7 m long and 5.4 m wide, strips running 
 across the room ? 
 
 26. How many meters of carpet 80 cm wide will be re- 
 quired for a room 6 m long and 5.47 m wide, strips running 
 across the room ? 
 
 27. How many meters of carpet 90 cm wide will be re- 
 quired for a room 5 m long and 4.5 m wide, strips running 
 lengthwise ? How much will it cost, at $1,875 a meter ? 
 
 28. How many meters of carpet 75 cm wide will be re- 
 quired for a room 5.25™ long and 4.75 m wide, strips running 
 across the room? Find the cost, at $2,125 a meter. 
 
METRIC MEASURES. 73 
 
 29. How many meters of carpet 75 cm wide will be re- 
 quired for a room 5.6 m square ? How wide a strip will 
 have to be turned under ? How much will the carpet cost, 
 at $1.25 a meter ? 
 
 Papering and Plastering. 
 
 151. The area of the four walls of a room is equal to 
 that of a rectangle whose length is the perimeter of the 
 room, and whose breadth is the height of the room. 
 
 Perimeter = twice the length -4- twice the breadth. 
 Area = height X perimeter. 
 
 30. Find the area of the walls of a room whose length 
 is 6.12 m , breadth 5.05 m , and height 3.5 m . 
 
 Perimeter = 2 X (6.12 m + 5.05 m )= 22.34 m . 
 Area = (3.5 X 22.34) v*=- 78.19<i m . 
 
 31. How many rolls of paper 45 cm wide and 8 m long, 
 allowing 11. l^™ 1 for doors and windows, will be required 
 to paper the room of^Ex. 30 ? 
 
 32. Find the cost of papering a room 8 m long, 5.5 m wide, 
 and 4.5 m high, with paper 50 cm wide and 7.5 m in a roll, at 
 $1.25 a roll, put on; if there is a baseboard 25 cm wide 
 running round the room, and an allowance of ll qm is made 
 for doors and windows. 
 
 33. Find the cost of plastering the room of Ex. 32, at 
 $0.50 a square meter. 
 
 34. Find the cost of papering a room 5.5 m long, 4.8 m 
 wide, and 3.2 m high, with paper 45 cm wide, 7.5 m in a roll, 
 at $0,875 a roll, put on, allowing 12<i m for baseboard, 
 doors, etc. 
 
 35. Find the cost of plastering the room of Ex. 34, at 
 $0.45 a square meter. 
 
 36. Find the cost of papering a room 6 m square and 
 3.5 m high, with paper 45 cm wide and 7.5 m in a roll, at 
 
74 METRIC MEASURES. 
 
 $0.75 a roll, put on ; and of putting on a border, at 5 
 cents per running meter. 
 
 37. Find the cost of plastering the room of Ex. 36, at 
 $0.36 a square meter. 
 
 38. Find the cost of papering a room 13™ long, 12 m 
 wide, and 7 m high, with paper 45 cm wide and 7.5 m in a roll, 
 at $1.50 a roll, put on ; and of putting on a border, at 
 $0.30 a running meter, allowing 115 qm for baseboard, 
 doors, etc. 
 
 39. Find the cost of plastering the room of Ex. 36, at 
 $0.60 a square meter. 
 
 Board Measure. 
 
 152. Boards 25 m,n or less in thickness are sold by the 
 square meter. 
 
 Boards more than 25 mm in thickness and squared lumber 
 are sold by the number of square meters of boards 25 mm in 
 thickness to which they are equivalent. 
 
 Thus, a board 4 m long, 25 cm wide, and 25 mm thick contains l m 
 board measure; if less than 25 mm thick it still contains l m ; but if 
 76 mm thick, it contains 3 m board measure, for it is equivalent to 
 three boards 4 m long, 25 cm wide, and 25 mm thick. Hence, 
 
 153. To Find the Board Measure of Boards more than 
 2 gmm thick and of Squared Lumber, 
 
 Express the length and width in meters, and the thickness 
 in millimeters ; divide the product of these three numbers by 
 25 for the number of meters board measure. 
 
 If a board tapers regularly, its average width is found 
 by taking one-half the sum of its end widths. 
 
 How many meters, board measure : 
 
 40. In a board 8 m long, 20 cm wide, and 20 mm thick ? 
 
 41. In a joist 5 m long, 25 cm wide, and 75 mm thick ? 
 
 42. In a stick of timber 15 m long and 40 cm square ? 
 
 43. In 2 joists 5 m long, 27.5 cm wide, and 50 mm thick ? 
 
METRIC MEASURES. 75 
 
 44. How many meters, board measure, in 10 planks, 
 each 4 m long, 45 cm wide, and 10 cm thick? What is the 
 value of these planks, at $25 a hundred meters ? 
 
 45. How many meters, board measure, in 25 box boards, 
 each 4 ra long, 42 cm wide, and 20 mm thick ? What is their 
 value, at $14 a hundred meters ? 
 
 Find the cost of : 
 
 46. Ten joists 4.5 m long, 10 cm wide, and 7.5 cm thick, at 
 $11 a hundred meters. 
 
 47. Thirty-six planks, each 4 m long, 27.8 cm wide, and 
 75 mm thick, at $16 a hundred meters. 
 
 48. Three sticks of timber, each 8 m long, 22.5 cm wide, 
 and 20 cm thick, at $17.50 a hundred meters. 
 
 49. A board 8.25 m long, 28 cm wide at one end and 35 cm 
 at the other, and 31.25 mm thick, at $0.30 a meter ? 
 
 50. A stick of timber 10 m long, 25 cm thick, 30 cm wide at 
 one end and 25 cm wide at the other, at $14 a hundred 
 meters. 
 
 51. The floor boards, 32 mm thick, for a two-story build- 
 ing 16 m by 10.5 m , at $30 a hundred meters. 
 
 52. The floor timbers, 25 cm by 50 mm , for the building of 
 Ex. 51, if the timbers run lengthwise and are placed on 
 edge 30 cm apart, and are worth $11.50 a hundred meters. 
 
 53. The fencing to enclose a field 150 m long and 75 m 
 wide ; the posts are set 2.5 m apart, and cost $0.25 apiece ; 
 the fence is 5 boards high ; the bottom board is 30 cm , the 
 top board 25 cm , and the other three each 22.5 cm wide, and 
 the boards cost $13.25 a hundred meters. 
 
 154. Kound logs are sold by board measure after 0.25 
 is deducted for slabs. 
 
 Large and heavy timber is sold by the ton. 0.20 is de- 
 ducted for slabs, and the amount is reckoned in cubic meas- 
 ure instead of board measure. 
 
76 METRIC MEASURES. 
 
 Volumes. 
 
 155. A solid has three dimensions, length, breadth, and 
 thickness (height or depth). 
 
 156. The unit of volume is a cube, each dimension 
 of which is a unit of length. 
 
 157. The volume of a solid is the number of cubic units 
 it contains. 
 
 158. The boundary of a solid is called its surface. If 
 the surface consists of planes, they are called the faces of 
 the solid. 
 
 The faces of a solid meet in lines called the edges of the 
 solid. 
 
 159. A rectangular solid is a solid bounded by six 
 rectangles. 
 
 160. Find the volume of a rectangular solid whose 
 length is 5 cm , breadth 3 cm , and height 7 cm . 
 
 The face on which the solid rests may be 
 divided into square centimeters ; there will 
 be three rows of 5 qcm each ; in all 15 qcm . 
 Upon each square centimeter may be 
 placed a pile of 7 ccm , so that the solid will 
 contain 15 X 7 ccm ; that is, 3 X 5 X 7 ccm . 
 
 161. To Find the Volume of a Rectangular Solid, 
 Express the length, breadth, and height of the solid in the 
 
 same linear unit ; take the product of these numbers for its 
 volume in cubic units of the same name as the linear unit. 
 
 If the number of cubic units in the volume is divided by 
 the product of the numbers of linear units in any two dimen- 
 sions, the quotient is the number of linear units in the third 
 dimension. 
 
METRIC MEASURES. 77 
 
 Exercise 33. 
 
 1. How many cubic centimeters in a block 9 cm long, 
 7 cm wide, and 6 cm deep ? 
 
 2. If wood is cut into 120 cm lengths, and a pile is 43.7 m 
 long and 1.4 m high, how many sters of wood are there in 
 the pile ? 
 
 3. How many hektoliters of grain will a bin hold, 
 11.2 m long, 4.34 m wide, and 2.83 m deep ? 
 
 4. If a liter of grain weighs 0.81 of the weight of a 
 liter of water, find the weight of the grain in the bin of 
 Ex. 3. 
 
 5. A bin 16 m by 9.7 m , and 2.8 m deep, is full of oats, 
 worth $0.98 a hektoliter. What is the whole worth ? 
 
 6. How many liters does a vat 197 cm long, 87 cm wide, 
 and 63 cm deep hold ? What weight of water will be 
 required to fill it ? 
 
 7. Add 1341 ccm , 231 1 , and 2.13 M , and give the sum in 
 terms of each of the three units. 
 
 8. If a spring delivers 467. 8 1 each minute, how many 
 hektoliters will it deliver in 60 minutes ? in 37 minutes ? 
 in 78 minutes ? 
 
 9. If 67. 3 1 of oil in a vat with perpendicular sides fills 
 it to a depth of 173 mm , how deep will 13.7 times that quan- 
 tity fill it ? How many hektoliters will there be ? 
 
 10. One cask contains 171.4 1 of oil ; another, 209.3 1 ; a 
 third, 73.8 1 ; while a square vat, 137 cm each way, is filled 
 to a depth of 69 cm . Eind in liters and in hektoliters the 
 amount of oil in the four vessels together. 
 
 11. How many liters of air in a room 7.8 m long, 6.23 m 
 wide, and 3 m high ? 
 
 12. If a person's breathing spoils the air at the rate of 
 0.2175 cbm a minute, how long will it take three persons 
 sitting in the closed room of Ex. 11 to spoil the air ? 
 
78 METRIC MEASURES. 
 
 13. How long, at the same rate as in Ex. 11, will the air 
 in a hall 22 m long, 16 m wide, and 7 m high last 280 persons ? 
 
 162. To Find the Volume of a Sphere, 
 
 Multiply the cube of the diameter by 0.5236 (£ of 8.1 416). 
 
 14. How many cubic centimeters in a ball 10 cm in diam- 
 eter? 
 
 15. Into a cubical box 20 cm on an edge, and full of water, 
 an iron ball 20 cm in diameter is gently lowered until it 
 touches the bottom. Find in liters and in cubic centimeters 
 the volume of the water left in the box. 
 
 16. If cast iron weighs 7.207 times as much as water, 
 what is the weight of a cast iron ball 5 cra in diameter? 
 
 17. A rubber ball is 6.2 cm in diameter. What is the 
 amount of rubber in the ball ? 
 
 18. If the circumference of a cannon ball is 52 cm , find 
 the volume of the ball. 
 
 163. A cylinder is a solid bounded by two equal and 
 parallel circles called its buses, and a uniformly 
 curved surface called its lateral surface. 
 
 Note. Two circles are parallel if all points of 
 one are equally distant from the other. 
 
 164. To Find the Volume of a Cylinder, 
 Multiply the number of square units in its 
 
 base by the number of linear units hi its height. 
 
 19. How many cubic centimeters of oil are there in a 
 cylindrical cup 10 cm across when the oil is 38 mm deep? 
 
 20. What is the capacity of a cylindrical cup 95 mm 
 across and 11.08 cm deep ? 
 
 21. What is the capacity of a cylindrical vessel 16.24 cm 
 across and 19.95 cm deep ? 75.4 mm across and 87.9 nun deep ? 
 
METRIC MEASURES. 79 
 
 22. How many cubic meters of wood in a round stick 
 of equal size throughout, 37 cm in diameter and 8.4 m long ? 
 
 23. A cylindrical stand-pipe whose diameter is 12 m and 
 whose height is 22 m is filled with water. Find the weight 
 of the water. 
 
 24. Find the number of liters of water in a well, if its 
 diameter is 1.2 m and the depth of the water is 2 m . 
 
 25. A cylindrical cup 90 mm in diameter is partly filled 
 with water. Into the cup is dropped a piece of iron, and 
 the water rises 63 mm . What is the volume of the piece of 
 iron? 
 
 Exercise 34. 
 
 1. What is the weight, in kilograms, of a hektoliter of 
 water ? of 73. 8 1 of water ? of a cubic meter of water ? of 
 a cubic centimeter of water ? 
 
 2. If a man buys half a ton of potatoes for $20, and 
 retails them all, without waste, at 5 cents a kilogram, what 
 profit does he make on the whole ? 
 
 3. What is the weight of water required to fill a vat 
 98 cm long, 71 cm wide, and 38 cm deep ? 
 
 4. If the vat of the last example is filled with brine 
 weighing 1.04 kg to the liter, what is the weight of the 
 brine ? 
 
 5. If the vat of Ex. 3 is filled with wine weighing 
 0.981 kg to the liter, what is the weight of the wine ? 
 
 6. What is the total weight of 13 men averaging 73.48 kg 
 each ? 
 
 7. How many kilograms, and how many tons, will 
 3.6175 cbm of brick weigh, at 2 tons to a cubic meter ? at 
 2.34 tons ? 
 
 8. From a barrel containing 67 kg of granulated sugar 
 there are taken three parcels of 2.75 kg each, and four 
 parcels of 7.50 kg each. How much is left in the barrel ? 
 
80 METRIC MEASURES. 
 
 9. Into how many pills of 325 mg each tan a mass of 
 7.8* be divided? 
 
 10. A mass of 21.8* is divided into 60 pills. What is 
 the weight of each pill ? 
 
 11. A bag, when empty, weighs 213*; when full of 
 silver five-franc pieces, 20* 8 5 hg 13*. A five-franc piece 
 weighs 25 g . How many five-franc pieces will the bag hold? 
 
 1 2. A vessel, when empty, weighs 2.7 kg ; and when full 
 of water 4235 dkg . What would it weigh if filled with 
 milk, which is 1.03 times as heavy as water ? 
 
 Specific Gravity. 
 
 165. The specific gravity of a substance is the number 
 found by dividing the weight of the substance by the 
 weight of an equal bulk of water. 
 
 Thus, if a sample of quicksilver has a specific gravity of 13.6, it is 
 13.6 times as heavy as water; a cubic centimeter of it would weigh 
 13.61 ; a liter of it would weigh 13.6 k « ; and a cubic meter of it would 
 weigh 13.6t. 
 
 Again, if the specific gravity of a certain alcohol is 0.827, that is, 
 if the alcohol weighs 0.827 as much as an equal bulk of water, then a 
 cubic centimeter of it would weigh 0.827s; a liter, 0.827 k «; and a 
 cubic meter, 0.827*. 
 
 166. The specific gravity of a substance, therefore, is 
 the number that expresses the weight of a cubic centimeter 
 of it in grams ; of a liter in kilograms ; of a cubic meter in 
 tons. 
 
 167. If a substance is in water, the water buoys it up by 
 just the weight of the water displaced by it. 
 
 168. Examples. 1. A lump of coal weighed in air is 
 found to weigh 1017* ; weighed in water it is found to 
 weigh 321*. What is the specific gravity of the coal ? 
 
METRIC MEASURES. 81 
 
 Solution. 1017s — 321s = 696s. Since the lump of coal weighs 
 696s less in water than in air, 696s is the weight of the water dis- 
 placed by the coal, and the lump contains 696 ccm of coal. 
 
 Therefore, the specific gravity of coal is 1017s -f- 696s, or 1.461; 
 that is, the number found by dividing the weight of the lump of coal 
 by the weight of an equal bulk of water. 
 
 2. A stone, weighing l.l kg in air and 0.6 kg in water, is 
 tied to a block of wood ; the two together weigh 1.28 kg in 
 air and 0.54 kg in water. What is the specific gravity of 
 the wood ? 
 
 Solution. The weight of the wood in air = 1.28 k s — 1.1*8 r= 0.18 k s. 
 
 The weight of water displaced by stone and wood = 1.28 k s — 0.54 k * 
 = 0.74 k *. 
 
 The weight of water displaced by stone alone = l.l k e— 0.6 k « = 
 0.5 k «. 
 
 The weight of water displaced by wood, therefore, = 0.74 k s — 0.5 k « 
 = 0.24 k s. 
 
 Hence, the specific gravity of the wood = 0.1 8 k s -f 0.24 k s = 0.76. 
 
 Exercise 35. 
 
 1. If a stone weighs 1.3 kg in air and 0.68 kg in water, and 
 the stone and a block of wood together weigh l-.55 kg in air 
 and 0.63 kg in water, what is the specific gravity of the 
 block of wood ? 
 
 2. What is the weight of 8.17 M of 
 alcohol, specific gravity 0.83 ? 
 
 3. What will 97 1 of alcohol weigh, 
 of specific gravity 0.817 ? of specific 
 
 Hektoiiter. gravity 0.819 ? of specific gravity 
 
 0.823 ? 0.838 ? 0.847 ? 
 
 4. A bar of aluminum 113 mm long, 17 mm wide, and 
 13 mra thick is said to be of specific gravity 2.57. What 
 does it weigh ? If it really is of specific gravity 2,67, 
 what does it weigh ? 
 
82 METRIC MEASURES. 
 
 5. What would be the specific gravity of the aluminum 
 in Ex. 4 if the bar weighed 65.1378 ? 
 
 6. What is the weight of a bar of aluminum 371 mm by 
 63 mm by 84 mm , specific gravity being 2.63 ? 
 
 7. An irregular mass of copper, gently lowered into a 
 pail brimful of water, caused 1.374 1 to run over. What 
 did it weigh if of specific gravity 8.91 ? if 8.89 ? 
 
 8. What would be the specific gravity of the copper in 
 Ex. 7 if the mass weighed 12.3016 k « ? 
 
 169. To Find the Specific Gravity of a Substance, 
 
 Divide the weight in grants by the bulk in cubic centi- 
 meters, the weight in kilograms by the bulk in liters, or the 
 weight in tons by the bulk in cubic meters. 
 
 9. A plate of iron 137 cm long, 64.3 cm wide, and 4.31 cm 
 thick weighs 277.54 k «. What is its specific gravity? 
 What would the same mass weigh at specific gravity 7.47 ? 
 at 7.79 ? 
 
 10. What is the specific gravity of sea water when a 
 hektoliter weighs 102.58 k « ? when 3 1 weighs 3077* ? 
 
 11. What is the specific gravity of a substance of which 
 7.3 ccm weighs 31.5* ? 
 
 12. If a cubic meter of sand weighs 1723 kg , what is its 
 specific gravity ? If 3.4 cbm of gravel weighs 7.134 tons, 
 what is its specific gravity ? 
 
 13. If a cubic centimeter of metal weighs 7.3 g , what is 
 its specific gravity ? 
 
 14. What is the specific gravity of a fluid weighing 
 2.317 k « to a liter ? 
 
 15. If a body weighs 3.71 k * in air and 2.38 k * in water, 
 what is its specific gravity ? 
 
 16. A piece of ore weighing 3.77 kg weighs in water only 
 2.53 kg . What is its specific gravity ? 
 
METRIC MEASURES. 83 
 
 17. How many cubic centimeters in a stone which loses 
 17.8 g of its weight when weighed in water ? What is its 
 specific gravity if it weighs 33.7 g in air ? 
 
 18. In a wrought-iron bottle I find 2.63 1 of quicksilver, 
 weighing 35.81 kg ; in another, 2.59 1 , weighing 35.193 kg ; in 
 a third, 2.617 1 , weighing 35.571 kg What is the specific 
 gravity of each ? What would be the specific gravity of 
 the mixture if the three were emptied into one vessel ? 
 
 19. A plate of iron 89 cm by 17 cm by 7 cm weighs 79.43 kg . 
 What is its specific gravity ? 
 
 20. What is the specific gravity of a rectangular block of 
 wood 1.6 m long, 0.3 m wide, and 0.15 m thick, if, floating in 
 water on its face 0.3 m wide, it sinks to a depth of 0.12 m ? 
 
 Exercise 36. 
 
 1. If 3 men eat 8 kg of bread a week, how much will 1 
 man eat at the same rate ? How much will 7 men ? How 
 much will the 3 men eat in 1 day ? How much will 1 man 
 eat in 1 day ? How much will 7 men eat in 1 day ? in 1 
 week ? in 5 weeks ? 
 
 2. At the same rate as in Ex. 1, how much will 17 
 men eat in 3 weeks and 4 days ? 
 
 3. If 1 M of oats is enough for 5 horses 1 week, how 
 much is enough for 1 horse 1 week ? for 1 horse 7 weeks ? 
 for 11 horses 17 weeks ? 
 
 4. If 2 hl of grain is enough for 3 horses 5 days, how 
 much is enough for 3 horses 1 day ? for 1 horse 1 day ? 
 for 7 horses 6 days ? 
 
 5. Mix 17 1 of vinegar, costing 6 cents a liter, with 
 39 1 at 5 cents, 21 1 at 7 cents, and 13 1 of water costing noth- 
 ing. Find the number of liters and the cost. 
 
 6. For how much a liter must I sell the mixture of 
 of Ex. 5 to gain 96 cents ? to gain $1.41 ? 
 
84 METRIC MEASURES. 
 
 7. A grocer sold 421 kegs of butter for $4995.25 ; 56 
 kegs brought $12.50 a keg, 91 brought $11.75 a keg, and 
 100 kegs brought $12.25 a keg. For how much a keg 
 were the other kegs sold ? 
 
 8. If 3 tons of coal cost $15.75, how many tons will 
 $36.75 buy ? 
 
 9. If 5 ra of cloth cost $18.75, what will 7 m cost ? 
 
 10. If a tap running 3.5 1 a minute fills a tub in 16 
 minutes, how long will a tap delivering 5 1 a minute be 
 in filling the same tub ? 
 
 11. If both taps of the last example are opened at once, 
 how soon will they fill the tub ? 
 
 12. If 3 men can dig 378 m of ditch in 2 days, how long 
 will it take 5 men, at the same rate, to dig 787 m ? 
 
 13. Into a tub that will hold 48 1 one tap is delivering 
 water at the rate of 3.7 1 a minute ; while out of it, by an- 
 other tap, the water is running at 2.5 1 a minute. How 
 long will it take to fill the tub, beginning with it 
 empty ? 
 
 14. A tap discharges into a tub 4.2 1 a minute ; from the 
 tub water is also running, by a second tap ; the water in 
 the tub gains 30 1 in 18 minutes. How fast is the second 
 tap discharging ? 
 
 15. If a wheel is 1.2 m across, how many times will it 
 turn in going one kilometer ? 
 
 16. How many times in a minute does the wheel of the 
 last example turn, when the carriage is driven at the rate 
 of 14 km an hour ? 
 
 17. What is the weight of the water in a tank if it 
 takes 1 hour and 38 minutes at the rate of 8.7 1 a minute 
 to empty the tank ? 
 
 18. If we replace the water of Ex. 17 with oil worth 
 $18.75 a hektoliter, what will the contents of the tank be 
 worth ? 
 
METRIC MEASURES. 85 
 
 Exercise 37. 
 
 1. A train leaves Paris at 11 o'clock a.m. and reaches 
 Lyons at 10 o'clock p.m. How many meters does it travel 
 in an hour, the distance from Paris to Lyons being 512. 7 km ? 
 
 2. A railroad has a single track 11.450 km long. How 
 many rails 4.569 m in length did it require to lay the 
 track ? 
 
 3. A book is 2.1 cm in thickness ; each leaf is 0.05 mm 
 thick. Find the number of pages in the book. 
 
 4. If the cost of opening a canal amounts to $25,400 
 a kilometer, how much will a canal cost which is 113.253 km 
 in length ? 
 
 5. The expense of laying out a paved road is $12,500 
 a kilometer. How much will a road cost which is 72.053 km 
 long? 
 
 6. The cost of building a railroad is about $78,000 a 
 kilometer in France, and only $25,000 in the United 
 States. How much -will it cost in each country to make 
 a road 295.671 km long ? 
 
 7. If you must go up 211 steps to reach the top. of a 
 tower, and each step is 195 mm high, what is the height of 
 the tower ? 
 
 8. A house has 5 stories, each story has 19 stairs, each 
 stair is 16 cm in height. Find the height of the floor of 
 the fifth story from the ground. 
 
 9. A ream of paper contains 20 quires, each quire has 
 24 sheets, the ream is 13.5 cm in thickness. Find the thick- 
 ness of each sheet. 
 
 10. The equator on a terrestrial globe measures 0.80 m 
 in circumference. By the aid of a tape measure we find 
 that the distance between two cities on this globe is 0.046' 11 . 
 What is really the distance in kilometers between the two 
 cities ? (The earth's equator is 40,075.45 km .) 
 
86 METRIC MEASURES. 
 
 1 1 . Upon a military map we find that the distance from 
 Paris to St. Denis is 78 mm . What is the distance in kilo- 
 meters from Paris to St. Denis ? The map is made on the 
 scale of 1 to 80,000 ; that is, l m on the map represents 
 80,000 m of actual measurement upon the ground. 
 
 12. Find the number of revolutions made by the wheels 
 of a carriage in traveling 82*". The wheels are lSo^"" 11 
 in diameter. 
 
 13. How many hektars in a square kilometer? how 
 many ars ? how many square meters ? 
 
 14. France has about 542,000 qkm . How many hektars 
 does it measure ? 
 
 15. A piece of land 1224.5 m square is sold at $140 a 
 hektar. How much does the land bring ? 
 
 16. The total surface measurement of the glass in the 
 windows of a house is 182 qm . How many panes of 53 cm 
 by 48 cra will it take to supply the windows ? 
 
 17. How many square slabs of marble 150 qcm on the 
 surface will it require to pave a court whose area is 
 25.35 qm ? 
 
 18. A speculator bought 31.0728 ha of land for $1296 a 
 hektar. For how much a square meter must he sell it to 
 realize a profit of $1937 ? 
 
 19. A man is offered $6000 for 2.5 a of land. He de- 
 clines to sell ; and soon after the town gives him $25.20 a 
 square meter. How much did he make by refusing the 
 first offer ? 
 
 20. A man surveys a piece of land and finds that it 
 measures 14.0715 ha . He afterwards discovers that his 
 chain was too short by 0.03 m . How can he calculate the 
 real superficial measurement of the land without surveying 
 it again ? (A surveyor's chain is 10 m long.) 
 
 21. A pile of wood is 4.25 m long, 1.33 m thick, and 2.60 m 
 high. How many sters are there in it ? 
 
METRIC MEASURES. 87 
 
 22. The railroad from Paris to Orleans has a double 
 track ; each rail is 4 m long, and the distance from Paris to 
 Orleans is 121 km . What is the number of rails used in 
 laying the track ? If the width of the road is 15 m , how 
 many hektars of land does the road include ? 
 
 23. Find the number of ars in a surface which a 
 ream of paper (480 sheets) will cover. The sheets are 
 30.3 cm long and 195 mm wide. 
 
 24. A beam is 7.070 m long ; its two other dimensions 
 are 0.258 m and 87 mm . Find its volume. 
 
 25. A bar of iron 3 m long measures 45 mm square on the 
 end where it has been evenly cut. The bar is heated and 
 drawn out to a greater length by being passed through an 
 orifice 24 mm square. What is the length of the bar after 
 the operation ? 
 
 26. A reservoir is 1.50 m wide, 2.80 m long, and 1.25 m 
 deep. Find how many liters it contains when full, and to 
 what height it would be necessary to raise it that it might 
 contain 10 cbm . 
 
 27. Suppose a box to be 3.75 m long, 3.50 m wide, and 
 0.50 m high. How much lime would it take to fill it with 
 mortar, reckoning that l cbm of lime after being slaked be- 
 comes 1.80 cbm of mortar ? 
 
 28. A chest has the following dimensions : 1.17 m , 0.90 m , 
 1.04 m . If 0.12 of the volume of the chest is deducted for 
 packing, how many cakes of soap 13 cm square on the 
 bottom and 29 cm thick could be put in it ? 
 
 29. A cubic meter of dry plaster makes 1.18 cbm when 
 tempered ; tempered plaster increases 1 in every 100 
 twenty-four hours after it is mixed. What volume of 
 tempered plaster would be obtained from 55 sacks of 25 1 
 each of dry plaster ? 
 
 30. A reservoir is 2.80 m long, 1.50 m wide, and 1.25 m 
 deep. How many liters will be required to fill 0.80 of it ? 
 
88 METRIC MEASURES. 
 
 31. A man buys 1415 m of wheat for $3.50 a hektoliter ; 
 but the measure used proves too small, the mistake 
 amounting to 3 1 in every hektoliter. What was the quan- 
 tity of wheat delivered to the purchaser, the cost, and the 
 reduction which ought to be made to him on account of 
 the error ? 
 
 32. The dimensions of a tile are as follows : length 
 22 cm , width ll cm , thickness 55 mm . Find the volume of 
 the tile, and the number of tiles in a pile of 25 cbm . 
 
 33. The measurement of a pile of wood shows that a 
 ster could be filled from it 25.68 times. Find the volume 
 of the pile in cubic meters, reckoning the length of the 
 logs to be 1.1 5 m . 
 
 Note. A ster is a frame, as represented on page 64, one meter 
 high and one meter between the upright posts. The ster may be 
 filled with wood of any length, and the volume will be as many 
 cubic meters as the sticks of wood are meters long. 
 
 34. A liter of air weighs 1.273*. How much does a 
 cubic meter of air weigh ? How many times as heavy as 
 air is water ? 
 
 35. A package of candles that weighs 465 g is sold for 
 28 cents. At the same rate what is the price of a kilogram 
 of candles ? 
 
 36. How many times will 3.243* of water fill a liter 
 measure ? 
 
 37. Express in kilograms the weight of 43.4578 ccm of 
 pure water. 
 
 38. The volume of the axle of an engine is 0.245 cbm . 
 Find its weight, if the specific gravity of the iron is 7.8. 
 
 39. Find the volume of a gram of the following sub- 
 stances : proof spirit, specific gravity 0.865 ; tin, specific 
 gravity 7.291 ; lead, specific gravity 11.35 ; copper, spe- 
 cific gravity 8.85 ; silver, specific gravity 10.47 ; cork, 
 specific gravity 0.240. 
 
METRIC MEASURES. 89 
 
 40. Olive oil costs 60 cents a kilogram. What is the 
 price of a liter ? The specific gravity of olive oil is 0.914. 
 
 41. Pure alcohol costs $1.87 a kilogram. What is the 
 price of a liter ? The specific gravity of alcohol is 0.792. 
 
 42. A man wishes to build a shed large enough to hold 
 135 8t of wood ; if the shed is to be 3 m high and 5 m wide, 
 how long must it be ? 
 
 43. In a country where firewood is cut 1.16 m long, what 
 must be the height of the sides of the ster that it may hold 
 a cubic meter ? 
 
 44. If a ster of cork costs $20.00, how much would 
 100 kg cost, the cork weighing 0.25 as much as water ? 
 
 45. A liter of powder weighs 825 g . What will be the 
 volume in cubic centimeters of a charge for a gun if the 
 charge weighs 5 g ? 
 
 46. Out of gold which weighs 19.362 times as much as 
 water sheets of gold foil are made which are 0.010 mm in 
 thickness. What surface will 3 g of gold cover ? 
 
 47. Find the weight of an oak board 3.25 m long, 0.31 ra 
 wide, and 0.04 m thick, if the specific gravity of the oak 
 is 0.808. 
 
 48. Find the weight of a bar of iron having the follow- 
 ing dimensions : length 3.6 m , width 6 cm , thickness 2 cm , if 
 the specific gravity of the iron is 7.8. 
 
 49. How many lead balls, each weighing 27 g , can be 
 obtained by melting a cubic mass of lead 0.356 m on an 
 edge, if the specific gravity of the lead is 11.35 ? 
 
 50. Marble costs $ 30.95 a cubic meter, and the specific 
 gravity of marble is 2.73. If a block of marble weighs 
 1260 kg , what is its volume and what is it worth ? 
 
 51. Sea water contains 28 parts, by weight, of salt in 
 1000. A liter of sea water weighs 1.025 kg . How many 
 kilograms of salt can be obtained from 126.276842 cbm of 
 sea water ? 
 
90 METRIC MEASURES. 
 
 52. An empty cask weighs 17.06 kg ; when filled with 
 water it weighs 275. 8 kg . How many liters does it hold ? 
 How many casks of this size will it take for the wine 
 from a vat containing 3.008 cbm ? 
 
 53. If it takes 2.048 m of wheat to 
 sow a hektar, how many cubic meters 
 will it take to sow a square kilo- 
 meter ? 
 
 54. A piece of road l km long and 
 7 m wide is to be macadamized to the 
 depth of 33 cm . What will the work 
 
 Hektoiiter. cost at 43 cents a cubic meter ? 
 
 55. A gasometer holds 28,000 cbm of gas. How many 
 jets will this gasometer feed for an evening, when each jet 
 burns 125 1 an hour, and is used 4 hours ? 
 
 56. The city of Venice is situated in the midst of a 
 great lake of salt water, communicating with the sea, and 
 all the rain water is caught for the cisterns. Ordinary 
 years the fall of rain in Venice is 82 cm ; the surface of the 
 city, after the canals have been deducted, is 520** . Reck- 
 oning the population at 115,530, how many liters a day of 
 rain water can each inhabitant- have ? 
 
 57. Find the weight of a bar of iron 5.35 m long, 4.56 cm 
 thick, and 3.54 cm wide. Find, also, the width of an oak 
 beam 4.30 m long, 9.12 cm thick, which has the same weight. 
 The specific gravity of the oak to be reckoned at 1.026, 
 that of the iron 7.788. 
 
 58. Find the specific gravity and volume of a body 
 weighing 35 kg in air and 30 kg in water. 
 
 59. A ster of piled oak wood weighs 425 kg ; the specific 
 gravity of the wood is 0.74. What is the volume occupied 
 by the spaces between the logs ? For how much must 
 lOO 1 * of separate sticks be sold to bring the same amount 
 as when sold at $2.20 a ster ? 
 
METRIC MEASURES. 91 
 
 60. Wrought iron sells for $7.00 per 100 kg . A bar of 
 iron 4.5 cm wide, 3.3 cm thick costs $5.08 ; what is its 
 length, reckoning the specific gravity of the iron at 7.4 ? 
 
 61. Experiment shows that water weighs 770 times as 
 much as air ; and the specific gravity of mercury is 13.6. 
 How many liters of air will it take to weigh as much as a 
 liter of mercury ? 
 
 62. A mass of lead weighing 753 kg is made into sheets 
 0.1 mm thick. Find, in square meters, the surface which 
 can be covered by the sheets thus obtained. The specific 
 gravity of the lead is 11.3. 
 
 63. A rectangular sheet of tin of uniform thickness is 
 85 cm wide, 1.35 m long, and weighs 268 g . What is its thick- 
 ness, if the specific gravity of tin is 7.3 ? 
 
 64. The fine coal which collects about the shafts of the 
 mines and in the coalyards was for a long time wasted, 
 because it could not be burned in stoves and grates. Now 
 this dust is mixed with tar in the proportion of 92 kg of 
 dust and 8 kg of tar ; Jihe mixture is heated, and afterwards 
 pressed in rectangular moulds 14.75 cm by 18.5 cm by 29 cra ; 
 each one of these blocks weighs 10 kg . They are sold at 
 $3.00 a ton, and make excellent fuel for heating steam 
 boilers. Find the specific gravity of this fuel ; also the 
 sum which would be realized in thus utilizing 800,000* of 
 coal dust, the cost of tar, mixing, etc., being $0.50 a ton. 
 
 65. A bar of iron a millimeter square on the end will 
 break under a tension of 30 kg . Find the length at which a 
 suspended bar of iron will break from its own weight, if 
 the specific gravity of the iron is 7.8. 
 
 66. Fifty-three kilograms of starch are obtained from 
 100 kg of wheat. A hektar of land produces 1363 1 of wheat ; 
 a hektoliter of wheat weighs 78 kg . If the wheat harvested 
 from a field measuring 2 ha and 33 qm is taken to a starch 
 factory, how much starch will be made from it ? 
 
92 METRIC MEASURES. 
 
 67. A gardener wishes to provide glass for his hotbeds. 
 The beds cover 2.65 a ; the panes will cover 0.75 of the 
 whole surface, the rest being taken up by the frames and 
 alleys. First find how many panes measuring 45 cm by 37 cm 
 it will take to cover the beds ; then find the price of the 
 glass, at a cost of 95 cents a square meter. 
 
 68. A jar full of water weighs 1.325**; filled with mer- 
 cury it weighs 12.540 kg . Find the capacity and the weight 
 of the jar, if the specific gravity of the mercury is 13.59. 
 
 69. A hektoliter of rape seed weighs 63**, and 32 1 of oil 
 can be extracted from it. How many kilograms of the 
 seed will it take to make a hektoliter of oil ? 
 
 70. Common burning gas is 0.97 of the weight of air, 
 and a liter of air weighs 1.293 g . In a shop there are 65 
 jets, each one of which burns 123 1 an hour, and is used 5 
 hours in the winter evenings. Find the weight of the gas 
 used in a month of 26 days, and the expense of lighting 
 the shop, when gas costs 6 cents a cubic meter. 
 
 71. A merchant buys one kind of wine at 30 cents a 
 liter, another kind at 21 cents a liter ; he mixes the two 
 kinds by putting 5 1 of the first with 8 l of the second. For 
 how much a liter must he sell the mixture in order to gain 
 $3.75 a hektoliter ? 
 
 72. If it requires 360 tiles to drain an ar of land, what 
 will it cost to drain 17.784 ha , when the tiles cost $20 a 
 thousand, and the expense of laying is the same as the 
 cost of the tiles ? 
 
 73. Hewn stone of medium durability ought not to 
 support, as a permanent weight, more than 0.07 of the 
 weight that is required to crush it. A certain kind of 
 stone used for building will be crushed under a weight of 
 250** a square centimeter. What is the greatest height to 
 which a wall constructed of this material can be safely 
 carried, if the specific gravity of the stone is 2.1 ? 
 
METRIC MEASURES. 93 
 
 74. Several different kinds of wine are mixed as follows : 
 245 1 at 20 cents a liter, 547 1 at 15 cents a liter, 344 1 at 25 
 cents a liter. How much does the mixture cost a liter ? 
 
 75. A farmer wishes to drain a field of 8.75 ha . Each 
 hektar requires 750 m of ditches. The opening of these 
 ditches costs 10 cents a running meter ; the tiles are 30 cm 
 long and cost $15 a thousand. He pays 2 cents a meter 
 for laying the tiles, and 4 cents a meter for filling the 
 ditches. What is the cost of draining the field ? 
 
 76. A silver five-franc piece weighs 25 g , and is com- 
 posed of 9 parts of pure silver and 1 part of pure copper. 
 A silver two-franc piece weighs 10 g , and is composed of 835 
 parts of pure silver and 165 parts of pure copper. A 
 silver twenty -centime piece weighs l g , and has the same 
 composition as the two-franc piece. Find the total weight 
 of pure silver and of pure copper contained in 272 five- 
 franc pieces, 145 two-franc pieces, and 179 twenty-centime 
 pieces. 
 
 77. The dimensions of the bottom of a rectangular box 
 are 70 cm by 50 cm . If the box contains exactly an hektoliter 
 of wheat when full, what is the height of the box ? 
 
 78. If a stick of oak timber. 54 centimeters wide and 65 
 centimeters thick costs $25 at $16 a cubic meter, what is 
 the length of the stick ? 
 
 79. A rectangular box whose bottom is a square 28 cm on 
 a side, and whose height is 19.2 cm , is exactly filled with 
 gold twenty-franc pieces, in piles touching each other. If 
 a twenty-franc piece is 35 mm in diameter, and 1.28 mm thick, 
 what is the value of the gold in the box ? 
 
 80. If l m of coal yields 1854 cbm of gas, and one burner 
 consumes 140 1 of gas in an hour, how many hektoliters of 
 coal are required to supply 2800 burners for 144 hours ? 
 
 8 1 . How many liters of water in a cylindrical well 1.96 m 
 in diameter, if the water is 2.84 ra deep ? 
 
CHAPTER VI. 
 
 MEASURES AND MULTIPLES OF NUMBERS. 
 
 170. Factors of a Number. The factors of a number 
 are the numbers whose product is that number. 
 
 171. Prime Numbers. A prime number is a number 
 that has no integral factors , except itself and one. 
 
 Thus, 2, 3, 5, 7, 11, 13, 17, 19 are prime numbers. 
 
 172. Composite Numbers. A composite number is a 
 number that is the product of two or more integral factors. 
 
 Thus, 10, 21, 143 are composite numbers, for 10 is 2 X 6 ; 21 is 3 X 
 7; 143 is 11 X 13. 
 
 Note. In speaking of the integral factors of a number we exclude 
 the number itself and one. 
 
 173. Prime Factors. A prime factor is a factor that is 
 a prime number. 
 
 174. A composite number can have but one set of prime 
 factors. 
 
 Thus, 12 cannot be expressed as the product of any set of prime 
 factors except 2X2X3. It is the product of 2 X 6, and of 3 X 4, 
 but one of the factors of 2 X 6 and one of 3 X 4 is composite. 
 
 175. A number that can be divided by another without 
 a remainder is said to be exactly divisible by that number ; 
 and the divisor is called an exact divisor. 
 
 176. Even Numbers. An even number is a number that 
 is exactly divisible by 2. 
 
 177. Odd Numbers. An odd number is a number that 
 is not exactly divisible by 2. 
 
MEASURES AND MULTIPLES OF NUMBERS. 95 
 
 178. To Determine Prime Numbers. Write the series 
 of integral numbers in order, beginning with the smallest ; 
 then cancel the even numbers, and place a dot over each 
 multiple of 3 : we have 
 
 1, % 3, 4, 5, $, 7, $, 9, Xd>, 11, it, 13, U> 15, X$, 17, X$ } 
 
 19, n, 21, U, 23, U, 25, M, 27, U, 29, $>, 31, etc. 
 
 Each multiple of 6 is cancelled, and also has the dot 
 over it, and the only numbers without the cancelling line 
 or the dot come just before or just after a multiple of 6. 
 Therefore, 
 
 The only numbers greater than 6 that can be prime num- 
 bers are one less than or one greater than a multiple of 6. 
 
 179. Examples. 1. Find the prime factors of 144. 
 
 144 
 
 _72 
 _36 
 
 9 
 3 
 
 That is, 144 = 2 X 2 X 2 X 2 X 3 X 3 = 2 4 X 3 2 . 
 
 Note. Divide as many times as possible by the smallest prime 
 number that will exactly divide the given number before taking the 
 next larger prime number for a divisor. 
 
 2. Find the prime factors of 233. 
 
 By actual trial we find that the prime numbers 2, 3, 5, 7, 11, 13, 
 and 17 are not factors of 233. We need not try any higher prime 
 number than 17, as the quotient when 17 is tried is less than 17. 
 Therefore, no prime number greater than 17 can be a factor ; and we 
 have found by trial that no prime number less than 17 is a factor. 
 Therefore, 233 is a prime number. 
 
 Notice that 233 is one less than 234, or 39 X 6. 
 
96 MEASURES AND MULTIPLES OF NUMBERS. 
 
 180. From these two examples we have the following 
 
 Rule. Divide the given number by any prime number 
 that exactly divides it ; then the quotient by any prime nit tu- 
 ber that exactly divides it ; and so on until the quotient is 
 itself a prime number. The several divisors and the last 
 quotient are the prime factors. 
 
 If no prime factor is found before the quotient becomes 
 equal to or less than the divisor, the number is a pHme 
 number. 
 
 181. The following tests are very useful for determining 
 without actual division whether a number contains certain 
 factors : 
 
 1. A number is divisible by 2 if its last digit is even. 
 
 2. A number is divisible by 4 (2 2 ) if the number 
 denoted by the last two digits is divisible by 4. 
 
 3. A number is divisible by 8 (2 s ) if the number 
 denoted by the last three digits is divisible by 8. 
 
 4. A number is divisible by 3 if the sum of its digits is 
 divisible by 3. 
 
 5. A number is divisible by 6 if its last digit is even 
 and the sum of its digits is divisible by 3. 
 
 6. A number is divisible by 9 (3 2 ) if the sum of its 
 digits is divisible by 9. 
 
 7. A number is divisible by 5 if its last digit is either 
 5 or 0. 
 
 8. A number is divisible by 25 (5*) if the number 
 denoted by the last two digits is divisible by 25. 
 
 9. A number is divisible by 125 (5 8 ) if the number 
 denoted by the last three digits is divisible by 125. 
 
 10. A number is divisible by 11 if the difference between 
 the sum of the digits in the even places and the sum of the 
 digits in the odd places is either or a multiple of 11. 
 
MEASURES AND MULTIPLES OF NUMBERS. 
 
 97 
 
 182. Other prime factors, 7, 13, 17, 19, etc., sometimes 
 betray their presence to one familiar with the subject; 
 but, practically, the best way to detect them is by actual 
 division. 
 
 183. If we divide any number less than 121 (ll 2 ) by 
 11, or by a number greater than 11, it is plain that the 
 quotient is less than 11. 
 
 If we divide any number between 121 and 143 (11 X 13) 
 by 11, the quotient will evidently lie between 11 and 13 ; 
 and, since there are no prime numbers between 11 and 
 13, the quotient, if a whole number, must be composite, 
 and contain factors smaller than 11. 
 
 What is true of 11 and 13 is evidently true of any two 
 adjacent prime numbers ; namely, that, excepting the 
 second power of the smaller prime, every composite number 
 less than the product of two adjacent prime numbers contains 
 prime factors less than the smaller of these two numbers. 
 
 Thus, every composite number less than 4087 (61 X 67), except 
 3721 (61 2 ), contains prime factors less than 61. 
 
 184. The value of the following table, in discovering 
 the prime factors of a given number, will be apparent. 
 
 Primes . . 
 
 7 
 
 11 
 
 13 
 
 17 
 
 19 
 
 23 
 
 29 
 
 31 
 
 37 
 
 Powers . . 
 
 49 
 
 121 
 
 169 
 
 289 
 
 361 
 
 529 
 
 841 
 
 961 
 
 1369 
 
 Products . 
 
 77 
 
 143 
 
 221 
 
 323 
 
 437 
 
 667 
 
 899 
 
 1147 
 
 1517 
 
 Primes . . 
 
 41 
 
 43 
 
 47 
 
 53 
 
 59 
 
 61 
 
 67 
 
 71 
 
 73 
 
 Powers . . 
 
 1681 
 
 1849 
 
 2209 
 
 2809 
 
 3481 
 
 3721 
 
 4489 
 
 5041 
 
 5329 
 
 Products . 
 
 1763 
 
 2021 
 
 2491 
 
 3127 
 
 3599 
 
 4087 
 
 4757 
 
 5183 
 
 5767 
 
 Primes . . 
 
 79 83 
 
 89 
 
 97 
 
 101 
 
 103 
 
 107 
 
 109 
 
 113 
 
 Powers . . 
 
 6241 
 
 6889 
 
 7921 
 
 9409 
 
 10201 
 
 10609 
 
 11449 
 
 11881 
 
 12769 
 
 Products . 
 
 6557 
 
 7387 
 
 8633 
 
 9797 
 
 10403 11021 
 
 11663 
 
 12317 
 
 14351 
 
 Opposite ' ' Powers ' ' are placed the squares of the primes from 7 to 
 113; and opposite "Products" are placed the products of the suc- 
 cessive pairs of adjacent primes from 7 to 127. 
 
2 2 ,610,764 
 
 3 
 
 152,691 
 
 7 
 
 50,897 
 
 11 
 
 7,271 
 
 98 MEASURES AND MULTIPLES OF NUMBERS. 
 
 185. Example. Find the prime factors of 610,764. 
 
 As 64 is divisible by 4, but 764 is not divisible by 8, 2 2 is the high- 
 est power of 2 contained in 610,764. 
 
 As the sum of the digits 152,691 is divisible by 3, 
 but not by 9, 3 1 is the highest power of 3 contained in 
 152,691. 
 
 The next quotient, 50,897, does not contain 5 ; but 
 661 divided by 7 gives 7271. 7271 does not contain 7; 
 but, since 7 + 7 — (2+1)= 11, it is divisible by 11. 
 The quotient 661 when divided by 6 gives a remainder of 1, which 
 shows that it may be a prime number. It cannot be divided by 11, 
 13, 17, or 19, and is seen by the table to be less than 667 (23 X 29), 
 and not equal to 529 (23 2 ) ; therefore it is a prime number. 
 Therefore, 610,764 = 2 a X 3 X 7 X 11 X 661. 
 
 Exercise 38. 
 
 Find the prime factors of : 
 
 1. 148. 18. 179. 35. 65. 
 
 2. 264. 19. 83. 36. 76. 
 
 3. 178. 20. 2125. 37. 86. 
 
 4. 183. 21. 2353. 38. 88. 
 
 5. 173. 22. 2333. 39. 142. 
 
 6. 187. 23. 165. 40. 326. 
 
 7. 346. 24. 168. 41. 368. 
 
 8. 343. 25. 2148. 42. 464. 
 
 9. 210. 26. 16,662. 43. 292. 
 
 10. 353. 27. 321. 44. 362. 
 
 11. 5280. 28. 1551. 45. 365. 
 
 12. 231. 29. 38. 46. 730. 
 
 13. 31,416. 30. 82. 47. 42. 
 
 14. 1369. 31. 129. 48. 868. 
 
 15. 1368. 32. 72. 49. 999. 
 
 16. 247. 33. 66. 50. 822. 
 
 17. 327. 34. 68. 51. 1346. 
 
MEASURES AND MULTIPLES OF NUMBERS. 99 
 
 52. 7641. 59. 128. 66. 78,309. 
 
 53. 6234. 60. 8192. 67. 25,179. 
 
 54. 234. 61. 8190. 68. 61,600. 
 
 55. 579. 62. 8197. 69. 48,048. 
 
 56. 577. 63. 3125. 70. 401,478. 
 
 57. 212. 64. 2401. 71. 278,208. 
 
 58. 126. 65. 1331. 72. 493,185. 
 
 186. A number is not only divisible by each of its 
 prime factors, but by every possible combination of them. 
 
 Thus, 120 is 23 x 3 X 5, and is divisible by 2, 4, 8, 6, 12, 24, 30, 60, 
 10, 20, 40, or 15. 
 
 187. The number 14.21 may be put in the form of 
 1421 X 0.01 ; and thus be resolved into 7 2 X 29 X 0.01. 
 But 0.01 is not properly a factor, it is a divisor ; it is the 
 reciprocal of 2 2 X 5 2 . Nevertheless, it is frequently of 
 great practical advantage to separate mixed decimals in 
 this way, by first taking out the apparent factors, 0.1, 
 0.01, etc. 
 
 Thus, the factors of 142.1 may be said to be 7, 7, 29, and 0.1 ; of 
 1.421, 7, 7, 29, and 0.001. 
 
 
 Exercise 39. 
 
 
 
 Find the 
 
 prime factors of : 
 
 
 
 1. 8.4. 
 
 9. 2.61. 
 
 17. 
 
 5.04. 
 
 2. 7.6. 
 
 10. 21.2. 
 
 18. 
 
 1.485. 
 
 3. 1.08. 
 
 11. 78.54. 
 
 19. 
 
 0.216. 
 
 4. 0.144. 
 
 12. 0.5236. 
 
 20. 
 
 34.87. 
 
 5. 0.036. 
 
 13. 0.00052. 
 
 21. 
 
 32.4. 
 
 6. 0.037. 
 
 14. 8.67. 
 
 22. 
 
 5.115. 
 
 7. 21.45. 
 
 15. 48.3. 
 
 23. 
 
 71.2. 
 
 8. 14.6. 
 
 16. 99.99. 
 
 24. 
 
 2.993. 
 
100 MEASURES AND MULTIPLES OF NUMBERS. 
 
 Greatest Common Measure. 
 
 188. Measures of a Number. The measures of a num- 
 ber are the exact divisors of the number. 
 
 Thus, a man with five-dollar bills can make up a sum of $20, but 
 not of .$18. A wood chopper with the two-foot mark on his axe-handle 
 can measure off lengths of 2, 4, 6, or 8 feet, but not of 3, 5, or 7 feet. 
 A man with scales and a four-ounce weight can weigh out 16 ounces 
 of tea, but not 22 ounces. A man with a four-quart measure can 
 measure out 4, 8, or 12 quarts of molasses, but not 5, 6, or 7 quarts. 
 
 189. Common Measures. A common measure of two 
 or more numbers is a number that exactly divides each of 
 them. 
 
 Thus, $5 is a common measure of $35 and $40, being contained 
 exactly 7 times in $35 and 8 times in $40. 3 feet is a common meas- 
 ure of 21 feet, 15 feet, and 12 feet. 1 yard is a common nuasure of 
 2 yards, 3 yards, and 6 yards. 4 is a common measure of 12, 16, 
 and 20. 
 
 190. Greatest Common Measure. The greatest com- 
 mon measure of two or more numbers is the greatest num- 
 ber that exactly divides each of them. 
 
 Thus, the measures of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 
 the measures of 36 are 1, 2, 3, 4, 6, 9, 12, 18. 
 
 We see from the two series of measures that 1, 2, 3, 4, 6, 12 are the 
 only measures of both 84 and 36, and that 12 is the greatest ; there- 
 fore, 12 is the greatest common measure of 84 and 36. 
 
 191. The letters G. C. M. stand for the words Greatest 
 Common Measure. 
 
 Note. Greatest Common Divisor is sometimes used instead of Great- 
 est Common Measure, and then G. C. D. is used instead of G. C. M. 
 
 192. If two integral numbers have no common measure 
 except 1, they are said to be prime to each other. 
 
 Thus, 27 and 32 are prime to each other, though both are compos- 
 ite numbers. 
 
MEASURES AND MULTIPLES OF NUMBERS. 101 
 
 193. Examples. 1. Find the G. C. M. of 84, 126, and 
 210. 
 
 Resolve each of the numbers into its prime factors. 
 
 84 
 
 2 
 
 126 
 
 42 
 
 21 
 
 7 
 
 3 
 3 
 
 63 
 21 
 
 7 
 
 7. 
 
 126 = 2 X 3 2 X 7 
 
 21210 
 
 105 
 _35 
 
 7 
 
 210 = 2 X 3 X 5 X 7. 
 
 84 = 2 2 X 3 X 7 
 
 The factor 2 occurs once in all the numbers. 
 The factor 3 occurs once in all the numbers. 
 The factor 7 occurs once in all the numbers. 
 No other factor occurs in all the numbeis. 
 Therefore, the G. C. M. is 2 X 3 X 7 = 42. 
 
 2. Find the G. C. M. of 40 and 72. 
 
 40 = 2 3 X 5. 72 = 2 3 X 3 2 . 
 
 Hence, 2 occurs three times as a factor in 40, and three times as a 
 factor in 72. No other factor is common to 40 and 72. Therefore, 
 2 3 , or 8, is the G. C. M. Hence, 
 
 194. To Find the G. C. M. of Two or More Numbers, 
 
 Separate the numbers into their prime factors. Select 
 the lowest power of each factor that is common to the given 
 numbers, and find the product of these powers. 
 
 195. The common factors of two or more numbers may 
 be taken out at the same time as follows : 
 
 2 
 
 84 
 
 126 
 
 210 
 
 3 
 
 42 
 
 63 
 
 105 
 
 7 
 
 14 
 
 21 
 
 35 
 
 2 3 5 
 
 As all the numbers are even, 2 is a common factor. As 3 is an 
 exact divisor of 42, 63, and 105, 3 is a common factor. 
 
 As 7 is an exact divisor of 14, 21, and 35, 7 is a common factor. 
 
 The quotients 2, 3, and 5 have no common factor. Therefore, the 
 only common factors are 2, 3, and 7 ; and the G. C. M. is 2 X 3 X 7 = 42. 
 
102 MEASURES AND MULTIPLES OF NUMBERS. 
 
 
 
 Exercise 40. 
 
 
 Find the G. C. 
 
 M. of: 
 
 
 1. 
 
 27 and 33. 
 
 7. 4,6,10. 
 
 13. 96, 36,48. 
 
 2. 
 
 13 and 39. 
 
 8. 9, 12, 21. 
 
 14. 84, 105, 63. 
 
 3. 
 
 8 and 28. 
 
 9. 10, 15, 25. 
 
 15. 24, 60,84,128 
 
 4. 
 
 27 and 45. 
 
 10. 14, 98, 42. 
 
 16. 45, 81, 27, 90. 
 
 5. 
 
 81 and 108. 
 
 11. 30, 18, 54. 
 
 17. 78, 18, 54,42. 
 
 6. 
 
 4, 10, 12. 
 
 12. 14, 56,42. 
 
 18. 98, 28,70, 42. 
 
 
 19. 96, 112 : 
 
 , 80, 32. 23. 252 
 
 , 315, 420, 504. 
 
 
 20. 24, 96, 
 
 48, 120. 24. 128 
 
 , 192, 320, 368, 432. 
 
 21. 84, 252, 168, 210. 25. 136, 204, 357, 459. 
 
 22. 33, 88, 77, 55. 26. 909, 1414, 2323, 4242. 
 
 196. Example. Find the G. C. M. of 69 and 184. 
 
 Solution. We divide the greater number 
 
 69)184(2 by the smaller, and the last divisor by the last 
 
 138 remainder, and so on until there is no re- 
 
 46)69(l mainder. The final divisor, 23, is the greatest 
 
 46 common measure. 
 
 23)46(2 This method of finding the G. C. M. can be 
 
 46 . employed when the numbers cannot readily 
 
 be separated into their prime factors. 
 
 This method depends upon two principles : 
 
 1 . Every factor of a number is also a factor of every multiple of that 
 number. 
 
 2. Every common factor of two numbers is also a factor of their 
 sum and of their difference. 
 
 Thus 4, which is a factor of 12, is also a factor of 24, 36, etc.; and 
 6, which is a common factor of 24 and 36, is also a factor of 60 and 
 12. 
 
 Let us apply these principles to this example : 
 
 Since 23 is a factor of itself and of 46, it is, by (2), a factor of 69. 
 
 Since 23 is a factor of 69, it is, by (1), a factor of 2 X 69, or 138 ; 
 and therefore, by (2), it is a factor of 138 + 46, or 184. 
 
 Hence, 23 is a common factor of 69 and 184. 
 
MEASURES AND MULTIPLES OF NUMBERS. 103 
 
 Again, every common factor of 69 and 184 is, by (1), a factor of 
 2 X 69, or 138 ; and, by (2), a factor of 184 — 138, or 46. 
 
 Every such factor, being now a common factor of 69 and 46, is, by 
 (2), a factor of 69 - 46, or 23. 
 
 Therefore, the greatest common factor of 69 and 184 is contained 
 in 23, and cannot be greater than 23. And 23, which we have shown 
 to be a common factor of 69 and 184, must be their G. C. M. 
 
 197. In the course of this operation every remainder 
 contains, as a factor of itself, the G. C. M. sought ; and 
 this G. C. M. is the greatest factor common to that re- 
 mainder and the preceding divisor. Therefore, 
 
 If the remainder from any division is found to contain a 
 factor that is not a factor of the preceding divisor, the re- 
 mainder may be divided by that factor, and the quotient 
 used as the next divisor. 
 
 If a factor common to any remainder and the preceding 
 divisor is found, both remainder and divisor may be divided 
 by the common factor, and that factor must be reserved as a 
 factor of the G. C. JH. sought. 
 
 198. We will illustrate by two examples. 
 
 1. Find the G. C. M. of 4627 and 8593. 
 
 4627)8593(1 t^ factor 6 is thrown out of the first 
 
 4627 remainder 3966, for it is prime to 4627, 
 
 6 1 3966 and therefore is not a factor of the G. C. M. 
 
 661)4627(7 sought. 
 
 4627 Therefore, the G. C. M. is 661. 
 
 2. Find the G. C. M. of 72,471 and 134,589. 
 
 3172471 134589 The common factor 3 is first 
 
 24157 ) 44863(l taken out of both numbers. From 
 
 24157 the remainder 20,706 the factor 6, 
 
 6 1 20706 which is prime to 24,157, is ejected. 
 
 345l)24157(7 The *** ^' M ' is ' there ^ ore ' 3 x 
 
 24157 3451, or 10,353. 
 
104 MEASURES AND MULTIPLES OF NUMBERS. 
 
 
 Exercise 41. 
 
 Find the G. C. M. of : 
 
 
 
 1. 
 
 2479 and 3589. 
 
 11. 
 
 44,323 and 61,087. 
 
 2. 
 
 3045 and 6195. 
 
 12. 
 
 232,353 and 39,699 
 
 3. 
 
 568 and 712. 
 
 13. 
 
 33,853 and 35,017. 
 
 4. 
 
 11,023 and 6493. 
 
 14. 
 
 5115 and 7254. 
 
 5. 
 
 1485 and 2160. 
 
 15. 
 
 2268 and 3348. 
 
 6. 
 
 7040 and 7392. 
 
 16. 
 
 1003 and 2419. 
 
 7. 
 
 2760 and 4485. 
 
 17. 
 
 419 and 52.301. 
 
 8. 
 
 1177 and 2675. 
 
 18. 
 
 30,072 and 133,784. 
 
 9. 
 
 78,473 and 94,653. 
 
 19. 
 
 4257 and 10.836. 
 
 10. 
 
 35,143 and 10,283. 
 
 20. 
 
 17,104 and 27,794. 
 
 199. To find the G. C. M. of several large numbers, we 
 find the G. C. M. of two of the numbers; then of that 
 result and a third number ; then of that result and a 
 fourth ; and so on. The last G. C. M. is the one required. 
 
 The work can often be very much shortened by removing from each 
 of the numbers all factors less than 13. 
 
 Find the G. C. M. of 3555, 4977, and 6636. 
 
 3 
 
 8666 
 
 4077 
 
 6636 
 
 Hence, the G. C. M. 
 
 6 
 
 1185 
 
 3 | 1659 
 
 4 | 2212 
 
 required is 3 X 79, or 
 
 3|237 
 
 7J553 
 
 7|553 
 
 237. 
 
 79 
 
 79 
 
 79 
 
 
 
 Exercise 42. 
 
 
 Fin 
 
 d the G. 
 
 C. M. of : 
 
 
 
 1. 855, 1197, 1596. 5. 1177, 1391, 1819. 
 
 2. 3864, 3404, 3657. 6. 4939, 1347, 3143. 
 
 3. 15,561, 11.115, 13.585. 7. 740, 333, 296. 
 
 4. 2943, 2616, 4578. 8. 833, 1785, 1309. 
 
 9. 4994, 7491, 9988, 12,485, 16,571. 
 
MEASURES AND MULTIPLES OF NUMBERS. 105 
 
 Least Common Multiple. 
 
 200. Multiples. If a number is multiplied by an in- 
 teger, the product is called a multiple of the number. 
 
 Thus, $20 is a multiple of $5, since 4 times $5 is $20. 
 
 201. A series of multiples of a number is found by 
 multiplying the number by the integers, 1, 2, 3, 4, 5, etc. 
 
 Since a composite number is the product of only one set 
 of prime numbers (§ 174), every multiple of a number con- 
 tains all the prime factors of the number. 
 
 202. Common Multiple. A multiple of two or more 
 numbers is called a common multiple of the numbers. 
 
 Thus, 6 X $2 = $12 ; 4 X $3 = $12 ; 3 X $4 = $12 ; 2 X $6 = $12. 
 Therefore, $12 is a common multiple of $2, $3, $4, and $6. 
 
 203. Least Common Multiple. The smallest common 
 multiple of two or more numbers is called their least com- 
 mon multiple ; and it- is the smallest number that is exactly 
 divisible by each of them. 
 
 Thus, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, etc.; and 
 the multiples of 4 are 4, 8, 12, 16, 20, 24, etc. The common multiples 
 of 3 and 4 are 12, 24, etc. ; and the smallest of these is 12. Therefore, 
 the least common multiple of 3 and 4 is 12. 
 
 204. The letters L. C. M. stand for the words Least 
 Common Multiple. 
 
 205. The L. C. M. of two or more numbers is a number 
 that contains all the prime factors of each of these 
 numbers. Every prime factor, therefore, must occur in 
 the L. C. M. the greatest number of times it occurs as a 
 factor in any one of them. 
 
 Thus, $20 = 2 X 2 X 5 X $1, and $30 = 2 X 3 X 5 X $1. 
 
 The L. C. M. of $20 and $30 is, therefore, 2 X 2 X 3 X 5 X $1, or $60. 
 
106 MEASURES AND MULTIPLES OF NUMBERS. 
 
 206. Find the L. C. M. of 84, 168, 252, and 420. 
 
 Solution. Resolve each of the numbers into its prime factors. 
 
 2 
 2 
 3 
 
 84 
 
 2 
 
 168 
 
 2 
 
 252 
 
 2 
 
 420 
 
 42 
 
 2 
 
 84 
 
 2 
 
 126 
 
 2 
 
 210 
 
 21 
 
 2 
 
 42 
 
 3 
 
 63 
 
 3 
 
 106 
 
 7 
 
 3 
 
 21 
 
 7 
 
 3 
 
 21 
 
 7 . 
 
 5 
 
 35 
 
 7 
 
 The factor 2 occurs three times in 168 ; the factor 3 occurs twice in 
 262 ; the factor 6 occurs once in 420 ; and the factor 7 occurs once in 
 all the given numbers. 
 
 Therefore, the L. C. M. is 2 8 X 3 2 X 5 X 7 = 2520. Hence, 
 
 207. To Find the L. C. M. of Two or More Numbers, 
 Separate each number into its prime factors. Find the 
 
 product of these factors, taking each factor the greatest num- 
 ber of times it occurs in any one of the given numbers. 
 
 208. Examples. 1. Find the L. C. M. of 18, 24, 27, 45. 
 
 Arrange the numbers in line, and divide by the smallest prime 
 number that will divide two or more of the numbers. 
 
 2 
 
 18 
 
 24 
 
 27 
 
 46 
 
 8 
 
 9 
 
 12 
 
 27 
 
 45 
 
 :) 
 
 
 4 
 
 9 
 
 15 
 
 We first divide by 2, and write the quotients and midivided num- 
 bers in a line below. In the first line of quotients we cancel 9, as it 
 is an exact divisor of 27, and, therefore, 27 contains all the factors of 
 9. We next divide by 3, and the quotients by 3, and obtain the num- 
 bers 4, 3, 6 in the last line. No two of the numbers 4, 3, and 5 have a 
 common factor. Hence, the L. C. M. is 2 X 3 X 3 X 4 X 3 X 5 = 1080. 
 
 2. Find the L. C. M. of 3, 9, 27, 54. 
 
 £, ?, 2T, 54. 
 
 We cancel the 3, which is contained in 9 ; then the 9, which is 
 contained in 27 ; then the 27, which is contained in 54, and have 54 
 for the L. C M. of the numbers. 
 
MEASURES AND MULTIPLES OF NUMBERS. 107 
 
 3. Find the L. C. M. of 13, 15, 26, 39 : 
 
 S) X? 15 26 39 
 
 26 
 
 n 
 
 We cancel the 13 of the first line and divide by 3, getting 5, 26, 13. 
 We cancel the 13 of this line. The L. C. M. is 3 X 6 X 26 = 390. 
 
 
 
 Exercise 43. 
 
 Find the L. C. M. of : 
 
 
 
 1. 
 
 6, 14, 21. 
 
 
 26. 
 
 30, 42, 105, 70. 
 
 2. 
 
 8, 12, 3, 24. 
 
 
 27. 
 
 36, 24, 35, 20. 
 
 3. 
 
 6, 10, 15. 
 
 
 28. 
 
 7, 11, 14, 15. 
 
 4. 
 
 9, 12, 18, 4. 
 
 
 29. 
 
 12, 18, 27, 63, 28. 
 
 5. 
 
 15, 21, 35. 
 
 
 30. 
 
 34, 26, 65, 85, 51, 39. 
 
 6. 
 
 12, 20, 24. 
 
 
 31. 
 
 12, 18, 96, 144. 
 
 7. 
 
 14, 24, 28. 
 
 
 32. 
 
 84, 156, 63, 99. 
 
 8. 
 
 12, 15, 20. 
 
 
 33. 
 
 17, 51, 119, 210. 
 
 9. 
 
 16, 24, 32. 
 
 
 34. 
 
 16, 30, 48, 56, 72. 
 
 10. 
 
 21, 33, 77. 
 
 — 
 
 35. 
 
 27, 33, 54, 69, 132. 
 
 11. 
 
 27, 33, 99. 
 
 
 36. 
 
 15, 26, 39, 65, 180. 
 
 12. 
 
 7, 11, 13. 
 
 
 37. 
 
 44, 126, 198, 280, 330, 
 
 13. 
 
 77, 55, 35. 
 
 
 38. 
 
 50, 338, 675, 975. 
 
 14. 
 
 16, 18, 27, 72. 
 
 
 39. 
 
 552, 575, 920. 
 
 15. 
 
 10, 12, 22, 33, 
 
 60. 
 
 40. 
 
 228, 304, 342. 
 
 16. 
 
 15, 16, 18, 20, 
 
 22, 24. 
 
 41. 
 
 1080 and 1260. 
 
 17. 
 
 56, 64, 70, 84, 
 
 112. 
 
 42. 
 
 600 and 480. 
 
 18. 
 
 48, 54, 81, 144 
 
 :, 162. 
 
 43. 
 
 1564 and 1932. 
 
 19. 
 
 75, 100, 120, 150, 180. 
 
 44. 
 
 2530 and 1760. 
 
 20. 
 
 112, 168, 196, 
 
 224. 
 
 45. 
 
 936 and 2925. 
 
 21. 
 
 7, 14, 15, 21, 45. 
 
 46. 
 
 3432 and 4032. 
 
 22. 
 
 16, 25, 81. 
 
 
 47. 
 
 1875 and 2425. 
 
 23. 
 
 26, 39, 52, 65. 
 
 
 48. 
 
 1632 and 2976. 
 
 24. 
 
 80, 72, 225, 48 
 
 
 49. 
 
 1001 and 2233. 
 
 25. 
 
 10, 20, 30, 40, 
 
 50, 60. 
 
 50. 
 
 539 and 1463. 
 
108 MEASURES AND MULTIPLES OF NUMBERS. 
 
 209. If the given numbers are large and contain no 
 prime factors that can readily be detected, it is best to 
 obtain the common factors by the process for finding the 
 G. C. M. under like circumstances. 
 
 Example. Find the L. C. M. of 1247 and 1769. 
 
 1247)1769(1 
 1217 
 
 2j 
 9 1 
 
 622 
 
 261 
 29)1247(43 
 116 
 87 
 87 
 
 Hence, the G. C. M. of 1247 and 1769 is 29 ; 
 and 1247 = 29 x 43, 
 and 1769=29 X <il. 
 Therefore, the L. C. M. of 1247 and 1769 
 is 29 X 43 X 61 = 1247 X 61, or 1769 X 43, 
 that is, 76,067. 
 
 210. From this process it will be seen that : 
 
 The L. C. M. of two numbers may be found by dividing 
 either of the numbers by their G. C. M. and multiplying the 
 quotient by the other number. 
 
 211. The L. C. M. of two prime numbers, or of two 
 numbers prime to each other, is their product. 
 
 
 Exercise 44. 
 
 Find the L. C. M. 
 
 of: 
 
 
 
 1. 
 
 424 and 583. 
 
 
 11. 
 
 3864, 3404, 3657. 
 
 2 
 
 319 and 407. 
 
 
 12. 
 
 539 and 253. 
 
 3. 
 
 1679 and 1932. 
 
 
 13. 
 
 2943, 2616, 4578. 
 
 4. 
 
 1003 and 2419. 
 
 
 14. 
 
 2863 and 1151. 
 
 5. 
 
 1003 and 1357. 
 
 
 15. 
 
 1177, 1391,1819.. 
 
 6. 
 
 899 and 961. 
 
 
 16. 
 
 5317 and 2863. 
 
 7. 
 
 407, 703, 444. 
 
 
 17. 
 
 12,703 and 12,879. 
 
 8. 
 
 411, 959, 2055. 
 
 
 18. 
 
 23,309 and 10,753. 
 
 9. 
 
 221 and 351. 
 
 
 19. 
 
 4939 and 3143. 
 
 10. 
 
 1426 and 989. 
 
 
 20. 
 
 4199 and 6137. 
 
CHAPTEE VII. 
 
 COMMON FEACTIONS. 
 
 212. What is the name of one of the parts when a unit 
 is divided into : 
 
 1. 
 
 2. 
 3. 
 4. 
 5. 
 
 Two equal parts ? 6. Eight equal parts ? 
 Three equal parts ? 7. Ten equal parts ? 
 Four equal parts ? 8. Twelve equal parts? 
 Five equal parts ? 9. Twenty equal parts ? 
 Six equal parts ? 10. One hundred equal parts ? 
 
 
 213. A unit contains how many : 
 
 1. 
 
 Halves? 7. Ninths? 13. Twentieths? 
 
 2. 
 
 Thirds? 8. Sevenths? 14. Twenty-fourths? 
 
 3. 
 
 Fourths? 9r Tenths? 15. Thirtieths? 
 
 4. 
 
 Fifths? 10. Twelfths? 16. Fortieths? 
 
 5. 
 
 Sixths? 11. Elevenths? 17. Fiftieths? 
 
 6. Eighths? 12. Fifteenths? 18. Hundredths? 
 
 214. When a unit is divided into twelve equal parts, 
 what is the name of : 
 
 1; One part? 
 
 2. Three parts ? 
 
 3. Six parts ? 
 
 4. Two parts ? 
 
 5. Five parts ? 
 
 6. Four parts ? 
 
 7. Eight parts ? 
 
 8. Ten parts ? 
 
 9. Twelve parts ? 
 
 215. Equal parts of a unit are called fractional parts of 
 the unit. 
 
 216. In three quarters of a yard, the unit counted is a 
 quarter of a yard. 
 
110 COMMON FRACTIONS. 
 
 217. A unit that is a fractional part of a whole unit is 
 called a, fractional unit. 
 
 218. Fractions. Numbers that count fractional units 
 are called fractions. 
 
 219. Common Fractions. A fraction expressed by two 
 numbers one under the other with a line between them 
 is called a common fraction. 
 
 220. Simple Fractions. If the two numbers are whole 
 numbers, the fraction is called a simple fraction. 
 
 Thus, three fifths, written \ , is a simple fraction. 
 
 221. The lower number is called the denominator (name- 
 giver), the upper number is called the numerator (number- 
 giver) ; and the numerator and denominator together are 
 called the terms of the fraction. 
 
 222. The fraction $ means 3 times J, where £ is the 
 fractional unit and 3 is the number of them. 
 
 223. The fractional unit is expressed by 1 divided by the 
 denominator, and the number of fractional units taken is 
 expressed by the numerator. 
 
 Name the fractional unit and the integral unit of : 
 
 1. f of an inch. 5. £ of a week. 
 
 2. $ of a dollar. 6. T 7 5 of a foot. 
 
 3. | of a bushel. 7. ^ of a pound. 
 
 4. § of a yard. 8. f of an acre. 
 
 224. To Read a Common Fraction, 
 
 Read the numerator and thert the denominator. 
 
 Thus, £, |, f , f , \, T 8 T are read one half, two thirds, three fourths 
 or three quarters, two fifths, one sixth, eight elevenths. 
 
COMMON FRACTIONS. Ill 
 
 Bead: |j |; |j Vs A* if; Ai Ari Hi AV 
 
 Express in figures : 
 
 1. Two thirds. 5. Eleven sixteenths. 
 
 2. Five sevenths. 6. Seventeen twentieths. 
 
 3. Seven ninths. 7. One twenty-fifth. 
 
 4. Eight tenths. 8. Thirty hundredths. 
 
 225. A proper fraction is a fraction whose numerator 
 is less than its denominator ; as J. 
 
 226. An improper fraction is a fraction whose numerator 
 is not less than its denominator ; as §, J^ 7 -. 
 
 Note. When the numerator is greater than the denominator, 
 more than one unit must be regarded as divided into equal parts. 
 
 Thus, £ means that three units have been divided each into four 
 equal parts, and that all the parts of two units and one part of the 
 third unit are taken ; or, £ means that nine units, considered as one 
 quantity, have been divided into four equal parts, and that one of 
 these parts has been taken. 
 
 227. How many integral units must be divided into 
 equal parts that we may have f of the unit ? § ? y 5 ? 
 £ ? 2£ ? y ? ^7 ? yt ? 
 
 228. A mixed number is a whole number and a frac- 
 tion ; as 4f , 5.35. These are read four and three sevenths, 
 five and thirty-five hundredths. 
 
 Note. Every mixed number means that some entire units are 
 taken and the fraction of another unit. 
 
 229. Name the proper fractions, the improper fractions, 
 and the mixed numbers of the following expressions : 
 
 3*1 § ; *&\ I; tV; ft; u»; 3« ; T y^ ; v; if; \%h 
 
 18H; «; 27^; fj; V 1 -; «; 3*; T \; V- 
 
112 COMMON FRACTIONS. 
 
 Reduction of an Improper Fraction to a Whole or a Mixed 
 
 Number. 
 
 230. An improper fraction represents a quantity which 
 can also be represented by a whole number, or else by a 
 mixed number. 
 
 Thus, ^= 2f; for, if we suppose several units to be divided each 
 into seven equal parts, and we take 17 of these parts, 14 (that is, 2X7) 
 will make two units, and the three parts remaining will be three 
 sevenths of another unit. Hence, 
 
 231. To Reduce an Improper Fraction to a Whole or a 
 Mixed Number, 
 
 Divide the numerator by the denominator. 
 
 The quotient will be the whole number, and the remainder, if any, 
 will be the numerator of the fractional part, of which the denominator 
 will be the denominator of the improper fraction. 
 
 Exercise 45. 
 Reduce to a whole or a mixed number : 
 
 1. if.. 8. V- »• V- 22 - W- 
 
 2. 
 
 ¥• 
 
 9. 
 
 1 
 
 16. 
 
 ¥• 
 
 23. 
 
 w- 
 
 3. 
 
 ¥• 
 
 10. 
 
 ¥• 
 
 17. 
 
 * 
 
 24. 
 
 w- 
 
 4. 
 
 W- 
 
 11. 
 
 «• 
 
 18. 
 
 w- 
 
 25. 
 
 w- 
 
 5. 
 
 V* 
 
 12. 
 
 ft. 
 
 19. 
 
 w- 
 
 26. 
 
 A IP. 
 
 6. 
 
 V 
 
 13. 
 
 «. 
 
 20. 
 
 w- 
 
 27. 
 
 m*- 
 
 7. 
 
 ¥« 
 
 14. 
 
 «• 
 
 21. 
 
 w- 
 
 28. 
 
 ¥t¥« 
 
 Reduction of a "Whole or a Mixed Number to an Improper 
 Fraction. 
 
 232. A whole number may be expressed as a fraction 
 with any given denominator. 
 
 Thus, 7 = A, 8 - ; for, as each unit contains 9 ninths, 7 units contain 
 7X9 ninths, that is, 63 ninths. Hence, 
 
COMMON FR ACTIONS. 113 
 
 233. To Reduce a Whole Number to a Fraction, 
 
 Multiply the whole number by the denominator of the required 
 fraction, and under this product write the denominator. 
 
 Exercise 46. 
 
 Reduce to an improper fraction : 
 
 1. 
 
 4 = -*-. 
 
 5. 
 
 U= r- 
 
 9. 
 
 9 = „. 
 
 2. 
 
 5 = - T -. 
 
 6. 
 
 7 = - 7 -. 
 
 10. 
 
 18 = TT . 
 
 3. 
 
 « = -»- 
 
 7. 
 
 3= 5 - 
 
 11. 
 
 12 = „. 
 
 4. 
 
 8=*- 
 
 8. 
 
 14 = T *- 
 
 12. 
 
 16=„. 
 
 234. A mixed number represents a quantity that can 
 also be represented by an improper fraction. 
 
 Thus, 6/ T = || ; for each unit contains 13 thirteenths ; therefore 5 
 units contain 6 X 13 thirteenths, or 65 thirteenths; which, together 
 with the 7 thirteenths, make 72 thirteenths. Hence, 
 
 235. To Reduce a Mixed Number to a Fraction, 
 
 Multiply the whole number by the denominator of the frac- 
 tion, and to the product add the numerator ; under this sum 
 write the denominator. 
 
 Exercise 47. 
 Reduce to an improper fraction : 
 
 1. 3|. 10. 44f. 19. 14}$. 28. 36f£. 
 
 2. 5 T V 11. 2^|. 20. 21^V 29. 11^. 
 
 3. 12 T V 12. 3||. 21. 6U. 30. 3H- 
 
 4. 8f 13. 10 T V 22. 16^. 31. 16H- 
 
 5. 25f. 14. 121-f. 23. lljf. 32. 15 ? V 
 
 6. 17 J. 15. 84^. 24. 8 T V 33. 108^. 
 
 7. 8 A- 16 - 16£f. 25. 12f. 34. 51 T V 
 
 8. 9^. 17. 17 T V 26. 27£. 35. 40}f 
 
 9. 162 T \. 18. 19f. 27. 111$. 36. 864^. 
 
114 COMMON FRACTIONS. 
 
 Reduction of a Fraction to Lower Terms. 
 
 a \ I 1 1 I I 1 1 I 1 1 1 I 1 I 1 * 
 C D E F 
 
 236. If AB is divided into 5 equal parts, how many of 
 these parts are there in AC ? What part of AB is AC? 
 
 If AB is divided into 15 equal parts, how many of these 
 parts are there in AC? What part of AB is AC? 
 Which is the greater fraction, {§ or $ ? 
 What must be done to \$ to make f ? 
 Which is greater, \ or £ ? ^ or \ ? \\ or f ? Hence, 
 
 237. To Reduce a Fraction to Lower Terms, 
 
 Divide both numerator and denominator by any common 
 factor. 
 
 238. A fraction is expressed in its lowest terms when 
 the numerator and denominator are prime to each other. 
 
 239. Examples. 1 . Reduce $££ to its lowest terms. 
 
 Ht=lf=A ss tJ 
 
 the common divisors used being 9, 4, and 3. 
 
 2. Reduce §£§ to its lowest terms. 
 
 We first find the prime factors of 333 to be 3, 3, and 37. 
 Since the factor 3 of 333 will not exactly divide 259 we try 37, and 
 find it is contained 7 times in 259. 
 
 Dividing 259 and 333 each by 37, we have f f| = $. 
 
 3. Reduce Iff £ to lts lowest terms. 
 
 Since no common factor can be readily detected, we find the G. C. M. 
 of 1261 and 1649 to be 97. 
 
 Dividing 1261 and 1649 each by 97, we have \\\\ = jf Hence, 
 
 240. To Reduce a Fraction to its Lowest Terms, 
 Cancel all the factors common to the numerator and denomi- 
 nator ; or divide both terms by their G. C. M. 
 
COMMON FRACTIONS. 115 
 
 V 
 
 Exercise 48. 
 Reduce to lowest terms : 
 i. B*. 8 - t¥AV 15. m%. 22. iffff. 
 
 2. 
 
 «f 
 
 3. 
 
 T3 2 2~V 
 
 4. 
 
 Hit- 
 
 5. 
 
 Htf- 
 
 6. 
 
 MIS- 
 
 7. 
 
 iltl- 
 
 9. 
 
 MM- 
 
 16. 
 
 T§Mtf- 
 
 23. 
 
 *»lf 
 
 10. 
 
 tVsV 
 
 17. 
 
 Hit- 
 
 24. 
 
 tM&- 
 
 11. 
 
 MH- 
 
 18. 
 
 T§M' 
 
 25. 
 
 Ht£- 
 
 12. 
 
 tVsV 
 
 19. 
 
 51 6 
 STOT' 
 
 26. 
 
 Hill- 
 
 13. 
 
 •HI- 
 
 20. 
 
 3 8 7 2 
 ¥2"F(TT* 
 
 27. 
 
 ««*• 
 
 14. 
 
 *WA- 
 
 21. 
 
 7 8 4 7 3 
 
 28. 
 
 mm 
 
 241. In practice, in the answers to all examples, unless 
 the problem expressly demands the contrary, every frac- 
 tion is left in its lowest terms, and every improper fraction 
 is reduced to a whole or a mixed number. 
 
 Reduction of a Fraction to Higher Terms. 
 
 242. How many quarters are there in -£-? Which is 
 greater, -J- of an apple or f of an apple ? 
 
 How many sixths of an orange can be cut from -J of an 
 orange ? -J- of an orange ? f of an orange ? 
 
 Which is greater, £ or T \ ? f r T e ¥ ? f or £| ? } or §§ ? 
 By what must we multiply both terms to change T 6 T to 
 
 H* ftoM? *to«? 
 
 243. To Reduce a Fraction to Higher Terms, 
 
 Divide the required denominator by the denominator of the 
 given fraction, and multiply both terms of the fraction by the 
 quotient. 
 
 Exercise 49. 
 
 Reduce : 
 
 
 
 
 
 1. fto20ths. 
 
 4. 
 
 T 7 s to 39ths. 
 
 7. 
 
 T \ to 144ths. 
 
 2. f to 24ths. 
 
 5. 
 
 T \ to 90ths. 
 
 8. 
 
 T 7 ¥ to 144ths. 
 
 3. § to 50ths. 
 
 6. 
 
 § to 108ths. 
 
 9. 
 
 T V to 156ths. 
 
116 COMMON FRACTIONS. 
 
 Multiplication of Fractions. 
 
 244. Compound Fractions. A fraction of a whole num- 
 ber, of a mixed number, or of a fraction, is called a com- 
 pound fraction. 
 
 245. The expression | X i means the same as $ of £, 
 and the sign X should be read of or multiplied by when it 
 follows a fraction. 
 
 246. 7 times 3 horses are how many horses ? 
 
 7 times three fifths ($ ) are how many fifths ? 
 £ of 12 men are how many men ? 
 12 X $£ are how many dollars ? 
 
 247. To Find the Product of a Whole Number and a 
 Fraction, 
 
 Divide the product of the numerator and the whole number 
 by the denominator. 
 
 Note. A factor common to the whole number and the denominator 
 should be cancelled. Thus, 
 
 o 8 
 
 Exercise 50. 
 Find the product of : 
 
 1. £X2. 10. £$X2. 19. ££Xl5. 28. Jf X 90. 
 
 2. |X9. 11. f$X3. 20. £§X20. 29. f of 434. 
 
 3. 10 X£. 12. ^|X4. 21. ,&of324. 30. 468 X y. 
 
 4. 15 X§. 13. 5 X}}. 22. ^ of 273. 31. 30 X If 
 
 5. 3&X7. 14. 6X^|. 23. | T of 242. 32. 100 X |f . 
 
 6. 16 X j. 15. 7 X ft 24. 340 X T V 33. ?$ X 54. 
 
 7. |X2. 16. 8 X^. 25. 450 X^. 34. §£ X 48. 
 
 8. fVX5\ 17. ^X10. 26. T ^X1000. 35. 72 X |$. 
 
 9. 27 Xf 18. JgXl2. 27. & X 210. 36. Jj of 128. 
 
COMMON FRACTIONS. 117 
 
 248. To Multiply a Fraction by a Fraction. 
 Multiply f by |. 
 
 M \ 1 I I I I I I I I I I 1 1 \B 
 C D E F 
 
 To multiply f by § means to find f of i. 
 
 If the line AB is divided into 5 equal parts at the points C, D, E, 
 and F, AF will be 4 of these parts, or 4 of AB. 
 
 If each part is subdivided into three equal parts, there will be 15 
 of these smaller parts in the whole line, and each part will be y 1 ^ of the 
 line. Therefore, | of A is T a s of the whole line. 
 
 Now | of f is 4 times as much as } of |, or r 4 5 of the whole line. 
 
 And f of f is twice as much as £ of f , or T 8 5 of the whole line. 
 
 249. Therefore, we have the following 
 
 Kule. Find the product of the numerators for the required 
 numerator, and the product of the denominators for the 
 required denominator. 
 
 Note. Mixed numbers and whole numbers are brought under this 
 rule by first reducing them to improper fractions. 
 
 Any factor common to a numerator and a denominator should be 
 cancelled before multiplying. 
 
 Example. Find the product of T f X 2 T % X {%. 
 
 2 2 
 
 3 
 
 We first reduce the mixed number 2-fc to an improper fraction and 
 obtain f f . 
 
 We cancel the 13 of the numerators and the 13 of the denominators. 
 We cancel the common factor 17 of the 34 and the 17, and the 
 common factor 5 of the 10 and the 15. 
 
 There remains in the numerators 2X2, and in the denominators 3, 
 from which we obtain the improper fraction |. 
 
118 COMMON FRACTIONS. 
 
 Exercise 51. 
 Find the product of : 
 
 1. ioift. 10- *X4f 
 
 2. $ of 2^. 11. fofAof^ofiof JoflSj. 
 
 3. } off 12. 5JX8|. 
 
 4. 2}X2f 13. fX|XAX7f 
 
 5. 4f X2f 14. }of«of Aof8f 
 
 6. 4f X9f 15. A-XHXJIX2H. 
 
 7. i of J of 10. 16. JJX^XM,. 
 
 8. J of J of $. 17. | X Iff X ff X 17. 
 
 9. * X | X | X 4f 18. 3| X « X H X 1||. 
 
 19. i of f of f of # of f of | of | of A of 10. 
 
 20 - A 0^ A of 30. 36. 37* X H X M X Jf 
 
 21. HI X ,flfr X H X If. 87. A X ft X A X 2f 
 
 22. |XiXAXfof|ofiof8. 38. 8* X ft X 1^ X f 
 
 23. A of |» of rfV 39. 62^ X A X | X 15. 
 
 24. ^ T X T V X | § X 48. 40. ft X 87* X ft X f 
 
 25. || of ^ of |f of 12. 41. 1|X1^X 3ft X ft. 
 
 26. If- X 4* X |. 42. 6| X H X | X |. 
 
 27. 2| X If X 1« X 8. 43. ft of ft of || of lOf 
 
 28. 3f X 2i of 1 A X 1 A- 44. || X 2J| X If X 27. 
 
 29. H X 5f X 4i X ft X5. 45. 2^ X Iff X T J V X 2^. 
 
 30. fofAX8»XAoflH-*«- H X l^fr X |J X 12|. 
 
 31. H X || X 1||. 47. HI X lft X || X ft. 
 32- |f| X «f X tfft. 48. 3| X 2 3 * ff X lft X J|. 
 
 33. IflJof AWofWI- 49 - HiX||XAXlA. 
 
 34. A X n X 6| X |f 50. 15f X ft X\%X f f 
 
 35. 12J X ft X 16» X A- ■«• m X V X A X Jf 
 
 52. lrf T Xl|jXi||XHf 
 
 53. f X ^ T X 6f X 9| X 2* X 63 X jft. 
 
 54. 6£ X 11 J X 16 T \ X ft of ft of A- 
 
 55. 2| X 7ft X 2 X 1J X A X 5 7 7 X |f 
 
COMMON FRACTIONS. 
 
 119 
 
 250. To Find the Product of a Mixed Number and a 
 Whole Number, 
 
 Find the product of the whole number and the fractional 
 part of the mixed number, then the product of the whole 
 number and the integral part, and add these products. 
 
 1. Multiply 7£ by 9. 
 
 n 
 
 9 
 64J 
 
 In Ex. 1, 9 times i equals f, or 1|. 
 1 to the product of 9 X 7, making 64. 
 
 In Ex. 2, | of 8 equals J£, or 5] 
 5 to the product of 2 X 8, making 21. 
 
 2. Multiply 8 by 2§. 
 8 
 
 n 
 
 We write the |, and add the 
 , We write the i, and add the 
 
 Exercise 52. 
 
 Find the product of : 
 
 1. 9X6|. 
 
 2. 8 X 17i. 
 
 3. 19 X 5J. 
 
 4. 7X12J. 
 
 5. 10X154. 
 
 6. 6Xlf 
 
 7. 12X2|. 
 
 8. 17X6£. 
 
 9. 19Xl T V- 
 
 10. 24 X 16f 
 
 11. 32X22§. 
 
 12. 40X8£. 
 
 13. 41x9£. 
 
 14. 18 X 7f . 
 
 15. 19X6^. 
 
 16. 20X5£. 
 
 17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 23. 
 24. 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 32. 
 
 15 X $9*. 
 6X8}. 
 HX8f. 
 100 X 6§. 
 5X34. 
 6 X 17f 
 32 X 6f. 
 13 X 3f 
 12 X 6} . 
 8 T 2 T X 12. 
 20i X 5. 
 6§ X 18. 
 
 11 x 114. 
 18 X 12$. 
 36 X 44. 
 
 12 X 20f . 
 
 33. 12X48||. 
 
 34. 11X24|. 
 
 35. 7 Xl9f. 
 
 36. 8 X 16J. 
 
 37. 5 X 29|, 
 
 38. 16 X3f 
 
 39. 19X12^. 
 
 40. 23 X 42f . 
 
 41. 18 X I24. 
 
 42. 22 X 22,V 
 
 43. 12 X 161}}. 
 
 44. 9X144*. 
 
 45. 10 X II24. 
 
 46. 14X42|. 
 
 47. 161 X 4§. 
 
 48. 140 X 5 T V 
 
120 
 
 COMMON FRACTIONS. 
 
 Division of Fractions. 
 
 251. The reciprocal of a fraction is the fraction with its 
 terms interchanged. 
 
 Thus, the reciprocal of f is J, for J X f = 1. 
 
 To write the reciprocal of a whole or a mixed number, we 
 reduce the number to an improper fraction, and write the 
 improper fraction with its terms interchanged. 
 
 Thus, the reciprocal of 4 is i; the reciprocal of 2| is f. 
 
 Multiplying by the reciprocal of a number gives the same 
 result as dividing by the number (§101). Hence, 
 
 252. To Divide by a Whole Number or a Fraction, 
 
 Multiply the dividend by the reciprocal of the divisor. 
 
 
 Exercise 53. 
 
 
 Divide : 
 
 
 
 1. §£by6. 
 
 15. 5 by 4f. 
 
 29. lt| by f . 
 
 2. H by 5. 
 
 16. 4§ by }. 
 
 30. 100 by 83£. 
 
 3. 9 by 8. 
 
 17. 8} by 6f 
 
 31. 50 by 16§. 
 
 4. 18$ by 7. 
 
 18. St by 1^. 
 
 32. H by If 
 
 5- | by f. 
 
 19. 100 by 6$. 
 
 33. liibylA- 
 
 6- « by f 
 
 20. H by If. 
 
 34. 20± by 5. 
 
 7. If by 3f 
 
 21. 3£by 5. 
 
 35. 16f by |. 
 
 8. 5* by 4§. 
 
 22. 100 by 33£. 
 
 36. 22f by 16jj. 
 
 9. 8} by 4f 
 
 23. 100 by 37£. 
 
 37. 20f by ltf. 
 
 10. 7Jby4f. 
 
 24. 7} by 6f 
 
 38. 16$ by 111. 
 
 11. 6f by9f 
 
 25. J by T V 
 
 39. 33J by 28| . 
 
 12. 8§by4§. 
 
 26. 6f by 32. 
 
 40. 471 by 171- 
 
 13. 3jby«. 
 
 27. 3} by 3f 
 
 41. 18*byl 5 V 
 
 14. 4$ by 6f. 
 
 28. 1A by 1J. 
 
 42. 37$ by 1^. 
 
 43. 3J of 2± by 1£ 
 
 of 2J. 45. 2 T \ ( 
 
 >f 51 by 7|. 
 
 44. 2f by 3iof 1 T V 46. 5$ of 
 
 81ofl^by2 T Vof5|. 
 
COMMON FRACTIONS. 121 
 
 A Short Method of Dividing a Mixed Number by a 
 Whole Number. 
 
 253. 1 
 
 Divide 16£ by 4. 
 
 2. Divide 16§ by 7. 
 
 4)_16| 
 
 7)18! 
 
 H 
 
 2/t 
 
 In the first problem we simply divide the whole number 16 by 4, 
 and then the fraction | by 4, and obtain the result at once, 4i. 
 
 In the second problem we divide the 16 by 7, and obtain the 
 quotient 2 and a remainder 2. The remainder 2 is joined with the f , 
 making 2|, or f , and § —• 7 = ^ 8 T . 
 
 This method is the shortest method of dividing a mixed number by 
 a whole number. 
 
 Exercise 54. 
 Find the quotient of : 
 
 1. 31£-^5. 5. 42f-^6. 9. 48||4-12. 
 
 2. 16^^-6. 6. 49J-T-7. 10. 24f-f-ll. 
 
 3. 14f^-2. 7. 52f-r-8. 11. 19f~7. 
 
 4. 33£-h7. r 8. 44 T \4-12. 12. 29J-f-8. 
 
 254. Examples. 1. Find the value of (2£ «rf) X f. 
 
 3 
 
 (2 1 t f) x f = 2 x g x I — I ~ li- 
 
 We have to divide 2£ by { , and to multiply the result by f. But 
 2^ is divided by § by multiplying 2$ by f. Hence, we invert the 
 divisor £ and find the product of the three fractions. 
 
 2. Find the value of 2£-f- f of f. 
 
 We have to divide 2\ by | of f ; hence we may find the product of 
 f and |, and invert this product, or we may invert both factors of this 
 product and multiply by the inverted factors f and f . 
 
122 COMMON FRACTIONS. 
 
 Exercise 55. 
 Find the value of : 
 
 1. 2jof2*-r-A<>f3|. 9. | of lA-r-Hofff 
 
 2. |of6§of A"^*. 10. » of |J-=- A of 4. 
 
 3. A -^1 of 2* of If 11. A of H p r^|ofl T V 
 
 4- A^-(*X2iXlf). 12. fofffof A-KiXJoft). 
 
 5. Jofil-rlAoflH. 13. |off ofif-rjof AoflA- 
 
 6. |of|-=-(|XA). 14 - tt-s-H)-^(6AH-4«). 
 
 7. I of H-T-Hofjf. 15. (14|-r4|) + (3H-^9|). 
 
 8- fofff-rHof if 16- fof JJof8i-r-3Aof Aofoi. 
 
 255. Example. If f of a barrel of flour costs $3, 
 what is the cost of ± of a barrel ? What is the cost of a 
 barrel ? 
 
 Solution. Since £ of a barrel of flour costs $3, \ of a barrel will 
 cost \ of $3, or $1. If £ of a barrel costs $1, a barrel will cost 4 X $1, 
 or $4. Hence, 
 
 256. To Find the Whole when a Fractional Part is 
 Given, 
 
 Divide the given part by the numerator of the fraction and 
 multiply the quotient by the denominator. 
 
 Exercise 56. 
 
 1. If | of a ton of hay costs $15, what is the cost of 
 one ton ? 
 
 2. 15 is | of what number ? 
 
 3. If $ of a roll of carpeting is worth $75, what is the 
 whole roll worth ? 
 
 4. A man sold 6f yards of cloth, which was A of tne 
 whole piece. How many yards were there in the piece ? 
 
 5. A farmer sold $ of his hay for $195.60. What was 
 the value of his entire crop of hay ? 
 
COMMON FRACTIONS. 123 
 
 6. 21} is jf of what number ? 
 
 7. 6f is ^ of what number ? 
 
 8. 2^| is |f of what number ? 
 
 9. If f of an acre of land is worth $32, what is the 
 value of an acre ? 
 
 10. If f of a bushel of wheat is worth 48 cents, what is 
 the value of 2 T 7 ^ bushels of wheat ? 
 
 11. If f of a ton of hay is worth $15, what is the value 
 of 7£ tons of hay ? 
 
 12. If | of a cord of wood is worth $4, find the value 
 of 7 cords of wood ? 
 
 13. If T 4 T of a barrel of apples is worth 44 cents, what 
 is the value of 12 barrels of apples ? 
 
 14. $125 is i more than (that is, | of) what sum of 
 money ? 
 
 15. $132 is £ less than what sum of money ? 
 
 16. 495 is -J- more than what number ? 
 
 17. 217 is -J- less than what number ? 
 
 18. 495 is T 2 jj less than what number ? 
 
 19. 495 is -j^- more than what number ? 
 
 20. If | of a yard of silk is worth $1, find the value of 
 4 yards of silk. 
 
 21. If | of a yard of linen is worth 60 cents, what is 
 the value of 2£ yards of linen ? 
 
 22. If a man who owned -J- of a schooner sold f of his 
 share for $1200, what was the value of the schooner ? 
 
 23. One fourth of one third of three sevenths of a num- 
 ber is 60. What is the number ? 
 
 24. Three fourths of two ninths of six sevenths of a 
 number is 12f . What is the number ? 
 
 25. If T 5 ^ of the goods in a store were sold for $1000, 
 what was the value of the whole stock of goods ? 
 
 26. If 3% of a farm is worth $1200, what is the value 
 of the whole farm ? 
 
124 COMMON FRACTIONS. 
 
 Least Common Denominator. 
 
 257. Similar Fractions. Fractions that have a common 
 denominator are called similar fractions. 
 
 258. Least Common Denominator. The L. C. M. of 
 
 the denominators of a series of fractions is called their 
 least common denominator. The letters L. C. D. stand for 
 the words Least Common Denominator. 
 
 259. Example. Change §, j, $ to similar fractions. 
 
 The L. C. D. of the fractions is 12. 
 
 If both terms of $ are multiplied by 4, the value of the fraction 
 will not be altered, but the form will be changed to ^. 
 
 If both terms of J are multiplied by 3, the equivalent fraction will 
 be ft. 
 
 And if both terms of $ are multiplied by 2, the equivalent fraction 
 will be |£. 
 
 The multipliers, 4, 3, and 2, are obtained by dividing 12, the L. C. D. 
 of the fractions, by the respective denominators of the given fractions. 
 Therefore, 
 
 260. To Change Fractions to Similar Fractions, 
 Divide the least common denominator of the fractions by 
 
 the denominator of the first fraction, and multiply both terms 
 of this fraction by the quotient. Proceed in the same way 
 with each of the other given fractions. 
 
 Change J, $, \% to similar fractions. 
 
 4 = 22 ; 8 = 2« ; 12 = 2* X 3. 
 
 Hence, the L. C. D. is 2 8 X 3, or 24. 
 
 
 
 and 
 
 5. : 
 
 = ih 
 
 result 
 
 may 
 
 be written as 
 
 follows : 
 
 
 
 18 
 
 21 
 
 22 
 
 24 
 
COMMON FRACTIONS. 125 
 
 By this operation the parts represented by the given fractions have 
 been subdivided into smaller parts all of one size, and the numerators 
 of the resulting fractions show the number of these smaller parts con- 
 tained in the given fractions. Thus, the quantities denoted by f , |, 
 and |£ are each subdivided into 24ths of the unit, and contain respec- 
 tively 18, 21, and 22 of these subdivisions. 
 
 261. Fractions may be compared by first reducing them to 
 equivalent fractions having a common denominator. 
 
 Determine the greater of the fractions f and |£. 
 In this case the least common denominator is 112. 
 
 Hence, f = T 8 T \, and ft = f&. 
 
 Therefore, f is greater than |£. 
 
 Exercise 57. 
 Change to similar fractions : 
 
 1. h h f 10- b H> H» tt- 
 
 11- f> f* tVtI- 
 
 13 - h h A> A> vfa- 
 
 14. h A, *>«>*■»• 
 
 is- i> h &Wi h fr- 
 it, ih a, H,i, «, ». 
 
 1^' £' |> I' A> A> A- 
 
 9. *>A>H>«- i 8 - A> A> ti» H> i> H- 
 
 19. Which is the greater, ££ or £| ? f or | ? § or $ ? 
 
 20. Arrange the fractions T \, |£, £§ in order of magni- 
 tude. 
 
 21. Arrange the fractions ^, ^, T \, T 7 ^ in order of 
 magnitude. 
 
 22. Arrange the fractions f, J, ^, £§ in order of mag- 
 nitude. 
 
 2. 
 
 f > f> i> A* 
 
 3. 
 
 fj t> A» it* 
 
 4. 
 
 A» A* A> A* 
 
 5. 
 
 H. «. H. II- 
 
 6. 
 
 |j A» A> A> ¥¥' 
 
 7. 
 
 tt> A» H> 1*. H- 
 
 8. 
 
 h h H> «• 
 
126 COMMON FRACTIONS. 
 
 Addition of Fractions. 
 262. Examples. 1. Find the sum of fa fa fa. 
 ixi+j _ 5+l + 7 _ ,,-j 
 
 2. Find the sum of J, |, £, ^. 
 
 Denominators . 
 
 4 = 22, 
 6 = 2X3, 
 9 = 32, 
 12 = 2 2 X 3. 
 
 Hence, the L. C. D. = 2 2 x 3 2 , or 36. 
 
 t + f + t+A**** 1 **^**! 
 
 = W* 
 
 = 2$. 
 
 3. Find the sum of 4^, 2 fa 5^. 
 
 f20=2 2 X 5, 
 
 Denominators . . .W 15 = 3 x 5, 
 
 [12 = 2 2 X3. 
 
 Hence, the L. C. D. = 2 2 X 3 X 5 = 60. 
 
 4H + 2 T ^ + 5 t \ = ll*a±M±ai, 
 
 = 12», 
 
 = 13|§» Hence, 
 
 263. To Add Fractions, 
 
 Change the fractions to similar fractions, if they are not 
 similar, and write the sum of their numerators over their 
 common denominator. 
 
 If any of the expressions to be added are whole or mixed 
 numbers, add together separately the fractions and the in- 
 tegers, and find the sum of the results. 
 
COMMON FRACTIONS. 127 
 
 Exercise 58. 
 Find the sum of : 
 
 1. i + |. 6. 3i + j. 11. | + f 
 
 2. £ + § + £. 7. 2f + 3|. 12.1 + 1. 
 
 3. i+i+|, 8. l| + f. 13. i+J. 
 
 4. li + 2£. 9. T * T + T 3 T + i f + H- 14. A + «■ 
 
 5. li+2|. 10. 8JV+6A + 5« + H. 15. A+H- 
 
 16. 12f+7A- IS- i + i + i + i- 
 
 17. 85A + 27«. 19. f+J+-|+|. 
 
 20. f + tt+A + A + M- 
 
 21. 5H + 11H + 24H + A + 17A + 14 + 11A. 
 
 22. 9t + 15ii + 163H + lH+10i. 
 
 23. 3f + 4f + lf + 2. 26. 4f + 3f + 2f + 1J + A- 
 
 24. 1A + 2A + 5A + A- 27. H+^ + 10 + fJ. 
 
 25. f + l| + 2+3f+4 T V 28. li + ff+B+Afc + AV 
 
 29. 2 + § + l} + 4f + 5H. 
 
 30. 3f + 6+.A + 2A + 5A + A- 
 
 31- A + A + 3H + 1H + 2TWF- 
 
 32. A + A + 9f 34. Jl + H + H. 
 
 33. 20A + HA + 5i + 305. 35. A+ii + M + H- 
 
 36. 317f + 17/ T + 4A + A + 6§ + T V 
 
 37. 4A + 8A + 4A+5? + 6* +'§•■ 
 
 38. 3a + 5A + 8^ + f* + Mjfo. 
 
 39. 4A + 7A + 6« + 275A^ + 2«. 
 
 40. H + 7A+6A + 400A + 51»f 
 
 41. 3§ + lJ + 2J + 3| + 107A + 2A. 
 
 42. 5A + 5f + 2| + 7A + 12A. 
 
 43. 4i + 2j + 3| + 7J + 8H- 
 
 44. 6^ + 7§ + 8|+9| + 8H- 
 
 45. 7j + 8} + 5M + 7H + 9i- 
 
 46. 5i + 6| + 7H + 9H + 3« + 2i. 
 
 47. 9} + 10f + ll§ + 5jJ+7A + 18f 
 
 48. H + A + if + H + * + A + H. 
 
128 COMMON FRACTIONS. 
 
 Subtraction of Fractions. 
 
 264. Example. From \\ take ft. 
 
 The L. C. D. of the fractions is 72. 
 
 U-T 5 ff =iJ TlP=H- Hence, 
 
 265. To Subtract One Fraction from Another, 
 
 Change the fractions to similar fractions, if they are not 
 similar. 
 
 Subtract the numerator of the subtrahend from the numer- 
 ator of the minuend, and write the result over their common 
 denominator. 
 
 266. Examples. 1. Subtract 3^ from 4|£. 
 The L. C. D. of the fractions is 120. 
 
 If the terms are mixed numbers, subtract separately the integers 
 and the fractions, and unite the results. 
 
 2. Subtract 2£ from 5ft. 
 
 6 A " 2J = 31^ = 2H1* 1 = 2H- 
 
 The difference between 5^ and 2 J is S^^ 1 . Since we cannot 
 subtract § \ from ££, we take 1, that is, f J, from the 3, and add it to 
 £$, making f J. 
 
 3. From 8 take 2§±. 
 
 8 m 7ff, and 7}| - 2ft = 6t^i = 6 &. 
 
 4. Subtract 5J from 15f . 
 
 5J + £ = 6, and 16| + | = 15J. 
 Then 15J - 6 = 9J. 
 
 Since adding the same number to both the minuend and the sub- 
 trahend does not alter the difference between them, we may add 
 to the subtrahend such a fraction as will make it a whole number, pro- 
 vided we add the same fraction to the minuend. 
 
COMMON FRACTIONS. 129 
 
 
 Exercise 59. 
 
 
 
 Find the value of : 
 
 
 
 1. 52^-46. 
 
 15. 7f-4f. 
 
 29. 
 
 *-!»«• 
 
 2. f-f. 
 
 16. 6§ — 2j. 
 
 30. 
 
 1473-279H- 
 
 3. !-§• 
 
 17. 9f-4f 
 
 31. 
 
 1473ft -279H 
 
 4- ft -ft- 
 
 18. 4}-i. 
 
 32. 
 
 1473ft -279H- 
 
 5. «-ft. 
 
 19. 6] — 4f. 
 
 33. 
 
 278H -30ft. 
 
 6. 4 — £. 
 
 20. 7£-2f. 
 
 34. 
 
 125ft -10iJ. 
 
 7. 7-f. 
 
 21. 8|-4f. 
 
 35. 
 
 118ft -17ft. 
 
 8. 3-f 
 
 22. 85ft — 27^. 
 
 36. 
 
 94ft -91«. 
 
 9. 8-f 
 
 23. 8ft -2«. 
 
 37. 
 
 7ft -2«. 
 
 10. 5-f. 
 
 24. 10 — 3|. 
 
 38. 
 
 »*-«. 
 
 11. 5-}. 
 
 25. 120JJ — 110H- 
 
 39. 
 
 H-tW 
 
 12. 6J-5f 
 
 26. 5«-|J. 
 
 40. 
 
 ft 3&V 
 
 13. 4f — 3f. 
 
 27. 13ft -2jf 
 
 41. 
 
 ttf-lff- 
 
 14. 74 -2ft. 28. 2HJ-1HJ. 42. §«-«»• 
 
 Plus and Minus Terms. 
 
 267. Example. Simplify 5| - 4} + 3§ - 2ft. 
 
 5f + 8f =8i5 r V-^ = 8!f = 9 T v 
 
 4| + 2ft = 6lftV± = 6§f = 7 ft, 
 
 and Oft - 7ft = 2**f#L = 2ft. 
 
 We first add the two plus terms and obtain Oft. 
 Then we add the two minus terms and obtain 7 ft. 
 Then we subtract the sum of the minus terms from the sum of the 
 plus terms and obtain 2ft. Hence, 
 
 268. To Simplify an Expression of Plus and Minus 
 Terms, 
 
 Subtract the sum of the minus terms from the sum of the 
 plus terms. 
 
130 common fractions. 
 
 Exercise 60. 
 Simplify : 
 
 1. 3|-2|+4^ + lJ-5A- 
 
 2- lA-tt + 7*-2i-lH- 
 
 3. 12-3»-l^-4/„ + 2iS-4S. 
 
 4. 43A-li-M*-lH-2*i-2A-2H-8TV 
 
 6- (8A + U» + 17«+«>)-(30j§ + ll«). 
 
 7. (m« + 93AV) + (172« - 93^)- 
 
 8. (172H + 93AV)- ("2« -93^V). 
 
 »• (ft-A) + (A + !*«)• 
 io i-A-2S+Ȥ + 7A-i*-A- 
 
 12. 9| — 7 — f — f. 
 
 13. 5§ + 8f — If-4J. 
 
 14. 6|-5§4-4§-4^. 
 
 15. HA + 9J-6i-12*-3». 
 
 16. 20§-2f-9| + 10ft-14 1 V 
 
 17. 95§-9A-8»-14f + 74j. 
 
 18. 12} + 23|-(4A + 12f + 7H). 
 
 19. 16A + 18A-(5| + 9A + 14A). 
 
 20. 97f-(20 + 9| + 18ft + 24fJ). 
 
 21. 2« + ajf-(l« + lJi + H). 
 
 22. «8 + «J*-*«Wfc- 
 
 Complex Fractions. 
 
 269. Complex Fractions. A fraction that has a frac- 
 tion in one or both of its terms is called a complex fraction. 
 
 Thus, i i ?i 2±j*f <L±JJ are complex fractions. 
 7' f 4 ' 1 - | ' 4 X A 
 
 The simplest meaning to give to a complex fraction is 
 that of an indicated division. Hence, 
 
3. 
 
 COMMON FRACTIONS. 131 
 
 270. To Simplify a Complex Fraction, 
 
 Multiply its numerator by the reciprocal of its denominator. 
 
 271. Examples. 1. | = f -20 = f x± = §>. 
 
 4f 6*7 6 U 12' 
 2 
 
 „2_ 5_ 6-5 _ 1. 5_ 1 _ 20 - 3 __ 17 , 
 3 h l 9 V *6 h 24 24' 
 
 X 9 24 ft 17 51 51 
 3 
 
 4j — 3J-2J + 1A 86 — 63-42 + 25 = 6 = 1 
 3j — 2f + 2i — A 64-48 + 45-7 54 9* 
 
 A complex fraction may be simplified by multiplying both its terms 
 by the smallest number that will make them integral. This multiplier 
 will always be the L. C. D. of the fractions contained in the terms of 
 the given fraction. 
 
 Here each term of the numerator and denominator is made integral 
 by multiplying it by 18, the L. C. D. of the fractions contained in the 
 numerator and denominator of the given fraction. 
 
 6}-lt 63-lt = 243 — 68 175 - 
 * ftoflf H 35 35 v 
 
 Here the compound term T \ of If is first reduced to the simple 
 term §§, and then the numerator and denominator of the resulting 
 fraction are multiplied by 36, the L. C. D. of the small fractions. 
 
 Compound terms must first be reduced to simple terms. 
 
132 COMMON FRACTIONS. 
 
 Exercise 61. 
 Simplify : 
 
 ,24 4 ± 7 2 J^zl3 1Q 6|-lft 
 
 SJ- 8J- • l|-lf • 2}+lf 
 
 2 A 5 I* « 10 ^~^ 11 5 * + 2 * 
 
 7J- • 8 A" ' 7} -3ft' ' 4§-3H 
 
 17} lit of 3} fl ?Qf2A 12 §}_i 
 
 * 13J- "■ 4J of ft - ■ l|-i-2f" • 14 If 
 
 „ 3i of 5i 20 HXl f + iof2 t -HX2 
 
 ,S - Ui 0I 2j- Hof2 + Jof2J-lJoflJ 
 
 •SI-2H' 1X 6ft+7 S X l*X9ft- 
 
 16 2| + 2j 22 8 t -7f + 5|-4) 
 
 " H-H' »ft-8H+7l-«» 
 
 "■StS- M.JxJLxa x ^LxAxi|. 
 
 „ M + lt + A + t 24 27_ x 87i J 89ft 
 H-H + A-f 374 X 98i X 2i X 128- 
 
 18 41-2} . 26 iA x 170^12j 
 18 ' 6J -2f 6ft* 399 * 7{ * 
 
 19 m -4^ + 31 /i_126 + 21V 31 
 19 5f-4l + r 26 - (/ 697 + 8lJ-5i- 
 
 27. 
 28. 
 29. 
 80. 
 
 }ofl}| + l}of61-llof5l 
 
 Jof2|of5§. 
 HXtfrX m x w 
 5} 
 ft X 9ft X 31 X 9ft 
 ft X 3ft X 12} X m X ft 
 
 21 X 7ft 
 } X I X 18f 
 
COMMON FRACTIONS. 133 
 
 To Express One Number as the Fraction of Another. 
 
 272. Examples. 1 . What fraction of 8 is 7 ? 
 
 Since 1 is £ of 8, 
 
 7 is 7 times \ of 8 ; 
 that is, 7 is | of 8. 
 Here the number denoting the part is the numerator and the number 
 denoting the whole is the denominator of the required fraction. 
 
 2. What fraction of | is f ? 
 
 Taking the number denoting the part for the numerator, and the 
 number denoting the whole for the denominator, we have 
 
 3 3 2 2 
 
 | ; and this becomes - X - = -. Therefore, 
 
 § 16 1 
 
 273. To Find the Fraction that One Number is of 
 Another, 
 
 Take the number denoting the part for the numerator, and 
 the number denoting the whole for the denominator. 
 
 Exercise 62. 
 What fraction of : 
 
 1. 8 is 3 ? li. 2J is 7| ? 21. $10 is $§ ? 
 
 2. 3 is 8 ? 12. 7^ is 2\ ? 22. $100 is $6 ? 
 
 3. 9 is 7 ? 13. 3£ is 8} ? 23. $100 is $4£ ? 
 
 4. 7 is 9 ? 14. $2 is $l£? 24. $4 is $25 ? 
 
 5. 8 is 12 ? 15. $2£ is $5 ? 25. lOOf is 8$ ? 
 
 6. 12 is 8 ? 16. $| is $J ? 26. 21 is {f of 3| ? 
 
 7. 2£ is | ? 1 7. $| is $| ? 27. 18JJJ is f of 33| ? . 
 
 8. £ is 2£ ? 18. $2| is $* ? 28. 3£ is § X l£ ? 
 
 9. 2j is 11 ? 19. $£ is $ T V? 29. 3 T l T X 5^ 7 is 1720 ? 
 10. li is 2| ? 20. $1 is $| ? 30. 3£ X f of $ is If ? 
 
 What part of : 
 
 31 - U X £! is £ X 4 x V 
 
 32. 13f X § X e \ is } of 1|| of 1 J ? 
 
 33. H + « + A + t i» H - H + tV ~ * ? 
 
134 
 
 COMMON FRACTIONS. 
 
 34. 4 * - 2J is 6£ - 2\ ? 
 
 35. 17|-12fi85-A-A-*? 
 
 36. 24-17Ai»7 + A-A-a? 
 
 37. ^ of 2^ is 1§ -f- 2$ ? 
 
 («-F*Mi4-) 
 
 13 ? 
 
 Conversion of Fractions. 
 
 274. A decimal may be changed to a common fraction. 
 Examples. 1. Reduce 0.527 to a common fraction. 
 
 0.527 means A + T ^ + T^ = ^m ±i = iWTT- 
 
 2. 0.525 = ^^ = 1*. 
 
 3. 18.375 = 18 AW r = 18j# = 181. Hence, 
 
 275. To Reduce a Decimal to a Common Fraction, 
 For the numerator, write the figures of the decimal ; for 
 
 the denominator, write 1 with as many zeros after it as there 
 are figures in the given decimal. 
 
 
 
 Exercise 63. 
 
 
 
 Reduce to a common fraction or to a 
 
 mixed number : 
 
 1. 
 
 0.125. 
 
 11. 
 
 10.012575. 
 
 21. 
 
 0.603125, 
 
 2. 
 
 0.625. 
 
 12. 
 
 104.235. 
 
 22. 
 
 6.03125. 
 
 3. 
 
 0.675. 
 
 13. 
 
 50.0004. 
 
 23. 
 
 60.3125. 
 
 4. 
 
 10.864. 
 
 14. 
 
 100.001. 
 
 24. 
 
 7.0315. 
 
 5. 
 
 50.84. 
 
 15. 
 
 8.00725. 
 
 25. 
 
 12.0625. 
 
 6. 
 
 3.00025. 
 
 16. 
 
 20.018375. 
 
 26. 
 
 4.7168. 
 
 7. 
 
 8.1075. 
 
 17. 
 
 125.6048. 
 
 27. 
 
 0.0425. 
 
 8. 
 
 35.01024. 
 
 18. 
 
 0.128. 
 
 28. 
 
 6.46875. 
 
 9. 
 
 7.015625. 
 
 19. 
 
 0.73125 
 
 29. 
 
 0.00256. 
 
 10. 
 
 20.100256. 
 
 20. 
 
 1.1875. 
 
 30. 
 
 0.000375. 
 
COMMON FRACTIONS. 135 
 
 276. A common fraction may be reduced to a decimal. 
 
 Example. Reduce f to a decimal. 
 
 8 )5.000 
 0.625 
 
 We annex zeros to the numerator of the fraction, inserting a deci- 
 mal point before the zeros ; and then divide the numerator by the 
 denominator. 
 
 By this operation the form of the quotient is changed from f to 
 0.625, but the value remains unchanged. Hence, 
 
 277. To Reduce a Common Fraction to a Decimal, 
 
 Divide the numerator by the denominator. 
 
 
 
 Exercise 64. 
 
 
 
 Reduce to a decimal 
 
 : 
 
 
 
 i. j. 
 
 6. t&* 
 
 M. Mv 
 
 16. 
 
 w- 
 
 2- «• 
 
 '• ^^ninS- 
 
 12. iM- 
 
 17. 
 
 foflf 
 
 3- A- 
 
 8- Mffo,. 
 
 13- *¥V- 
 
 18. 
 
 foffofA 
 
 4- *• 
 
 »• 1MJ*- 
 
 14- AV- 
 
 19. 
 
 3f of 4*. 
 
 6. «V 
 
 10- xfc- 
 
 IS- rtr 
 
 Exercise 65. 
 
 20. 
 
 II of |f. 
 
 Simplify by common fractions, then by reducing the 
 common fractions to decimals, and show that the results 
 in each example agree : 
 
 1 • n + 4f + 9« + llff 8. f | - 4|. 
 
 2. 84JJ + 19 H + H . 9. 8 2 1 - 37«. 
 
 3. m + 13« + 42fJ + 2« + H- 10. 100 - 17J# 
 
 4. 5 J + 13* + 19 A + 7 A- 1 1 • 5* ~ 1* of 1&. 
 
 5. 5rf r + »ofl$ + iof2* + |off 12. H-H- 
 
 6. lAof2|. 13. 8J-Hof A- 
 
 7. 3^ + 2«. 14. if X 1000, 
 
136 COMMON FRACTIONS. 
 
 Repeating Decimals. 
 
 278. If the denominator of a common fraction in its 
 lowest terms contains other factors than 2 and 5 (the prime 
 factors of 10), the fraction can be expressed exactly by a 
 decimal ; otherwise it cannot. 
 
 Thus, if we take the fraction ^ r to express as a decimal, we have 
 
 0.27272727 and the division will never end, however far it is 
 
 carried. 
 
 279. A decimal that contains a constantly recurring 
 figure or series of figures is called a repeating decimal or 
 a circulating decimal. 
 
 Thus, the decimal 0.27272727 is a repeating decimal, the series 
 
 of figures constantly recurring being 27. 
 
 280. Repetend. The figure or series of figures that 
 constantly recurs is called the repetend. 
 
 281. If the repetend begins at the first place in the 
 decimal, the decimal is called a pure repeating decimal. 
 If the repetend begins at any place except the first, it is 
 called a mixed repeating decimal. 
 
 282. In writing a repeating decimal, we place dots 
 over the first and last figures of the repetend. 
 
 Thus, we write 0.272727 0.27, and we write 0.333 0.3. 
 
 283. Examples. 1. Reduce £ to a decimal. 
 
 The denominator contains neither 2 nor 5 ; the first figure of 
 the decimal begins the repetend ; and we reach the 
 0.857142 end of the repetend when this figure appears as a 
 7)6.000000 remainder. 0.857142 Ans. 
 
COMMON FRACTIONS. 137 
 
 2. Reduce 12^f to a decimal. 
 
 0.535714 28 
 
 28) 15.0 We reach the end of the repetend when a 
 
 140 previous remainder reappears. 
 
 100 When 2 or 5 is a factor of the denominator, 
 
 84 the number of decimal places preceding the repe- 
 
 160 (1) tend is equal to the highest exponent which 
 
 140 either of these factors has. Thus, 28 is equal 
 200 to 2 2 X 7, and the quotient contains two deci- 
 le mal places preceding the repetend. 
 40 The remainder at the beginning of the repe- 
 28 tend is the same as the remainder at the end 
 120 of the repetend, and when this remainder 
 112 reappears we need carry the division no 
 80 further. 
 
 56 Hence, 12|f = 12.53571428. 
 240 
 224 
 16 the same remainder as (1). 
 
 We may shorten the work by cancelling the factor 4 common to 
 28 and the second dividend 100, giving 25 divided by 7. 
 
 Exercise 66. 
 Reduce to a decimal : 
 
 If 5. 3Jf 9. 9 T W 13. if. 
 
 2. T « T . 6. 2 A- 10. llsY 14. f7. 
 
 3. 3 T V 7. ^. 11. it . 15. 2jfr. 
 
 4. JJ. 8. HiJ. 12. ft. 16. 5A- 
 
 rM 117 . 
 
 17. If w 7 03 1S expressed as a decimal, how many 
 
 decimal places will the quotient contain ? 
 119 
 
 18. If o* y is * s ex P resse( i as a decimal, how many 
 
 decimal places will precede the repetend ? 
 
 57 
 
 19. If - 2 _ is reduced to a decimal, how many 
 
 decimal places will precede the repetend ? 
 
138 
 
 COMMON FRACTIONS. 
 
 To Reduce a Repeating Decimal to a Common Fraction. 
 
 284. Examples. 1. Reduce 0.27 to a common fraction. 
 
 From 100 X 0.27, or 27.2727 
 
 take 1 X 0.27, or 0.2727 
 
 Then 99 X 0.27 = 27. 
 Therefore, 0.27 = |J = T 3 r . 
 
 2. Reduce 0.5243 to a common fraction. 
 
 From 10,000 X 0.5243, or 5243.243243 
 
 take _10 X 0.5243, or 5.243243 
 
 Then 9990 X 0.5243 = 5243 — 6. 
 Therefore, 0.5243 = ^^ = |||| = ^. 
 
 We multiply the decimal by such a number as will put the 
 decimal point at the end of the repetend, then by such a number as 
 will put the decimal point at the beginning of the repeiend; and divide 
 the difference of the products by the difference of the multipliers. 
 Hence, 
 
 285. To Reduce a Repeating Decimal to a Fraction, 
 For the numerator, write the difference between two num- 
 bers, one expressed by the figures of the repetend, and the 
 other by the figures that precede the repetend. 
 
 For the denominator, write a 9 for each figure of the repe- 
 tend, and annex a for each figure that precedes the repe- 
 tend. 
 
 Exercise 67. 
 
 Reduce to a common fraction or to a mixed number : 
 
 1. 
 
 0.245. 
 
 9. 
 
 1.416. 
 
 17. 
 
 0.2368. 
 
 2. 
 
 0.425. 
 
 10. 
 
 0.5575. 
 
 18. 
 
 1.136. 
 
 3. 
 
 53.00243. 
 
 11. 
 
 2.08i. 
 
 19. 
 
 1.53i. 
 
 4. 
 
 7.20li. 
 
 12. 
 
 5.12297. 
 
 20. 
 
 3.28963. 
 
 5. 
 
 2.5306. 
 
 13. 
 
 0.3590. 
 
 21. 
 
 5.8783. 
 
 6. 
 
 0.00426. 
 
 14. 
 
 4.3i62. 
 
 22. 
 
 1.69408. 
 
 7. 
 
 31.203. 
 
 15. 
 
 0.7283. 
 
 23. 
 
 0.48324. 
 
 8. 
 
 0.351. 
 
 16. 
 
 5.142857. 
 
 24. 
 
 0.0012213. 
 
COMMON FRACTIONS. 139 
 
 The Greatest Common Measure and the Least Common 
 Multiple of Fractions. 
 
 286. If we divide } by fa, we obtain the quotient 12. 
 If we divide f by }, we obtain the quotient 5f . 
 
 If we divide f by ^ 8 5 , we obtain the quotient l£. 
 
 We see from these three examples that in dividing one 
 fraction by another the quotient is integral only when the 
 numerator of the divisor is a measure of the numerator of 
 the dividend, and the denominator of the divisor is a mul- 
 tiple of the denominator of the dividend. 
 
 Therefore, that a fraction may be a common measure of 
 a series of fractions, its numerator must be a measure of 
 each numerator, and its denominator a multiple of each 
 denominator ; and that a fraction may be the greatest 
 common measure of a series of fractions, its numerator 
 must be the greatest common measure of the numerators 
 of the fractions, and its denominator the least common 
 multiple of the denominators of the fractions. Hence, 
 
 287. To Find the G. C. M. of a Series of Fractions, 
 
 Find the G. C. M. of the numerators for the numerator of 
 the required measure, and the L. C. M. of the denominators 
 for the denominator of the required measure. 
 
 288. Conversely, that a fraction may be a common mul- 
 tiple of a series of fractions, its numerator must be a 
 multiple of each numerator, and its denominator a measure 
 of each denominator; and, that a fraction may be the 
 least common multiple of a series of fractions, its numerator 
 must be the least common measure of the numerators of 
 the fractions, and its denominator the greatest common 
 measure of the denominators of the fractions. Hence, 
 
140 COMMON FRACTIONS. 
 
 289. To Find the L. C. M. of a Series of Fractions, 
 
 Find the L. C. M. of the numerators for the numerator of 
 the required multiple, and the G. C. M. of the denominators 
 for the denominator of the required multiple. 
 
 290. Examples. 1. Find the G. C. M. of ^, y, }f 
 
 The G. C. M. of 5, 25, 35 = 5 ; 
 and the L. C. M. of 36, 9, 99 = 396. 
 Hence, j$ 5 is the G. C. M. required. 
 
 2. Find the L. C. M. of ^, y, g j. 
 
 The L. C. M. of 5, 25, 35 = 175; 
 and the G. C. M. of 36, 9, 99 = 9. 
 Hence, a J* is the L. C. M. required. 
 
 Exercise 68. 
 Find the G. C. M. and L. C. M. of : 
 
 1. J, «, Jf. 7. 50^,67^,441,841,707. 
 
 2. 2| f 2f,A. »• hhhhhfv 
 
 3. 33?, 50f. 9. 1A, 1», 4f, }f 
 
 4. A,Jf |f 10 - 18f,57f 
 
 5. 5f 7J f 81, 4f, 91, 6A- It. 134f, 128J, 115f 
 6- fJ,J,i, £, A, tV 12 - %%* 
 
 13. A, B, and C start together to walk in the same 
 direction round a circular island. It takes A 2% days, B 
 2$ days, C 2$ days to walk round the island. They walk 
 until they all meet at the point of starting. In how many 
 days will they be together at the point of starting ? 
 
 14. If the step of a man is 2^ feet, and that of a horse 
 is 2| feet, find the smallest number of feet which is an 
 exact number of steps for a man and for a horse. 
 
 15. Find the largest number that is contained without 
 remainder in 2f 6 T 7 ¥ , 11£, and 19£. 
 
COMMON FRACTIONS. 141 
 
 Exercise 69. — Review. 
 
 1. Simplify JJ(|, fifjjj, T |«f F> t»f J. 
 
 2. Which is greater, and by how much, { or J J ? 
 
 3. Find the sum of 3f, 2 T * r , 6J, 7ft, lft, 
 
 4. Simplify 5J - 3 7 + 2ft ~ If- 
 
 5. Simplify If + 3| - 2ft + 4 A - 3ft. 
 
 7 95^ 16 
 
 7. Simplify 7-^2|; j-J jgj 15 ^f; ^5 7ft -4-95 
 
 6 A #of4l 17 
 
 8. Simplify 7J| X 8; 43*§ X 6f ; 6*-t-8J; 5 T V X 51 ; 
 »0f«i jjofftofjofjofj; «0fHfj JXfXftXt 
 Xf. 
 
 9. By what must £ be multiplied to obtain ^ ? ^ to 
 obtain § ? £ to obtain £ ? § to obtain $ ? f to obtain | ? 
 
 10. By what must £ be divided to obtain £ ? § to obtain 
 J ? { to obtain $ ? f to obtain |? 8 to obtain 7Jf ? 
 
 11. What number exceeds 5§ by 4} ? 
 
 12. Erom what must 6f be subtracted to leave ^ of 3£ ? 
 
 13. What fraction falls short of ft by ft ? 
 
 14. What fraction must be added to ft to make £} Y 
 
 15. Reduce to decimals: £; J; J; j; |; |; |; jj ft; 
 
 fti Tff> TffJ A> H> II J t|5 &> |> 7' |> ft J ft- 
 
 16. Reduce to common fractions : 0.16; 0.016; 0.125; 
 0.13 ; 0.725 ; 0.625 ; 0.00625 ; 0.8125 ; 0.03125 ; 0.08 ; 0.54 ; 
 0.016; 0.5437; 0.027; 0.277; 0.68494; 1.345. 
 
 a . fl . 2.8 of 2.27 
 
 17. Simplify - — • 
 
 * J 1.136 
 
 18. Multiply 6.954 by 5.303, and express the result as a 
 whole number and common fraction. 
 
142 COMMON FRACTIONS. 
 
 19. Simplify 1 J of 2% -f 6$ -=- 2§ and reduce the result 
 to a decimal. 
 
 20. From what number can 4££ be taken 9 times and 
 leave no remainder ? 
 
 21. Of what fraction is 17£ the 7th part ? 
 
 22. Add f , 0.35, J, |, 0.112, 45.28. 
 
 23. Reduce to decimals : JJ; ftj gV; JJ; Jf; ^. 
 
 24. What part of }$ is T2 \ T ? 
 
 25. Divide 0.0015 by 0.012, and express the result as a 
 common fraction in its lowest terms. 
 
 26. Reduce to decimals : fc ssibin H> I* 
 
 27. The product of two factors is j, and one factor is 1 J ; 
 find the other factor. 
 
 28. The dividend is }£, the quotient 6^ ; find the divisor. 
 
 29. The dividend is 12}£, quotient 3, remainder 1 T ^ ; 
 find the divisor. 
 
 30. Find the G. C. M. and the L. C. M. of 833, 1127, 
 1421, 343. 
 
 31. Arrange in order of magnitude -ft-, ££, £$, JJ, §§. 
 
 32. Find the L. C. M. of jf, $ f , ffs- 
 
 33. Find the G. C. M. of f f , ^-, |J, and 6£. 
 
 34. Reduce to common fractions 7.20li; 6.954. 
 
 37. Simplify ^i—Al±^L. 
 
 38. Simplify (3-71-1.908) X 7.03 
 
 2-2 -^A 
 
 / 0.24 + 0.53\ 
 40. Simplify li of 2| + 6J^ 21+^ + ^3^1 
 
COMMON FRACTIONS. 143 
 
 41. Simplify 0.9 of f of $ of 15f . 
 
 42. What part of § is } ? 
 
 43. What part of 0.390625 is 0.05 ? 
 
 44. What fraction of 0.2045 is 0.09 ? 
 
 45. Reduce to decimals : ff ; ^f ; f§. 
 
 46. The G. C. M. of three numbers is 15, and their 
 L. C. M. is 450. What are the numbers ? 
 
 47. A merchant, after selling 5£ yards and 3^ yards from 
 a remnant of calico, found that he had 7§ yards left. What 
 was the entire length of the remnant ? 
 
 48. If 3| yards of cloth are required for a coat, how 
 many coats can be made from 56^ yards of cloth ? 
 
 49. A grocer bought a hogshead of sugar weighing 744 
 pounds at 4} cents per pound, and sold it at 5^ cents per 
 pound. How much did he gain ? 
 
 50. A man, after selling § of his field, sold f of the 
 remainder and then had 13£ acres left. How many acres 
 did he own at first ? 
 
 51. A railroad train passed over T 7 ^ of its route in 3£ 
 hours. In how many hours would it pass over the entire 
 route ? In how many hours over § of the route ? | ? T 9 ¥ ? 
 
 52. A boy, being asked to find the value of 8-^ + 2^ + 
 3§ -f-4§, gave as his answer 20. How great was his error? 
 
 53. The meter is equal to 3^ feet, very nearly. Express 
 in centimeters the value of 4 T ^ feet. 
 
 54. For a piano cover a lady bought 2f yards of plush 
 at $3^ per yard, the same amount of lining flannel at $ J 
 per yard, 1^ yards of satin at #1J per yard, and l£ yards 
 of fringe at $1^ per yard. If the making cost $5, what 
 was the cost of the piano cover ? 
 
 55. A mason built 6£ yards of wall on Monday, 4| 
 yards on Tuesday, if yards on Wednesday, and 7f yards 
 on Thursday. If he is paid $0.80 per yard, how much 
 has he earned in the four days together ? 
 
144 COMMON FRACTIONS. 
 
 56. A coal dealer sold 100 tons of coal. If he shipped 
 by six cars 14£. 14^, 14-^, 14 3 8 5 , 14 T 7 ff , 14^ tons respec- 
 tively, how many tons must he load on the seventh car to 
 complete his shipment ? 
 
 57. The moon's diameter is -ft that of the earth, and 
 the sun's diameter is 110 times that of the earth. What 
 fraction of the sun's diameter is the moon's diameter? 
 
 58. If a silver rupee in Calcutta is worth $£§, what is 
 the value in dollars and cents of a fan costing 4$ rupees ? 
 
 59. If a man can do -ft- of a piece of work in 25 days, 
 what fraction of the work can he do in 62f days ? 
 
 60. I paid a tailor $3£ a yard for 5£ yards of broad- 
 cloth. On measuring it, I found that there were only 4| 
 yards. How much money ought the tailor to return ? 
 
 61. From a tank full of water § of the water was drawn 
 otf. Then 35 gallons were added, and the tank was just 
 half full. What is the capacity of the tank ? 
 
 62. What number exceeds the sum of its fourth, fifth, 
 sixth, and seventh parts by 101 ? 
 
 63. A trader bought wheat at 75 cents a bushel, and 
 sold it at 71 cents a bushel. How many cents did he lose 
 on every dollar he paid ? 
 
 64. How many bushels of potatoes at $£ per bushel will 
 pay for 16 bushels of wheat at $£# per bushel ? 
 
 65. From a piece of calico containing 35$ yards, there 
 have been sold at different times 12| yards, 2\ yards, 2^\ 
 yards, and 8| yards. How many yards remain ? 
 
 66. If gun metal is composed of 90£ parts of copper to 
 9^ parts of tin by weight, how many ounces of tin are there 
 in one pound (16 ounces) of gun metal ? how many ounces 
 of copper in one pound ? 
 
 67. One man mows \ of a field, a second ^ of it, and a 
 third ^ of it. What fraction of the field remains to be 
 mowed ? 
 
COMMON FRACTIONS. 145 
 
 68. Bell metal by weight consists of 4 parts of copper 
 to 1 part of tin. What is the cost of a bell weighing 12,400 
 pounds, if the copper costs 19 cents per pound, the tin 22^ 
 cents per pound, and the cost of making is $500 ? , 
 
 69. If an ore loses i£ of its weight in roasting, and -^ 
 of the remainder in smelting, how many tons of ore must 
 be mined to obtain 466 tons of pure metal ? 
 
 70. The amount of starch in potatoes is J J- of their 
 weight, but the amount that can usually be extracted is 
 only T ^. How many pounds of starch can be obtained 
 from 100 pounds of potatoes, and how many pounds of 
 starch will be left in the potatoes ? 
 
 71. How many pairs of trousers, each pair requiring 2| 
 yards, can be made from 33£ yards of cloth ? 
 
 72. If 3^ yards of cloth are required for a shirt, how 
 many shirts can be made from 12 pieces of cloth, each 
 piece measuring 47£ yards ? 
 
 73. Green coffee when roasted loses £ of its weight. If 
 a dealer buys green coffee at 22\ cents a pound, and sells 
 it roasted at 30 cents a pound, what will be his gain in 
 selling 1000 pounds of roasted coffee, the cost of roasting 
 the whole quantity being $ 2.25 ? 
 
 74. If an iron bar, when heated 1 degree, expands 
 T^sfftf of its length, what is the length at 212 degrees of 
 a bar whose length at 32 degrees is 10$ feet ? 
 
 75. If a horse eats -^ of a ton of hay in 30 days, how 
 long will 4^ tons of hay last 5 horses ? 
 
 76. If 4 is added to both terms of the fraction |£, by 
 how much is the value of the fraction increased ? 
 
 77. If 4 is subtracted from both terms of the fraction 
 \%, by how much is the value of the fraction decreased ? 
 
 78. Find the least number of apples that arranged in 
 groups of 8, 9, 10, or 12 will have just 6 over in each 
 case. 
 
146 COMMON FRACTIONS. 
 
 79. The diameter of a bicycle wheel is 2$ feet, and the 
 circumference is 3\ times the diameter. How many times 
 does the wheel turn in going 1 mile (5280 feet)? 
 
 80. What is the least number of yards of carpet in a 
 roll that can be cut into lengths of exactly 13£ yards, 8 
 yards, or 11^ yards ? 
 
 81. What is the length of the longest pole that will 
 exactly measure the sides of a field whose lengths are 
 respectively 135| yards, 118$ yards, 152 yards, and 202§ 
 yards ? 
 
 82. Find the least multiplier of £, tf, and f f that will 
 make each product an integral number. 
 
 83. Find the least integral number that is exactly divisi- 
 ble by 5±, 3£, and 7. 
 
 84. Four bells commence tolling together, and toll at 
 intervals of 1, 1|, 1^, and l-^- seconds, respectively. In 
 how many seconds will all four toll again at the same 
 instant ? 
 
 85. What number multiplied by f T of ft of 29$ will 
 give 102$ for the product ? 
 
 86. How many miles an hour must a man walk to go 
 28 miles in 7 T 7 5 hours ? 
 
 87. If the rent of 5^ acres of land is $21§, what will 
 be the rent of 19^ acres at the same rate ? 
 
 31i 
 
 88. If the English acre is -r-r of an Irish acre, how 
 
 4y 
 
 many English acres are there in 218f Irish acres ? 
 
 89. Resolve the denominator of ff into its prime fac- 
 tors ; from the result state the number of figures the 
 equivalent decimal will have, and the number that will 
 precede the repetend. 
 
 90. Find the greatest common measure of 9083, 9207, 
 8897. 
 
CHAPTER VIII. 
 COMPOUND QUANTITIES. 
 
 291. A quantity expressed in units of one denomination 
 is called a simple quantity. A quantity expressed in units 
 of two or more denominations is called a compound quan- 
 tity, or a compound denominate number. 
 
 Thus, 4£ gallons is a simple quantity ; but its equivalent, 4 gallons 
 1 quart, is a compound quantity. 
 
 292. Reduction. The process of changing the denomina- 
 tion in which a quantity is expressed, without changing 
 the value of the quantity, is called reduction. 
 
 Pleasures of Capacity. 
 
 293. Liquid Measure. Liquid measure is used in 
 measuring liquids, as water, milk, etc. 
 
 Table. 
 
 4 gills (gi.) = 1 pint (pt.). 
 2 pints = 1 quart (qt.). 
 4 quarts = 1 gallon (gal.). 
 1 gal. = 4 qt. = 8 pt. = 32 gi. 1 gal. contains 231 cubic inches. 
 
 31£ gal. = 1 barrel (bbl.). 
 63 gal. = 1 hogshead (hhd.). 
 
 Note. Casks holding from 28 gal. to 43 gal. are called barrels, 
 and casks holding from 54 gal. to 63 gal. are called hogsheads. 
 
 Whenever barrels or hogsheads are used as measures, a barrel 
 means 31£ gallons, and a hogshead means 63 gallons. 
 
148 COMPOUND QUANTITIES. 
 
 294. Dry Measure. Dry measure is used in measuring 
 dry articles, as grain, seeds, fruit, vegetables. 
 
 Table. 
 
 2 pints (pt.) = 1 quart (qt.). 
 8 quarts = 1 peck (pk.). 
 4 pecks = 1 bushel (bu.). 
 
 1 bu. = 4 pk. = 32 qt. 1 bu. contains 2150.42 cubic inches. 
 
 Note. In measuring grain, seeds, and small fruits, the measure 
 must be even full. In measuring apples, potatoes, and other large 
 articles, the measure must be heaping full. 
 
 The quart of liquid measure contains 67 f cubic inches, and the 
 quart of dry measure 671 cubic inches. 
 
 In Great Britain, the quart of liquid measure is of the same size as 
 the quart of dry measure, and contains 69.3185 cubic inches. There- 
 fore, the imperial gallon of Great Britain contains 277.274 cubic 
 inches, and the imperial bushel contains 2218.192 cubic inches. 
 
 The quarter contains 8 imperial bushels. 
 
 Reduction of Compound Quantities. 
 
 295. If the reduction is from a higher denomination to 
 a lower, it is called reduction descending. If the reduction 
 is from a lower denomination to a higher, it is called reduc- 
 tion ascending. 
 
 Thus, 2 pecks = 16 quarts is an example of reduction descending ; 
 and 24 quarts = 6 gallons is an example of reduction ascending. 
 
 296. Example. Reduce 12 gal. 2 qt. 1 pt. to pints. 
 
 gal. qt. pt. Solution. 12 gal. = 12 X 4 qt. = 48 qt, and 48 
 
 12 2 1 qt. with the 2 qt. added are 50 qt. 
 
 _4 50 qt. = 50 X 2 pt. = 100 pt., and 100 pt. with 
 
 60 the 1 pt. added are 101 pt. 
 
 2 The multiplicand and multiplier are interchanged 
 
 101 in the operation, Hence, 
 
COMPOUND QUANTITIES. 149 
 
 297. In Reduction Descending, 
 
 Multiply the given number of units of the highest denomi- 
 nation by the number of units of the next lower denomination 
 required to make one of this higher ; and add to the -product 
 the given number of units of this lower denomination. 
 
 Proceed in this way with each successive result until the 
 required denomination is reached. 
 
 298. Example. Reduce 221 pt. to higher units. 
 
 2 221 pt. Solution. 221 pt. = 110 qt. and 1 pt. 
 
 4 110 qt. . . . 1 pt. over. 110 qt. = 27 gal. and 2 qt. over. 
 27 gal. . . 2 qt. Therefore, 221 pt. = 27 gal. 2 qt. 1 pt. 
 Hence, 
 
 299. In Reduction Ascending, 
 
 Divide by the number of units required to make one of the 
 next higher denomination. 
 
 Divide this quotient and each successive quotient in like 
 manner until the required denomination is reached. 
 
 The last quotient with the several remainders, arranged in 
 order, is the answer sought. 
 
 Exercise 70. 
 
 Reduce : ■ 
 
 1. 3 pk. 5 qt. 1 pt. to pints. 
 
 2. 4234 pt. (dry measure) to higher units. 
 
 3. 24 gal. 2 qt. 1 pt. 2 gi. to gills. 
 
 4. 3047 gills to higher units. 
 
 5. 1715^ bu. to pints. 
 
 6. 508 dry quarts to higher units. 
 
 7. 1016 liquid pints to higher units. 
 
 8. 44 gal. 3 qt. 1 pt. to pints. 
 
 9. 44 bu. 3 pk. 7 qt. 1 pt. to pints. 
 
 10. 272 liquid quarts to dry quarts. 
 
 1 1 . 429 dry quarts to liquid quarts. 
 
Solution. 
 
 V 
 
 gal. qt. 
 
 pt 
 
 3 2 
 
 1 
 
 5 3 
 
 
 
 13 1 
 
 1 
 
 9 
 
 1 
 
 150 COMPOUND QUANTITIES. 
 
 Addition and Subtraction of Compound Quantities. 
 
 300. Examples. 1. Add 3 gal. 2 qt. 1 pt. ; 5 gal. 3 qt. ; 
 13 gal. 1 qt. 1 pt.; 9 gal. 1 pt. 
 
 We write units of the same name in the same column, 
 and add the columns, beginning with the pints. 
 3 pt. = 1 qt. and 1 pt. over. We write the 1 pt. 
 under the pints, and add the 1 qt. to the quarts. 
 7 qt. = 1 gal. and 3 qt. over. We write the 3 qt. 
 under the quarts, and add the 1 gal. to the gallons. 
 31 3 1 The required sum is, therefore, 31 gal. 3 qt. 1 pt. 
 
 2. From 5 bu. 2 pk. 3 qt. take 3 bu. 3 pk. 3 qt. 
 
 Solution. 3 qt. — 3 qt. = qt. We write under the quarts. 
 
 Since we cannot take 3 pk. from 2 pk., we take 
 bu. pk. qt. 1 bu. (4 pk.) from the 6 bu. and add it to the 2 pk. 
 5 2 3 4 pk. + 2 pk. = 6 pk., and 6 pk. - 3 pk. = 3 pk. 
 3 3 3 We write 3 under the pecks. Then 4 bu. — 3 bu. 
 1 3 =1 bu. The required difference is, therefore, 
 
 1 bu. 3 pk. 
 
 Exercise 71. 
 
 1. Add 5 bu. 3 pk. 6 qt. 1 pt.; 6 bu. 2 pk. 7 qt.; 7 bu. 
 1 pk. 1 qt. 1 pt. ; 1 pk. 7 qt. ; 2 bu. 3 pk. 1 pt. 
 
 2. Add 50 gal. 3 qt. 1 pt. 3 gi.; 12 gal. 1 qt. 1 pt. 1 gi.; 
 5 gal. 2 qt. 1 pt. 2 gi.; 75 gal. 3 qt. 1 pt. 3 gi.; 80 gal. 3 
 qt. 1 gi. ; 17 gal. 1 qt. 1 pt. 3 gi. 
 
 3. Add 4 gal. 3 qt. 1 pt.; 3 gal. 2 qt. 1£ pt.; 12 gal. 3 
 qt.; 14 gal. l£j>t.; 5 gal. 2 qt. 1 pt. 
 
 4. Subtract 5 bu. 1 pk. 6 qt. 1 pt. from 5 bu. 3 pk. 3 qt. 
 
 5. Subtract 22 gal. 3 qt. 1 pt. from 30 gal. 2 qt. 
 
 6. Add 6 bu. 1 pk. 7 qt. 1 pt.; 2 bu. 2 pk. 5 qt. -j- pt.; 
 19 bu. 3 pk. qt. 1 pt.; 14 bu. 2 pk. 4 qt. 1£ pt.; 10 bu. 1 
 pk. 3 qt. 1 pt.; 5 bu. 3 pk. 2 qt. 
 
 7. Find the difference between 2 bu. and 5 qt. 1 pt 
 
COMPOUND QUANTITIES. 151 
 
 Multiplication and Division of Compound Quantities. 
 
 301. Examples. 1. Multiply 15 gal. 3 qt. 1 pt. by 5. 
 
 Solution. 5 X 1 pt. = 5 pt. = 2 qt. 1 pt. We write the 1 pt. under 
 
 the pints, and reserve the 2 qt. to be added to 5 X 
 
 gal. qt. pt. 3 qt. 5X3 qt. = 15 qt., and this with the 2 qt. 
 
 15 3 1 reserved = 17 qt. = 4 gal. 1 qt. We write the 1 qt 
 
 5 under the quarts, and add the 4 gal. to 5 X 15 gal., 
 
 79 1 1 having 79 gal. The required product is, therefore, 
 79 gal. 1 qt. 1 pt. 
 
 2. Divide 122 bu. 2 pk. 7 qt. 1 pt. by 5. 
 
 Solution. 122 bu. -f- 5 = 24 bu. and 2 bu. over. We write 24 
 
 under the bushels. 2 bu. =2x4 pk. = 8 
 
 bu. pk. qt. pt. pk. , and 8 pk. + 2 pk. = 10 pk. 10 pk. -r 5 
 
 5 )122 2 7 1 = 2pk. We write 2 under the pecks. 7 qt. 
 
 24 2 1 1 -=- 5 as 1 qt. and 2 qt. over. We write 1 
 
 under the quarts. 2 qt. = 2 X 2 pt. = 4 pt., 
 
 and 4 pt. + 1 pt. = 5 pt. 5 pt. -J- 5 = 1 pt. We write* 1 under the 
 
 pints. The required quotient is, therefore, 24 bu. 2 pk. 1 qt. 1 pt. 
 
 Exercise 72. 
 
 1. Multiply 19 gal. 3 qt. 1 pt. by 70. 
 
 2. Multiply 43 bu. 2 pk. 6 qt. 1 pt. by 63. 
 
 3. Multiply 17 bu. 3 pk. 6 qt. by 8. 
 
 4. Multiply 26 gal. 2 qt. 1 pt. 3 gi. by 12. 
 
 5. Multiply 12 bu. 3 pk. 7 qt. 1 pt. by 25. 
 
 6. Divide 34 gal. 3 qt. 1 gi. by 7. 
 
 7. Divide 147 gal. 2 qt. 1 pt. 2 gi. by 17. 
 
 8. Divide 54 bu. 3 pk. 2 qt. 1 pt. by 11. 
 
 9. Divide 34 bu. 3 pk. 5 qt. 1 pt. by 15. 
 
 302. All compound quantities are reduced, added, sub- 
 tracted, multiplied, divided, by the methods given for 
 measures of capacity. 
 
152 COMPOUND QUANTITIES. 
 
 Measures of Weight. 
 
 303. Troy Weight. Troy weight is used in weighing 
 gold, silver, and precious stones. 
 
 Table. 
 
 24 grains (gr.) = 1 pennyweight (dwt.). 
 20 pennyweights — 1 ounce (oz.). 
 12 ounces = 1 pound (lb.). 
 
 The pound troy contains 5760 grains. 
 
 304. Avoirdupois Weight. Avoirdupois weight is used 
 in weighing all articles except gold, silver, and precious 
 stones. 
 
 Table. 
 
 16 ounces (oz.) = 1 pound (lb.). 
 100 pounds = 1 hundredweight (cwt.). 
 2000 pounds = 1 ton (t.). 
 
 112 pounds = 1 long hundredweight. 
 2240 pounds = 1 long ton. 
 
 The long ton is used in the United States Custom Houses, and in 
 wholesale transactions in iron and coal. 
 
 The pound avoirdupois contains 7000 troy. grains. 
 
 Exercise 73. 
 
 1. Reduce 27,587 gr. to higher troy units. 
 
 2. Reduce 34,652 pounds avoirdupois to long tons, etc. 
 
 3. Reduce 136,851 ounces avoirdupois to higher units. 
 
 4. Reduce 864,205 gr. to higher troy units. 
 
 5. Reduce 864,205 gr. to higher avoirdupois units. 
 
 6. Reduce 5 lb. 7 oz. 6 dwt. 12 gr. to grains. 
 
 7. Reduce 745 lb. avoirdupois to troy measures. 
 
 8. Reduce 745 lb. troy to avoirdupois measures. 
 
 9. Reduce 1,440,445 oz. avoirdupois to higher units. 
 
COMPOUND QUANTITIES. 153 
 
 10. Reduce 5,640,773 oz. avoirdupois to higher units. 
 
 11. Add 48 t. 13 cwt. 75 lb. 6 oz.; 25 t. 12 cwt. 27 lb. 
 8 oz.; 51 t. 10 cwt. 44 lb.; 80 t. 5 cwt. 6 oz.; 19 cwt. 27 
 lb.; 25 1b. 8 oz.; 5 t. 5 cwt. 5 1b. 
 
 12. Add 13 lb. 4 oz. 8 dwt. 6 gr.; 25 lb. 8 oz. 13 dwt. 
 20 gr.; 8 lb. 11 oz. 14 gr.; 20 lb. 16 dwt. 8 gr.; 15 lb. 9 
 oz. 12 dwt. ; 4 oz. 3 dwt. 
 
 13. Subtract 23 lb. 8 oz. 19 dwt. 10 gr. from 58 lb. 6 oz. 
 17 dwt. 21 gr. 
 
 14. Subtract 17 t. 7 cwt. 17 lb. 6 oz. from 25 t. 13 cwt. 
 15 lb. 12 oz. 
 
 15. Multiply 3 lb. 4 oz. 8 dwt. 10 gr. by 10. 
 
 16. Multiply 5 t. 10 cwt. 67 lb. 4 oz. by 15. 
 
 17. Divide 17 t. 19 cwt. 79 lb. 8 oz. by 8. 
 
 18. Divide 60 lb. 6 oz. 10 dwt. 20 gr. by 7. 
 
 19. How many bags each holding 2 bu. 1 pk. 3 qt. are 
 required to hold 234 bu. 1 pk. 4 qt. of corn ? 
 
 Note. Reduce both quantities to quarts. 
 
 20. What is the value at 4 J cents a pound of a calf 
 weighing 184 lb. 6 oz.? 
 
 21. How many tablespoons each weighing 2 oz. 17 dwt. 
 12 gr. can be made from 155 oz. 5 dwt. of silver ? 
 
 305. In compounding medicines, apothecaries make use 
 of the following : 
 
 Apothecaries' Weight. 
 
 20 grains (gr.) = 1 scruple O). 
 
 3 scruples = 1 dram (5). 
 
 8 drams = 1 ounce (§ ). 
 
 12 ounces = 1 lb. troy. 
 
 Apothecaries' 1 Measure. 
 60 minims (m) = 1 dram (m lx.). 
 8 drams = 1 ounce (fl. drm. viij.). 
 
 16 ounces = 1 pint (fl. oz. xvj.). 
 
154 COMPOUND QUANTITIES. 
 
 Measures of Length. 
 
 306. The unit of measure for lengths is the yard. From 
 the yard are derived the units of surface and volume. 
 
 307. The standard yard of Great Britain, as defined by 
 Act of Parliament, is the distance between the centres of 
 two cylindrical holes in a certain bar of gun metal, when 
 the metal has a temperature of 62 degrees Fahrenheit. 
 
 The standard yard of the United States conforms as 
 nearly as possible to that of Great Britain. 
 
 308. Measures of length are used in measuring lines 
 or distances. 
 
 Table. 
 
 12 inches (in.) = 1 foot (ft.). 
 
 3 feet = 1 yard (yd.). 
 
 5£ yards, or l(ty feet = 1 rod (rd.). 
 320 rods = 1 mile (mi.). 
 
 1 mi. = 320 rd. = 1760 yd. = 6280 ft. 
 
 Note. A hand (used in measuring the height of horses) = 4 in. ; 
 a knot (used in navigation) = 6086 ft. ; a league = 3 knots ; a fathom 
 = 6 ft. ; a cable length =120 fathoms ; a line = ^ in. ; a barleycorn 
 = \ in. ; a palm = 3 in.; a span s 9 in. ; a cubit = 18 in. ; a military 
 pace = 28 in. ; a furlong = \ mi. 
 
 309. Examples. 1. Change 106,760 ft. to higher de- 
 nominations. 
 
 Solution. There are 16£ ft., or 33 half-feet, in a rod ; so we 
 
 change the 106,760 ft, to 
 16^)106760 ft. half-feet, and these to rods, 
 
 2 by dividing by 33. The 
 
 33 )213520 half-feet. remainder is 10 half-feet, 
 
 320 )6470 rd. ... 10 half-feet = 5 ft. or 5 ft. 6470 rd. = 20 mi. 
 
 20 mi. ... 70 rd. 70 rd. Therefore, 106,760 
 
 ft. = 20 mi. 70 rd. 5 ft. 
 
COMPOUND QUANTITIES. 155 
 
 2. Add 4 mi. 110 rd. 5 yd. 1 ft. 8 in. and 6 mi. 25 rd. 
 4 yd. 1 ft. 6 in. 
 
 mi. 
 
 rd. 
 
 yd. 
 
 ft. in. 
 
 4 
 
 110 
 
 5 
 
 1 8 
 
 6 
 
 25 
 
 4 
 
 1 6 
 
 Solution. We have for the sum 10 mi. 
 
 136 rd. 3^ yd. ft. 2 in. We reduce the \ 
 
 10 136 M 2 yd. of this sum and add its value 1 ft. 6 in. 
 
 16 to the ft. 2 in. We have for the sum, 
 
 10 136 3 1 8 therefore, 10 mi. 136 rd. 3 yd. 1 ft. 8 in. 
 
 Exercise 74. 
 
 1. Keduce 3 yd. 2 ft. to inches. 
 
 2. Eeduce 4 mi. 124 rd. 3 yd. 2 ft. to feet. 
 
 3. Eeduce 27 rd. 4 yd. 9 in. to inches. 
 
 4. Eeduce 290 leagues to feet. 
 
 5. Eeduce 82,976,432 in. to higher units. 
 
 6. Eeduce 7 mi. 3 yd. 1 ft. 6 in. to inches. 
 
 7. Eeduce 22 mi. 222 rd. 4 ft. 8 in. to inches. 
 
 8. Eeduce 712 mi. to feet. 
 
 9. Eeduce 540,451 ft. to higher units. 
 
 10. Eeduce 271,256 in. to higher units. 
 
 11. Eeduce 723,964 ft. to higher units. 
 
 12. Eeduce 233,205 in. to higher units. 
 
 13. How many feet high is a horse 16 hands high ? 
 
 14. Add 6 mi. 120 'rd. 3 yd. 2 ft. 2 in.; 18 mi. 15 rd. 
 1 yd. 1 ft. 6 in.; 3 mi. 215 rd. 2 yd. 2 ft. 3 in.; 7 mi. 
 95 rd. 1 yd. 1 ft. 8 in. 
 
 15. Subtract 3 mi. 217 rd. 4 yd. 1 ft. 3 in. from 4 mi. 
 100 rd. 3 yd. 2 in. 
 
 16. Multiply 5 mi. 126 rd. 9 ft. 6 in. by 7125. 
 
 17. Divide 54 mi. 124 rd. 1 yd. 2 ft. 6 in. by 33. 
 
 18. If a man builds 1 rd. 1 yd. 1 ft. 6 in. of stone wall 
 in one day, how much will he build in 26 days ? 
 
 19. A man builds 25 rd. 2 yd. 1 ft. 6 in. of wall in 20 
 days. How much does he build per day ? 
 
156 COMPOUND QUANTITIES. 
 
 Measures of Surface. 
 
 310. The area of a surface is the number of square 
 units it contains. 
 
 Table 
 
 144 square 
 
 A square inches (sq. in.) = 1 square foot (sq. ft.). 
 9 square feet = 1 square yard (sq. yd.). 
 
 304; square yards, or ) , _ t _ % 
 
 272* square feet \ = 1 «l^e rod (sq. rd.). 
 
 160 square rods or ) 
 43,560 square feet } - 1 acre (A.). 
 
 640 acres = 1 square mile (sq. mi.). 
 
 Note. In measures of surface the scale ascends and descends by 
 the squares of the units of length ; thus, 144 = 12 2 ; 9 = 3 2 ; 30± = 
 
 (5£)2;272±=(16£)2. 
 
 Exercise 75. 
 
 1. Reduce 92,638 sq. yd. to square inches. 
 
 2. Reduce 1,223,527 sq. in. to higher units. 
 
 3. Reduce 721 sq. mi. to square rods. 
 
 4. Reduce 34,729 sq. yd. to higher units. 
 
 5. Reduce to square inches 3 A. 107 sq. rd. 27 sq. yd. 
 7 sq. ft. 23 sq. in. 
 
 6. Reduce 99,894,712 sq. in. to higher units. 
 
 7. Reduce 15,376 sq. yd. to higher units. 
 
 8. Reduce 562,934 sq. in. to higher units. 
 
 9. Add 74 A. 21 sq. rd. 5 sq. yd. 4 sq. ft. 100 sq. in.; 
 123 A. 23 sq. rd. 13 sq. yd. 5 sq. ft. 83 sq. in.; 112 A. 106 
 sq. rd. 17 sq. yd. 8 sq. ft. 7 sq. in.; 541 A. 50 sq. rd. 
 23 sq. yd. 24 sq. in. 
 
 10. From 20 A. take 13 A. 150 sq. rd. 98 sq. ft. 10 sq. in. 
 
 11. Multiply 27 A. 76 sq. rd. 22 sq. yd. 5 sq. ft. by 90. 
 
 12. Divide 74,128 sq. mi. 517 A. 80 sq. rd. by 10,000. 
 
COMPOUND QUANTITIES. 157 
 
 Surveyors' Measure. 
 
 311. Surveyors use a chain, called Gunter's chain, 
 which is 4 rods, or 66 feet, long. The chain has 100 links, 
 and therefore links are written as hundredths of a chain. 
 
 Surveyors' Table of 
 
 Measures of Length. 
 7.92 in. = 1 link (1.). 
 100 links = 1 chain (ch.) 
 
 80 chains = 1 mile (mi.). 
 
 Measures of Surface. 
 16 sq. rd. = 1 sq. ch. 
 10 sq. ch. = 1 A. 
 640 A. =1 sq. mi. 
 
 1 sq. mi. = 1 section (sec). 
 36 sec. = 1 township (tp.). 
 
 Exercise 76. 
 
 1. Reduce 10 ch. to inches. 
 
 2. Reduce 3168 in. to chains. 
 
 3. How many acres are there in a township ? 
 
 4. Reduce 6400 sq. ch. to acres ; to square miles. 
 
 5. Reduce 82,426 sq. ch. to higher units. 
 
 6. Add 4 sq. mi. 412 A. 6 sq. ch. 8 sq. rd.; 7 sq. mi. 
 88 A. 2 sq. ch. 11 sq. rd.; 3 sq. mi. 367 A. 7 sq. ch. 2 sq. rd.; 
 
 11 sq. mi. 344 A. 9 sq. ch. 15 sq. rd. 
 
 7. Subtract 1 mi. 75 ch. 85 1. from 4 mi. 44 ch. 38 1. 
 
 8. What is the area of a field if it can be divided into 
 
 12 lots each containing 2 sq. ch. 7 sq. rd.? 
 
 . 9. Multiply 3 sq. mi. 172 A. 5 sq. ch. 7 sq. rd. by 11. 
 
 10. Divide 6 sq. mi. 422 A. 2 sq. ch. 13 sq. rd. by '5. 
 
 11. A field is divided into 47 gardens each containing 
 1 sq. ch. 9 sq. rd. What is the area of the field ? 
 
 12. A field containing 5 A. 4 sq. ch. 11 sq. rd. is divided 
 into 25 equal lots. What is the area of each lot ? 
 
 13. Find the rent of 8 sq. ch. 10 sq. rd. at $2 an acre. 
 
 14. If a field contains 3 A. 6 sq. ch. 12 sq. rd., what is 
 it worth at 14 cents a square foot ? 
 
158 COMPOUND QUANTITIES. 
 
 Measures of Volume. 
 
 312. The volume of a body is the number of cubic units 
 it contains. 
 
 Table. 
 
 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 
 27 cubic feet = 1 cubic yard (cu. yd.). 
 
 Note. In measures of volume, the scale ascends and descends by 
 the cubes of the units of length ; thus 1728 = 12* ; 27 = 3 8 . 
 
 313. In measuring wood and small, irregular stones the 
 following is the 
 
 Table. 
 
 16 cubic feet = 1 cord foot (cd. ft.). 
 
 8 cord feet, or > , , , , . 
 , OQ , • r > = 1 cord(cd.). 
 
 128 cubic feet > x ' 
 
 Note. A cord is a pile 8 ft. long, 4 ft. wide, and 4 ft. high. One 
 foot of the length of such a pile is called a cord foot. 
 
 Exercise 77. 
 
 1. Reduce 25 cu. yd. 5 cu. ft. 143 cu. in. to cubic 
 inches. 
 
 2. Reduce 921,730 cu. in. to higher units. 
 
 3. Wood cut in lengths of 4 ft. is piled 3£ ft. high. 
 How long must the pile be to contain 2 cords ? 
 
 4. How many cords in a pile of 4-ft. wood 43 ft. long 
 and 6 ft. high ? 
 
 5. Add 130 cu. yd. 5 cu. ft. 820 cu. in.; 56 cu. yd. 20 
 cu. ft. 304 cu. in.; 37 cu. yd. 4 cu. ft. 86 cu. in.; 8 cu. yd. 
 10 cu. ft. 129 cu. in.; 12 cu. yd. 19 cu. ft. 175 cu. in. 
 
 6. Subtract 32 cu. yd. 13 cu. ft. 1600 cu. in. from 
 39 cu. yd. 17 cu. ft. 1400 cu. in. 
 
 7. Multiply 12 cd. 4 cd. ft. by 14. 
 
 8. Divide 5 cu. yd. 10 cu. ft. 371 cu. in. by 6. 
 
COMPOUND QUANTITIES. 159 
 
 Measures of Value. 
 
 314. Money is the measure of value. 
 
 315. Currency is the medium of exchange employed in 
 buying and selling. 
 
 316. Coins or Specie are stamped pieces of metal of 
 fixed purity and weight issued by governments as money. 
 
 317. Bullion is uncoined gold or silver, of standard 
 purity, usually in the shape of bars. 
 
 318. Paper Money is stamped paper containing the 
 promise of the government or of a bank of issue to pay 
 the holder on presentation a specified sum of standard 
 coined money. 
 
 319. United States Money. The unit of value in the 
 United States is the dollar. 
 
 Table. 
 10~mills (m.) = 1 cent (ct.). 
 10 cents = 1 dime (d.). 
 
 10 dimes or \ , . „ .„,,. 
 
 -iaa * ( = 1 dollar ($)• 
 
 100 cents ) ' 
 
 Cents are made of bronze; half-dimes of nickel; dimes, quarter- 
 dollars, half-dollars, and dollars of silver; pieces of two and a half 
 dollars, five dollars, ten dollars, and twenty dollars of gold. 
 
 A 10-dollar gold coin is called an eagle, and a 20-dollar gold coin 
 is called a double eagle. 
 
 Note. The standard gold dollar weighs 25.8 gr. and contains 
 23.22 gr. of pure gold. 
 
 320. In common use dimes and cents are read together 
 as cents ; figures to the right of mills are read as the deci- 
 mal of a mill ; mills are used only in computation. 
 
 Thus, $4.27765 is read four dollars and twenty-seven cents, seven 
 and sixty-five hundredths mills; $4,273 is reckoned $4.27, and $4,277 
 is reckoned $4.28. 
 
160 COMPOUND QUANTITIES. 
 
 321. English Money. The unit of value in Great 
 Britain is the pound sterling, which is equivalent in United 
 States money to $4.8665. 
 
 Table. 
 
 4 farthings = I penny (d.). 
 12 pence = 1 shilling (s.). 
 20 shillings = 1 pound (£). 
 
 A guinea = 21s. ; a sovereign = 20s. ; a half-sovereign = 10s. ; a 
 crown = 5s. ; a half-crown = 2s. 6d. ; a florin = 2s. 
 
 Note 1. The penny and half-penny are made of copper ; the three- 
 penny piece, the six-penny piece, the shilling, the florin, the half- 
 crown, and the crown are made of silver ; the half-sovereign and the 
 sovereign of gold. 
 
 Note 2. Farthings are generally written as the common fraction 
 of a penny. Thus, 1, 2, and 3 farthings are written £d., $d., and |d., 
 respectively. 
 
 Exercise 78. 
 
 1. Eeduce £583 6s. 8d. to pence. 
 
 2. Reduce £79 18s. ll£d. to farthings. 
 
 3. Reduce 28,572d. to higher units. 
 
 4. Reduce 27,281 crowns to guineas. 
 
 5. Reduce 1,716,114 guineas to pounds. 
 
 6. Reduce 706,126d. to higher units. 
 
 7. Add £35 2s. 6fd.; £18 5s. 4d.; £27 3s. 10d.; £12 
 5d.; £6 7s. 8d.; £14 19s. lid.; £29 16s. 2d. 
 
 8. Subtract £92 15s. l£d. from £120 13s. 4d. 
 
 9. Multiply £31 2s. 6^d. by 8. 
 
 10. Divide £394 2s. 10^d. by £5 2s. 4*d. 
 
 11. Divide £108 15s. 4d. by 13. 
 
 12. Find the value in United States money of the money 
 in a box containing 35 sovereigns, 27 half-sovereigns, 13 
 crowns, 41 half-crowns, and 85 shillings. 
 
COMPOUND QUANTITIES. 
 
 161 
 
 322. Values of Foreign Coins. Oct. 1, 1897. 
 
 Country. 
 
 Standard. 
 
 Monetary Unit. 
 
 Value in 
 
 terms of 
 
 U. S. gold 
 
 dollar. 
 
 
 Gold and silver.... 
 Gold . .. 
 
 Peso 
 
 $0,965 
 
 
 Crown 
 
 0.203 
 
 
 Gold and silver.... 
 
 Silver 
 
 Gold 
 
 Franc 
 
 0.193 
 
 Bolivia 
 
 Boliviano 
 
 Milreis „ 
 
 Dollar 
 
 0.412 
 
 Brazil 
 
 0.546 
 
 
 Gold.... 
 
 1.000 
 
 (except Newfoundland). 
 Central Amer. States — 
 
 Gold 
 
 Gold 
 
 Silver 
 
 Colon 
 
 0.465 
 
 
 Dollar 
 
 1.000 
 
 Guatemala "] 
 
 Honduras I 
 
 Peso 
 
 0.412 
 
 Nicaragua f 
 
 Salvador J 
 
 Chile 
 
 Gold 
 
 Peso 
 
 0.365 
 
 
 Silver 
 
 Silver 
 
 
 0.664 
 
 China 
 
 
 0.608 
 0.412 
 
 
 Peso 
 
 
 Gold and silver.... 
 Gold 
 
 Peso 
 
 0.926 
 
 
 Crown 
 
 0.268 
 
 Ecuador.... 
 
 Silver 
 
 Sucre 
 
 0.412 
 
 Egypt 
 
 Gold 
 
 Pound (100 piasters)... 
 Mark 
 
 4.943 
 
 Finland ... 
 
 Gold 
 
 0.193 
 
 France 
 
 Gold and silver.... 
 
 Gold 
 
 Gold 
 
 Franc 
 
 0.193 
 
 German Empire ^. 
 
 Great Britain 
 
 
 0.238 
 
 Pound sterling 
 
 4.866^ 
 
 
 Gold and silver 
 
 Gold and silver ... 
 Silver 
 
 
 0.193 
 
 Haiti 
 
 Gourde 
 
 0.965 
 
 India 
 
 Rupee 
 
 0.196 
 
 Italy 
 
 Gold and silver 
 
 Gold 
 
 Gold 
 
 Lira 
 
 0.193 
 
 Japan 
 
 Yea 
 
 0.498 
 
 Liberia 
 
 Dollar 
 
 1.000 
 
 Mexico , 
 
 Silver. 
 
 Dollar 
 
 0.446 
 
 Netherlands 
 
 Gold and silver 
 
 Gold 
 
 
 0.402 
 
 Newfoundland 
 
 Dollar.... 
 
 1.014 
 
 Norway 
 
 Gold . 
 
 
 0.268 
 
 Persia 
 
 Silver 
 
 Kran 
 
 0.076 
 
 Peru 
 
 Silver 
 
 Sol ... 
 
 0.412 
 
 Portugal 
 
 Gold . 
 
 Milreis 
 
 1.080 
 
 Russia 
 
 Gold 
 
 Gold and silver..... 
 
 Gold 
 
 Gold and silver....". 
 
 Gold 
 
 Gold .. 
 
 Ruble 
 
 0.772 
 
 Spain 
 
 
 0.193 
 
 Sweden 
 
 
 0.268 
 
 Switzerland 
 
 Franc 
 
 0.193 
 
 Turkey 
 
 Piaster 
 
 0.044 
 
 Uruguay 
 
 
 1.034 
 
 Venezuela 
 
 Gold and silver.... 
 
 
 0.193 
 
 
 
 
 The " British dollar " has the same legal value as the Mexican dollar in Hong- 
 kong, the Straits Settlements, and Labuan. 
 
 By Imperial ukase, January 3/15, 1897, l£ paper rubles = 1 gold ruble, giving 
 paper ruble a value of 51 A cents U. S. money. 
 
162 COMPOUND QUANTITIES. 
 
 Measures of Time. 
 
 323. The unit of time is the day. 
 
 324. The interval of time, measured from the instant 
 the sun is due south until it is due south the next day, is 
 called a solar day. The length of the solar day varies 
 slightly, and its average length, called the mean solar day, 
 is the unit of time. 
 
 Table. 
 
 60 seconds (sec.) = 1 minute (min.). 
 
 60 minutes = 1 hour (hr.). 
 
 24 hours = 1 day (dy.). 
 
 7 days = 1 week (wk.). 
 
 366 days = 1 common year (yr.). 
 
 366 days = 1 leap year. 
 
 100 years = 1 century. 
 
 325. A year is the time in which the earth performs one 
 revolution round the sun, and consists of 365.242218 mean 
 solar days. 
 
 Note. Before the time of Julius Caesar the year was reckoned as 
 365 days. On the supposition that 365} days was the true length, he 
 introduced a calendar in which every fourth year {every year which 
 will give an integral quotient when its number is divided by 4) was to 
 consist of 366 days. The year of 366 days is called a leap year. 
 
 The error of the Julian calendar, 365.25 — 365.242219, or 0.007781 
 of a day, would amount to 3.1124 days in four centuries. To correct 
 this error Pope Gregory XIII, in 1582, introduced a calendar in which 
 three leap years in every four centuries were reckoned as common 
 years. Hence, the centuries are not leap years unless the number of 
 the century divided by 4 gives an integral quotient. Thus, 1600 and 
 1884 were leap years ; 1800 and 1885 were not ; 1900 will not be a 
 leap year ; 2000 will be a leap year. 
 
 The present, or Gregorian, calendar leaves a slight error equal to 
 one day in about 3300 years. 
 

 
 dy. 
 
 7. 
 
 July 
 
 . 31 
 
 8. 
 
 August (Aug.) . . 
 
 . 31 
 
 9. 
 
 September (Sept.) . 
 
 . 30 
 
 10. 
 
 October (Oct.) . . 
 
 . 31 
 
 11. 
 
 November (Nov.) . 
 
 . 30 
 
 12. 
 
 December (Dec.) . 
 
 . 31 
 
 COMPOUND QUANTITIES. 163 
 
 326. The year is divided into twelve calendar months. 
 
 The names of the months (mo.) and the number of days in each are : 
 
 dy. 
 
 1. January (Jan.) ... 31 
 
 2. February (Feb.) 28 or 29 
 
 3. March (Mar.) ... 31 
 
 4. April (Apr.) .... 30 
 
 5. May 31 
 
 6. June 30 
 
 February has 28 days in common years and 29 days in leap years. 
 
 Note. The number of days in each month may be easily remem- 
 bered by committing to memory the following lines : 
 
 " Thirty days hath September, 
 April, June, and November ; 
 All the rest have thirty-one, 
 Except the second month alone, 
 Which has but twenty-eight, in fine, 
 Till leap year gives it twenty-nine." 
 
 A lunar month is the time between two new moons, and is a little 
 more than 29 dy. 12 hr. 44 min. 
 
 Exercise 79. 
 
 1. Reduce 6 hr. 17 min. 25 sec. to seconds. 
 
 2. Reduce 1 yr. 13 dy. 8 hr. 4 min. to minutes. 
 
 3. Reduce 48,567 min. to higher units. 
 
 4. Reduce 7,423,922 sec. to higher units. 
 
 5. How many minutes are there from midnight of 
 March 7 to midnight of June 20 ? 
 
 6. Find the number of seconds from eight o'clock Mon- 
 day morning till six o'clock the next Saturday evening. 
 
 7. Which of the years 1600, 1656, 1700, 1734, 1800, 
 1818, 1880, 1900, 1924, 2000 are leap years ? 
 
 8. Add 8 dy. 14 hr. 21 min. 37 sec; 44 dy. 17 hr. 13 
 min. 32 sec; 208 dy. 9 hr. 47 min. 43 sec; 161 dy. 12 hr. 
 53 min. 54 sec. ; 88 dy. 22 hr. 17 min. 50 sec. 
 
164 COMPOUND QUANTITIES. 
 
 9. Subtract 2 yr. 213 dy. 17 tor. 48 niin. 48 sec. from 
 3 yr. 147 dy. 14 hr. 14 min. 32 sec. 
 
 10. Multiply 34 dy. 10 hr. 13 min. 12 sec. by 108. 
 
 11. Divide 16 yr. 357 dy. 17 hr. 20 min. 48 sec. by 18. 
 
 12. Divide 22 wk. 2 dy. by 11 hr. 31 min. 12 sec. 
 
 Difference between Two Dates. 
 
 327. Examples. 1. Find the difference in time between 
 July 4, 1897, and December 25, 1848. 
 
 Solution. In finding the period of time between long dates, 30 
 
 yr. mo. dy. days are considered a month. As July is the 
 
 1897 7 4 seventh and December the twelfth month, we 
 
 1 848 12 26 write 7 and 12 instead of the names of the months. 
 
 48 6 9 The required difference is 48 yr. 6 mo. 9 dy. 
 
 2. Find the number of days from July 25 to September 5. 
 
 Solution. The number of days in July = 6 
 
 The number of days in Aug. = 31 
 
 The number of days in Sept. = _6 
 
 Total number of days = 42 
 
 In finding the period of time between short dates the exact number 
 of days is counted ; and the last day named is included. 
 
 Exercise 80. 
 
 1. Napoleon was born Aug. 15, 1769, and died at the 
 age of 51 yr. 8 mo. 20 dy. What was the date of his 
 death ? 
 
 2. Daniel Webster was born Jan. 18, 1782, and died 
 Oct. 24, 1852. How old was he when he died ? 
 
 3. A note dated July 14, 1897, has 63 days to run. 
 When is the note due ? 
 
 4. A note dated Feb. 11, 1896, has 93 days to run. 
 When is the note due? 
 
COMPOUND QUANTITIES. 
 
 165 
 
 5. A note dated Feb. 11, 1897, has 63 days to run. 
 When is the note due ? 
 
 6. In the morning of July 5 a man went into the coun- 
 try for his vacation, and returned in the evening of Sept. 26. 
 Express in weeks and days the length of his vacation. 
 
 7. Find the difference in time between Oct. 12, 1492, 
 and July 4, 1776. 
 
 8. Jan. 1, 1859, fell on Saturday. What day of the 
 week was Jan. 1, 1860 ? Jan. 1, 1861 ? 
 
 Circular and Angular Measures. 
 
 328. Any portion of the circumference of a circle 
 is called an arc. 
 
 329. If a straight line fixed at one end is revolved in a 
 plane, the other end describes the arc of a circle ; and the 
 straight line in moving from its original position to any 
 other given position describes an angle. 
 
 Thus, if OA revolves on a fixed point 0, the end A makes the cir- 
 cumference ABC. When OA has reached the position OB, the arc 
 
 AB has been made by A, and the angle 
 AOB between OA and OB has been 
 made by OA. The angle A OB is such 
 a part of the angular magnitude about 
 as AB is of the circumference. 
 
 The circumference of every circle is 
 divided into 360 equal parts, called 
 degrees (arc-degrees), and correspond- 
 ing to every one of these equal parts 
 is an angle at the centre of the circle. 
 Hence, the whole angular magnitude 
 about any point in a plane is divided 
 into 360 equal parts called degrees (angle-degrees), and the number of 
 degrees in the angle formed by two lines drawn from the centre of a 
 circle is the same as the number of degrees in the arc which is inter- 
 cepted between these two lines. 
 
166 COMPOUND QUANTITIES. 
 
 330. An angle described by a line making one fourth 
 of a revolution contains 90 degrees, and is called a right 
 angle, as AOB ; and OA and OB are said to be perpe?idicu- 
 lar to each other. An angle less than a right angle is 
 called an acute angle; an angle greater than a right angle 
 and less than two right angles is called an obtuse angle. 
 
 Table. 
 
 60 seconds (") = 1 minute C). 
 60 minutes = 1 degree (°). 
 360 degrees = 1 revolution or circumference. 
 
 Note. A degree of the circumference of the earth at the equator 
 contains 60 geographical miles, or 69.16 statute miles. 
 
 Exercise 81. 
 
 1. Reduce 2°30'25" to seconds. 
 
 2. Keduce 15° 3' 22" to seconds. 
 
 8. Reduce 56,760" to higher units. 
 
 4. Reduce 212,221" to higher units. 
 
 5. Add 60° 50' 50"; 20° 41' 52"; 30° 25' 20"; 20° 32' 43" 
 
 6. Subtract 58° 33' 36" from 90° 11' 21". 
 
 7. Multiply 12° 14' 32" by 48. 
 
 8. Divide 321° 49' 24" by 22. 
 
 9. Divide 38° 37' 42" by 5° 31' 6". 
 
 331. 
 
 Miscellaneous Tables. 
 
 Numbers. 
 
 12 units = 1 dozen. 
 12 dozen = 1 gross. 
 12 gross = 1 great gross. 
 20 units = 1 score. 
 
 Papeb. 
 
 24 sheets = 1 quire. 
 20 quires = 1 ream. 
 
 2 reams = 1 bundle. 
 
 6 bundles = 1 bale. 
 
COMPOUND QUANTITIES. 
 
 167 
 
 Weights. 
 
 1 bu. of corn or rye = 56 lb 
 
 1 bu. of corn meal, rye } _ 
 
 meal, or cracked corn 
 1 bu. of wheat 
 
 \ 
 
 501b. 
 = 60 lb. 
 
 1 bu. of potatoes, beets, ) _ 
 
 etc. } 
 
 1 bu. of beans or peas = 60 lb. 
 1 bu. of oats = 32 lb. 
 
 = 48 lb. 
 451b. 
 
 1 bu. of barley 
 
 1 bu. of timothy- ) _ 
 
 seed > " 
 
 1 stone of iron or > __ 
 
 lead > 
 
 1 pig of iron or lead = 21| stone. 
 1 fother of iron or ) _ . 
 
 lead | = 8 ^ 
 
 lb. 
 
 The weight of a bushel of barley, oats, etc. , varies slightly in differ- 
 ent States, but the weights here given are those generally adopted in 
 business transactions. 
 
 Weights. 
 
 1 bbl. of flour = 196 lb. 
 
 1 bbl. of pork or beef = 200 lb. 
 1 cask of lime == 240 lb. 
 
 1 cental of grain , = 100 lb. 
 1 quintal of fish " = 100 lb. 
 
 Books. 
 
 A book of sheets folded in : 
 2 leaves is a folio. 
 4 leaves is a quarto. 
 8 leaves is an octavo. 
 
 12 leaves is a duodecimo. 
 
 16 leaves is a 16mo. 
 
 . Denominate Fractions. 
 
 332. Examples. 1. Express § rd. in yards, feet, and 
 
 inches. 
 
 Solution. § rd. = § of b\ yd. = 3§ yd. 
 2 y( j. = 2 f 3 f t . - 2 ft. 
 
 Hence, f rd. = 3 yd. 2 ft. 
 
 2. Find the value of } of £3 2s. 4d. 
 
 8[3_ 
 
 J_ Solution. We divide by 8 to get £ of £3 2s. 
 
 9| 4d. and multiply the quotient by 3 to get f of 
 
 _3_ £3 2s. 4d. Hence, | of £3 2s. 4d. is £1 3s. 4£d. 
 
 H 
 
168 
 
 COMPOUND QUANTITIES. 
 
 3. Find the value of 0.3975 of a mile. 
 
 0.3975 
 320 
 79500 
 
 11925 
 
 127.2 
 16* 
 3.3 
 
 12 
 3.6 
 
 Solution. 0.3975 mi. = 0.3975 of 320 rd. 
 = 127.2 rd. 
 0.2 rd. = 0.2 of 16 £ ft. = 3.3 ft. 
 0.3 ft. = 0.3 of 12 in. = 3.6 in. 
 Hence, 0.3975 mi. = 127 rd. 3 ft. 3.6 in. 
 
 Exercise 82. 
 
 Find the value of : 
 
 1. $ of a mile. 
 
 2. T \ of an acre. 
 
 3. | of a hundred weight. 
 
 4. § of a pound sterling. 
 5. 
 6. 
 
 ft of a mile. 
 ^ T of an acre. 
 
 7. | of a degree. 
 
 8. £ of a year. 
 
 9. 0.15625 of a bushel. 
 
 10. 0.625 of a gallon. 
 
 11. 0.875 of a leap year. 
 
 12. 0.325 of a pound troy. 
 
 13. 6 £ of 3 A. 101 J sq. rd. 
 
 14. l^of 7 hr. 21 min. 27 sec. 
 
 15. 10.0175 of 1 dy. 13 hr. 
 
 16. 17^ of 10 yd. 2 ft. 3£ in. 
 
 17. 0.01284 of 14 mi. 
 
 18. 0.42776 of 12 t. 10 cwt. 
 
 Find the value of : 
 
 19. f of 1 lb. + 3$ oz. + 5§ dwt. 
 
 20. 0.35 of 4 lb. 5 oz. 6 dwt. 16 gr. 
 
 21. 3.726 mi. -33.57 rd. 
 
 22. T 3 3 of a year -}- ^ of a week -f- t 7 * °f an hour. 
 
 23. 5.268 of 2 dy. + 2.829 of 16 hr. + 0.9528 of 25 min. 
 
 24. ^ of a mile + f of 40 rd. -f f of a yard. 
 
 25. f ,of 2 cwt. 84 lb. + f of 5 cwt. 98 lb. + £ of 7£ lb. 
 
 26. f of 21 ft. 7 in. + 0.855 of 16 ft. 2 in. + 0.365 of 1 ft. 
 
 27. 0.9 of 4 A. 17 sq. rd. - \\ of 3 A. 15 sq. rd. 
 
 28. 0.652 of 2 cu. yd. 7 cu. ft. — 0.888 of 1 cu. yd. 2 cu. ft. 
 
 29. 0.456 of 12 bu. 3 pk. — 0.654 of 5 bu. 2 pk. 
 
COMPOUND QUANTITIES. 169 
 
 333. Examples. 1. Express 10 hr. 33 min. 36 sec. as 
 the fraction of a day. 
 
 Solution. 36 sec. = f # min. = f min. 
 33f min. = ^ hr. = iff hr. = if hr. 
 
 1014 
 
 10tt hr. = ^f dy. = f ft dy. = H dy. 
 Hence, 10 hr. 33 min. 36 sec. = \\ dy. 
 
 2. Express 127 rd. ft. 7.92 in. as the decimal of a mile. 
 12 
 
 161 
 320 
 
 7.92 in. Solution. 7.92 -fl2 =0.66. 
 
 0.66 ft. 0.66 -M6| = 0.04. 
 
 127.04 rd. 127.04 + 320 = 0.397. 
 
 0.397 mi. Hence, 127 rd. ft. 7.92 in. = 0.397 mi. 
 
 3. Express 1 yd. 2 ft. 4 in. as the common fraction, and 
 as the decimal, of 5 yd. 1 ft. 4 in. 
 
 Solution. 1 yd. 2 ft. 4 in. = 64 in. ; 5 yd. 1 ft. 4 in. = 196 in. 
 -64 in. _ 64 _ 16 _„, 
 
 Hence, 1 yd. 2 ft. 4 in. is £f, or 0.32653+ of 5 yd. 1 ft. 4 in. 
 
 Exercise 83. 
 Express : 
 
 1 . A pound avoirdupois as the fraction of a pound troy. 
 
 2. An ounce avoirdupois as the fraction of an ounce 
 troy. 
 
 3. 363 sq. yd. as the fraction of an acre. 
 
 4. f of £2 Is. 3d. + T 5 T of £1 4s. 9d. as the fraction of 
 £2 14s. 
 
 5. 2 mi. 138 rd. 1 yd. as the fraction of 3 mi. 265 rd. 
 3 yd. 1 ft. 6 in. 
 
 6. f of 560 lb. as the fraction of 5 long tons. 
 
 7. § of 200 rd. as the fraction of 4 mi. 
 
 8. tf of 2 dy. 2 hr. 24 min. as the fraction of 2 wk. 1 dy. 
 
170 COMPOUND QUANTITIES. 
 
 9. $ of the difference between 3 yd. 2 ft. 11 in. and 10 
 yd. 7 in. as the fraction of 8 yd. 
 
 10. £f of the difference between | of 7 hr. and ^ of 15 
 min. as the fraction of 12 hr. 18 min. 
 
 11. £ pt. as the fraction of a gallon. 
 
 12. 16s. 3|d. as the decimal of a pound. 
 
 13. 233 rd. 9 ft. 10.8 in. as the decimal of a mile. 
 
 14. 71 sq. rd. 54 sq. ft. 64.8 sq. in. as the decimal of an 
 acre. 
 
 15. 15 hr. 14 min. 6 sec. as the decimal of 2 days. 
 
 16. 38 sq. rd. 21 sq. yd. 5 sq. ft. 108 sq. in. as the deci- 
 mal of an acre. 
 
 17. 3 mi. 242 rd. 2 yd. 2 ft. 3 in. as the decimal of 7 
 mi. 160 rd. 
 
 18. 5 hr. 13 min. 30 sec. as the decimal of a week. 
 
 19. 27° 14' 45" as the decimal of 90°. 
 
 20. 54 dy. 2 hr. 40 min. as the decimal of 365 J days. 
 
 21. 2 lb. avoirdupois as the decimal of 10 lb. troy. 
 
 22. 44,920.9025 hr. as the decimal of a year. 
 
 23. 14.52 sq. yd. as the decimal of a square chain. 
 
 24. 8 cwt. 77 lb. 9.6 oz. as the decimal of a ton. 
 
 25. What part of 4 lb. 1 oz. 8 dwt. 15 gr. is 1 lb. 1 oz. 
 9 dwt. 15 gr.? 
 
 26. What part of 2 mi. is § of 6 rd. 3 yd. 2 in.? 
 
 27. What part of a bushel is 1 pk. 2 qt. 1 pt.? 
 
 28. What part of 20 acres is 19 A. 3.5 sq. ch.? 
 
 29. What part of 5 tons is 3 t. 240 lb.? 
 
 30. What part of an acre is 38 sq. rd. 194 sq. ft. 
 108 sq. in.? 
 
 31. Express 2 lb. 9 oz. 21 dwt. as the decimal of 4 lb. 
 7 oz. 19 dwt. 
 
 32. Express 17 wk. 6 dy. 22 hr. 39 min. as the decimal 
 of 35 wk. 3 dy. 15 hr. 25 min. 
 
 33. What part of 61 ft. 3 in. is 8 ft. 7 in,? 
 
COMPOUND QUANTITIES. 171 
 
 Longitude and Time. 
 
 334. Meridians are imaginary lines drawn straight 
 around the earth through both poles. 
 
 335. Longitude is reckoned in degrees, minutes, and 
 seconds east or west from a standard meridian, as the 
 meridian of Greenwich, near London. Since longitude 
 is reckoned east and west from a given meridian the 
 longitude of a place is never greater than 180°, half the 
 distance round the earth. 
 
 336. When two places are both east or both west of 
 the standard meridian, the difference of their longitudes 
 is found by subtracting the one from the other. 
 
 When one place is east and the other west of the 
 standard meridian, the difference of their longitudes is 
 found by adding the two longitudes. 
 
 If the sum Of the two longitudes is greater than 180°, 
 this sum must be subtracted from 360° to obtain the cor- 
 rect difference of longitude. 
 
 Thus, if one longitude is 130° west and another 120° east, the dif- 
 ference of their longitudes is 360° - (130° + 120°), or 110°. 
 
 337. As the earth turns upon its axis once in twenty- 
 four hours, a point on the earth's surface will describe 
 a circumference (360°) in twenty-four hours. Therefore, 
 longitude may be reckoned in time as well as in degrees. 
 
 In one hour a point on the earth's surface describes -fa 
 of 360° = 15°; in one minute, ^ of 15°= 15'; and in one 
 second, ^o of 15'= 15". 
 
 Again, since it requires one hour (60 min.) for a point to 
 pass over 15°, to pass over 1° it requires T V of 60 min. = 
 4 min.; and to pass over 1' it requires ^ of 4 min. = 
 4 sec. 
 
172 COMPOUND QUANTITIES. 
 
 338. Examples. 1. Express 20° 36 f 15" of longitude 
 in time. 
 
 Solution. Since 15° longitude give 1 hr. in time, 15' longitude 
 1 min., and 15" longitude 1 sec, divide 
 
 15 )20° 36 / 15" 20° 36' 16" by 15, as in compound divi- 
 
 1 hr. 22 min. 25 sec. sion, and the quotient will be the time 
 required, 1 hr. 22 min. 25 sec. 
 
 2. Express 1 hr. 4 min. 4 sec. in degrees. 
 
 Solution. Since 1 hr. of time equals 15° of 
 1 hr. 4 min. 4 sec. longitude, 1 min. of time 15', and 1 sec. of time 
 
 15 15", multiply 1 hr. 4 min. 4 sec. by 15, as in 
 
 16° r 0" compound multiplication, and the product will 
 
 be the longitude required. Hence, 
 
 339. If longitude is expressed in degree-measures, 
 divide by 15 ; the quotient gives the longitude in time- 
 measures. 
 
 If longitude is expressed in time-measures, multiply by 
 15 ; the product gives the longitude in degree-measures. 
 
 Exercise 84. 
 
 Find the difference in longitude between two cities, if 
 the difference in time is : 
 
 1. 1 hr. 15 min. 5. 6 hr. 12 min. 30 sec. 
 
 2. 2 hr. 11 min. 6. 4 hr. 8 min. 12 sec. 
 
 3. 5 hr. 10 min. 10 sec. 7. 18 hr. 10 min. 
 
 4. 3 hr. 25 min. 35 sec. 8. 15 hr. 15 min. 15 sec. 
 
 Find the difference in time between two cities, if the 
 difference in longitude is : 
 
 9. 9° 20'. 13. 120° 14' 30". 
 
 10. 70° 30'. 14. 100° 45' 54". 
 
 11. 56° 36' 12". 15. 2° 2' 2". 
 
 12. 108° 32' 36". 16. 75° 10'. 
 
COMPOUND QUANTITIES. 173 
 
 17. Find the difference in time between New York, 
 longitude 74° 0' 3" West, and San Francisco, longitude 
 122° 26' 15" West. 
 
 18. The difference in time between Berlin and New York 
 is 5 hr. 49 min. 35 sec. What is the difference in longi- 
 tude? 
 
 340. Since the sun appears to move from east to west, 
 sunrise will' occur earlier at all points east and later at all 
 points west of a given place. Hence, clock-time will be 
 later in all places east and earlier in all places west of a 
 given meridian. 
 
 Therefore, if the time of a place is given, 
 
 To find the time of a place east, add to the given time 
 the difference of time between the two places. 
 
 To find the time of a place west, subtract from the given 
 time the difference of time between the two places. 
 
 341. To Find the Difference in Clock-time when the 
 Difference in Longitude is Known. 
 
 When it is noon at Boston (long. 71° 3' 30" West), what 
 is the time at Paris (long. 2° 20' 22" East) ? 
 
 71° 8' 30" W. 
 
 2° 20' 22" E. 
 
 23' 52". . . difference in longitude. 
 
 15)73° 23' 52' 
 
 4 hr. 53 min. 35 T 7 ? sec. difference in time. 
 
 Solution. Since Boston is west and Paris is east of the meridian 
 of Greenwich, the difference between their longitudes is found by 
 taking the sum of their longitudes. 
 
 The difference between their longitudes, 73° 23' 52", is equivalent 
 to 4 hr. 53 min. 35^ sec, and as Paris is east of Boston, the time at 
 Paris is found by adding the 4 hr. 53 min. 35 T ^ sec. to the time at 
 Boston, making 53 min. 35 T 7 5 sec. past 4 p.m. 
 
174 COMPOUND QUANTITIES. 
 
 Exercise 85. 
 The longitude of some public building in : 
 
 (1) Berlin is 13° 23' 43" E. (7) Jerusalem, 35° 32' E. 
 
 (2) Rome, 12° 27' 14" E. (8) Bombay, 72° 54' E. 
 
 (3) Constantinople, 28° 59' E. (9) Calcutta, 88° 19' 2" E. 
 
 (4) Pekin, 116° 23' 45" E. (10) Chicago, 87° 35' W. 
 
 (5) San Francisco, 122° 26' 15" W. (11) New York, 74° 0' 3" W. 
 
 (6) St. Louis, 90° 15' 15" W. (12) Montreal, 73° 25' W. 
 
 What is the clock-time at each of the above cities : 
 
 1. When it is noon at Greenwich ? 
 
 2. When it is half-past four p.m. at Chicago ? 
 
 3. When it is eight o'clock a.m. at Constantinople ? 
 
 When it is noon at Greenwich the time at : 
 
 (1) Boston, Mass., is 7 hr. 15 min. 46 sec. a.m. 
 
 (2) Columbia, S. C, 6 hr. 35 min. 32 sec. a.m. 
 
 (3) Salt Lake, 4 hr. 30 min. a.m. 
 
 (4) Albany, N. Y., 7 hr. 5 min. 1 sec. a.m. 
 
 (5) Harrisburg, Penn., hr. 52 min. 40 sec. a.m. 
 
 (6) New Orleans, La., 6 hr. a.m. 
 
 (7) Columbus, O., 6 hr. 27 min. 48 sec. a.m. 
 
 (8) Washington, D. C, 6 hr. 51 min. 44 sec. a.m. 
 
 (9) Springfield, 111., 6 hr. 1 min. 48 sec. a.m. 
 
 4. What is the longitude of each of the above cities ? 
 
 Note. Standard time is the clock-time of some selected meridian. 
 Eastern standard time is the clock-time of the meridian 75° west of 
 Greenwich, and is five hours slower than Greenwich time. Central 
 standard time is the clock-time of 90° west of Greenwich, and is just 
 one hour slower than Eastern standard time. Mountain standard time 
 is the clock-time of the meridian of 105°, and is one hour slower than 
 that of 90°. Western standard time is the clock-time of the meridian 
 of 120°, and is one hour slower than that of 105°. The railroads and 
 many cities and towns of the United States have adopted standard time. 
 
 Places not more than 7|° east or west of the meridians of 75°, 90°, 
 105°, 120° are reckoned to have the same time respectively as places 
 on these meridians. 
 
COMPOUND QUANTITIES. 175 
 
 Exercise 86. — Eeview. 
 
 1. Reduce 7 gal. 3 qt. 1 pt. to gallons and the decimal 
 of a gallon. 
 
 2. Reduce £4.375 to pounds, shillings, and pence. 
 
 3. Reduce 7.6875 gal. to gallons, quarts, and pints. 
 
 4. If $4.85 is equal to a pound, reduce to pounds, shil- 
 lings, and pence $5,875 ; $7.38 ; $17.85; $21.75. 
 
 5. How many square yards in 3.7156 A.? 
 
 6. If 2 qt. of linseed oil are mixed with \ pt. spirits of 
 turpentine, what fraction of the mixture is turpentine ? 
 How much turpentine in one pint of the mixture ? 
 
 7. Reduce 5.1732 mi. to yards, feet, and inches. 
 
 8. If a man walks 88 mi. in 26 hr., how many feet does 
 he walk in a second ? 
 
 9. Of a mixture of sand and lime 0.27 of the weight is 
 lime. How many ounces of lime in a pound of the mix- 
 ture ? How many troy grains of lime in an avoirdupois 
 pound of the mixture ? 
 
 10. A gill of water is put into a quart measure, and the 
 measure then filled with milk. What part of the mixture 
 is water ? 
 
 11. Reduce 555 ft. to the decimal of a mile. 
 
 12. Reduce 1 mi. 13 rd. 2 yd. 2 ft. 6 in. to inches. 
 
 13. How many cubic inches in 2 \ cu. ft. ? 
 
 14. How many pounds avoirdupois does a cubic yard of 
 water weigh if a cubic foot weighs 1000 oz. ? 
 
 15. Express the weight of a cubic yard of water as the 
 decimal of a ton. 
 
 16. What is the weight of 7 bu. 3^ pk. of potatoes ? 
 
 17. A farmer sowed 5 bu. 1 pk. 1 qt. of seed, and har- 
 vested from it 103 bu. 3 pk. 5 qt. How much did he 
 raise from a bushel of seed ? 
 
 18. How many bushels in 5 tons of oats ? 
 
176 COMPOUND QUANTITIES. 
 
 19. How many bottles, each holding 1 pt. 3 gi., can be 
 filled from a barrel of cider ? 
 
 20. If a steamer makes 13 mi 6 rd. an hour, how far 
 will she go between 6 a.m. and 6 p.m.? How many hours 
 will she require to make 113 mi. ? 
 
 21. If a train runs at the average rate of 111 rd. a 
 minute, how many hours will it require to run from Boston 
 to Buffalo, 498 mi. ? 
 
 22. What is the cost of 12 A. 146 sq. rd. of land at 
 $16.25 an acre ? 
 
 23. What is the cost of 8 t. 3 cwt. 27 lb. of coal at $5.75 
 a ton? 
 
 24. What is the cost of 7 t. 1560 lb. of hay at $15.50 a 
 ton? 
 
 25. What is the cost of a car load of wheat weighing 
 20,000 lb. at $1.05 a bushel ? 
 
 26. Reduce 5 rd. 4 yd. 2 J ft. to the decimal of a 
 mile. 
 
 27. Reduce 9 sq. ch. 11.25 sq. rd. to the decimal of an 
 acre. 
 
 28. Reduce 0.09375 bu. to quarts. 
 
 29. Reduce 7560 chains to miles. 
 
 30. How many gross are 2000 pens ? 
 
 31. Find the cost of 27.248 A. at $93.75 an acre. 
 
 32. Which is the greater, 2.8 of 3 ft. 11 in. or 3.11 of 2 
 ft. 8 in., and by how much ? 
 
 33. Reduce 171 lb. 6 oz. troy to the decimal of a ton 
 avoirdupois. 
 
 84. Express 14.52 sq. yd. as the decimal of a square 
 chain. 
 
 35. If a sovereign is equal to 25.22 francs, or to $4.85, 
 what decimal of a dollar is a franc ? 
 
 36. If 0.327 of some work is done in 3 hr. 38 min., how 
 long will the whole work require ? 
 
COMPOUND QUANTITIES. 177 
 
 37. A can run a mile in 7.68 min.; B can run at the 
 rate of 7.68 mi. an hour. Which is the faster runner ? 
 
 38. How many miles an hour does a person walk who 
 takes 2 steps a second and 1900 steps in a mile ? 
 
 39. If an ounce troy of gold is worth $20, what is the 
 value of a pound avoirdupois ? 
 
 40. Two stars cross the meridian at 6 hr. 4 min. 42.3 
 sec. and 7 hr. 2 min. 57.21 sec, respectively. What is the 
 interval between the observations ? 
 
 41. How long will it take to fill %% of a cistern, when 
 the whole requires 6 hr. 10 min.? 
 
 42. The circumference of a circle is 6 yd. 1 ft. 5.1 in. 
 What is the length of an arc of 55° ? 
 
 43. Multiply 2 t. 16 cwt. 63f lb. by If. 
 
 44. Into how many shares has £120 been divided when 
 each share is £3 8s. 6f d.? 
 
 45. If || of one line is equal to § of another line, which 
 is the greater t m What fraction of the greater is the less ? 
 
 46. Multiply 5 mi. 206 rd. 2 ft. 2 in. by 786. 
 
 47. The returns of a gold mine are 241 1. of ore yielding 
 
 2 oz. 1 dwt. 15 gr. of fine gold a ton, and 193" t. yielding 
 1 oz. 12 dwt. 9 gr. a ton. Find the value of the whole 
 yield, at $19.45 an ounce. 
 
 48. Divide 93 long tons 56 lb. by 23 lb. 5 oz. 
 
 49. Telegraph poles on railroads are generally erected 
 at intervals of 88 yd. Show that if a passenger counts 
 the number of poles which the train passes in three 
 minutes, that number will express the number of miles an 
 hour the train is going. 
 
 50. If Greenwich time is 5 hr. 8 min. 16 sec. later 
 than Washington time, and Chicago is 87° 35' W., what is 
 the difference between Washington time and Chicago time ? 
 
 5 1 . What fraction of 21 cu. yd. 11 cu. ft. 1215 cu. in. is 
 
 3 cu. yd. 1 cu. ft. 1161 cu. in.? 
 
178 COMPOUND QUANTITIES. 
 
 52. How many minutes in the first three months of 
 1895 ? How many in the first three months of ^896 ? 
 
 53. A knot is ^ of a degree, and a mile is 0.01477 of a 
 degree. Find in miles the value of a knot to five decimals. 
 
 54. The captain of a steamer, sailing from Liverpool, 
 found on taking an observation that the sun crossed his 
 meridian at 42 min. 5 sec. past one o'clock p.m. by Green- 
 wich time. Find his longitude. 
 
 55. If a walk 6 ft. wide is made round a park 600 ft. 
 square within the enclosure, how many square yards will 
 the walk contain ? 
 
 56. How many pickets 3 in. wide, placed 3 in. apart, 
 will be required to fence a rectangular lot 231 ft. long and 
 99 ft. wide ? What will they cost at $3.25 per hundred ? 
 
 57. The length of a year is 365.242218 mean solar days. 
 Express the length of a year in days, hours, minutes, and 
 seconds. 
 
 58. The Flying Dutchman Express runs from London to 
 Exeter, a distance of 193£ mi., in 4J hr., making one stop 
 of 10 min., two of 5 min. each, and one of 3 min. What 
 is its average speed per hour when in motion ? 
 
 59. The Scotch Express runs from London to Edin- 
 burgh, a distance of 393f mi., in 9 hr., making one stop of 
 30 min., three of 5 min. each, and one of 3 min. What is 
 its average speed per hour when in motion ? 
 
 60. The Empire State Express runs from New York to 
 Buffalo, a distance of 439 mi., in 8 hr. 15 min., making two 
 stops of 3 min. each and two stops of 2 min. each. What 
 is its average speed per hour when in motion ? 
 
 61. How many dollars worth 4s. 2d. each will pay a 
 bill of £11 17s. 6d.? 
 
 62. The lunar month is 29.53059 days. Express the 
 length of a lunar month in days, hours, minutes, and 
 seconds. 
 
. CHAPTER IX. 
 
 PROBLEMS. 
 
 342. Arithmetical Analysis. If the value of any 
 number of units is given, we may by division find the 
 value of one unit of the same kind, and by multiplication 
 the value of any number of units of the same kind. The 
 solution which combines these two processes is called 
 analysis. 
 
 Example. If 18 yards of cloth cost $45, what will be 
 the cost of 27 yards of cloth ? 
 
 Solution. If 18 yd. of cloth cost $45, 1 yd. will cost T \ of .$45, 
 or $2|. If 1 yd. of cloth costs $2£, 27 yd. will cost 27 X $2£, or $67£. 
 
 Exercise 87. 
 
 1. If 15 yards of silk cost $18.75, what will be the cost 
 of 20£ yards ? 
 
 2. If 3f pounds of tea cost $3.80, how many pounds can 
 be bought for $21.89? 
 
 3. If T \ of a ton of coal costs $1.12, what is the price 
 of 5£ cwt.? 
 
 4. If T 2 T of a piece of work is done in 25 days, what 
 fraction of the work will be done in llf days ? 
 
 5. A bankrupt's debts are $2520, and the value of his 
 property is $1890. How much can he pay on a dollar ? 
 
 6. If a bankrupt's debts are $4264, and he pays 62£ 
 cents on a dollar, what are his assets ? 
 
180 PROBLEMS. 
 
 7. If an ounce of gold is worth $20 67, what is the 
 value of 0.04 of a pound ? 
 
 8. A man spent £ of his money for dry goods, \ of the 
 remainder for groceries, and had $15 left. How much had 
 he at first ? 
 
 Solution. After spending f of his money he had \ left. After 
 spending \ of f of his money he had left \ of f = / 5 . Then, $15 = & 
 of the money he had at first. 
 
 9. Sampson & Reed sold f of a lot of wheat to one 
 man, f of the remainder to another, and had 93 bushels 
 left. How much had they at first ? 
 
 10. In a certain school ^ of the scholars are girls ; \ of 
 the boys are over 16 years old, and 6 boys are under 16. 
 How many girls and how many scholars are there in the 
 school ? 
 
 11. In a certain school \\ of the scholars are boys; 
 fa of the girls are under 16, and 13 girls are over 16. How 
 many boys and how many girls are there in the school ? 
 
 12. If from a certain number f of it is subtracted, then £ 
 of the remainder, then } of that remainder, 6 still remains. 
 What is the number ? 
 
 13. A ship's cargo sold for $45,000 belongs to three 
 partners. A owns $ of $ of it, B's share is equal to 3^ 
 of | of A's share, and C owns the remainder. What does 
 each receive from the sale ? 
 
 14. A man bequeathed T \ of his property to A, £ of it 
 to B, £ to C, i to D, and the remainder, $550, to E. What 
 was the value of his whole property ? 
 
 15. A farmer raised 321 bu. 3 pk. of corn from 9 acres 
 of land. At the same rate, what would be the yield from 
 25 acres ? 
 
 16. If 7 horses eat 21 bushels of oats in 16 days, how 
 many days will 99 bu. 3 pk. last them ? 
 
PROBLEMS. 181 
 
 17. If 12 horses can plow 96 acres in 6 days, how many 
 horses will plow 64 acres in 8 days ? 
 
 Solution. In 6 days 96 acres can be plowed by 12 horses. 
 In 1 day 96 acres can be plowed by 6 X 12 horses. 
 
 In 1 day 1 acre can be plowed by -tt — horses. 
 
 9o 
 
 6 X 12 
 
 In 8 days 1 acre can be plowed by — horses. 
 
 o X \jo 
 
 In 8 days 64 acres can be plowed by — — horses. 
 
 o X 9o 
 
 18. If 40 acres of grass is mowed by 8 men in 7 days, 
 how many acres will be mowed by 24 men in 28 days ? 
 
 Solution. 24 men will mow three times as much as 8 men in the 
 same time ; the same number of men will mow four times as much in 
 28 days as in 7 days. Hence, 24 men in 28 days will mow 3x4 
 or 12 times as much as 8 men in 7 days. 
 
 19. How many bushels of wheat will serve 72 people 
 8 days when 4 bushels serve 6 people 24 days ? 
 
 20. If 2 horses eat 8 bushels of oats in 16 days, how 
 many horses will eat 3000 bushels in 24 days ? 
 
 21. If a man travels 150 miles in 5 days of 12 hours, 
 in how many days of 10 hours will he travel 500 miles ? 
 
 22. If 939 soldiers consume 351 bu. of wheat in 21 days, 
 how many soldiers will consume 1404 bu. in 7 days ? 
 
 23. If 5 men, working 16 hours a day, can reap a field 
 of 12£ acres in 3-J- days, in how many days can 7 men, 
 working 12 hours a day, reap a field of 15 acres ? 
 
 24. If 7 men in 8 days of 11 hours mow 22 acres, in how 
 many days of 10 hours will 12 men mow 360 acres ? 
 
 25. If 44 cannon, firing 30 rounds an hour for 3 hours 
 a day, use 300 barrels of powder in 5 days, how many days 
 will 400 barrels last 66 cannon, firing 40 rounds an hour 
 for 5 hours a day ? 
 
182 PROBLEMS. 
 
 Areas. 
 
 343. If the length and breadth of a rectangle are 
 expressed in the same linear unit, the product of these 
 two numbers will express its area in square units of the 
 same name (§ 145). 
 
 Also, the number of square units in a rectangle divided 
 by the number of linear units in one dimension gives the 
 number of linear units in the other dimension (§ 145). 
 
 Exercise 88. 
 
 1. Find the area of a floor 16 ft. 3 in. long and 12 ft. 
 6 in. wide. 
 
 Solution. 16 ft. 3 in. = 16± ft-; 12 ft. 6 in. = 12* ft. 
 16* X 12* = 203*, the area of the floor in square feet. 
 
 2. A rectangle contains 672 sq. ft. 108 sq. in., and is 
 19 ft. 6 in. wide. Find its length. 
 
 Solution. 672 sq. ft. 108 sq. in. = 672f sq. ft. ; 19 ft. 6 in. - 19* ft. 
 672f -r 19* = 34*, the length of the rectangle in feet. 
 
 3. What length of board 15 in. wide will contain 
 11 sq. ft. 36 sq. in.? 
 
 4. What length of road 44 ft. wide will contain an acre ? 
 
 5. Find the area of a rectangular field 13.12 chains long, 
 10.35 chains broad. 
 
 6. A path 216 ft. long measures 72 sq. yd. Find its 
 breadth. 
 
 7. A rectangular field of 21.66 A. is 250.8 yd. broad. 
 Find its length. 
 
 8. What is the area of a table, if its length and breadth 
 are 4 ft. 3f in. and 2 ft. 9f in., respectively ? 
 
 9. From each corner of a square, each side of which is 
 2 ft. 5 in. long, a square measuring 5 in. on a side is cut 
 out. Find the area of the remainder of the figure. 
 
PROBLEMS. 183 
 
 10. The length and breadth of a map are 4-J- ft. and 
 3£ ft., respectively. If the map represents 77,760 sq. mi. 
 of country, how many square miles are there to a square 
 inch? 
 
 11. In rolling a grass plot that is 24 yd. long and con- 
 tains 400 sq. yd., how many times must a roller 3 ft. 4 in. 
 wide be drawn over it lengthwise that the whole plot may 
 be rolled ? 
 
 12. How many sods, each 2 ft. 3% in. long and 8£ in. 
 broad, will be required to turf an acre of ground ? 
 
 13. Find the area of a picture frame 2^ in. broad, if 
 the outside measurement is 4 ft. 6^ in. in length and 2 ft. 
 8 in. in width. 
 
 14. Find the expense of glazing four windows, each con- 
 taining 12 panes, if the panes are each 1 ft. long and 
 10 in. wide, and the price of the glass is 38 cents per 
 square foot. 
 
 15. A field 76 yd. long and 56 yd. broad, enclosed by a 
 wall, has a border 4 ft. wide within the wall, and within 
 this a path 5 ft. wide. If the remainder of the field is grass, 
 find the area of the border, of the path, and of the grass. 
 
 16. A square plot of land 127 yd. long has a path 1 yd. 
 wide running round the inside of it. Find the cost of 
 graveling this path at 15 cents per square yard. 
 
 17. A street f of a mile long has on each side a side- 
 walk 7-J- ft. wide. What will it cost to pave the sidewalks 
 with stones, each measuring 2 ft. 9 in. by 1 ft. 8 in., if the 
 stones cost, including the laying, 75 cents each ? 
 
 18. How many planks 11 ft. by 9 in. are needed to cover 
 a platform 27 ft. 6 in. long and 8 yd. wide ? What will 
 be the cost at 20 cents a square foot ? 
 
 19. How many tiles 9 in. long and 4 in. wide will be 
 required to pave a walk 8 ft. wide that surrounds a rect- 
 angular court 60 ft. long and 36 ft, wide ? 
 
184 PROBLEMS. 
 
 344. If the diameter of a circle is multiplied by 3.1416, 
 the product is the length of the circumference (§ 138). 
 
 If the circumference of a circle is multiplied by 0.31831, 
 the product is the length of the diameter (§ 139). 
 
 345. If the square of the radius of a circle is multiplied 
 by 3.1416, or if the square of the diameter is multiplied by 
 0.7854 (£ of 3.1416), the product is the area of the circle 
 (§ 146). 
 
 20. How many times will a wheel 2£ ft. in diameter 
 turn in going a distance of 110 yards ? 
 
 21. What distance will a wheel ft yd. in diameter pass 
 over in making 4£ revolutions ? 
 
 22. Find the diameter of a wheel that makes 9 revo- 
 lutions in going 7£ yards ? 
 
 23. If the circumference of a wheel is y of 1 yd. 1| ft., 
 how many times will the wheel turn in going 3^ miles ? 
 
 24. If the wheel of a locomotive is 3} times 5.52 ft. in 
 circumference, how many times does it turn in a minute 
 when the locomotive is running at the rate of 13.34 mi. 
 an hour ? 
 
 25. Find the area of a circle that has a radius of 3 feet. 
 
 26. What is the area of a circular field that has a radius 
 of 400 yards ? 
 
 27. The radius of the rotunda of the Pantheon at Rome 
 is 71 ft. 6 in. Find the area of the floor. 
 
 28. The diameter of a cylindrical cistern is 13 ft. What 
 is the area of the bottom ? 
 
 29. The two dials of the clock of St. Paul's, London, 
 are each 18} ft. in diameter. What is the area of each in 
 square feet ? 
 
 30. At 20 cents a square yard, what will it cost to gravel 
 a walk 6 ft. wide running round a circular fish pond 70 yd. 
 in diameter ? 
 
PROBLEMS. 185 
 
 346. If the square of the diameter of a sphere is multi- 
 plied by 3.1416, the product is the area of the surface of 
 the sphere (§ 149). 
 
 31. How many square inches on the surface of a ball 
 3 inches in diameter ? 
 
 32. How many square inches of surface on a spherical 
 blackboard 12 inches in diameter ? 
 
 33. What is the interior surface of a hemispherical vase 
 whose interior diameter is 20 inches ? 
 
 34. Find the external and the internal surface of a 
 spherical shell whose external and internal diameters are 
 8 in. and 5 in., respectively. 
 
 35. How many square feet of tin are required to make 
 16 hemispherical bowls, each 2 ft. 4 in. in diameter ? 
 
 347. The line joining the centres of the bases of a 
 cylinder is called the axis of the cylinder (§ 163). 
 
 348. Right Cylinder. A cylinder whose axis is per- 
 pendicular to the bases is called a right cylinder. 
 
 349. To Find the Area of the Lateral Surface of a 
 Right Cylinder, 
 
 Multiply the height of the cylinder by the perimeter of its 
 base. 
 
 36. Find the lateral surface of a right cylinder if its 
 height is 10 in. and the radius of its base is 7 in. 
 
 37. Find the lateral surface of a right cylinder if its 
 height is 12 ft. and the diameter of its base is 9 ft. 4 in. 
 
 38. At 32 cents a square foot, what is the cost of cement- 
 ing a cylindrical cistern 20 ft. deep and 18 ft. in diameter ? 
 
 39. The diameters of two right cylinders of the same 
 height are as 6 to 1. Compare the lateral surfaces. 
 
186 PROBLEMS. 
 
 Carpeting Rooms. 
 
 350. Carpeting is of various widths, and is sold by the 
 yard. Oilcloth and linoleum are sold by the square yard. 
 
 To find the number of yards of carpeting required for 
 a room, we decide whether the strips shall run length- 
 wise or across the room, and then find the number of strips 
 required (§ 150). The number of yards in a strip, includ- 
 ing the waste in matching the pattern, multiplied by the 
 number of strips will give the number of yards required. 
 
 Exercise 89. 
 
 1. How many yards of carpeting 27 in. wide will be 
 required for a floor 26 ft. long, 15£ ft. wide, if the strips 
 run lengthwise ? How many if the strips run across the 
 room ? How much will be turned under in each case ? 
 
 2. How many yards of carpeting £■ yd. wide will be 
 required for a room 8£ yd. by 17 ft., if the strips run 
 lengthwise, and if there is a waste of ^ yd. a strip ? 
 
 3. How many square yards of oilcloth will be required 
 for a hall floor 5£ yd. long and 10 ft. wide ? 
 
 4. At $0.92 a yard, what is the cost of a carpet 27 in. 
 wide for a room 28 J ft. by 18f ft., if the strips run 
 lengthwise ? 
 
 5. Find the cost of carpet 30 in. wide, at $1.25 per yard, 
 for a room 18 ft. by 14 ft., if the strips run lengthwise. 
 
 6. Find the cost of carpeting 27 in. wide, at $1.12£ per 
 yard, for a room 29 ft. 9 in. by 23 ft. 6 in., if the strips 
 run across the room. 
 
 7. Find the cost of carpeting 27 in. wide, at $2.75 per 
 yard, for a room 34 ft. 8 in. by 13 ft. 3 in., if the strips 
 run lengthwise, and if there is a waste of £ yd. a strip. 
 
 8. Which way must the strips of carpet 27 in. wide run 
 to carpet most economically a room 20£ ft. by 19£ ft.? 
 
PROBLEMS. 187 
 
 Papering Rooms. 
 
 351. Wall paper is 18 in. wide, and is sold in single 
 rolls 8 yd. long, or in double rolls 16 yd. long. 
 
 In estimating the number of rolls of paper required for 
 a room of ordinary height, we find the number of feet in 
 the perimeter of the room, leaving out the widths of the 
 doors and windows, and allow a double roll or two single 
 rolls for every seven feet. 
 
 Note. A room is considered of ordinary height if the distance 
 from the base board to the border is not more than 8 ft. 
 
 9. How many double rolls of paper will be required for 
 a room of ordinary height, 15 ft. long and 12 ft. wide, if 
 the room has one door and three windows, each 3-J ft. wide ? 
 
 Solution. Perimeter of room = 2 X (15 + 12) ft. = 54 f t. 
 
 Width of door and windows = 4 X 3± ft. = 14 ft. 
 
 Perimeter less door and windows = 40 ft. 
 
 40 -S- 7 = 5^. Hence, 6 double rolls will be required. 
 
 10. At $2.25 a double roll, put on, what is the cost of 
 papering a room of ordinary height, 16 ft. by 14 ft., if the 
 room has two doors each 4 ft. wide, and four windows each 
 
 3 ft. 6 in. wide ? 
 
 11. At 75 cents a single roll, put on, what is the cost of 
 papering a room of ordinary height, 20 ft. 6 in. long and 
 17 ft. 4 in. wide, if the room has two doors each 3 ft. 6 in. 
 wide, and five windows each 3 ft. 3 in. wide ? 
 
 12. What is the cost of the border for the room of Ex. 11 
 at $0.45 a running yard ? 
 
 Note. Border is sold by the yard, and no allowance is made for 
 doors or windows. 
 
 13. At $1.75 a double roll, put .on, what is the cost of 
 papering a room of ordinary height, 18 ft. 6 in. by 14 ft. 
 
 4 in., if the room has three doors 4 ft. wide, and three 
 windows 3 ft. 9 in. wide ? 
 
188 PROBLEMS. 
 
 Plastering, Painting, and Paving. 
 
 352. The unit of plastering, painting, and paving is the 
 square yard. 
 
 The rule for estimating these kinds of work is : 
 
 Measure the total area ; deduct from this total area half 
 the area of the doors, windows, and other openings, and 
 express the result to the nearest square yard. 
 
 14. Find at 20 cents a square yard the cost of plaster- 
 ing the walls and ceiling of a room 18 ft. by 16 ft. by 10 ft., 
 if the room has two doors 7 ft. 6 in. by 4 ft., three windows 
 6 ft. 6 in. by 4 ft., and a base board of 10 in. 
 
 15. Find at 25 cents a square yard the cost of plaster- 
 ing the walls and ceiling of a room 16 ft. by 15 ft. by 10 ft., 
 if the room has two doors 7 ft. by 3 ft. 9 in., three windows 
 
 5 ft. 6 in. by 3 ft. 6 in., and a base board of 10 in. 
 
 16. Find at 20 cents a square yard the cost of plaster- 
 ing the walls and ceiling of a room 15 ft. by 14 ft. by 9 ft. 
 
 6 in., if the room has two doors 7 ft. 4 in. by 4 ft., two 
 windows 5 ft. 6 in. by 3 ft. 6 in., and a base board of 
 9 in. 
 
 17. Find at 15 cents a square yard the cost of painting 
 the outside of the walls of a cottage-roofed house 36 ft. by 
 32 ft. by 13 ft., if the house has three doors 7 ft. 6 in. by 
 4 ft., and eleven windows 6 ft. by 4 ft. 
 
 18. Find at 20 cents a square yard the cost of painting 
 the walls of a room 16 ft. by 15 ft. by 10 ft., if the room 
 has two doors 7 ft. 6 in. by 4 ft., four windows 6 ft. by 
 3 ft. 9 in., and a base board of 9 in. 
 
 19. How many bricks 8 in. long and 4 in. wide will be 
 needed to pave a rectangular court 60 ft. by 30 ft.? 
 
 20. How many bricks 8 in. long and 2\ in. thick, laid 
 on edge, will be needed to pave the court of Ex. 19 ? 
 
PROBLEMS. 189 
 
 Clapboards and Shingles. 
 
 353. Clapboards. Clapboards are usually cut 4 ft. long 
 and 6 in. wide, and laid 3^- in. to the weather. Therefore, 
 each clapboard covers l£ sq. ft. of surface. 
 
 Note. In estimating the number of clapboards required, we 
 deduct the area of all openings. 
 
 354. Shingles. Shingles are 16 in. long, and are esti- 
 mated to average 4 in. wide, so that a shingle laid 4-J- in. to 
 the weather will cover 18 sq. in., and 8 shingles will cover 
 1 sq. ft. At this rate, 800 shingles would cover a square, 
 or 100 sq. ft. It is found, in practice, that 1000 shingles, 
 laid 4-j- in. to the weather, will cover about 120 sq. ft. 
 
 Shingles are put up in bunches of 250, and therefore it 
 takes four bunches to make a thousand. 
 
 2 1 . How many clapboards will be required for the front 
 of a house 40 ft. long and 20 ft. high, allowing 120 sq. ft. 
 for doors and windows ? 
 
 22. How many clapboards will be required for a house 
 44 ft. long, 35 ft. wide, and 22 ft. high to the eaves, if the 
 gables extend 14 ft. above the end walls, the two gables to 
 be reckoned as one full wall, and 500 sq. ft. to be allowed 
 for doors and windows ? 
 
 23. Allowing 1000 shingles for 120 sq. ft., how many 
 thousand will be required for the pitched roof of a house 
 60 ft. long, if the width of each side of the roof is 24£ ft.? 
 
 24. Allowing 1000 shingles for 110 sq. ft., how many 
 thousand will be required for the pitched roof of a barn 
 40 ft. long, if the width of each side of the roof is 24 ft.? 
 
 25. Allowing 1000 shingles for 120 sq. ft., how many 
 thousand will be required for the pitched roof of a house 
 28 ft. long, if the width of each side of the roof is 18 ft.? 
 
190 PROBLEMS. 
 
 Board Measure. 
 
 355. Boards one inch or less in thickness are sold by the 
 square foot. 
 
 Boards more than one inch in thickness, and all squared 
 lumber, are sold by the number of square feet of boards 
 one inch in thickness to which they are equivalent. 
 
 Thus, a board 16 ft. long, 1 ft. wide, and 1 in. thick contains 16 ft. 
 board measure. If only $, f, or \ of an inch thick, it still con- 
 tains 16 ft. ; but if l± in. thick, it contains 1£ X 16, or 20 ft. board 
 measure. 
 
 356. To Find the Number of Feet Board Measure in 
 Boards an Inch or More in Thickness, and Squared Lumber, 
 
 Express the length and width in feet, and the tlnckness 
 in inches ; take the product of these three numbers for the 
 number of feet board measure. 
 
 Note 1. In practice, the width of a board, unless sawed to order, 
 is reckoned only to the next smaller half-inch. Thus, a width of 
 llf in. is reckoned 11 in.; of 13| or 13f in. is reckoned 13$ in. 
 
 Note 2. If a board tapers regularly, its average width is found by 
 taking one half the sum of its end widths. 
 
 How many feet board measure in : 
 
 26. A board 18 ft. long, 9 in. wide, £- in. thick ? 
 
 27. A board 16 ft. long, 11 in. wide, 1 in. thick ? 
 
 28. Twenty boards averaging 14 ft. long, 10 in. wide, 
 1£ in. thick ? 
 
 29. Three joists 13 ft. long, 8 in. wide, 3 in. thick ? 
 
 30. A stick of timber 8 in. by 9 in., and 27 ft. long ? 
 
 31. Two beams, each 6 in. by 9 in., and 23 ft. long? 
 
 32. Three joists, each 3 in. by 4 in., and 11 ft. long ? 
 
 33. Five joists, each 6 in. by 4 in., and 14 ft. loug? 
 
 34. A stick of timber 10 in. square, and 36 ft. long ? 
 
 35. Ten planks, each 13 ft. long, 15 in. wide, 2 in. thick ? 
 
PROBLEMS. 191 
 
 Find the cost of : 
 
 36. Nine joists, each 15 ft. long, 3£ in. by 5 in., at $12 
 per M. 
 
 Note. The abbreviation per M means by the thousand. 
 
 37. Thirty planks, each 12 ft. long, 11 in. wide, 3 in. 
 thick, at $15 per M. 
 
 38. Four sticks of timber, each 8 in. by 9 in. and 23 ft. 
 long, at $18 per M. 
 
 39. A board 24 ft. long, 23 in. wide at one end and 
 17 in. at the other, and 1£ in. thick, at $30 per M. 
 
 40. A stick of timber 29 ft. long, 10 in. by 12 in., at 
 $13.50 per M. 
 
 41. The flooring for two floors, each 23 ft. by 17 ft., 
 each floor double, and of boards j- in. thick; the under 
 floors at $18, and the upper at $24, per M. 
 
 42. The flooring timbers for a room 23 ft. by 17 ft., at 
 $18 per M, if they are 2 in. by 10 in., 17 ft. long, and are 
 placed, on edge, one close to each wall and the others with 
 spaces || ft. wide between them. 
 
 357. Round Logs. Round logs are sold by the number 
 of feet board measure that can be cut from them. If a log 
 is not more than 16 ft. long, we measure the length of the 
 log and the diameter of the smaller end, and find the number 
 of feet board measure as follows : 
 
 Subtract twice the diameter expressed in inches from the 
 square of the diameter, and take f i of the remainder for the 
 number of feet board measure in a log 10 ft. long. 
 
 43. Find the number of feet board measure in a log 12 ft. 
 long, and 20 in. in diameter at the smaller end. 
 
 Solution. 20 2 - 2 X 20 = 400 - 40 = 360. 
 
 f J- of 360 = 189. 
 As the log is 12 ft. long, we must take f$ of 189 ft. to obtain the 
 number of feet in the whole log ; that is, 226.8 ft. 
 
192 PROBLEMS. 
 
 By this rule find the number of feet board measure in : 
 
 44. A log 14 ft. long, smallest diameter 17 in. 
 
 45. A log 11 ft. long, smallest diameter 13 in. 
 
 46. A log 16 ft. long, smallest diameter 20 in. 
 
 47. A log 12 ft. long, smallest diameter 15 in. 
 
 Find the value at $9 per M of : 
 
 48. A log 15 ft. long, smallest diameter 11 in. 
 
 49. A log 16 ft. long, smallest diameter 13 in. 
 
 50. A log 13 ft. long, smallest diameter 16 in. 
 
 51. A log 14 ft. long, smallest diameter 12 in. 
 
 358. Large, heavy timber of hard wood is generally 
 sold by the ton, signifying 50 cu. ft., or 600 ft. board 
 measure. 
 
 Volumes. 
 
 359. If the length, breadth, and height of a rectangular 
 solid are expressed in the same linear unit, the product of 
 these three numbers will express its volume in cubic units 
 of the same name (§ 161). 
 
 Also, the number of cubic units in a rectangular solid 
 divided by the product of the numbers of linear units in 
 any two dimensions gives the number of linear units in the 
 third dimension (§ 161). 
 
 Exercise 90. 
 
 1. Find the volume of a rectangular solid 7 ft. long, 
 2 ft. 6 in. wide, and 11 in. thick. 
 
 2. How many cubic feet of air in a hall 54 ft. long, 
 33 ft. wide, and 21 ft. 4 in. high ? 
 
 3. Find the volume of a cube whose edge is 2-j- yd. 
 
 4. A cellar was dug 21 ft. long, 17 ft. 3 in. wide, and 
 9 ft. deep. How many cubic yards of earth were taken 
 out? 
 
PROBLEMS. 193 
 
 5. Find the volume of a brick 8 in. long, 3£ in. wide, 
 and ( 1\ in. thick. 
 
 6. How many cubic feet of water will a rectangular 
 cistern hold whose length, breadth, and height are 5 ft. 
 4 in., 3 ft. 6 in., and 2 ft. 10 in., respectively ? 
 
 7. Find the volume in cubic inches of a bar of iron 
 21 ft. long, 3 in. wide, and 2 in. thick. 
 
 8. What is the value at $190 a cubic inch of a bar of 
 gold 8 in. long and § of an inch square ? 
 
 9. A rectangular reservoir 15 yd. long, 12 yd. wide, 
 holds 330 cu. yd. of water. What is its depth ? 
 
 10. What length must be cut off a beam 9 in. by 15 in. 
 that the part cut off may contain 2£ cu. ft.? 
 
 11. How high is a room, if it is 31 ft. 3 in. long, 24 ft. 
 broadband contains 10,000 cu. ft. of air? 
 
 12. A piece of wood 5 ft. long, 1 ft. broad, and 9 in. 
 thick is cut up into matches 2-J- in. long and 0.1 of an inch 
 square. How many matches will there be, if no allowance 
 is made for waste in cutting ? 
 
 13. How long a wall 6 ft. high, 12f in. thick, can be 
 built with the bricks forming a rectangular pile 17 ft. 6 in. 
 long, 5 ft. wide, and 4 ft. 3 in. high ? 
 
 14. Find the surface of a cube whose edge is 3 ft. 5f in. 
 
 15. Find the surface of a rectangular block of stone 
 4 ft. long, 2\ ft. broad, and \\ ft. thick. 
 
 16. A lake whose area is 45 A. is covered with ice 3 in. 
 thick. Find the weight of the ice in tons, if a cubic foot 
 of ice weighs 920 oz. 
 
 17. How many bricks will be required to build a wall 
 75 ft. long, 6 ft. high, and 16 in. thick, if each brick is 
 8 in. long, 4 in. wide, and 2\ in. thick ? 
 
 18. The ceiling of a room 27 ft. long, 24 ft. broad, and 
 10 ft. high is to be raised so as to increase the space by 
 84 cu. yd. What will then be the height of the room ? 
 
194 PROBLEMS. 
 
 19. Find the cost of making a road 110 yd. long and 
 18 ft. wide, if the soil is first removed to the depth of 1 ft. 
 at a cost of 25 cents a cubic yard, rubble then laid 8 in. 
 deep at 25 cents a cubic yard, and gravel placed on top 
 9 in. thick at 62 J cents a cubic yard. 
 
 20. If a rectangular block of wood 5 ft. 4.8 in. long, 
 1 ft. 9 in. wide and thick, weighs 7.56 cwt., find in pounds 
 its weight per cubic foot. 
 
 360. A cord of wood or stone is a pile 8 ft. long, 4 ft. 
 wide, and 4 ft. high, making 128 cu. ft. 
 
 A cord foot is a pile 1 ft. long, 4 ft. wide, and 4 ft. high, 
 and is therefore one eighth of a cord, or 16 cu. ft. Hence, 
 
 Cord. Cord Foot. 
 
 361. To Find the Number of Cords in a Pile of Wood, 
 
 Divide the product of the length, breadth, and height, 
 expressed in feet, by 8 X ^ X 4- 
 
 21. How many cords of wood in a pile 40 ft. long, 4 ft. 
 wide, and 5 ft. 4 in. high ? 
 
 22. A pile of wood containing 67£ cords is 270 ft. long 
 and 4 ft. wide. How high is it ? 
 
 23. What will be the cost of a pile of wood 25 ft. long, 
 4 ft. wide, and 4 ft. 8 in. high, at $3.75 a cord ? 
 
PROBLEMS. 195 
 
 24. What must be the length of a load of wood 3£ ft. 
 high and 5 ft. wide to contain a cord ? 
 
 25. How high must manure be piled in a cart 6 ft. by 
 4 ft., that the. load may contain half a cord ? 
 
 26. How many cords of wood in a pile 32 ft. long, 8 ft. 
 wide, and 6 ft. high ? 
 
 27. How many cords of wood in a pile 40 ft. long, 4 ft. 
 wide, and 8 ft. high ? 
 
 28. Find the cost of the wood at $3.75 a cord that can 
 be piled in a shed 18 ft. long, 16 ft. wide, and 7 ft. high. 
 
 362. By § 162, to find the volume of a sphere, we mul- 
 tiply the cube of the diameter by 0.5236 (J of 3.1416). 
 
 29. Find the number of cubic inches in a sphere 11 in. 
 in diameter. 
 
 30. How many cubic inches of water can be poured into 
 a hollow sphere whose inner diameter is 16J in.? 
 
 31. What is the volume of the ball on top of St. Paul's 
 in London, which is 6 ft. in diameter ? 
 
 32. If 30 cu. in. of powder weigh 1 lb., how many 
 ounces of powder will just fill a shell, inner diameter 3 in.? 
 
 363. To find the volume of a cylinder, we multiply the 
 number of square units in its base by the number of linear 
 units of the same name in its height (§ 164). 
 
 33. Find the volume of a cylinder whose height is 5 ft. 
 and the radius of whose base is 1 ft. 2 in. 
 
 34. Find the volume of a cylinder whose height is 4 ft. 
 6 in. and the diameter of whose base is 8 ft. 2 in. 
 
 35. How many cubic yards of earth must be excavated 
 to make a well 3 ft. in diameter and 20 ft. deep ? 
 
 36. How many cubic yards in a tunnel 800 ft. long, if a 
 cross section is a semicircle with a radius of 10 ft.? 
 
196 PROBLEMS. 
 
 Capacity of Bins and Cisterns. 
 
 37. Find the number of cubic feet in a bushel. 
 
 Solution. Since a bushel contains 2160.42 cu. in. (§ 294), and a 
 
 2160 42 
 cubic foot contains 1728 cu. in., therefore, a bushel contains , ' 
 
 1728 
 
 cu. ft., or 1.24445 cu. ft. 
 
 If we add i of 0.01 of 1.24446 to 1.24445, we have 1.25+ . Hence, 
 
 364. To Find the Approximate Number of Bushels a 
 Bin will Hold, 
 
 Take $ of the number of cubic feet in the bin, and add to 
 the product £ of 0.01 of the product. 
 
 365. To Find the Number of Cubic Feet Required for 
 a Given Number of Bushels, 
 
 Take | of the number of bushels, and subtract from the 
 product ^ of 0.01 of the product. 
 
 38. Find the number of bushels a bin will hold that is 
 6 ft. long, 5 ft. wide, and 4 ft. deep. 
 
 Solution. f of 6x5x4 = 96. 
 
 £ of 0.01 of 96 s 0.48 
 96.48 
 Therefore, the bin will hold 96.48 bu. 
 
 39. Find the number of cubic feet required for 1000 bu. 
 
 Solution. £ of 1000 = 1250. 
 
 £ of 0.01 of 1250 = 6.25 
 1243.75 
 Hence, 1243.75 cu. ft. are required for 1000 bu. 
 
 40. Find the number of bushels a bin will hold that is 
 
 8 ft. long, 4 ft. wide, 3 ft. deep. 
 
 41. Find the number of bushels a bin will hold that is 
 
 9 ft. long, 6 ft. 6 in. wide, 3 ft. 4 in. deep. 
 
 42. Find the depth of a bin that will hold 360 bu., if its 
 length is 12 ft. and its width 6 ft. 
 
PROBLEMS. 197 
 
 43. Find the length of a bin that is 6 ft. wide and 5 ft. 
 deep, if it will hold 400 bu. 
 
 44. Find the number of bushels that will fill a bin 8.5 ft. 
 long, 4.5 ft. wide, 3.5 ft. deep. 
 
 45. A bin 20 ft. long, 12 ft. wide, and 6 ft. deep is full 
 of wheat. What is its value at $0.75 a bushel ? 
 
 46. If a ton of coal occupies 40 cu. ft., how many tons 
 of coal will fill a bin 21 ft. long, 10 ft. wide, 5 ft. deep ? 
 
 47. If a ton of Lehigh coal occupies 35 cu. ft., how 
 many tons of Lehigh coal will fill a bin 8 ft. long, 5 ft. 
 9 in. wide, 3 ft. 6 in. deep ? 
 
 48. How many bushels will a bin hold that is 22 ft. long, 
 12 ft. 6 in. wide, 9 ft. 9 in. deep ? 
 
 49. Find the number of gallons in a cubic foot. 
 
 Solution. Since a cubic foot contains 1728 cu. in., and a gallon 
 contains 231 cu. in., therefore, a cubic foot contains -Vit 5 " S^m or 
 7.480&2 gal. 
 
 If we add J of 0.01 of 7.48052 to 7.48052, we have 7.5 nearly. 
 Hence, 
 
 366. To Find the Approximate Number of Gallons a 
 Cistern will Hold, 
 
 Multiply the number of cubic feet by 7J, and from the 
 product subtract \ of 0.01 of the product. 
 
 367. To Find the Exact Number of Gallons a Cistern 
 will Hold, 
 
 Divide the number of cubic inches in the contents of the 
 cistern by 231. 
 
 50. Find the exact number of gallons a cistern will hold 
 that is 5 ft. square, and 6 ft. deep. 
 
 51. Find the exact number of gallons a cistern will hold 
 that is 13 ft. long, 6 ft. wide, 7 ft. 4 in. deep. 
 
 52. Find the exact number of gallons a tank will hold 
 that is 4 ft. long, 2 ft. 8 in. wide, 1 ft. 8 in. deep. 
 
198 PROBLEMS. 
 
 53. Find the capacity in cubic feet of a cistern that will 
 hold 200 bbl. of water. 
 
 54. Find the approximate number of gallons a cylindri- 
 cal cistern will hold that is 6 ft. in diameter and 7 ft. deep. 
 
 55. Find the approximate number of gallons a cylindri- 
 cal vessel will hold that is 12 in. in diameter and 10 in. deep. 
 
 56. How many quarts will a cylindrical vessel hold 
 5£ in. in diameter and 6 in. deep ? 
 
 57. How many quarts will a hollow sphere hold whose 
 interior diameter is 12 in.? 
 
 58. What part of a bushel will a hemispherical bowl 
 hold that is 13 in. in diameter ? 
 
 59. If a cubical box 2 ft. on an edge contains a solid 
 sphere 2 ft. in diameter, how many gallons of water can be 
 poured into the box ? 
 
 60. If 64 qt. of water are poured into a vessel that will 
 hold 2 bu. of wheat, what part of the vessel will be filled ? 
 
 Specific Gravity. 
 
 368. The specific gravity of a substance is the number 
 found by dividing the weight of the substance by the 
 weight of an equal bulk of water (§ 165). 
 
 Note. If a substance is in water, the water buoys it up by just 
 the weight of the water displaced by it. 
 
 Exercise 91. 
 
 1 . Find the number of cubic inches in 1 oz. (av.) of water. 
 
 2. Find the weight in ounces (av.) of 1 on. in. of water. 
 
 3. Find the weight in ounces (av.) of 1 pt. of water. 
 
 4. Find the number of pints in 1 lb. of water. 
 
 5. Find the weight in grains of 1 cu. in. of water. 
 
 6. A bar of iron 5 in. long and 2 in. square weighs 5 lb. 
 What is the specific gravity of the iron ? 
 
PROBLEMS. 199 
 
 7. If a bar of iron 18 in. long, 2-J- in. wide, If in. thick, 
 weighs 18 lb. 9 oz., what is the specific gravity of the iron ? 
 
 8. If the specific gravity of iron is 7.48, find the num- 
 ber of cubic inches of iron to the pound. 
 
 9. If the specific gravity of gold is 19.36, find the num- 
 ber of cubic inches in 2 lb. 6£ oz. of gold. 
 
 10. How many pounds does a boy lift in raising a cubic 
 foot of stone under water, if its specific gravity is 2£ ? 
 
 11. A square-built scow 12 ft. long, 6 J ft. wide, sinks 
 
 5 in. into the water. What does it weigh, and how many 
 pounds will be required to sink it 7 in. deeper ? 
 
 12. A square-built scow 11 ft. long, 5^ ft. wide, weighs 
 320 lb., and is loaded with 750 lb. of stone. How deep 
 does it sink in the water ? 
 
 13. How many tons of ice, specific gravity 0.93, can be 
 packed in a building 50 ft. long, 40 ft. wide, 20 ft. high ? 
 
 14». If the specific gravity of an iceberg is 0.9, how many 
 cubic yards does an iceberg contain that is 40 rd. long, 
 
 6 yd. wide, and rises 160 ft. out of the sea ? 
 
 15. If a cubic foot of brick wall weighs 90 lb. and con- 
 tains 22 bricks, with the mortar, what is the weight and 
 the specific gravity of a brick and its share of mortar ? 
 
 16. What is the weight of a brick wall 40 ft. long, 20 ft. 
 high, and 1 ft. thick, if the specific gravity of a brick with 
 its mortar is 1.46 ? How many thousand bricks will be 
 required for the wall, allowing 22 for a cubic foot ? 
 
 17. If the specific gravity of iron is 7.48, what is the 
 weight of a cylindrical iron shell 1 in. thick and 2 ft. long, 
 whose inner radius is 7 in.? 
 
 18. If a piece of marble weighs 37.78 oz. in air, and 
 23.89 oz. in water, what is its volume and its specific 
 gravity ? 
 
 19. If a mass of lead weighs 1986^ lb. in air, and 1811 J lb. 
 in water, what is its volume and its specific gravity ? 
 
200 
 
 PROBLEMS. 
 
 210— 
 
 1~^1 
 
 200— 
 
 
 190— 
 
 —90 
 
 180— 
 
 —80 
 
 170— 
 
 
 160— 
 
 —70 
 
 150— 
 
 
 140— 
 
 —60 
 
 130— 
 
 
 120— 
 
 —50 
 
 110 — 
 
 
 100 — 
 
 —40 
 
 90 — 
 
 —30 
 
 80— 
 
 
 70— 
 
 —20 
 
 60— 
 
 
 50— 
 
 —10 
 
 40— 
 
 
 30— 
 
 —0 
 
 20— 
 
 —10 
 
 10— 
 
 
 0— 
 
 —20 
 
 10— 
 
 
 20— 
 
 —30 
 
 30— j 
 
 
 Temperature. 
 
 369. A thermometer is an instrument 
 for registering temperature. 
 
 There are three scales for registering tem- 
 perature by means of the thermometer. 
 
 Fahrenheit's has the freezing point of 
 water marked 32°, and boiling point 212°. 
 
 The Centigrade lias the freezing point 0°, 
 and the boiling point 100°. 
 
 Reaumur's has the freezing point 0°, and 
 the boiling point 80°. 
 
 Temperature below 0° is indicated by pre- 
 fixing the minus sign. 
 
 Thus, — 20° means 20° below zero. 
 
 370. Examples. 
 
 Reaumur's scale. 
 
 1. Express 80° C. in 
 
 Therefore, 
 
 100° C. = 80° R. 
 
 = 64° R. 
 
 Thermometer. 
 
 2. Express 50° F. in Centigrade scale. 
 
 The number of degrees F. between the freezing 
 and boiling points is 180. 
 
 Therefore, 1° F. = j§°° C. = $° C. 
 
 But 50° F. is 18° F. above freezing point, 
 and $ of 18° = 10°. 
 
 That is, 50° F. = 10° C. 
 
 3. Express 60° C. in Fahrenheit's scale. 
 
 100° C. = 180° F. 
 Therefore, 60° C. = ^ of 180° F. 
 
 = 108° F. 
 This is the height above the freezing point, and 
 is marked 32° + 108° = 140°. 
 
PROBLEMS. 201 
 
 Exercise 92. 
 Express : 
 
 1. 59° F. in Centigrade scale ; in Reaumur's scale. 
 
 2. 77° F. in Centigrade scale j in Reaumur's scale. 
 
 3. 950° F. in Centigrade scale ; in Reaumur's scale. 
 
 4. — 40° F. in Centigrade scale ; in Reaumur's scale. 
 
 5. — 4° F. in Centigrade scale ; in Reaumur's scale. 
 
 6. 10° C. in Fahrenheit's scale ; in Reaumur's scale. 
 
 7. 22° C. in Fahrenheit's scale ; in Reaumur's scale. 
 
 8. — 30° C. in Fahrenheit's scale ; in Reaumur's scale. 
 
 9. — llf ° C. in Fahrenheit's scale ; in Reaumur's scale. 
 
 Time and Work Problems. 
 Exercise 93. 
 
 1, If one man can do a piece of work in 9 days and 
 another man can do the same work in 8 days, in how many 
 days can the men working together do the work ? 
 
 Solution. If a man can do a piece of work in 9 days, in 1 day he 
 can do £ of the work ; and if another man can do the same work in 
 8 days, in 1 day he can do f of it. 
 
 Both men together in 1 day can do ^ + }, or \\ of the work. 
 
 Therefore, if the whole work is considered as divided into 72 equal 
 parts, they together can do 17 of these parts in 1 day, and the number 
 of days required to do the whole work will be $f, or 4 T 4 r . 
 
 2. A cistern can be filled by a water-pipe in 30 min. 
 and emptied by a waste-pipe in 20 min. If the cistern is 
 full and both pipes are opened, in how many minutes will 
 the cistern be emptied ? 
 
 Solution. In 1 min. the waste-pipe empties fa of the cistern. 
 In 1 min. the water-pipe fills fa of the cistern. 
 When both are open fa — fa, or fa of the cistern, is emptied in 
 1 min. 
 
 Therefore, the cistern will be emptied in 60 min. 
 
202 PROBLEMS. 
 
 3. If A can mow a certain meadow in 4 days, and B in 
 
 3 days, how long will it take both together ? 
 
 4. If A can lay a certain wall in 4£ days, and B in 5 J 
 days, how long will it take both together ? 
 
 5. If one pipe will fill a cistern in 4 J hr., and another 
 pipe in 3£ hr., how long will it take both together to fill 
 the cistern ? 
 
 6. If A can go from Boston to Albany in 9 J hr., and B 
 from Albany to Boston in 11^- hr., and they start at the 
 same time, in how many hours will they meet ? 
 
 7. If it takes A working alone 4 days, B 3 days, and 
 C 4£ days to do a piece of work, how long will it take to 
 do the work if all three work together ? 
 
 8. A can mow | of a field in 3 days ; and B £ of it in 
 
 4 days. How long will it take both together to mow the 
 field? 
 
 9. One pipe can fill a cistern half full in £ of an hour, 
 and another can fill it three quarters full in £ an hour. 
 How long will it take both pipes together to fill the cis- 
 tern? 
 
 10. A pipe can fill a cistern one third full in £ of an 
 hour ; a waste-pipe can empty one fourth of the cistern in 
 20 minutes. If both pipes are opened, in what time will 
 the cistern be filled ? 
 
 11. A cistern that will hold 100 gallons can be filled 
 by a pipe in 25 minutes and emptied by a waste-pipe in 
 45 minutes. If the cistern is empty and both pipes are 
 opened, how long will it take to fill the cistern, and how 
 much water will be wasted ? 
 
 12. If water runs into a cistern by one pipe at the rate 
 of 2 gal. in 3 min., by another at the rate of 5 gal. in 
 
 4 min., and runs out by a third at the rate of 4 gal. in 
 
 5 min., how long will it take to gain 71 gal. in the cis- 
 tern ? 
 
PROBLEMS. 203 
 
 13. A can do a piece of work in 6 days, and B can do it 
 in 7 days. If they work together 2 days, and A then 
 leaves, how long will it take B to finish the work ? 
 
 14. A cistern that will hold 200 gal. has two pipes ; one 
 will supply 0.15 gal. a second, the other If qt. a second. 
 If the first is turned on for 10 minutes and afterwards both 
 run together, in what time will the cistern be filled ? 
 
 15. A and B together can do a piece of work in 15 days. 
 After working together 6 days, A leaves and B finishes the 
 work in 30 days more. In how many days can each alone 
 do the work ? 
 
 16. A and B together can do a piece of work in 12 days. 
 After working together 9 days, however, they call in C to 
 help them, and the three finish the work in 2 days. In 
 how many days can C alone do the work ? 
 
 17. A and B together can do a piece of work in 2\ days ; 
 A and C in 3-J- days ; B and C in 3f days. How long will 
 it take the three working together to do the work, and how 
 long will it take each alone ? 
 
 Note. By working two days each they can do — + ^r + 7^ of the 
 
 2f o-j- of- 
 
 work, that is, f + T % + T \ or || of the work. Hence, by working 
 
 one day each, they can do \ of f § , or f $ of the work. 
 
 In one day A can do §£ — T \ of the work. 
 
 18. A and B together can do a piece of work in 48 days ; 
 l A and C together in 30 days ; B and C together in 26f 
 
 days. How long will it take each alone to do the work ? 
 
 19. A cistern has three pipes. The first and second will 
 fill it in 1 hr. 10 min. ; the first and third in 1 hr. 24 min. ; 
 the second and third in 2 hr. 20 min. How long will it 
 take each alone to fill the cistern ? 
 
 20. A, B, and C together can do a piece of work in 10 
 days ; A and B together in 12 days ; B and C together in 
 20 days. How long will it take each alone to do the work ? 
 
204 PROBLEMS. 
 
 Rate and Time Problems. 
 Exercise 94. 
 
 1. A train travels 24 miles in 0.8 of an hour. Find 
 its rate per hour. 
 
 Solution. If the question had read, a train travels 70 mi. in 2 hr., 
 its rate per hour would be found by dividing the whole distance, 
 70 mi., by 2. The application of the same method to this question 
 gives 24 mi. -f 0.8, or 30 mi., for the rate per hour. 
 
 2. A train runs from New York to Philadelphia, 90 
 miles, in 1 hr. 33 min. What is its rate per hour ? 
 
 Solution. 1 hr. 33 min. = 1££ hr. Therefore, the rate per hour 
 is 90 mi. -f 1£$, or 58 & mi. Hence, 
 
 371. To Find the Rate when the Distance and the 
 Time are Known, 
 
 Divide the distance by the number of units of time. 
 
 3. A train runs from New York to Philadelphia, 90 
 miles, in 2 hr. 5 min. What is its rate per hour ? 
 
 4. Winlock, in 1869, found that electricity went 
 through 7200 miles of wire in §- of a second. What 
 was its rate per second ? 
 
 5. If the time required for a signal to pass through 
 the cable from Brest to Duxbury, 3799 miles, is 0.816 of 
 a second, what is the rate per second ? 
 
 6. If the report of a gun 1\ miles distant is heard 
 in 5f seconds after the flash is seen, what is the velocity of 
 sound in feet per second ? 
 
 7. If a man walks 3 J miles in 46 minutes, what is 
 his rate per hour ? 
 
 8. If a horse goes 48 miles in 10 hr. 40 min., what is 
 his average rate per hour ? 
 
 9. If a stone on a glacier is carried 95 J feet in 188 
 days, what is its rate in inches per day ? 
 
PROBLEMS. 205 
 
 10. If a horse went 5£ miles in 33 minutes, how long 
 did it take him to go a mile ? 
 
 Solution. 33 min. -r 5£ = 6 min. Hence, 
 
 372. To Find the Rate for a Unit of Distance when 
 the Distance and the Time are Known, 
 
 Divide the time by the number of units of distance. 
 
 11. If a horse can trot | of a mile in 2 J minutes, in 
 what time can he trot a mile ? 
 
 12. If a train runs 18 miles in 39 minutes, how long 
 does it take to run one mile ? 
 
 13. If sound travels 1125 feet a second, how long will it 
 take to travel one mile ? 
 
 14. If a train requires 3 hours to run 104£ miles, find 
 its average time for running a mile. 
 
 15. If a man cuts 1\ A. of grass in 3 J- days, what part 
 of a day will it take him to cut an acre ? If 10 hr. makes 
 a day, what part of an acre will he cut in an hour ? 
 
 16. If a mower cuts 3^ square rods in -J of an hour, how 
 many acres will he cut in a day of 10 hours ? 
 
 17. If a fountain yields 117^- gallons of Water in f of 
 an hour, at what rate per hour is the water flowing ? 
 
 18. If a merchant's profits are $3147 in 1\ months, 
 what will be his profits at the same rate for a year ? 
 
 19. If a wheel turns 17° 30' in 35 minutes, in how many 
 hours does it make a complete revolution ? 
 
 20. If a man's expenditures are $4358 in 13-J- months, 
 what is his yearly rate of expenditure ? 
 
 21. If a cistern loses by leakage 7 gal. 1 pt. in 49 hr. 
 40 min., what is its hourly rate of loss ? 
 
 22. If a man travels 3f miles in 7£ minutes, how many 
 miles will he travel in 50 minutes ? How long will it take 
 him to travel 50 miles ? 
 
206 PROBLEMS. 
 
 Clock Problems. 
 Exercise 95. 
 
 1 . At what time between 5 and 6 o'clock do the hour 
 and minute hands of a clock coincide? 
 
 Solution. Since in one hour the hour hand moves through 5 
 minute-spaces, and the minute hand through 60 minute-spaces, the 
 minute hand moves 12 times as fast as the hour hand, and in moving 
 through 12 minute-spaces gains 11 minute-spaces. 
 
 When the hour hand is at V, the minute hand, being at XII, is 25 
 minute-spaces behind. Since to gain 11 minute-spaces the minute 
 hand must move through 12 minute-spaces, to gain 1 minute-space the 
 minute hand must pass through T f of 1 minute-space, and to gain 25 
 minute-spaces, it must pass through 25 X {f, or 27 T 3 r minute-spaces. 
 
 Hence, the hands coincide when the minute hand has moved 
 through 27 T 3 T minute-spaces ; that is, at 27 T 3 T min. after 5 o'clock. 
 
 2. At what time between 10 and 11 o'clock do the hour 
 and minute hands of a watch coincide ? 
 
 3. At what time between 1 and 2 o'clock do the hour 
 and minute hands of a clock coincide ? 
 
 4. At what time between 8 and 9 o'clock are the hands 
 of a clock exactly opposite each other ? 
 
 5. At what time between 11 and 12 o'clock are the hands 
 of a clock exactly opposite each other ? 
 
 6. At what time between 4 and 5 o'clock are the hands 
 of a clock exactly opposite each other ? 
 
 7. At what time between 2 and 3 o'clock do the hands 
 of a clock make right angles with each other ? 
 
 8. At what times between 6 and 7 o'clock do the hands 
 of a watch make right angles with each other ? 
 
 9. At what time between 7 and 8 o'clock do the hands 
 of a watch make an angle of 120° with each other ? 
 
 10. At what time between 12 and 1 o'clock do the hands 
 of a watch make an angle of 60° with each other ? 
 
PROBLEMS. 
 
 207 
 
 Bills. 
 
 373. Bills. A bill is a written statement of goods sold, 
 or services rendered, giving the price of each item and the 
 total cost, as well as the date of each transaction, and the 
 names of the parties concerned. 
 
 The party who owes is called the Debtor, and the party 
 to whom a debt is owed is called the Creditor. 
 
 Mr. George Brown, 
 
 (Specimen of a Bill.) 
 
 Boston, Mass. , March 9, 1896. 
 
 Bought of JAMES BATES. 
 
 1896 
 
 
 
 
 
 
 Jan. 
 
 15 
 
 10 lb. Coffee 
 
 @ 35/ 
 
 $3 
 
 50 
 
 
 22 
 
 11 lb. Lard 
 
 @ 9? 
 
 
 
 99 
 
 Feb. 
 
 5 
 
 25 lb. Sugar 
 
 @ 5? 
 
 1 
 
 25 
 
 
 12 
 
 2 lb. Tea 
 
 @ 65/ 
 
 1 
 
 $7 
 
 30 
 04 
 
 (Specimen of a Receipted Bill.) 
 
 Boston, Mass., March 17, 1896. 
 Mr. John Jones, 
 
 To JAMES BROWN, Dr. 
 
 1896 
 
 
 
 
 
 
 
 
 Jan. 
 
 22 
 
 To 40 t. Coal 
 
 @ $4-75 
 
 $190 
 
 00 
 
 
 
 
 29 
 
 To 20 cd. Wood 
 
 @ 3.25 
 
 65 
 
 00 
 
 
 
 
 $255 
 
 00 
 
 
 
 Cr. 
 
 
 
 
 
 
 Jan. 
 
 29 
 
 By JfO bbl. Apples 
 
 @ $3.50 
 
 140 
 
 00 
 
 
 
 Feb. 
 
 10 
 
 By 50 bu. Potatoes 
 
 @ 0.80 
 
 40 
 
 00 
 
 180 
 
 00 
 
 
 
 Balance due 
 
 
 
 $75 
 
 00 
 
 1896, March 26. 
 
 Received payment, 
 
 James Brown. 
 
208 PROBLEMS. 
 
 Exercise 96. 
 
 Make out receipted bills for the following accounts, sup- 
 plying dates : 
 
 1 . James Hardy bought of C. H. Mills 275 bbl. flour, at 
 $6.75 ; 324 bbl. flour, at $6.25 ; 300 bu. potatoes, at 48 
 cents ; 1578 lb. butter, at 32 cents ; 2000 bbl. apples, at 
 $1.25 ; a car-load (20,000 lb.) of oats, at 42 cents a bushel ; 
 a car-load (28,575 lb.) of corn, at 55 cents a bushel. 
 
 2. James Harlow bought of John Pike 12 bales, 480 lb. 
 each, Texas cotton, at 9£ cents ; 7 bales, 502 lb. each, 
 upland, at 10 J cents ; 3 bales, 492 lb. each, low middling, 
 at 9} cents ; 18 bales, 490 lb. each, good ordinary, at 
 9 cents. 
 
 3. Richard Rowe bought of John Doe 125 lb. sugar, at 
 5 cents ; 1 bag coffee, 115 lb., at 32 cents a pound ; 25 gal. 
 molasses, at 38 cents ; 8 lb. Japan tea, at 92 cents ; 28 lb. 
 crackers, at 8 cents ; 2 bbl. flour, at $7.50. 
 
 4. William Litchfield bought of John Garvin 8 bags 
 cracked corn, at 75 cents ; 4 bags oats, at 80 cents ; 16 lb. 
 sweet potatoes, at 3£ cents ; 2 bu. potatoes, at $1.10 ; 
 100 lb. wire nails, at 2£ cents ; 5 lb. coffee, at 35 cents. 
 
 5. Amos Tuck sold to Aaron Young 11 lb. ham at 
 15 cents, 22 lb. beefsteak at 24 cents, 18 lb. mutton at 
 13 cents, 14 lb. veal at 11 cents ; and took in exchange 
 5 dozen eggs at 18 cents, 15 lb. butter at 26 cents, 9 bu. 
 potatoes at 40 cents, and 2 bbl. apples at $1.35. 
 
 6. W. G. Fernald sold to John Waldron 35 lb. sugar at 
 5 cents, 18 lb. coffee at 35 cents, 20 lb. rice at 8 cents, 
 4 tons hay at $15.75, 3 cords pine wood at $2.75, 4 cords 
 hard wood at $3.50, 8 tons furnace coal at $6.75, 5 tons 
 stove coal at $7.25, 8 rolls wall paper at 35 cents ; and 
 took in exchange 25 bbl. apples at $1.15, 32 bu. pears at 
 60 cents, and 42 bu. blueberries at 8 cents a quart. 
 
PROBLEMS. 209 
 
 7. C. A. Colton bought of Green, Fisk & Co. 4 doz. 
 No. 7 teakettles, at 85 cents each ; 2 safety ash barrels, 
 at $2.50 ; 3 doz. common scrapers, at 50 cents a dozen ; 
 8 eagle shovels, at 10 cents ; -J- doz. 8 by 12 black registers, 
 at $ 1.50 each ; -J doz. spice boxes, at 55 cents each ; -J- doz. 
 14-qt. dish pans, at $ 6.00 a dozen ; 2 doz. common stove 
 lifters, at 50 cents a dozen ; -J- doz. 12 by 14 drip pans, 
 at $4.00 a dozen ; -J- gross retinned teaspoons, at 25 cents 
 a dozen ; 1 doz. ash sifters at $1.00 each. 
 
 8. E. M. Hanson bought of W. F. Fox & Co. 2 bbl. 
 flour, at $5.75 ; £ bbl. fine sugar, 153 lb., at $4.81 a 
 cwt.; 25 lb. coffee, at 33 cents; 3 lb. Oolong tea, at 
 50 cents ; 15 pint bottles olives, at 25 cents ; 2 boxes 
 graham wafers, at 40 cents ; \ doz. cans tomatoes, at $1.20 
 a dozen ; \ doz. cans J. H. F. peaches, at $3.50 a dozen ; 
 4 Ferris hams, 48 lb., at 12^- cents a pound ; 6 strips Ferris 
 bacon, 19 lb. 9 oz., at 13 cents a pound; 3 lb. rice, at 9 
 cents ; 3 lb. tapioca, at 5 cents ; 40 lb. rye meal, at 2-J- 
 cents ; 5 lb. boneless codfish, at 14 cents ; \ doz. cans 
 plums, at $2.90 a dozen. 
 
 9. G. B. Cook bought of Gray, Higginson & Co. 1 
 No. 8-20 Glenwood B range, at $35.00 ; 1 No. 12 Rock- 
 ford heater, at $20.00 ; 4 lb. Eng. stovepipe, at 15 cents ; 
 3 lb. Rus. stovepipe, at 25 cents ; 8 lb. sheet zinc, at 
 8- cents ; 1 stove board, at $2.00 ; 1 set kitchen knives 
 and forks, at $1.50 ; 2 wash tubs, at 85 cents ; 1 wash- 
 board, at 25 cents; 1 set Mrs. Potts' nickel sad-irons, at 
 75 cents ; 2 milk cans, at 35 cents ; 1 hand lamp com- 
 plete, at 30 cents ; 1 stand lamp, at $3.50 ; 1 granite iron 
 washbowl, at 50 cents ; 1 tea canister and 1 coffee can- 
 ister, at 20 cents each ; 1 carving knife and fork, at $2.00 ; 
 1 corn popper, at 25 cents ; 1 rolling-pin, at 20 cents ; 2 
 8-qt. porcelain kettles, at 70 cents ; 1 granite iron coffee- 
 pot, at 75 cents. 
 
CHAPTER X. 
 
 374. 
 
 METBIC AND COMMON SYSTEMS. 
 Table of Equivalents. 
 
 Length. 
 
 Meter 
 Kilometer 
 
 _ ( 39.37043 in., or 
 ~ t 1.09362 yd. 
 = 0.62138 mi. 
 
 Inch =2.63998"" 
 Yard = 0.91439™. 
 Mile = 1.60933k" 
 
 Surface. 
 
 ( 1550.031 sq. in., or 
 Sq ' meter= I 1.19601 sq. yd. 
 Hektar = 2.47110 A. 
 
 Sq. inch = 6.45148v m 
 Sq. yard = 0.8361 1*"'. 
 Acre = 0.40468 h ». 
 
 Volume. 
 
 Cu. centimeter ss 0.06103 cu. in. 
 Cu. meter = 1.30799 cu. yd. 
 
 Ster = 0.27590 cd. 
 
 Cu. inch = 16.38662 ccm . 
 Cu. yard = 0.76453<*">. 
 Cord = 3.62446'*. 
 
 Capacity. 
 
 Liter = 
 
 1.05671 liquid qt., or 
 0.90810 dry qt. 
 
 Liquid quart = 0.94633 1 . 
 Dry quart = 1.10119 1 . 
 
 Weight. 
 
 Milligram = 0.016432 gr. 
 Gram = 15.43235 gr. 
 
 Kilogram = 2.20462 lb. av. 
 Metric ton = 2204.62 lb. av. 
 
 Grain = 0.06480k. 
 
 Ounce av. = 28.349548. 
 Ounce troy = 31.103508. 
 Pound av. = 0.45359 k *. 
 
METRIC AND COMMON SYSTEMS. 
 
 211 
 
 375. 
 
 Table of Approximate Equivalents. 
 
 Meter 
 
 = 1.1 yd. 
 
 Yard 
 
 = 0.9 m . 
 
 Kilometer 
 
 = I mi. 
 
 Mile 
 
 = 1.6 k m. 
 
 Sq. meter 
 Hektar 
 
 = li sq. yd. 
 = 21 A. 
 
 Sq. yard 
 Acre 
 
 = 2ha # 
 
 Cu. centimetei 
 
 = t l cu. in. 
 
 Cu. inch 
 
 — \Qcem m 
 
 Cu. meter 
 Ster 
 
 = 1.3 cu. yd. 
 
 = A Cd. 
 
 Cu. yard 
 Cord 
 
 z= 1 ocbm 
 T3 
 
 = 8| sters. 
 
 Liter 
 Hektoliter 
 
 -/ixViiq-qt^or 
 
 l T Vdryqt. 
 = 2|bu. 
 
 Liquid quart 
 Dry quart 
 Bushel 
 
 = || liter. 
 = 1^ liters. 
 
 Gram 
 
 = 15| gr. 
 
 Pound av. 
 
 -**■ 
 
 Kilogram 
 
 = 2i lb. av. 
 
 Pound troy 
 
 = #* 
 
 Note. These tables are given for reference, and are not to be 
 committed to memory. It may be well, however, to remember that 
 a meter is 39.37 in.; a liter is 1.0567 liquid qt., or 0.908 dry qt.; 
 and that a gram is 15.432 gr.; as by these equivalents all measures 
 expressed in one system may be converted into the corresponding 
 measures of the other system. 
 
 Exercise 97. 
 
 In the following problems take the equivalents from the Table of 
 Equivalents, using three places of decimals, or four when the first 
 decimal figure is zero, and add one to the last decimal figure when the 
 next is 5 or more. 
 
 1. Reduce 25.55 kg to pounds avoirdupois. 
 
 Solution. Since l k e = 2.205 lb., 26.55 k & = 25.55 X 2.205 lb., or 
 56.33775 lb., that is, 56 lb. 5.4 oz. 
 
 2. Reduce 5 sq. yd. 6 sq. ft. 108 sq. in. to square meters. 
 
 Solution. 5 sq. yd. 6 sq. ft. 108 sq. in. = 5.75 sq. yd. Since 
 I sq. yd. = 0.8361", 5.75 sq . y d. = 5.75 X 0.836* m ; that, is 4.8074m. 
 
 3. Reduce 24 gal. to liters. 
 
 4. Reduce 10 lb. troy to kilograms. 
 
 5. Reduce 50.5 cu. yd. to cubic meters. 
 
212 METRIC AND COMMON SYSTEMS. 
 
 6. Reduce 69^^ mi. to kilometers. 
 
 7. Reduce 12 A. 12 sq. rd. to hektars. 
 
 8. Reduce 10 cd. to sters. 
 
 9. Reduce 4 cwt. 24 lb. to kilograms. 
 
 10. Reduce 25 bu. 2 pk. to hektoliters. 
 
 11. Express 15 km in the common system. 
 
 12. Express 3^ in the common system. 
 
 13. Express 12.125 cbm in the common system. 
 
 14. Express 101. 25 1 in the common system. 
 
 15. Reduce 20.25 M to liquid quarts ; to dry quarts. 
 
 16. Express 5 kg in troy weight. 
 
 17. Express 24 8t in the common system. 
 
 18. Express 62.5 qm in the common system. 
 
 19. Express 1001 kg in avoirdupois weight. 
 
 20. Express 42 A. 100 sq. rd. in the metric system. 
 
 21. Find in acres, etc., the area of a rectangular field 
 if it is 100 m long and 75 m broad. 
 
 22. Find the number of cubic meters in a rectangular 
 box 2 yd. long, 3 ft. wide, and 2£ ft. deep. 
 
 23. Find the number of cubic yards in a rectangular 
 box 2 m long, 75 cm wide, and 50 cm deep. 
 
 24. If a man walks 75 m a minute, what is his rate in 
 miles per hour? 
 
 25. If a cubic centimeter of cast iron weighs 7.113 g , 
 how many pounds does a cubic foot weigh ? 
 
 26. How many steps 2 ft. 6 in. long will a man take in 
 walking a kilometer ? 
 
 27. Find the value of a carboy (17 qt.) of sulphuric 
 acid, specific gravity 1.841, at 4| cents a kilogram. 
 
 28. Find the value of a carboy (17^) of nitric acid, 
 specific gravity 1.451, at 15 cents a pound. 
 
 29. If the specific gravity of sea water is 1.026, and that 
 of olive oil is 0.915, what is the weight of a hektoliter of 
 each in pounds and in kilograms? 
 
METRIC AND COMMON SYSTEMS. 213 
 
 30. Find the weight in pounds and in kilograms of 3l£ 
 gal. of the best alcohol, specific gravity 0.792. 
 
 31. Find the weight in pounds and in kilograms of the 
 air, specific gravity 0.00129206, in a room 7 m long, 5 m wide, 
 and 3.5 m high. 
 
 32. Find the weight in pounds and in kilograms of the 
 air, specific gravity 0.00129206, in a room 23 ft. long, 16 
 ft. wide, and 10 ft. high. 
 
 33. What is the lifting force in kilograms and in pounds 
 of a balloon that weighs 2 k s, and contains 10,000* of hydro- 
 gen gas, specific gravity 0.00008929 ? 
 
 Note. The lifting force is the weight of the air displaced by the 
 balloon diminished by the weight of the hydrogen and the balloon. 
 
 34. What is the value at $4.50 a cord of a pile of wood 
 1.2 m wide, 7 m long, and 2 m high ? 
 
 35. How many miles will a train run in 1 hr. 28 min. 
 21 sec, at the rate of 50 km an hour ? 
 
 36. Find the time it takes a train to run 31 mi. 180 yd. 
 at the rate of 1 min. 25 sec. per kilometer. 
 
 37. What is the weight of 12 cu. yd. 16 cu. ft. 720 cu. 
 in. of earth, if a cubic meter weighs 1 t. 17 cwt.? 
 
 38. Find the weight in grams of a liter of mercury, if a 
 cubic inch weighs 0.4925 of a pound avoirdupois. 
 
 39. How many yards of cloth, at $3.12^ a meter, should 
 be given in exchange for 15 m at $2.75 a yard ? 
 
 40. If a wine merchant buys 3 W of wine for 1600 francs, 
 what does a gallon cost him in United States money, if 
 25 francs are equivalent to $4,825 ? 
 
 41. A mill wheel is turned by a stream of water running 
 at the rate of a yard a second in a channel 5 ft. wide and 
 9 in. deep. Find the weight in metric tons and in tons 
 avoirdupois of the water supplied in 12 hr., if a cubic foot 
 of water weighs 1000 oz. 
 
214 METRIC AND COMMON SYSTEMS. 
 
 Exercise 98. 
 
 In the following problems take the equivalents from the Table of 
 Approximate Equivalents, and use *f- for 3.1410. 
 
 1. When water is heated from the freezing point to 
 the boiling point it expands J± in volume. Find in kilo- 
 grams the weight of a cubic foot of water at the freezing 
 point and at the boiling point. 
 
 2. A circular plate of lead 8 in. in diameter and 2 in. 
 thick is changed without loss into spherical shot each 
 1.25 mm in radius. How many shot does it make ? 
 
 3. If | of a yard of velvet costs $3, how many francs 
 will f of a meter cost ? 
 
 4. Water expands ^ in freezing, and a floating body 
 displaces an amount of water equal in weight to the body. 
 What is the volume in cubic meters and the weight in 
 metric tons of an iceberg floating in the ocean, if the spe- 
 cific gravity of sea water is 1.026, and the part of the iceberg 
 above the water is a rectangular solid 200 ft. long, 60 ft. 
 wide, and 12 ft. high? 
 
 5. How many hektoliters of wheat will a rectangular 
 bin hold 14 ft. long, 10 ft. wide, and 6 ft. high ? 
 
 6. How many hektoliters of water will a cylindrical 
 stand-pipe hold 70 ft. high and 35 ft. in diameter ? 
 
 7. How many bushels of wheat will a rectangular bin 
 hold 4 m long, 3 m wide, and 2.5™ high ? 
 
 8. How many gallons of water in a well 1.2 m in diam- 
 eter, if the depth of the water is 2 m ? 
 
 9. If 1 lb. troy of silver is worth $6.20, what is the 
 value of a lump of silver weighing 2.64 kg ? 
 
 10. A pound of brass contains 3.3 cu. in., and a pound 
 of antimony contains 6.27 cu. in. Find the weight in kilo- 
 grams of a mass of 313^ cu. in. that contains equal volumes 
 of the two metals. 
 
METRIC AND COMMON SYSTEMS. 215 
 
 11. If 2 cu. in. of mercury weighs 1 lb., and 100 cu. in. 
 of air weighs 31 gr., how many kilometers high must a 
 column of air be to weigh as much as a column of mercury 
 29.388 in. high, standing on a base of the same area ? 
 
 12. If a sprinter can run 0.00645 of a mile in 1.08 sec, 
 how many meters can he run in a second ? How many 
 seconds will it take him to run 100 m ? 
 
 13. Two trains going in opposite directions pass each 
 other in 3^ sec. If their lengths are 260 ft. and 200 ft., 
 respectively, and the first train is going at the rate of 80 km 
 an hour, what is the rate of the second train ? 
 
 14. If a cubic inch of water converted into steam will 
 produce mechanical force sufficient to raise a weight of 
 2200 lb. one foot high, how many meters high would the 
 conversion into steam of a cubic centimeter of water raise 
 a weight of one kilogram ? 
 
 15. If a man takes 100 steps of 0.7 m each in a minute, 
 how long will it take him to walk a distance of 28 km ? 
 
 16. A lot of land containing 63 a 21 ca , worth $0.35 a 
 square yard, is exchanged for a second lot containing V 3 * 
 5 a . What is the cost per ar of the second lot ? 
 
 17. Light travels in 8 min. 13 sec. from the sun to the 
 earth, 153,624,000 km . What is the velocity of light in 
 miles per second ? 
 
 18. How many square feet of surface has a rectangular 
 table that is l.l m long and 0.85 m wide ? 
 
 19. How many square meters of surface has a circular 
 table that is 3£ ft. in diameter ? 
 
 20. If sound travels 340 m a second, how many feet dis- 
 tant is a cannon from a man who hears the report 13 sec. 
 after he sees the flash ? 
 
 21. How many square meters of zinc will be required 
 to line a rectangular cistern, open at the top, 12 ft. long, 
 10 ft. wide, and 8 ft. deep ? 
 
216 METRIC AND COMMON SYSTEMS. 
 
 22. A rectangular tank is 3 m long, 2£ m wide, and l£ m 
 high, external measurement. If its sides are 0.1 m thick, 
 how many gallons of water will the tank hold ? 
 
 23. If a cube of pine wood 11.2 cm on an edge weighs 
 2 lb., what is the specific gravity of the pine ? 
 
 24. Find in kilograms the weight of water a cubical cis- 
 tern will hold, 6 ft. on an edge. 
 
 25. Rain has fallen to the depth of half an inch. How 
 many cubic meters of water lias fallen on an acre of land ? 
 
 26. How many centimeters will the water sink in a 
 cylindrical cistern 7 ft. in diameter, if 310 gallons of water 
 is pumped out ? 
 
 27. How many square yards of tin are required to cover 
 the roof of a hemispherical dome 12 m in diameter ? 
 
 28. If a cubic inch of iron weighs 4£ oz., what is the 
 weight in kilograms of an iron ball 10 cm in diameter ? 
 
 29. If a cubic inch of lead weighs 7 oz., what is the 
 weight in kilograms of a lead pipe 3 m long, 6 cm in external 
 diameter, if the pipe is l cm thick ? 
 
 30. Find the cost at $7.25 per meter of building a wall 
 around a rectangular garden 90 ft. long and 55 ft. wide. 
 
 31. The minute hand of a clock is 0.5 m long. How 
 many feet does its point move in an hour ? 
 
 32. A spherical shot 3 in. in diameter is melted and then 
 cast into a cylinder 9 cm in diameter. What is the height 
 in centimeters of this cylinder ? 
 
 33. What is the cost at $18 per 1000 ft. board measure 
 of 4 beams, each 4.5 m long, 7.5 cm wide, and 5 cm thick ? 
 
 34. The radius of a cylindrical roller is 0.4 m and its 
 length is 2.15 m . Find its volume in cubic feet. 
 
 35. A cylindrical cistern, the circumference of whose 
 base is 2.2 m , and whose depth is 2.1 m , is four fifths filled 
 with water. Find in gallons the volume of the water, and 
 in pounds the weight of the water. 
 
CHAPTER XI. 
 
 RATIO AND PROPORTION. 
 
 376. Ratio. The relative magnitude of two numbers 
 is called their ratio, when expressed by the fraction that 
 has the first number for its numerator and the second 
 number for its denominator. 
 
 Thus, the ratio of 2 to 3 is expressed by the fraction f . 
 
 377. Antecedent and Consequent. The terms of this 
 fraction are called the terms of the ratio. The first term 
 of a ratio is called the antecedent; the second term, the 
 consequent. 
 
 Thus, in the ratio of 2 to 3, commonly written 2 :3, the first term 
 2 is the antecedent, and the second term 3 is the consequent. 
 
 378. If both terms of a ratio are multiplied or both 
 divided by the same number, the value of the ratio is not 
 changed. 
 
 Thus, if the ratio 2\ : 3^ is multiplied by 6, the resulting ratio is 
 
 2 1 
 15 : 20, and the ratio 2\ : 3| is equal to 15 : 20 ; for ;r| = ]§. Since 
 
 |f reduced to its lowest terms = £, the simplest expression for the 
 ratio of 2| : S\ is 3 : 4. 
 
 379. If the numerator and denominator of a fraction 
 are interchanged, the fraction is said to be inverted; like- 
 wise, if the antecedent and consequent of a ratio are inter- 
 changed, the resulting ratio is the inverse of the given ratio. 
 
 Thus, if the fraction £ is inverted the resulting fraction is f ; and 
 the inverse of the ratio 3 : 4 is 4 : 3. 
 
218 RATIO AND PROPORTION. 
 
 380. If two quantities are expressed in the same unity 
 their ratio is the same as the ratio of the two numbers by 
 which they are expressed. 
 
 Thus, the quantity $5 is the same fraction of $11 as 5 is of 11 ; and, 
 therefore, the ratio $5 : $11 equals the ratio 6:11. 
 
 381. Since ratio is simply relative magnitude, two quan- 
 tities different in kind cannot form the terms of a ratio ; 
 and two quantities the same in kind must be expressed in 
 a common unit before they can form the terms of a ratio. 
 
 Thus, no ratio exists between 5 tons and 30 days ; and the ratio of 
 5 tons to 3000 pounds can be expressed only when both quantities are 
 written as tons or as pounds. 
 
 382. Since ratios are mere numbers, they may be com- 
 pared. 
 
 383. Example. Which is the greater ratio, 5 : 7 or 
 12 : 18 ? 
 
 Solution. 5 : 7 = f, and 12 : 18 = jf = § . 
 
 Now, » = #, and$'= \\. 
 
 As \\ is greater than \\ , 
 
 the ratio 5 : 7 is greater than the ratio 12 : 18. 
 
 Exercise 99. 
 Which is the greater ratio : 
 
 1. 5:8 or 6:9? 5. 10 cwt. : 15 cwt. or $7 : $9 ? 
 
 2. 7 : 10 or 9 : 12 ? 6. 5 dy. : 7 dy. or 8 ft. : 11 ft.? 
 3.8:9 or 10 : 12 ? 7. 9 yd. : 6 yd. or 5 : 3 ? 
 
 4. 6 : 12 or 8 : 14 ? 8. § lb. : \ lb. or £ yd. : f yd.? 
 
 9. Find the ratio of 3 dry quarts to 2 pecks. 
 
 10. Find the ratio of 2500 lb. to 1 ton. 
 
 11. Find the ratio of a rectangular field 16 rd. long, 14 
 rd. wide to a rectangular field 14 rd. long, 12 rd. wide. 
 
 12. Find the ratio of a circle 1 in. in diameter to a 
 circle 1 in. in radius. 
 
RATIO AND PROPORTION. 219 
 
 384. When two ratios are equal the four terms are said 
 to be in proportion, and are called proportionals. 
 
 Thus, 5, 3, 15, 9 are proportionals ; for f = -L 5 -. 
 
 385. Proportion. An expression of equality between 
 two ratios is called a proportion. 
 
 A proportion is written by putting the sign of equality 
 or a double colon between the ratios. 
 
 Thus, 5 : 3 = 15 : 9, or 5 : 3 :: 15 : 9, means, and is read, the ratio of 
 5 to 3 is equal to the ratio of 15 to 9. 
 
 386. Means and Extremes. The first and last terms 
 of a proportion are called the extremes, and the two middle 
 terms are called the means. 
 
 387. Test of a Proportion. When four numbers are in 
 proportion, the product of the extremes is equal to the product 
 of the means. 
 
 This is seen to be true by expressing the ratios in the form of frac- 
 tions, and multiplying both by the product of the denominators. 
 
 Thus, the proportion 5 : 3 = 15 : 9 may be written f = - 1 /- ; and if 
 both are multiplied by 3 X 9, the result is 5 X 9 = 3 X 15. 
 
 388. Either extreme, therefore, is equal to the product 
 of the means divided by the other extreme ; and either 
 mean is equal to the product of the extremes divided by 
 the other mean. Hence, if three terms of a proportion are 
 given, the fourth may be found. 
 
 389. Examples. 1. Find the missing term of the pro- 
 portion 18 : 32 = 45:?. 
 
 32 X 45 _ 
 
 Solution. 
 
 18 
 
 2. Find the missing term of 20 : 24 = ? : 30. 
 
 Solution. — — — = 25. 
 
 24 
 
220 RATIO AND PROPORTION. 
 
 Exercise 100. 
 Find the missing term of : 
 
 1. 24: 18:: 16:?. 6. 18:?:: 32: 45. 
 
 2. 35:?:: 15: 21. 7. ?:12::5:18. 
 
 3. 45: 40::?: 32. 8. 8:17 ::?:119. 
 
 4. 30: 27:: 40:?. 9. 9: 16:: 12:?. 
 
 5. ?:36::4 : 3. 10. 17: 3::?: 12. 
 
 390. Rule of Three. When three terms of a proportion 
 are given, the method of finding the fourth term is called 
 the Rule of Three. 
 
 It is usual to arrange the quantities (that is, to state the 
 question), so that the quantity required for the answer 
 may be the fourth term. Hence, the quantity which come- 
 sponds to that of the required answer is the third term. 
 
 391. Examples. 1. If 5 tons of hay cost $87.50, what 
 will 21 tons cost ? 
 
 Solution. Since cost is required, $87.50 is the third term. 
 
 Since 21 tons will cost more than 5 tons, 21 tons is the second terra 
 and 5 tons the first term. 
 
 That is, 6 t. : 21 t. :: $87.50 : ?. 
 
 A difficulty presents itself here, inasmuch as no meaning can be 
 given to the product of the means ($87.50 multiplied by 21 1.). Since, 
 however, the ratio of 5 t. : 21 t. = the ratio of 5 : 21, the ratio 6:21 
 may be substituted for the ratio 6 t. : 21 t. 
 
 Then, 5:21 :: $87.50: ?. 
 
 21 X $87 50 
 Therefore, the fourth term required is — '- — , or $367.60. 
 
 2. When a post 11.5 ft. high casts a shadow on level 
 ground 17.4 ft. long, a neighboring steeple casts a shadow 
 63.7 yd. long. How high is the steeple ? 
 
 Solution. Since height is required, the height 11.5 ft. is the third 
 term. 
 
RATIO AND PROPORTION. 221 
 
 Since the shadow of the steeple is the longer, the height of the 
 steeple must be the greater ; therefore, the second term must be the 
 greater of the two remaining quantities expressed in the same unit. 
 63.7 yd. = 191.1 ft. Therefore, 
 
 Shadow. Shadow. Height. Height. 
 
 17.4 ft. : 191.1 ft. :: 11.5 ft. : What ? 
 or, 17.4 : 191.1 :: 11.5 ft. : What ? 
 
 Hence, the height of the steeple is '■ — titt 1 \ or 126.3 ft. 
 
 392. To Solve Problems by the Rule of Three, 
 
 Make that quantity which is of the same kind as the re- 
 quired answer the third term. 
 
 Make the numbers by which the two remaining quantities 
 are represented when expressed in the same unit the first 
 and second terms. 
 
 If, from the nature of the question, the answer will be 
 greater than the third term, make the greater of these two 
 numbers the second term ; if the answer will be smaller than 
 the third term, make the smaller of these numbers the second 
 term, and the larger the first term. 
 
 Divide the product of the second and third terms by the 
 first term, and the quotient will be the answer required. 
 
 Exercise 101. 
 
 1. If 24 men can do a piece of work in 14 days, how 
 long will it take 21 men to do it ? 
 
 2. A well is dug in 13 days of 9 hours each. How many 
 days of 10 hours each wonld it have taken ? 
 
 3. A man who steps 2 ft. 5 in. takes 2480 steps in 
 walking a certain distance. How many steps of 2 ft. 7 in. 
 will be required for the same distance ? 
 
 4. If T % of a ton of hay costs $6, what will 7$ cwt. 
 cost, at the same rate ? 
 
222 RATIO AND PROPORTION. 
 
 5. If 42 yd. of carpet 2 ft. 3 in. wide are required for 
 a room, how many yards of carpet 2 ft. 4 in. wide will be 
 required ? 
 
 6. A court was paved with 950 stones, each containing 
 lg sq. ft., and is repaved with 836 stones of a uniform size. 
 Find the surface of each. 
 
 7. If a train, at the rate of *fc of a mile per minute, 
 requires 3£ hours to make a certain distance, how long will 
 it require at the rate of *fo of a mile a minute ? 
 
 8. When a post 4 ft. 8 in. high casts a shadow 7 ft. 
 3 in. long, how long a shadow will a post 11 ft. high cast ? 
 
 9. When a post 5 ft. 7 in. high casts a shadow 8 ft. 5 
 in. long, how high is a steeple that casts a shadow of 202 ft. ? 
 
 10. If 4 men can mow a certain field in 10 hours, how 
 many men will it take to mow it in 5 hours ? 
 
 11. If a tap discharging 4 gal. a minute empties a cis- 
 tern in 3 hours, how long will it take a tap discharging 7 
 gal. a minute to empty it ? 
 
 12. If a pipe discharging 3 gal. 1 pt. a minute fills a tub 
 in 4 min. 20 sec, how long will it take a pipe discharging 
 83 qt. a minute to fill it ? 
 
 13. If both pipes of Ex. 12 discharge at the same time 
 into the tub, how long will it take to fill it ? 
 
 14. How long will it take to fill a cistern of 165 gal. by 
 a pipe that fills one of 120 gal. in 7 min. 16 sec. ? 
 
 15. If a ship sails 1800 mi. in a fortnight, how long will 
 it take to make a voyage of 5000 mi. ? 
 
 16. The wheels of a carriage are 6 ft. 9 in. and 9 ft. 6 
 in., respectively, in circumference. How many times will 
 the larger turn while the smaller turns 3762 times ? 
 
 17. If & of a ship is worth $2167, what is T 7 T of it 
 worth ? 
 
 18. What is the weight of 18 cu. ft. 432 cu. in. of stone, 
 if 10 cu. ft. 864 cu. in. of the stone weighs 14 cwt. 7 lb. ? 
 
RATIO AND PROPORTION. 223 
 
 19. If 280 lb. of flour makes 360 lb. of bread, how many 
 four-pound loaves can be made from 1 cwt. of flour ? 
 
 20. If a column of mercury 27.93 in. high weighs 0.76 
 of a pound, what is the weight of a column of mercury of 
 the same diameter 29.4 in. high ? 
 
 21. How many francs will pay a bill of £100, when 
 £42 10s. 8d. is equivalent to 1090.98 francs ? 
 
 22. What is the weight of a cube of stone 2 ft. 2 in. on 
 an edge, if a cube 1 ft. 4 in. on an edge weighs 537.6 lb. ? 
 
 23. If a square field 50 yd. lOf in. on a side is worth 
 $2710}^, what is a square field 62 yd. 1 ft. on a side 
 worth ? 
 
 24. A gains 4 yd. on B in running 30 yd. How many 
 yards will he gain while B is running 97^ yd. ? 
 
 25. If 10 cu. in. of gold weighs as much as 193 cu. in. 
 of water, how many cubic inches are there in a nugget of 
 gold that weighs as much as a cubic foot of water ? 
 
 26. If a garrison of 1500 men has provisions for 13 
 months, how long will the provisions last if the garrison is 
 reenforced by 700 men ? 
 
 27. If a tree 38 ft. high is represented by a drawing 
 1J in. high, what height on the same scale will represent 
 a house 45 ft. high ? 
 
 28. If a country 630 mi. long is represented on a raised 
 map by a length of 5^ ft., by what height ought a moun- 
 tain of 15,750 ft. be represented on the map ? 
 
 29. A train travels \ of a mile in 18 sec. How many 
 miles an hour does it travel ? 
 
 30. If 4£ t. of coal fill a bin 9 ft. long, 5 ft. broad, 5 ft. 
 high, how many cubic feet are required for the coal of a 
 steamer that carries coal for 3 wk. at 20 t. a day ? 
 
 31. If 2 lb. of rosin are melted with 5 oz. of mutton 
 tallow to make a grafting wax, how many ounces of tallow 
 will 20 oz. of the wax contain ? 
 
224 RATIO AND PROPORTION. 
 
 Compound Proportion. 
 
 393. Compound Ratio. A ratio is said to be com- 
 pounded of two or more given ratios when it is expressed 
 by a fraction that is the product of the fractions repre- 
 senting the given ratios. 
 
 Thus, the ratios 2 : 3 and 7:11 are represented by the fractions \ 
 and T \ ; and the ratio 14 : 33, which is represented by ^f (the product 
 of $ and /y), is said to be compounded of the ratios 2 : 3 and 7:11. 
 
 394. Compound Proportion. A proportion which has 
 one of its ratios a compound ratio is called a compound 
 proportion. 
 
 In stating problems in compound proportion the quantity 
 that corresponds to the answer required is made the third 
 term. Each pair of the remaining quantities is then con- 
 sidered separately with reference to the answer required. 
 
 395. Example. If 4 men mow 15 A. in 5 dy. of 14 hr., 
 in how many days of 13 hr. can 7 men mow 19£ A.? 
 
 As the answer is to be in days, we make 5 dy. the third term. 
 
 I. It will require less days for 7 men to mow 16 A. than for 4 men. 
 Therefore, we make 7 the first term, and 4 the second. 
 
 II. It will require more days for the same number of men to mow 
 19J A. than to mow 15 A. 
 
 Therefore, we make 16 the first term, and 19^ the second. 
 
 III. It will require more days of 13 hr. than of 14 hr. for the same 
 number of men to mow the same number of acres. 
 
 Therefore, we make 13 the first term, and 14 the second. 
 Hence, the statement is 
 
 7:4 
 
 15 : 19.5 : : 5 dy. : ? 
 
 13 : 14. 
 
 Therefore, the fourth term, or the time required, is 
 
 4 X 19.5 X 14 X 5 dy. 
 
 7 X 15 X 13 *' 
 
RATIO AND PROPORTION. 225 
 
 Exercise 102. 
 
 1. In how many days of 8 hr. will 60 men do the 
 same work that 24 men can do in 15 dy. of 10 hr.? 
 
 2. What is the expense of covering a room with drug- 
 get 4 ft. wide, at 91 J cents a yard, if carpet 2 ft. 3 in. wide 
 for the room costs $70.50, at $1.37£ a yard ? 
 
 3. If 4418 tons of iron ore produce $36,190 worth of 
 metal, when iron is at $37.50 a ton, what will be the 
 value of the iron at $47 a ton from 2275 tons of ore ? 
 
 4. If a bar of iron 3£ ft. long, 3 in. wide, and 2f in. 
 thick weighs 93 lb., what will be the weight of a bar 3f ft. 
 long, 4 in. wide, and 2-J- in. thick ? 
 
 5. If 40 bu. of wheat can be grown on the same area as 
 48 bu. of barley, and 28 A. produce 840 bu. of wheat, how 
 mucn barley will 38 A. produce ? 
 
 6. If 18 men can dig a trench 150 ft. long, 6 ft. broad, 
 and 4 ft. 6 in. deep in 12 days, in how many days will 16 
 men dig a trench 210 ft. long, 5 ft. broad, and 4 ft. deep ? 
 
 7. A book of 810 pages, 40 lines to a page, and 60 
 letters to a line, is reprinted in pages of 50 lines, 72 
 letters to a line. How many pages will the new edition 
 contain ? 
 
 8. If 3280 42-lb. shot cost $3000, how many 32-lb. shot 
 can be bought for $4200 ? 
 
 9. What is the rate of wages, if 12 men earn in i0 dy. 
 as much as 9 men earn in 14 dy., at $1.50 a day ? 
 
 10. A rectangular reservoir 15 yd. long and 4 ft. deep 
 holds 32,500 gal. What quantity of water will it hold if 
 its length is increased by 18 ft. and its depth by 1 ft.? 
 
 11. What must be the length of a bar of silver f in. 
 square to weigh the same as a bar of gold £ in. square and 
 6£ in. long, if the weight of a cubic inch of silver to that 
 of a cubic inch of gold is in the ratio 47 : 88 ? 
 
226 RATIO AND PROPORTION. 
 
 12. How far can A, who takes 3.1 ft. each step, walk, 
 while B, who takes 2.3 ft. each step, walks 220 yd., if A 
 takes 7 steps while B takes 11 ? 
 
 13. If 6 hr. are needed to go a given distance at a given 
 rate, how many hours are needed when the distance is dimin- 
 ished by one fourth and the rate increased by one half ? 
 
 14. How many hours a day must 5 men work to mow a 
 field in 8 dy. that 7 men can mow in 6 dy. of 10 hr.? 
 
 15. If a bar of iron 10 ft. 6 J in. long, 3 J in. broad, and 
 3J in. thick weighs 4 cwt. 20.21 lb., what is the length of a 
 bar of iron that weighs a long ton, if its breadth and thick- 
 ness are 4 J in. and 4-J in., respectively ? 
 
 16. If 27 men in 28 dy. of 10 hr. dig a trench 126 yd. 
 long, 2£ yd. broad, 1£ yd. deep, how long a trench 2f yd. 
 broad and If yd. deep will 56 men dig in 25 dy. of 8£ hr.? 
 
 17. If 34 k * of wool makes 25 m of cloth 0.6 m wide, how 
 long a piece of cloth 0.8 m wide will 108.8^ of wool make ? 
 
 18. If an oak beam 5.40 m long, 0.63 m thick, and 0.57 nB 
 wide weighs 1469.25^, what is the weight of an oak beam 
 4.87 m long, 0.58 m thick, and 0.53 m wide ? 
 
 19. A certain quantity of air has a volume of 195.5 cu. ft. 
 at 27.8° C. What will be its volume at 100° C? 
 
 Note. The coefficient of expansion of a body is the increase of a 
 unit of volume of the body when the temperature is increased 1° C. 
 The coefficient of expansion for air is 0.00367. That is, a cubic foot 
 of air at 0° C. occupies at 1° C. 1.00367 cu. ft., if the pressure remains 
 the same. The increase in volume is found approximately by multi- 
 plying the increase of 1° by the number of degrees. 
 
 20. A quantity of air at a temperature of 15.6° C. has a 
 volume of 4 cu. ft. under a pressure of 12 lb. per square 
 inch. What will be its volume at 48.7° C. under a pres- 
 sure of 14 lb. per square inch ? 
 
 Note. The volumes occupied by the same quantity of air, the tem- 
 perature remaining unchanged, are in inverse ratio to the pressures. 
 
RATIO AND PROPORTION. 227 
 
 Cause and Effect. 
 
 396. Problems in compound proportion are readily 
 solved by the cause and effect method. 
 
 397. The cause and effect method depends upon the 
 following principle : 
 
 Like causes produce like effects; and the ratio between 
 any two causes equals the ratio between the effects produced. 
 
 Note. Examples of causes are men at work, time, and goods 
 bought or sold ; examples of effects are work done, wages, and cost 
 or selling price of goods. 
 
 398. Example. If 4 men mow 15 A. in 5 dy. of 14 hr., 
 in how many days of 13 hr. can 7 men mow 19£ A.? 
 
 Solution. 1st cause. 2d cause. 1st effect. 2d effect. 
 
 4 men >> 7 men 
 
 5 dy. I : ? dy. } 
 14 hr. J 13 hr. 
 
 : I 19.5 A. 
 
 Since the product of the means equals the product of the extremes, 
 
 the number of days required equals the product of the extremes divided 
 
 by the product of the remaining means. 
 
 v- * a -a- 4 X 5 X 14 X 19.5 . 
 
 Hence, the number of days required is — = 4. 
 
 7 X 18 X 15 
 
 Compare this solution with the solution of the example in § 395. 
 
 Exercise 103. 
 Solve the following problems by the cause and effect method. 
 
 1. If a man can mow ^ T of a field in a day, how long 
 will it take another man to mow £ of a field 5^ times as 
 large, if the second man works If times as fast as the first 
 but only $• as many hours each day ? 
 
 2. If 4 men or 7 boys can do a piece of work in 6 days, 
 how long will it take 6 men and 9 boys to do the work ? 
 
228 RATIO AND PROPORTION. 
 
 3. If 50 men working 9 hr. a day require 6 dy. to dig 
 a trench 100 yd. long, 2 yd. wide, and 3 yd. deep, how 
 many men working 10 hr. a day for 9 dy. will be required 
 to dig a trench 50 yd. long, 6 yd. wide, and 5 yd. deep, in 
 ground twice as hard to dig ? 
 
 4. If 12 men in 9 dy. can harvest 40 A. of wheat, how 
 many acres can 16 men harvest in 3 dy.? 
 
 5. If 120 men can make an embankment £ of a mile 
 long, 30 yd. wide, and 7 yd. high, in 42 dy., how many men 
 will it take to make an embankment 1000 yd. long, 36 yd. 
 wide, and 22 ft. high, in 30 dy.? 
 
 6. If 7 women in 8 dy. of 11 hr. each can make 22 
 dozen shirts, in how many days of 10 hr. each can 12 
 women make 360 dozen shirts ? 
 
 7. Twenty-five lamps used 5 hr. an evening for 40 dy. 
 required a quantity of oil that cost $4.25. How many 
 lamps used 4 hr. an evening for 30 dy. can be furnished 
 with oil at a cost of $7.65 ? 
 
 8. If 8 horses can be kept 12 dy. for a certain sum 
 when hay is worth $15 a ton, how many days can 6 
 horses be kept for the same sum when hay is worth $12 
 a ton ? 
 
 9. Twenty horses working 14 wk., 6 dy. a week and 
 8 hr. a day, transport the output of a mine to the nearest 
 wharf. In how many weeks will 24 horses do the same 
 work, if they work 5 dy. a week and 7 hr. a day ? 
 
 10. If 6 men can reap a field of rye 200 yd. long and 
 150 yd. wide in 4 dy. of 12 hr. each, in how many days of 
 10 hr. each will 8 men reap a field 300 yd. long and 250 yd. 
 wide ? 
 
 11. If a boy can do only half as much work as a man, 
 how many hours a day must 42 boys work to accomplish 
 as much in 45 dy. as 27 men, working 10 hr. a day, would 
 accomplish in 28 dy.? 
 
RATIO AND PROPORTION. 229 
 
 Proportional Parts. 
 
 399. If it is required to divide a quantity into parts 
 proportional to 3, 4, 5 ; the numbers 3, 4, 5 may be taken 
 to represent the parts, and then the whole will be repre- 
 sented by 3 + 4 + 5 ; that is, by 12. 
 
 400. Examples. 1. Divide $391 into parts propor- 
 tional to the numbers 5, 7, and 11. 
 
 Solution. The whole quantity will be represented by 5 + 7 + 1 1 =23. 
 Therefore, the respective parts will be £ Jt j/g, |£ of $391 ; that is, 
 $85, $119, and $187. 
 
 Or, the parts of the quantity may be found by the proportions : 
 
 23: 5:: $391:? 
 
 23: 7:: $391:? 
 
 23: 11:: $391:? 
 The required terms will be $85, $119, and $187, respectively. 
 
 2. Divide $248 into parts proportional to y 1 ^, y 1 ^, ^. 
 
 Solution. Multiply the fractions by 150, the L. C. M. of their 
 denominators. The results are 15, 10, 6. Hence; the parts will be 
 represented by the numbers 15, 10, 6, and the whole by 31. 
 
 Therefore, the respective parts will be i^, |f, -fa of $248 ; that is, 
 $120, $80, $48. 
 
 Exercise 104. 
 
 1. Divide $12,000 proportionally to the numbers 3, 4, 5. 
 
 2. Divide 815 tons proportionally to £, §, f , f . 
 
 3. Divide 6853 lb. of wool proportionally to If, 2f , 5| ; 
 also proportionally to the reciprocals of these numbers. 
 
 4. Two men purchase some property together, one pay- 
 ing $1250 and the other $1000. If the value of the prop- 
 erty rises to $3600, what will be the share of each ? 
 
 5. Gun metal is composed by weight of 3 parts of tin 
 to 100 parts of copper. What weight of each of these 
 metals is there in a cannon weighing 721 lb.? 
 
230 RATIO AND PROPORTION. 
 
 6. Bell metal contains by weight 78 parts of copper 
 and 22 parts of tin. What weight of each of these metals 
 is there in a bell weighing 937 lb.? 
 
 7. It takes 75^ of saltpetre, 12.5** of charcoal, and 
 12.5 kg of sulphur to make 100** of powder. .How many 
 kilograms of each will be required to make 10,000,000 
 cartridges, each containing 5* of powder ? 
 
 8. Yellow copper contains by weight 2 parts of red 
 copper and 1 part of zinc. How many ounces of red cop- 
 per in an article of yellow copper that weighs 1 lb.? 
 
 9. Type metal is an alloy containing by weight 39 parts 
 of lead to 11 parts of antimony. How many pounds of 
 each are required to make 957 lb. of type ? 
 
 10. Plumbers' solder contains by weight 2 parts of lead 
 and 1 part of tin. How many pounds of each are required 
 to make 100 lb. of solder ? 
 
 11. The air is composed of oxygen and nitrogen. In 
 100 volumes of air there are 21 volumes of oxygen and 79 
 of nitrogen. If the weight of a liter of oxygen is 1.4295 s , 
 and that of a liter of nitrogen is 1.2577 g , how many grams 
 of each gas does 100 g of air contain ? 
 
 12. At $20.67 an ounce for pure gold, what is the value 
 of the gold in a chain that weighs 3 oz. 4 dwt., if it is 18 
 carats fine (that is, 18 parts of pure gold out of 24) ? 
 
 13. Two men agree to do a piece of work for f 63. They 
 finish the work in 18 days, but one of them was absent 
 5 days of this time. How should the pay be divided ? 
 
 14. Five men working together do a piece of work in 
 20 days and receive as pay $ 253. One of the men was 
 absent 5 days, and another 2 days of this time. How 
 should the pay be divided ? 
 
 15. Standard silver consists of 37 parts of pure silver 
 to 3 parts of copper. What weight of pure silver in the 
 crown piece that weighs \<{ oz. troy ? 
 
RATIO AND PROPORTION. 231 
 
 Partnership. 
 
 401. Partnership. An association of two or more per- 
 sons for the purpose of conducting business is called a 
 partnership. 
 
 A partnership association is called a firm, company, or 
 house, and the persons associated in business are called 
 partners. 
 
 402. Assets and Liabilities. The property of all kinds 
 of a firm or company together with all amounts due it is 
 called its assets ; the debts of a firm or company are called 
 its liabilities. 
 
 403. Partnership is separated into simple and compound. 
 In simple partnership the capital of each partner is 
 
 invented for the same time. 
 
 In compound partnership the time for which the capital 
 of each partner is invested is taken into account, as well as 
 the amount of the capital. 
 
 404. The division of profits and losses is made proportion- 
 ally to the amount of the capital and the time it is invested. 
 
 405. Examples. 1. A and B entered into partnership, 
 A furnishing $4000, and B $5000. If they gained $1800, 
 what was each partner's share of the profits ? 
 
 Solution. $4000 + $5000 = $9000, the entire capital. 
 A's share of the profits was f£gg of $1800, or $800 ; and B's share 
 of the profits was g### of $1800, or $1000. 
 
 2. A and B entered into partnership, A furnishing $2000 
 for 2 yr., and B $3000 for 1 yr. If their profits are $1400, 
 what is the share of each ? 
 
 Solution. The use of $2000 for 2 yr. is equal to the use of 
 2 X $2000, or $4000, for 1 yr. Hence, the entire capital for 1 yr. 
 equals $4000 + $3000, or $7000. A's share of the profits is $$# of 
 $1400, or $800 ; and B's share is f£#£ of $1400, or $600. 
 
232 RATIO AND PROPORTION. 
 
 Exercise 105. 
 
 1. A, B, and C entered into partnership, A furnishing 
 $18,150 ; B, $19,360 ; and C, $10,890. If their profits 
 were $12,100, what was each man's share of the profits ? 
 
 2. Four men engaged in business together and made a 
 profit of $1200. How much of it should each man receive, 
 if the first put in $3000, the second $5000, the third $4200, 
 and the fourth $2400 ? 
 
 3. A man dies owing three creditors $8050, $2970, and 
 $7170, respectively. If his assets, after deducting ex- 
 penses, are $13,646, how much will each creditor receive? 
 
 4. Three heirs receive from an estate $4700, $3200, 
 and $12,500, respectively, on condition that they together 
 pay a debt of $2000. What amount will each have ? 
 
 5. Arnold and Baker enter into partnership. Arnold 
 puts in $6000 for 8 mo., and Baker $4000 for 6 mo. Their 
 profits are $2000. What is each man's share ? 
 
 6. Dobson furnishes the firm of Dobson & Fogg with 
 $5000 for 13 mo.; Fogg furnishes $7000 for 9 mo. Their 
 profits are $1700. What is the share of each ? 
 
 7. In a business partnership, A furnishes $800, and 
 after 3 mo. $250 more ; B furnishes $950, and at the end 
 of 2 mo. withdraws $200 ; C furnishes $650, and at the 
 end of 6 mo. $400 more. At the end of the year their 
 profit is $2516. How shall it be divided among them ? 
 
 8. Two partners, A and B, enter into partnership with 
 capitals of $3500 and $8700, respectively, and A is to have 
 0.12 of the profits for managing the business. How shall 
 a profit of $1906.25 be divided between them ? 
 
 9. A puts $2100 into a business, and B $1750. At 
 the end of a year each puts in $700 more, and C joins 
 them with $2500. How shall a profit of $2166.50 be 
 divided 18 months after C enters the firm ? 
 
RATIO AND PROPORTION. 233 
 
 10. Three graziers hire a pasture, for which they pay 
 $132.50. One puts in 10 oxen for 3 months, another 12 
 oxen for 4 months, and the third 14 oxen for 2 months. 
 How much of the rent ought each to pay ? 
 
 11. A begins business, with a capital of $2400, on the 
 19th of March; and on the 17th of July admits B as a 
 partner, with a capital of $1800. December 31 the profits 
 are $943. What is t^ie share of each ? 
 
 12. A and B join capitals in the ratio 7:11. At the 
 end of 7 months A withdraws \ of his, and B £ of his ; 
 and, after 11 months more, they divide a profit of $5148.50. 
 What is the share of each ? 
 
 13. Divide £65 9s. among three men, so that the first 
 may have as many half-crowns as the second has shillings ; 
 and the second as many guineas as the third has pounds. 
 
 14-. A and B begin business each with a capital of $2000. 
 A adds $500 at the end of 2 months, and $500 more at the 
 end of 7 months ; B adds $800 at the end of 3 months. 
 If the profits are $3605.25 at the end of a year, what is 
 the share of each ? 
 
 15. Three partners in a restaurant furnish respectively 
 $500 for 7 months, $600 for 8 months, and $900 for 9 
 months. If they lose $410, what is each one's share of 
 the loss ? 
 
 16. Two capitalists contribute, one $10,000, the other 
 $12,000, to an enterprise which continues in operation for 
 10 years. 10 months after starting, a third man becomes a 
 partner and contributes $15,000; and 2 years after this a 
 fourth man contributes $17,400. If the total profits are 
 $45,600, what amount does each partner receive ? 
 
 17. A began business with a capital of $2500. After 
 three years he invested $1250 more, and took as a partner 
 B, who invested $5000. At the end of four years more the 
 profits amounted to $9562.50. What was the share of each? 
 
234 RATIO AND PROPORTION. 
 
 Averages, or Alligation. 
 
 406. The average of several numbers is the number 
 that can be put in place of each of them without altering 
 their sum. 
 
 Thus, the average weight of four turkeys, weighing respectively 
 
 10 lb., 11 lb., 12 lb., and 13 lb., is 11| lb.; for the four turkeys 
 together weigh 10 lb. + 11 lb. + 12 lb. + 13 lb., or 46 lb., and the 
 average weight of the four turkeys is ± of 46 lb., or 11$ lb. 
 
 407. Alligation is the process of finding the average 
 value of a compound or mixture composed of quantities of 
 different values; or of finding the proportion of several 
 quantities of different values that must be used to form a 
 compound or mixture of a given average value. 
 
 Note. The first process is called alligation medial; the second 
 process, alligation alternate. 
 
 408. Examples. 1. A grocer mixed 11 lb. of coffee, 
 costing $0.25 a pound, with 6 lb. of chicory, costing $0.08 
 a pound. What was the cost of the mixture a pound ? 
 
 11 X $0.25 = $2.76 Solution. 11 lb. at $0.26 cost $2.76, and 
 6 X 0.08 = 0.48 6 lb. at $0.08 cost $0.48. Therefore, 17 lb. of 
 
 17 X $ ? = $3.23 the mixture cost $3.23, or $0.19 a pound. 
 
 2. In what proportion may a grocer mix syrups costing 
 respectively 42, 56, 65, and 75 cents a gallon to make a 
 mixture worth 60 cents a gallon ? 
 
 Solution. 1 gal. of the 42-cent syrup gains in value $0.18, and 
 1 gal. of the 75-cent syrup loses in value $0. 15. Hence, to make a mix- 
 ture of these two syrups worth $0.60 a gallon, the grocer must mix 
 them in the ratio 15 : 18 ; that is, 6 : 6. 
 
 1 gal. of the 66-cent syrup gains in value $0.04, and 1 gal. of the 
 65-cent syrup loses in value $0.05. Hence, to make a mixture of these 
 two syrups worth $0.60 a gallon, he must mix them in the ratio 5:4. 
 
 Therefore, the grocer may take 5 gal. at $0.42, 5 gal. at $0.66, 
 4 gal. at $0.65, and 6 gal. at $0.75. 
 
RATIO AND PROPORTION. 235 
 
 Exercise 106. 
 
 1. There were 125 pupils at school on Monday, 130 on 
 Tuesday, 128 on Wednesday, 132 on Thursday, and 125 on 
 Friday. What was the average daily attendance ? 
 
 2. A spring of water that yields 250 gal. an hour 
 supplies a town containing 360 families. What is the 
 average daily supply of water for each family ? 
 
 3. A wine merchant put into an empty cask 15 qt. of 
 brandy costing $1.10 a quart, 66 qt. costing $1.20 a quart, 
 and 43 qt. costing $ 1.40 a quart. At what price per quart 
 must he sell the brandy to gain one fifth of the cost ? 
 
 4. A grocer mixed 120 lb. of tea costing 50 cents a 
 pound with 180 lb. costing 40 cents a pound. At what 
 price per pound must he sell the mixture to make a profit 
 of $30 on the whole ? 
 
 5. A grocer buys two kinds of tea at 40 cents a pound 
 and 56 cents a pound, respectively, and mixes them in the 
 ratio of 5 to 3. What is his profit, if he sells 56 lb. of the 
 mixture at 84 cents a pound ? 
 
 6. The average length of ten sticks is 2 ft. 10£ in.; one 
 stick is 27£ in. long, another 37-J in. long, and the remain- 
 ing eight are of the same length. What is the length of 
 one of the remaining eight? 
 
 7. The average age of the boys in the four classes of 
 a school is 18.4 yr., 17.9 yr., 16.8 yr., and 15.7 yr. The 
 classes contain 29, 33, 34, and 33 boys, respectively. What 
 is the average age of the boys in the school ? 
 
 8. Seven boys weigh respectively 119.7 lb., 105 lb., 
 178.3 lb., 165.3 lb., 142.8 lb., 109 lb., 154.2 lb. What is 
 their average weight ? 
 
 9. In what proportion should tea costing 60 cents a 
 pound be mixed with tea costing 45 cents a pound that 
 the cost of the mixture should be 54 cents a pound ? 
 
236 RATIO AND PROPORTION. 
 
 10. A merchant has teas that cost 80 cents, 60 cents, 
 and 40 cents a pound, respectively. How many pounds of 
 each kind shall he take to make a mixture of 1000 lb., so 
 that in selling it at 70 cents a pound he may make a profit 
 of 8 cents a pound ? 
 
 11. A grocer mixed black tea that cost him 28 cents a 
 pound with green tea that cost him 42 cents, and by sell- 
 ing the mixture at 35 cents a pound he gained £ of its 
 cost. What was the actual cost of the mixture a pound ? 
 In what ratio were the teas mixed ? 
 
 12. A dealer has an order for 1000 bu. of wheat at 70 
 cents a bushel. In what proportion shall he mix three 
 kinds of wheat at 66, 69, and 72 cents a bushel to fill the 
 order ? 
 
 13. A wine merchant mixes wines that cost $0.95, $1.05, 
 $1.10, and $1.20 a gallon to make a mixture costing $1.00 
 per gallon. How many gallons of each kind of wine does 
 he take ? 
 
 14. A merchant wishes to fill a barrel that will hold 
 240 lb. of sugar with sugar costing 4 J, 4£, and 5 £ cents 
 a pound, respectively, so that the mixture may cost 4$- cents 
 a pound. How many pounds of each kind shall he take? 
 
 15. A grocer wishes to mix 12 lb. of coffee at 40 cents a 
 pound and 20 lb. at 35 cents a pound with coifee at 28 
 cents a pound, so that the mixture may be worth 30 cents 
 a pound. How many pounds at 28 cents must he use ? 
 
 16. A grocer mixed 14 lb. of coffee costing 32 cents a 
 pound, 18 lb. costing 35 cents a pound, 22 lb. costing 38 
 cents a pound, and 40 lb. costing 30 cents a pound. What 
 is the cost of the mixture per pound, and at what price 
 must he sell it to gain 0.25 of the cost ? 
 
 17. In what proportion may oils costing $1.20, $0.80, 
 and $0.60 a gallon be mixed that the mixture may cost 
 $0.70 a gallon ? 
 
CHAPTER XII. 
 PERCENTAGE. 
 
 409. A percentage of a number is the result obtained 
 by taking a stated number of hundredths of it. 
 
 One hundredth of a number is called one per cent of it ; 
 two hundredths, two per cent ; and so on. 
 
 410. A rate per cent is a fraction whose denominator 
 is 100, and whose numerator is the given number of hu?i- 
 dredths. The methods of common fractions or of decimals 
 are used in the solution of all examples in Percentage. 
 
 The shortest method is the best method. 
 
 411. The symbol % stands for the words per cent. 
 Thus, 13% is 0.13 ; 2£% is 0.02i; 867% is 8.67. 
 
 412. Example. Express 37-J-% as a common fraction. 
 
 37-i- q 
 Solution. 37£% = —J = '- • Hence, 
 
 413. To Express a Rate Per Cent as a Common Fraction, 
 
 Write the rate for the numerator and 100 for the denomi- 
 nator, and reduce this fraction to its lowest terms. 
 
 Exercise 107. 
 Reduce to a common fraction : 
 
 1. 
 
 20%. 
 
 6. 
 
 5%. 
 
 11. 
 
 62^%. 
 
 16. 
 
 18f%. 
 
 2. 
 
 80%. 
 
 7. 
 
 10%. 
 
 12. 
 
 87^%. 
 
 17. 
 
 95%. 
 
 3. 
 
 25%. 
 
 8. 
 
 12*%. 
 
 13. 
 
 66f%. 
 
 18. 
 
 70%. 
 
 4. 
 
 50%. 
 
 9. 
 
 16f%. 
 
 14. 
 
 37^%. 
 
 19. 
 
 144|% 
 
 5. 
 
 75%. 
 
 10. 
 
 114%. 
 
 15. 
 
 83£%. 
 
 20. 
 
 262£% 
 
238 
 
 PERCENTAGE. 
 
 414. Example. Express £ as a rate per cent. 
 
 Solution. 
 
 i = m = 100%. 
 
 | = | of 100% = 40%. Hence, 
 
 415. To Express a Common Fraction as a Rate Per 
 Cent, 
 
 Divide 100 by the denominator of the fraction and mul- 
 tiply the quotient by the numerator. 
 
 416. Examples. 1. Express 0.4 as a rate per cent. 
 Solution. 0.4 = 0.40, or 40%. 
 
 2. Express 0.4575 as a rate per cent. 
 Solution. 0.4575 = 0.45^0 = 0.45|, or 45|%. 
 
 3. Express 0.00375 as a rate per cent. 
 Solution. 0.00375 = O.OO&Vff = O.OOf = |%. Hence, 
 
 417. To Express a Decimal as a Rate Per Cent, 
 Write the decimal as hundredths, and the number that 
 
 exj>resses the hundredths is the rate per cent required. 
 
 Note. If the decimal has more than two places, the figures that 
 follow the hundredths' place signify a fraction of 1%. 
 
 
 
 Exercise 108. 
 
 
 
 lX[ 
 
 res£ 
 
 as a rate per 
 
 cent : 
 
 
 
 1. 
 
 i- 
 
 8- t- 
 
 15. 0.25. 
 
 22. 
 
 0.33333 
 
 2. 
 
 i- 
 
 9- A- 
 
 16. 0.6. 
 
 23. 
 
 0.16667 
 
 3. 
 
 i- 
 
 io. a- 
 
 17. 0.75. 
 
 24. 
 
 0.83333 
 
 4. 
 
 i- 
 
 11. 1. 
 
 18. 0.9. 
 
 25. 
 
 0.875. 
 
 5. 
 
 i 
 
 12. T V 
 
 19. 0.65. 
 
 26. 
 
 1.375. 
 
 6. 
 
 f 
 
 13. T*p 
 
 20. 0.45. 
 
 27. 
 
 2.66667. 
 
 7. 
 
 §• 
 
 14. a- 
 
 21. 0.2. 
 
 28. 
 
 4.2525. 
 
PERCENTAGE. 239 
 
 418. Problems in Percentage are conveniently divided 
 into three classes, as follows : 
 
 Class I. To find a certain fraction of a number. 
 
 Class II. To find the fraction that one number is of 
 another. 
 
 Class III. To find a number when a fraction of it is given. 
 
 The following examples illustrate the three classes : 
 
 Class I. What number is f of 300 ? 
 
 Class II. What fraction of 300 is 200 ? 
 
 Class III. What is the number, if 200 is | of it ? 
 
 The fraction in each class is expressed in hundredths for 
 the sake of a uniform standard ; the phrase per cent is used 
 for tjie word hundredths, and the symbol °J is written for 
 the phrase per cent. 
 
 Thus, in common fractions 8 is £ of 16 ; in decimals & is 0.5 of 16 ; 
 in percentage 8 is 50% of 16. 
 
 In common fractions 5 is £ of 40 ; in decimals 5 is 0.125 of 40 ; in 
 percentage 5 is 12|% of 40. 
 
 419. One hundred per cent of a number is the number 
 itself. 
 
 Thus, 100% of 40 is 40. 
 
 Class I. 
 
 420. Examples. 
 
 1. Findl6i% of 288. 
 
 Solution. 16£% = 0.16f 
 
 0.16i of 288 = 47.04. 
 Therefore, 16£% of 288 is 47.04. 
 
 2. Find 16J% of 288. 
 
 Solution. 16f-% = 0. 16$- * \. 
 
 \ of 288 = 48. 
 Therefore, 16£% of 288 is 48. 
 
 421. To Find a Percentage of a Number, 
 
 Multiply the number by the given rate per cent, expressed 
 as a common fraction or as a decimal. 
 
240 PERCENTAGE. 
 
 Exercise 109. 
 Find by using decimals : 
 
 1. 23% of 1728. 6. 2% of 846. 11. 0.5%ofl44. 
 
 2. 44% of 1861. 7. 9% of 24.87. 12. 8752% of 2645. 
 
 3. 87% of 14.22. 8. 122% of 12.5. 13. 0.02% of 52.36. 
 
 4. 63% of 2.832. 9. 287% of 48.2. 14. 2% of 3. 
 
 5. 72% of 841. 10. 1% of 7854. 15. 2.06% of 312. 
 
 Find by using common fractions : 
 
 16. 33£%of363. 21. 62i%of216. 26. 14f%of81.9. 
 
 17. 20% of 545. 22. 37£%of360. 27. 22§%of8.19. 
 
 18. 25% of 1728. 23. 83£%of486. 28. 168^% of 256. 
 
 19. 50% of 8642. 24. 66f % of 456. 29. 143| % of 288. 
 
 20. 75% of 432. 25. 12^% of 2.56. 30. 70% of 8432. 
 
 31. The population of a town in 1880 was 12,275, and 
 it increased 8% in the next ten years. Find the popula- 
 tion of the town in 1890. 
 
 32. How much metal will be obtained from 365 tons of 
 ore, if the ore contains 7% of metal? 
 
 33. If gunpowder contains 75% of saltpetre, 10% of 
 sulphur, 15% of charcoal, how many pounds of each are 
 there in a ton of powder ? 
 
 34. Air is composed by volume of 20.0265% of oxygen 
 and 79.9735% of nitrogen. How many cubic feet of oxy- 
 gen in 1750 cu. ft. of air ? 
 
 85. If 2% of a regiment of 750 men are killed in an 
 engagement, 6% are wounded, and 4% are missing, what 
 is the number still available for service ? 
 
 36. A man sold a bicycle that cost him $60, and lost 
 16f% 0I " the cost. For what price did he sell it ? 
 
 37. A merchant sold hats that cost him $1.50 each, and 
 gained 33£%. For what price did he sell them ? 
 
PERCENTAGE. 241 
 
 38. In a school of 80 children, 17£% are girls. Find 
 the number of boys. 
 
 39. The lead ore from a certain mine yields 60% of 
 metal, and of the metal f of 1% is silver. How much 
 silver and how much lead will be obtained from 1200 t. 
 of ore ? 
 
 40. If 13% of a population of 27,000,000 are foreign 
 born, how many of the population are foreign born ? 
 
 41. If iron expands J of l°/ when heated 185° F., what 
 will be the expansion of iron when heated from — 20° F. 
 to+120°F.? 
 
 42. A tubular iron bridge 740 ft. long has one end fast 
 to a pier. How much play must be allowed at the other 
 end for the expansion of the iron, if the climate varies 
 from — 30° F. in winter to + 130° F. in a July sun ? 
 
 43. How much longer is 100 miles of iron rails at 118° 
 F. than at 20° below zero ? 
 
 Class II. 
 
 422. Examples. 1. What per cent of 4 is 3 ? 
 
 Solution. 3 is f of 4, and £ = T 7 ^, or 75%. 
 Therefore, 3 is 75% of 4. 
 
 2. What per cent of 447 is 169.86 ? 
 
 Solution. 4.47 = 1% of 447. 
 
 Therefore, 169.86 = ^Tgr % of 447 = 38% of 447. Hence, 
 
 423. To Find the Rate Per Cent when a Number and 
 a Percentage of the Number are Given, 
 
 Write the number as the denominator and the percentage 
 as the numerator of a fraction, and reduce this fraction to 
 hundredths. Or, 
 
 Divide the percentage of the number by one per cent of the 
 number. 
 
 h T f 
 
242 PERCENTAGE. 
 
 Exercise 110. 
 
 1. What per cent of 64 is 16 ? 
 
 2. What per cent of 16 is 64 ? 
 
 3. What per cent of 450 lb. is 50 lb.? 
 
 4. What per cent of 50 lb. is 450 lb.? 
 
 5. What per cent of $465 is $130.20 ? 
 
 6. What per cent of $832 is $807.04 ? 
 
 7. What per cent of $987 is $2289.84 ? 
 
 8. A brick kiln contained 29,800 bricks, but after burn- 
 ing only 29,734 were found in good condition. What per 
 cent had been spoiled in burning ? 
 
 9. If a house worth $4000 rents for $360 a year, what 
 per cent of its value is the rent ? 
 
 10. If 75 bu. of corn are raised from 1 pk. of corn, what 
 per cent is the increase ? 
 
 11. Ten years ago the population of a city was 26,275 ; 
 its present population is 31,530. What is the increase per 
 cent ? 
 
 12. If 3f tons of sulphur are required to make 31 J- tons 
 of gunpowder, what per cent of gunpowder is sulphur ? 
 
 13. If a long ton of ore in a gold mine yields 5 oz. (troy) 
 of gold, what is the yield per cent ? 
 
 14. If 12£ tons of iron are obtained from 235 tons of 
 ore, what per cent of the ore is iron ? 
 
 Find the gain per cent in population in each of the fol- 
 lowing cities from 1880 to 1890 : 
 
 
 Cities. 
 
 1880. 
 
 1890. 
 
 15. 
 
 New York, 
 
 1,206,594, 
 
 1,513,501. 
 
 16. 
 
 Chicago, 
 
 503,304, 
 
 1,099,850. 
 
 17. 
 
 Philadelphia, 
 
 846,981, 
 
 1,046,964. 
 
 18. 
 
 Brooklyn, 
 
 566,689, 
 
 806,343. 
 
 19. 
 
 Boston, 
 
 362,535, 
 
 448,477. 
 
PERCENTAGE. 243 
 
 20. If 2 gal. of water are added to 25 gal. of alcohol, 
 what per cent of the mixture is water ? What per cent is 
 alcohol ? 
 
 21. If 5% of the present population of a town has been 
 the increase in the preceding ten years, what per cent of 
 the population ten years ago has been added ? 
 
 22. A man gained in weight in January 3%, and in 
 February lost 3%. What per cent of his weight the first 
 day of January is his weight the first day of March ? 
 
 23. If 7 lb. of a certain article loses 3 oz. in weight by 
 drying, what per cent of its original weight is water ? 
 
 24. If 7 lb. of a dry article has lost 3 oz. by drying, what 
 per cent of its original weight was water ? 
 
 25. If a dry article exposed to damp air absorbed 3 oz t 
 of water, and then weighed 7 lb., what per cent of its 
 present weight is water ? 
 
 26. If rosin is melted with 20% of its weight of tallow, 
 what per cent of tallow does the mixture contain ? 
 
 27. If 20% of a mixture of tallow and rosin is tallow, 
 what per cent of the weight of the rosin is the weight of 
 the tallow ? 
 
 28. Nitrogen gas, under standard pressure and tempera- 
 ture, is -J- of 1% of the weight of an equal volume of water. 
 What is the specific gravity of nitrogen ? How many gal- 
 lons of nitrogen will it take to weigh as much as a pint of 
 water ? 
 
 29. Oxygen gas is \ of l°/ of the weight of an equal 
 volume of water. What is its specific gravity ? How 
 many gallons of oxygen will it take to weigh, as much as a 
 pint of water ? 
 
 30. If common air consists of 4 volumes of oxygen to 
 13 of nitrogen, what is its specific gravity ? 
 
 31. How many gallons of air will it take to weigh as 
 much as a pint of water ? 
 
244 PERCENTAGE. 
 
 Class IIL 
 
 424. Examples. 1. If 17% of a number is 799, what 
 is the number ? 
 
 Solution. Since 17% of a number is 799, 1% of the number is T J 7 
 of 799, or 47, and 100% of the number is 100 X 47, or 4700. 
 
 2. If 16f % of a number is 432, what is the number ? 
 
 Solution. 432 is 16 1%, or £ of the required number. 
 Therefore, the required number is 6 X 432, or 2592. 
 
 3. 1400 is 16J% more than what number ? 
 
 Solution. 100 % of the number = the number. 
 
 16%% of the number = the increase. 
 
 116|%, or I of the number = 1400. 
 
 I of the number = \ of 1400, or 200. 
 Therefore, the number = 6 X 200, or 1200. 
 
 4. 1200 is 25% less than what number? 
 
 Solution. 100% of the number = the number. 
 
 25% of the number = the decrease. 
 
 75%, or £ of the number = 1200. 
 If 1200 is £ of the number, £ of the number is \ of 1200, or 400. 
 Therefore, the number is 4 X 400, or 1600. Hence, 
 
 425. To Find a Number when a Percentage of the 
 Number and the Rate Per Cent are Given, 
 
 Express the rate per cent as a fraction, divide the per- 
 centage by the numerator of this fraction and multiply the 
 quotient by the denominator. 
 
 Exercise 111. 
 
 1. 15 is f of what number? 15 is 75% of what 
 number ? 
 
 2. $500 is 4% of what sum of money ? 
 
 3. Find the number of which 324 is 27%. 
 
PERCENTAGE. 245 
 
 4. 288 is 20% more than what number ? 
 
 5. 145 is 25% more than what number ? 
 
 6. 1240 is 55% less than what number ? 
 
 7. 260 is 33-J% less than what number ? 
 
 8. 91 is 40% more than what number ? 
 
 9. 901 is 6£% more than what number ? 
 
 1 0. If 8f % of a number is 4140.15, what is the number ? 
 
 11. If 3% of a number is 2f, what is the number ? 
 
 12. If 140% of a number is 630, what is the number ? 
 
 13. If 6£% of a number is 33.25, what is the number? 
 
 14. A town, after decreasing 11%, has 4539 inhabitants. 
 Find its number at first. 
 
 15. In a certain school there are 200 girls, and the 
 number of girls is 40% of the whole number of pupils. 
 How many pupils in the school ? 
 
 16. A manufactory uses 24 tons of coal a day, 20% of 
 which is lost in smoke. How much coal would be needed 
 if this waste could be prevented ? 
 
 17. A town, after decreasing 25%, has 4539 inhabitants. 
 Find its number at first. 
 
 18. If the ore from a mine yields -fa of 1% of pure gold, 
 how many long tons of ore must be taken to obtain 7 lb. 
 (troy) of gold ? 
 
 19.^ Goods were sold, at a loss of 3%, for $2667.50. 
 What was the cost ? 
 
 20. A tradesman, in selling goods, deducts from the 
 marked price 5% for cash. What was the marked price 
 of goods for which he received $14.25 ? 
 
 21. If an ore loses 41^-% of its weight in roasting, and 
 43f % of the remainder in smelting, how much ore will be 
 required to yield 1000 tons of metal ? 
 
 22. How many pounds of tallow must be mixed with 
 8 J pounds of rosin that the mixture may contain 15% of 
 tallow ? 
 
246 PERCENTAGE. 
 
 Commercial Discount. 
 
 426. Commercial Discount. A reduction from the list 
 price of an article, from the amount of a bill of goods, or 
 from the amount of a debt is called commercial discount. 
 
 427. Discounts are reckoned at some common fraction 
 of, or at some rate per cent of, the amount from which the 
 discount is made. 
 
 If two or more discounts are quoted, the first denotes a 
 discount off the list price ; the second, a discount off the 
 remainder after the first discount is made ; the third, off the 
 remainder after the second discount is made ; and so on. 
 
 Thus, discounts of 20 and i mean that 20% is to be deducted from 
 the amount, and then from the remainder ^ of it is to be taken. 
 
 Note. By varying the rate of discount the manufacturer can raise 
 or lower the price of his goods without issuing a new catalogue. 
 
 428. Examples. 1. Find the net amount of a bill of 
 $150 after a discount of 33£% is made. 
 
 Solution. Discount is 33^%, or i of $160 = $30. 
 Net amount is $160 — $30 = $120. 
 
 2. Find the net amount of a bill of $527.10 with £ and 
 20% off. 
 
 $527.10 
 
 175.70 Solution. $527.10 less £ of $527.10 is $351.40. 
 
 $351.40 20% = £, and $361.40 less £ of $351.40 is $281.12, the 
 
 70.28 net amount of the bill. 
 
 $281.12 
 
 Exercise 112. 
 
 1. Find the net amount of a bill of $1550, if a dis- 
 count of 5% is made for cash. 
 
 2. Find the net amount of a bill of $88, if the dis- 
 counts are 20 and 10. 
 
PERCENTAGE. 247 
 
 3. Find the net cash amount of a bill of $ 800, if the 
 discounts are 75, 5, and 2\. 
 
 4. Find the net cash amount of a bill of $272, if the 
 discounts are -J-, 10, and 5. 
 
 5. Find the net cash amount of a bill of $1440, if the 
 discounts are 55, 10, and 5. 
 
 6. Find the net cash amount of a bill of $1125, if the 
 discounts are -J, 10, 10, 10, and 5. 
 
 7. Find the net amount of a bill of $872.29, if the 
 discounts are -J-, 20, and 25. 
 
 8. Find the difference between a single discount of 
 50% and two successive discounts of 25°f an( i 25% off a 
 bill of $1272.36. 
 
 9. An agent bought 25 sewing machines with 15, 10, 
 and 5 off the list price of $40 each, and sold them at a 
 discount of 10% off the list price. What was the net 
 amount he received for the sewing machines and his 
 profit ? 
 
 10. An agent bought a bicycle with 25 and 5 off the list 
 price of $100. If he received an additional discount of 
 2J% for cash, and sold the bicycle at a discount of 12-j-% 
 off the list price, what was the selling price and his profit ? 
 
 11. A collector collects 6b % of a debt of $727, and 
 charges 5% of the amount he collected. What was the 
 net amount for the creditor ? 
 
 Gain and Loss. 
 
 429. The gain or loss in business transactions is often 
 computed as a per cent of the cost. 
 
 430. In commercial discount the per cent is always 
 reckoned upon the price asked. In gain or loss the per cent 
 is always reckoned upon the price paid ; that is, upon the 
 cost. 
 
248 PERCENTAGE. 
 
 Exercise 113. 
 
 1. If goods are bought for $415, and sold for $500, 
 what is the gain per cent ? 
 
 2. If goods are bought for $415, and sold for $400, 
 what is the loss per cent ? 
 
 3. A farmer buys 24 head of cattle at $80 a head. 
 After losing 6 head, he sells the remainder at $105 a head. 
 What does he gain or lose per cent ? 
 
 4. Teas at 68 cents, 86 cents, and 96 cents a pound are 
 mixed in equal quantities, and sold at 90 cents a pound. 
 Find the gain per cent. 
 
 5. By selling goods for $1173.92 a merchant gains 
 $153.12. Find the gain per cent. 
 
 6. What was the cost, when 17£% was gained by sell- 
 ing goods for $253.80 ? 
 
 7. A wine merchant mixes 24 gal. of wine at $7 a gal- 
 lon, with 18 gal. at $5 a gallon, and sells the whole at $7 
 a gallon. What does he gain per cent ? 
 
 8. By selling a horse for $200, a dealer loses 12£%. 
 What would he have gained or lost per cent if he had sold 
 the horse for $250 ? 
 
 9. A spirit merchant buys 75 gal. of spirits at $3.25 a 
 gallon, and, after drawing off 10 gal., sells the remainder 
 so as to gain 5% on the cost of the whole. What is the 
 selling price per gallon ? 
 
 10. A man owns two city lots worth respectively $9845 
 and $12,155. If the first gains in value 32%, and the sec- 
 ond loses 13%, what is the gain or loss per cent in the value 
 of the two lots ? 
 
 11. A tradesman marks a hat $5, but takes off 5%. If 
 his profit is 14%, what was the cost of the hat? 
 
 12. What would a dishonest dealer gain per cent by 
 using a false weight of 15 oz. instead of a pound ? 
 
PERCENTAGE. 249 
 
 13. A dishonest dealer gains 12% by using false weights. 
 What is the real weight of his pound ? 
 
 14. What per cent above cost must a merchant mark 
 his goods that he may take off 20% from the marked price, 
 and still make 20% on the cost ? 
 
 Solution. Since the merchant is to make 20% on the cost of the 
 goods, the selling price is 120% of the cost price. 
 
 Since the selling price is to be 20% below the marked price, the 
 selling price is 80% of the marked price. 
 
 Therefore, the marked price will be -^ of 120% of the cost price, 
 or 150% of the cost price ; that is, the goods must be marked 50% 
 above cost. 
 
 15. What per cent above cost must a merchant mark 
 his goods to take off 10%, and still gain 17% ? 
 
 16. What per cent above cost must a merchant mark 
 his goods to take off 12£%, and still gain 12 J % ? 
 
 17. What per cent above cost must a merchant mark 
 his goods to take off 15%, and still gain 15% ? 
 
 18. What per cent above cost must a merchant mark 
 his goods to take off 33£%, and still gain 33£% ? 
 
 19. A man bought a horse for $70, and sold him for 
 $80. What per cent did he gain ? What per cent of the 
 selling price of the horse did he gain ? 
 
 2Q. If a merchant clears $800 by selling goods for 12^-% 
 profit, what was the cost of the goods, and for how much 
 were they sold ? 
 
 21. A man selling eggs at $0.40 a dozen gains 33J% ; 
 what was the cost ? Another, selling at the same price, 
 gains 33^-% of his receipts ; what did his eggs cost ? 
 
 22. A man lost 10% by selling a carriage for $117. 
 At what price should he have sold it to make 10% ? 
 
 23. If a real estate dealer gained $600 by selling a farm 
 for 20% profit, what was the cost of the farm, and for how 
 much did he sell it ? 
 
250 PERCENTAGE. 
 
 Commission. 
 
 431. Commission is the payment made by one person, 
 called the Principal, to another, called the Agent or Factor, 
 for the transaction of business. 
 
 432. Commission is usually a percentage of the money 
 involved in the transaction. If goods are bought, it is a 
 percentage of the amount paid; if sold, a percentage of the 
 amount received; if money is collected, a percentage of the 
 amount collected. 
 
 Exercise 114. 
 
 1. Find the commission on $2595, at 2£%. 
 
 2. An agent sells 200 bbl. of flour, at $6.25, and 600 
 gal. of molasses, at 65 cents, and charges a commission of 
 If %. What are the net proceeds ? 
 
 Note. The sum left after the payment of the commission and of 
 all other expenses is called the net proceeds. 
 
 3. A commission merchant received $1640 to buy corn, 
 and charged a commission of 2£%. What is his commis- 
 sion, and how many bushels of corn at 62£ cents a bushel 
 can he buy ? 
 
 4. An agent sells a consignment of cotton for $5216. 
 He pays $51 for storage, and charges a commission of 2£%. 
 What are the net proceeds ? 
 
 5. An agent sold butter for $1570, and remitted 
 $1546.45. What was the rate per cent of commission ? 
 
 6. What are the net proceeds from the sale of 2250 bbl. 
 of flour at $6.25 a barrel, if the charge for freight is 50 
 cents a barrel, the commission for selling 2%, and the 
 commission for guaranteeing payment 1£% ? 
 
 7. An agent sells 350 crates of peaches, at $2.60. If 
 the commission is 4-£-<j£, find the net proceeds. 
 
^ERCEtfTAGE. 251 
 
 8. An agent sells 420 acres of land, at $40 an acre, and 
 charges l\°f commission. What is his commission ? 
 
 9. An agent, charging k\°f commission, receives for 
 his services $313. Find the amonnt of his sales. 
 
 10. A merchant buys 730 yd. of carpeting at $1.25 a 
 yard, and pays his agent f of l°/ commission. If the 
 freight amounts to $23.58, at what price per yard must he 
 sell the carpeting to gain 20% ? 
 
 11. An agent sells a consignment of goods for $2100. 
 He pays $33.50 for freight, and remits $2024.50. Find 
 his rate of commission. 
 
 12. An agent sells 5000 lb. of cotton at 14 cents a pound, 
 charging 2% commission. With the net proceeds he buys 
 cotton cloth at 10 cents a yard, charging \\°f commission. 
 How many yards of cloth does he buy ? 
 
 13. An agent sold 500 bbl. of flour at $5.50 a barrel, and 
 charged 2\°j commission ; the expenses for freight, etc., 
 were $250. With the net proceeds he bought sugar at 4f 
 cents a pound, charging 2-J-% commission. How much 
 sugar did he buy, and what was his total commission ? 
 
 14. A collector's commission for collecting taxes, at 
 1£%, is $206.55. What sum did he collect ? 
 
 15. An agent received $2961 to purchase goods, and 
 charged 5% commission. What was his commission? 
 
 l£. An agent buys 3100 bbl. of flour at $4.50 a barrel, 
 and charges \\°j commission. What is his commission? 
 
 17. A broker receives $6150 to invest in cotton, at 7f 
 cents a pound. If his commission is 2£%, how many 
 pounds of cotton can he buy ? 
 
 18. An agent sells 1100 bbl. of flour at $4.50 a barrel, 
 and charges 2-j-% commission. He invests the proceeds in 
 steel at \.\ cents a pound, charging l-j-% commission. What 
 is his entire commission, and how many long tons of steel 
 does he buy ? 
 
252 PERCENTAGE. 
 
 Insurance. 
 
 433. Insurance is the guarantee by an insurance com- 
 pany of the payment of a stated amount to the person 
 insured, in the event of loss of property by fire, by storm 
 at sea, or by other specified disaster ; or in the event of 
 the death of the person insured, or of an accident to him. 
 
 434. The policy is the written agreement. 
 
 The face of the policy is the sum named therein which 
 is to be paid to the holder in case of loss. 
 
 435. The premium is the sum paid for the insurance. 
 It is reckoned as a percentage on the face of the policy. 
 
 436. Insurance companies usually insure £ or £ of the 
 value of the property ; and, in case of partial loss, pay 
 only the value of the property destroyed. 
 
 Note. Insurance agents are also called underwriters, and insur- 
 ing is often called underwriting. 
 
 Exercise 115. 
 
 1. Find the premium of the fire insurance on a house 
 for $2650 at£of 1%. 
 
 2. Find the premium for insuring a man's life for 
 $2500, at an age for which the rate is 2J%. 
 
 3. At 6£ %, what premium will be paid on a vessel 
 worth $36,400, insured for £ its value ? 
 
 4. A vessel worth $16,000 is insured for £ its value at 
 ?i%. What is the premium ? 
 
 5. The premium of insurance at 1£% * s $150. What 
 is the amount insured ? 
 
 6. A vessel valued at $128,000 is insured for £ its 
 value at 3 J °f . What is the net loss to the owners, if the 
 vessel is destroyed during the third year after it is insured ? 
 
PERCENTAGE. 253 
 
 7. A building worth $7500 is insured at f its value, at 
 ■J of 1% per annum. What is the annual premium ? 
 
 8. Four companies insure a store and contents for 
 $60,000. One company takes $20,000, at £ of 1% ; a 
 second takes $10,000, at f of 1% ; a third, $15,000, at { 
 of l°/ ; a fourth, the remainder, at ^ of 1%. What is 
 the premium ? 
 
 9. If the store of Ex. 8 is damaged to the extent of 
 $4500, what amount does each company pay ? 
 
 10. A man insures his life for $10,000, paying $350 
 a year in advance, and dies the day before the fifth pre- 
 mium is due. The company pays his widow $10,000. 
 How much has the company lost by him, if the interest 
 gained on the premiums paid amounts to $175 ? 
 
 11. A merchant shipped a cargo to London, and took a 
 policy of $100,800 at 3-J-%, to cover both the cargo and the 
 premium. Find the value of the cargo. 
 
 Hint. 100% of policy = policy (cargo and premium). 
 
 2»\% of policy = premium. 
 96£% of policy = cargo. 
 
 12. Three companies insure, at f its value, a building 
 worth $16,000. The first company takes \ the risk at f 
 of \of ; the second, § at { of 1% ; and the third, the 
 remainder at f of l°f . Find the total premium. 
 
 13. S. Williams pays $18.40 premium for insuring his 
 house for J its value at l-j-%. What is the value of his 
 house ? 
 
 14. Find the annual premium for an ordinary life policy 
 of $5000 issued to a man 30 years old, if the rate of insur- 
 ance is 1.93%. 
 
 15. What is the annual premium for an ordinary life 
 policy of $12,000 issued to a man 40 years old, if the rate 
 of insurance is 2.661% ? 
 
254 PERCENTAGE. 
 
 Direct Taxes. 
 
 437. Taxes are levied for the support of the govern- 
 ment, for the support of schools, and for other purposes. 
 
 438. Direct taxes are laid on the person or on the 
 value of the property he possesses. 
 
 439. A poll tax is a tax levied upon a person, and a 
 property tax is a tax levied on property. 
 
 440. Assessors are officers whose duty it is to appraise 
 the property belonging to each person to be taxed. 
 
 441. A tax collector is an officer who collects the taxes. 
 He receives a salary, or a per cent of the sum collected. 
 
 442. The treasurer receives the money collected, and is 
 usually paid a salary. 
 
 443. The tax to be raised on the assessed valuation of 
 property is the total amount of tax voted, diminished by 
 the poll taxes, and by such corporation taxes as are col- 
 lected by the state and distributed to the several towns. 
 
 A tax of $18,000 is levied upon a town which contains 
 800 polls, assessed at $1.50 each, and which lias taxable 
 property assessed at $1,200,000. The town receives from 
 the state $3600 as its share of corporation taxes. Find 
 the rate of taxation, and the tax paid by Mr. Brown, if his 
 property is assessed at $5960, and he pays for one poll. 
 
 Solution. The amount of poll taxes = 800 X §1.50 = $1200; and 
 the amount from state and polls = $3600 + $1200 = 84800. 
 
 The amount levied on property = $18,000 - $4800 = $13,200. 
 
 The rate = $13,200 -r 1,200,000 = $0,011, or $11 on $1000. 
 
 To provide for contingencies, such as abatement of taxes, cost of 
 collecting, etc., the assessors make the rate $12 on $1000. 
 
 Therefore, Mr. Brown's property tax is 0.012 of $5960 = $71.52, 
 and his total tax = $71.52 + $1.50 = $73.02. 
 
PERCENTAGE. 
 
 255 
 
 444. To facilitate the making of taxes, the assessors 
 usually prepare a table like the following, which is com- 
 puted at 12 mills on a dollar : 
 
 Table. 
 
 Prop. 
 
 Tax. 
 
 Prop. 
 
 Tax. 
 
 Prop. 
 
 Tax. 
 
 Prop. 
 
 Tax. 
 
 $1 
 
 $0,012 
 
 $10 
 
 $0.12 
 
 $100 
 
 $1.20 
 
 $1000 
 
 $12.00 
 
 2 
 
 0.024 
 
 20 
 
 0.24 
 
 200 
 
 2.40 
 
 2000 
 
 24.00 
 
 3 
 
 0.036 
 
 30 
 
 0.36 
 
 300 
 
 3.60 
 
 3000 
 
 36.00 
 
 4 
 
 0.048 
 
 40 
 
 0.48 
 
 400 
 
 4.80 
 
 4000 
 
 48.00 
 
 6 
 
 0.060 
 
 50 
 
 0.60 
 
 500 
 
 6.00 
 
 5000 
 
 60.00 
 
 6 
 
 0.072 
 
 60 
 
 0.72 
 
 600 
 
 7.20 
 
 6000 
 
 72.00 
 
 7 
 
 0.084 
 
 70 
 
 0.84 
 
 700 
 
 8.40 
 
 7000 
 
 84.00 
 
 8 
 
 0.096 
 
 80 
 
 0.96 
 
 800 
 
 9.60 
 
 8000 
 
 96.00 
 
 9 
 
 0.108 
 
 90 
 
 1.08 
 
 900 
 
 10.80 
 
 9000 
 
 108.00 
 
 445. Example. Find by the above table the tax of 
 Mr. Gay, who pays for one poll at $1.50, and has property 
 assessed at $6545. 
 
 Solution. 
 
 Tax on 
 Tax on 
 Tax on 
 Tax on 
 
 : $72.00 
 
 500 = 6.00 
 
 40 s= 0.48 
 
 5 = 0.06 
 
 Tax on poll = 1.50 
 Total tax = $80.04 
 
 Exercise 116. 
 Make a table for a tax rate of 16 mills, 
 
 1. Find the tax on property assessed 
 
 2. Find the tax on property assessed 
 
 3. Find the tax on property assessed 
 
 4. Find the tax on property assessed 
 
 5. Find the tax on property assessed 
 
 6. Find the tax on property assessed 
 
 7. Find the tax on property assessed 
 
 and : 
 
 at $7500. 
 at $4825. 
 at $9685. 
 at $10,727. 
 at $12,863. 
 at $16,458. 
 at $38,249. 
 
256 PERCENTAGE. 
 
 8. James Brown is assessed $2500 on his real estate 
 and $5200 on his personal property, and pays for two polls 
 at $ 1.50 each. If the rate is $12.18 on $1000, what is his 
 total tax ? 
 
 9. If the tax rate of a town is $12.25 on $1000, and 
 the amount of the levy $11,788.50, what is the assessed 
 valuation of the town ? 
 
 10. If the assessed valuation of a town is $1,777,000, 
 and the levy is $29,231.65, what is the rate on $1000 ? 
 
 1 1. What sum must be assessed that $15,000 may remain 
 after paying 2% commission for collecting the taxes ? 
 
 12. For building a schoolhouse a tax of $1857.60 was 
 levied upon a school district, assessed valuation $1,935,000. 
 What was the tax on property assessed at $6250 ? 
 
 13. In a certain town there are 1350 polls. The assessed 
 valuation of the real estate is $713,250, and of the personal 
 property is $738,954. The poll tax is $2 per poll, and 
 the tax on property is 1£%. Only 96% of the property 
 tax can be collected, and the collector is paid 2£% of the 
 amount collected. How much does the town receive from 
 the taxes ? How much does the collector receive for his 
 services ? 
 
 Indirect Taxes. 
 
 446. Indirect Taxes are taxes levied upon articles of 
 merchandise. 
 
 447. Indirect taxes are of two kinds : Customs or 
 Duties, taxes levied on certain imported goods ; Excises 
 or Internal Revenue, taxes levied on certain domestic 
 goods. 
 
 Note. These are called Indirect Taxes, because, while paid to the 
 government by the person in whose possession the goods are first 
 found, this tax forms a part of the price finally paid by the consumer, 
 and is, therefore, a tax paid by him. 
 
PERCENTAGE. 257 
 
 448. Duties are of two classes : Specific and Ad Valorem. 
 
 449. Specific Duties. A specific duty is a definite sum 
 levied on each unit by which the article is measured or 
 weighed, without regard to the value of the unit. 
 
 450. Ad Valorem Duties. An ad valorem duty is 
 levied as a certain per cent of the cost of the article in 
 the country where purchased. The cost of the article 
 includes the cost of transportation, commissions, etc. 
 
 451. On merchandise imported at a specific duty the 
 following allowances are made : 
 
 Tare is an allowance for the weight of the box, cask, 
 bag or other wrapping. 
 
 Breakage is an allowance of a certain per cent on liquors 
 in bottles ; also on glassware and china. 
 
 Leakage is an allowance of a certain per cent on liquors 
 in barrels or casks. 
 
 Duties are computed on the sum left after all deductions 
 are made. 
 
 Note. The long ton only is used in computing duties. 
 
 452. Excises are taxes or licenses for the manufacture 
 or sale of certain domestic articles, as tobacco, whiskey, 
 beer, etc. 
 
 Exercise 117. 
 
 In the following examples the rates of duty are taken from the 
 Dingley Tariff Law, in effect July 24, 1897. 
 
 1. What is the duty at 2 J- cents a pound on 320 boxes 
 of raisins each containing 40 pounds ? 
 
 2. What is the duty at 6 cents a gallon on 420 hhd. of 
 best molasses of 63 gal. each ? 
 
 3. What is the duty at $4 a dozen bottles on 50 cases 
 of champagne, each containing 24 pint bottles, if breakage 
 of 5 of is allowed ? 
 
258 PERCENTAGE. 
 
 4. Find the duty on 150 gross of spectacles, cost price 
 $1.20 a dozen ; specific duty 45 cents a dozen, breakage 
 allowed 2£% ; an <l 20% ad valorem. 
 
 5. Find the duty on 100 shotguns, cost price $8.50 
 each ; specific duty of $4 each, and 15% ad valorem. 
 
 6. Find the duty at $1 per M on 12,500 ft. of white- 
 wood boards, planed on one side, if an additional duty of 
 50 cents per M is collected for each side planed. 
 
 7. Find the duty on 500 boxes of cigars, gross weight 
 475 lb., tare 40%, costing 82-j- cents per box in Havana. 
 Specific duty $4.50 per pound ; and 25% ad valorem. 
 
 8. Find the duty on 400 pairs of woolen blankets, cost 
 price $1.75 per pair; weighing 7£ lb. per pair, tare 5%. 
 Specific duty 33 cents per pound, ad valorem 40%. 
 
 9. Find the duty on 12 boxes of skein silk, each box 
 weighing 40 lb.; cost price $2,125 per pound, tare 10%. 
 Specific duty 50 cents per pound, ad valorem 15%. 
 
 10. Find the duty on 150 gross of clay tobacco pipes, 
 cost price 55 cents a gross. Specific duty 15 cents a gross, 
 and 25% ad valorem. 
 
 11. A New York merchant bought in London 400 gal. 
 of cologne at $1.25 a gallon, and commission and other 
 expenses amounted to $56.25. At what price per pint 
 must he sell the cologne to gain 40% on the cost, if he 
 paid a specific duty of 60 cents a gallon, and an ad valorem 
 duty of 45% ? 
 
 12. Find the duty on 750 lb. of glue, cost price 40 cents ; 
 specific duty of 15 cents a pound, tare 2% ; and ad valorem 
 duty of 25%. 
 
 13. A Boston merchant bought in Sheffield 50 gross 
 of razors at a net price of $4.25 a dozen. At what price 
 per dozen must he sell the razors to gain 33£% on the net 
 cost, if he paid a specific duty of $1.75 a dozen, and an ad 
 valorem duty of 20% ? 
 
CHAPTER XIII. 
 
 INTEREST AND DISCOUNT. 
 
 Simple Interest. 
 
 453. Interest is money paid for the use of money. 
 
 454. Principal. The sum loaned is the principal. 
 
 455. Rate of Interest. The rate of interest is a specified 
 per cent, called rate per cent, of the principal for one year. 
 
 456. Amount. The sum of the principal and interest is 
 the amount. 
 
 Note. In the applications of percentage considered in Chapter 
 XII no reference has been made to time. In all applications of per- 
 centage that relate to the use of money the elements of time and rate 
 are considered. 
 
 457. Example. Find the interest on $1024 at 5£% for 
 2 yr. 8 mo. 
 
 Solution. 2 yr. 8 mo. = 2£ yr. 
 
 Interest on $1024 for 1 yr. =a &*% of $1024 = $56.32. 
 
 Interest on $1024 for 2f yr. = 2f X $56.32 = $150.19. 
 
 Exercise 118. 
 Find the interest on : 
 
 1. $125.65 for 1 mo. at 6%. 3. $1296.50 for 2 mo. at 5£%. 
 
 2. $1165 for 3 yr. at 5%. 4. $630.50 for 3 yr. at 4%. 
 
 5. $231.50 for 3 yr. 8 mo. at 4£%. 
 
 6. $580.40 for 2 yr. 4 mo. at 6%. 
 
 7. $285.85 for 1 yr. 7 mo. at 4%. 
 
 8. $1275.35 for 3 yr. 2 mo. at 3|%. 
 
260 
 
 INTEREST AND DISCOUNT. 
 
 Six Per Cent Method. 
 
 458. The interest at six per cent on one dollar for one 
 year is 6 cents ; for one month is T ^ of 6 cents, or £ cent ; 
 for one day is ^ of \ cent, or ^ cent, that is, £ mill. 
 
 The interest at six per cent on any number of dollars is 
 that number times the interest on one dollar. Hence, 
 
 459. To Find the Interest at Six Per Cent on any Prin- 
 cipal for Time expressed in Years, Months, and Days, 
 
 Multiply six cents by the number of years, one half cent by 
 the number of months, one sixth mill by the number of days. 
 Take the sum of these products, and multiply this sum by 
 the number of dollars in the principal. 
 
 460. Example. 
 9 mo. 20 dy., at 6 
 
 Int. on ft 1. 
 
 For 3 yr. = 3 X §0.06 
 For 9 mo. = 9 X 0.005 
 For 20 dy. = 20 X 0.000; 
 For 3 yr. 9 mo. 20 dy. 
 
 Find the interest on $320 for 3 yr. 
 
 = $0.18 
 
 Int. on $120. 
 $0,228* 
 320 
 106* 
 456 
 684 
 $73.07 
 
 119. 
 
 Condensed Work. 
 
 3 yr. 9 mo. 20 dy. 
 
 s= 0.045 
 = 0.003* 
 
 $0.18 0.045 0.003* 
 0.045 
 
 = $0,228* 
 
 0.003* 
 
 
 $0,228* 
 320 
 
 Exercise 
 
 
 Find the interest at 6% on : 
 
 1. $744.20 for 3 yr. 6 mo. 18 dy. 
 
 2. $625.44 for 6 yr. 7 mo. 12 dy. 
 
 3. $124.87 for 2 yr. 10 mo. 16 dy. 
 
 4. $847.64 from Jan. 12, 1896 to Aug. 7, 1899. 
 
 5. $84.84 from Mar. 22, 1895 to Jan. 1, 1898. 
 
 6. $1248.27 from Apr. 7, 1894 to May 17, 1897. 
 
 461. Short time notes generally run for 1, 2, 3, or 4 
 months ; or for 30, 60, or 90 days. 
 
INTEREST AND DISCOUNT. 261 
 
 462. To Find Interest at 6% for Time in Months, 
 
 Move the decimal point in the principal two places to the 
 left, and multiply by half the number of months. 
 
 463. To Find Interest at 6% for Time in Days, 
 
 Move the decimal point in the principal three places to the 
 left, and multiply by one sixth the number of days. 
 
 Thus, the interest at 6% on $600 for 4 mo. is 2 X $6.00, or $12 ; 
 and for 30 dy. is 5 X $0,600, or $3. 
 
 Exercise 120. 
 Find the interest at 6 % on : 
 
 1. $1278.75 for 1 mo.; 2 mo.; 3 mo.; 4 mo. 
 
 2. 1 2265.50 for 1 mo. ; 2 mo.; 3 mo.; 4 mo. 
 
 3. $1840.25 for 30 dy.; 60 dy.; 90 dy. 
 
 4. $1946.75 for 30 dy.; 60 dy.; 90 dy. 
 
 Six Per Cent Method for Other Rates. 
 
 464. Example. Find the interest on $73.42 for 5 yr. 
 8 mo. 16 dy., at l%°f . 
 
 Solution. 5 yr. 8 mo. 16 dy. $0,342$- 
 
 $0.30 0.04 0.002* 73.42 
 
 0.04 6 ) $25. 158 
 
 0.002 * $4,193 
 
 $0,342$ Multiply by 7£ 
 
 $31.45 
 
 465. To Find Interest at Rates other than 6%, 
 
 Divide the interest at six per cent by six and multiply the 
 quotient by the given rate. Or, 
 
 Take such a part of the interest at six per cent as the 
 given rate is of six per cent. 
 
 For 7%, we add \ of the interest at 6% to the interest at 6% ; for 
 7|%, we add \ ; for 8%, we add | ; for 9%, we add \ ; for 5%, we sub- 
 tract \ ; for 4|%, we subtract \ ; for 4%, we subtract \. 
 
262 INTEREST AND DISCOUNT. 
 
 Exercise 121. 
 Find the interest on : 
 
 1. $680.40 for 2 yr. 4 mo. 6 dy., at 6%. 
 
 2. $25.62 for 30 dy., at 6%. 
 
 3. $85.85 for 1 yr. 7 mo. 21 dy., at 6%. 
 
 4. $1100 for 3 yr. 4 mo., at 5%. 
 
 5. $1275 for 3 yr. 2 mo. 15 dy., at 8%. 
 
 6. $475.16 for 27 dy., at 4£%. 
 
 7. $1290.50 for 60 dy., at 6%. 
 
 8. $125 for 1 yr. 2 mo. 2 dy., at 9%. 
 
 9. $250.80 for 10 mo. 10 dy., at 3|%. 
 
 10. $258.85 from Mar. 6 to June 24, at 5%. 
 
 11. $380 for 2 yr. 11 mo. 27 dy., at 4£%. 
 
 12. $475.05 for 1 yr. 9 mo. 14 dy., at 7 T \%. 
 
 13. $725.40 for 11 mo. 24 dy., at 5±%. 
 
 14. $680.50 for 2 yr. 6 dy., at 5%. 
 
 15. $630.50 for 90 dy., at 6%. 
 
 16. $547.60 from Feb. 20 to Dec. 5, at 6£%. 
 
 17. $875 from May 5, 1897 to June 21, 1898, at 5£%. 
 
 18. $758.50 from Jan. 5 to July 1, at 4£%. 
 
 19. $342.42 from Feb. 5, 1897 to Mar. 15, 1899, at 7%. 
 
 20. $540 from Mar. 5 to Sept. 21, at 3£%. 
 Find the amount of : 
 
 21. $431.50 for 2 yr. 8 mo., at 4J%. 
 
 22. $476.50 from July 5, 1897 to Feb. 9, 1898, at 4%. 
 
 23. $319.20 from Apr. 7 to Aug. 31, at 3£%. 
 
 24. $6460 from June 15, 1897 to May 7, 1899, at 4J<&. 
 
 25. $150 from Aug. 5, 1897 to Mar. 17, 1899, at 7%. 
 
 26. $527.20 from Jan. 1 to Nov. 20, at 4£%. 
 
 27. $1250 from Nov. 15, 1897 to Mar. 1, 1898, at 5%. 
 
 28. $624.36 from Mar. 5 to Dec. 20, at 7^%. 
 
 29. $12,260 from May 6 to Oct. 24, at 3|%. 
 
 30. $11,216 from Oct. 20 to Dec. 31, at 1% a month. 
 
INTEREST AND DISCOUNT. 263 
 
 Other Interest Problems. 
 
 466. In interest problems four elements are considered : 
 principal, rate per cent, time, and interest or amount. 
 
 In the problems already considered the first three ele- 
 ments are given to find the fourth. 
 
 If p stands for the principal, r for the rate per cent, t for 
 the time expressed in years, i for the interest, and a for the 
 amount, we have 
 
 1. prt = i. 2. p -\-prt = a. 
 
 In either equation, if three of the elements are given, the 
 other can be found. 
 
 467. Examples. 1. What principal will in 2 yr. 7 mo. 
 24 dy. produce $1006.47 interest, at 4|% ? 
 
 Solution. Divide by rt both sides of the equation 
 
 prt = i; 
 
 then p = — • 
 
 rt 
 
 HereZ, 2 yr. 7 mo. 24dy., = 2.65yr.; r, 4£%, = 0.045; i = $1006.47. 
 
 Hence ' ^ = $ 0.0 1 4 5 06 x 4 J.65 = $8440 - 
 
 2.' What principal will in 3 yr. 6 mo. amount to $748.41, 
 at 4% ? 
 
 Solution. Divide by 1 + rt both sides of the equation 
 p + prt = a ; 
 a 
 
 then p 
 
 1 +rt 
 
 Here a = $748.41 ; r, 4%, = 0.04 ; *, 3 yr. 6 mo., = 3.5 yr.; and 
 1 '+ rt = 1 + 0.04 X 3.5 = 1 + 0.14 = 1.14. 
 
 Hence, p - $ ^yrr^ = $656. 50. 
 
264 INTEREST AND DISCOUNT. 
 
 3. At what rate per cent will $8440 produce $1006.47 
 interest in 2 yr. 7 mo. 24 dy.? 
 
 Solution. Divide by pt both sides of the equation 
 prt = i; 
 
 i 
 then r = — • 
 
 pt 
 
 Here i = $1006.47 ; p = $8440 ; t, 2 yr. 7 mo. 24 dy., = 2.65 yr. 
 
 1006.47 . nnA . 
 
 HenCG ' T = 8440 X 2.65 - 0045 ' 
 
 Therefore, the rate required is 4£%. 
 
 4. In what time will the interest on $8440 at 4£% 
 amount to $1006.47 ? 
 
 Solution. Divide by pr both sides of the equation 
 
 prt = i; 
 
 i 
 then t = — 
 
 pr 
 
 Here i = $1006.47 ; p = $8440 ; r, 4*%, = 0.045. 
 
 . 1006.47 onc 
 
 HeDCe ' ' = 8440 X 0.045 = 2M ' 
 
 Therefore, the time required is 2.65 yr., or 2 yr. 7 mo. 24 dy. 
 
 468. From these examples we have the following rule 
 for finding the principal, rate, or time : 
 
 Divide the given interest (or amount) by the interest (or 
 amount) obtained when the required element is represented by 
 a unit. 
 
 The unit of this rule is $1 in finding the principal ; 1% in finding 
 the rate ; 1 yr. in finding the time. 
 
 Exercise 122. 
 Find the rate per cent : 
 
 1. When the interest on $326 for 15 yr. is $220.05. 
 
 2. When the interest on $745 for 18 yr. is $603.45. 
 
 3. When $980 amounts to $1016.75 in 9 mo. 
 
INTEREST AND DISCOUNT. 265 
 
 « Find the rate per cent : 
 
 4. When the interest on $470.50 is $141.15 for 5 yr. 
 
 5. When $3631.25 amounts to $3715.98 for 7 mo. 
 
 6. When the interest on $997.75 is $199.55 for 5 yr. 
 4 mo. 
 
 7. When $350 amounts $406.70 for 3 yr. 7 mo. 6 dy. 
 
 8. When the interest on $6875 is $68.75 for 90 dy. 
 
 9. When the interest on $642 is $10.70 for 5 mo. 
 
 10. When the interest on $8432 for 2 yr. 7 mo. 23 dy. 
 is $1339.28. 
 
 11. When a sum of money is doubled in 14 yr. 
 
 12. When an investment for 4 yr. 2 mo. produces a sum 
 equal to / ¥ of the capital. 
 
 13. When an investment for 3 yr. 1 mo. 15 dy. produces 
 a sum equal to ^ of the capital. 
 
 Find the time in which the : 
 
 14. Interest on $450 will amount to $72, at 4%. 
 
 15. Interest on $487.50 will amount to $39, at 4%. 
 
 16. Interest on $238.75 will amount to $64.46, at 4£%. 
 
 17. Sum of $1587.75 will amount to $1611.68, at 5£%. 
 
 18. Sum of $1 will double itself at 4%. 
 
 19. Sum of $10 will amount to $17, at 6%. 
 
 20. Sum of $502.67 will amount to $578.07, at 4£%. 
 
 21. Interest on $537.50 will amount to $80.62, at 4%. 
 
 22. Interest on $6875 will amount to $75.05, at 4£%. 
 
 23. Interest on $8520 will amount to $1746.60, at 6%. 
 Find the principal that will : 
 
 24. Produce $90 interest in 3 yr., at 4%. 
 
 25. Produce $63 interest in 3 yr., at 6£%. 
 
 26. Produce $100 interest in 8 yr. 6 mo., at 5%. 
 
 27. Produce $1746.60 interest in 3 yr. 5 mo., at 6%. 
 
 28. Produce $12 interest in 7 mo., at 5%. 
 
 29. Produce $50 interest in 228 dy., at 4£%. 
 
266 INTEREST AND DISCOUNT. 
 
 Find the principal that will : 
 
 30. Produce $ 1339.28 interest in 2 yr. 7 mo. 24 dy., at 
 6%. 
 
 31. Produce $1312.65 interest in 2 yr. 3 mo., at 6%. 
 
 32. Produce $750 interest in 3 yr. 8 mo., at 5%. 
 
 33. Amount to $840 in 3 yr., at 4%. 
 
 34. Amount to $90,113.84 in 2 yr. 6 mo., at 4£%. 
 
 35. Amount to $6000 in 21 dy., at 5%. 
 
 36. Amount to $297.60 in 8 mo., at 6%. 
 
 37. Amount to $6378.75 in 1 .yr. 1 mo., at 5%. 
 
 38. Amount to $21,047.95 in 1 yr. 7 mo. 21 dy., at 4£%. 
 
 39. Amount to $185.09 in 2 yr. 3 mo. 18 dy., at 5%. 
 
 40. Amount to $659.40 in 2 yr. 11 mo. 15 dy., at 6%. 
 
 41. Amount to $94,375.16 in 2 yr. 7 mo. 24 dy., at 4£%. 
 
 42. Amount to $10,266.60 in 3 yr. 5 mo., at 6%. 
 
 43. Find the interest on $195 for 2 yr. 2 dy., at 6£%. 
 
 44. At what rate per cent will $1025.20 produce $25.72 
 interest in 4 mo. 9 dy.? 
 
 45. The principal is $653; the interest, $5.52; the 
 rate, 8%. Find the time. 
 
 46. Find the amount of $520 for 2 mo. 3 dy., at 4£%. 
 
 47. What sum bearing interest at \\°j will yield an 
 annual income of $1000 ? 
 
 48. In what time will $4000 amount to $4625, at 5£% ? 
 
 49. At what rate per cent will $3000 produce $250 
 interest in 1 yr. 10 mo. 7 dy.? 
 
 50. Find the interest on $1721.84 from April 1 to 
 Nov. 12, at 4£%. 
 
 51. How long must $3904.92 be on interest to amount 
 to $4568.76, at 5% ? 
 
 52. Find the interest on $137.60 from July 3 to Dec. 12, 
 at 7A%. 
 
 53. Find the interest on $680.20, at 7£%, for 73 dy., 
 reckoning 365 dy. for a year. 
 
INTEREST AND DISCOUNT. 267 
 
 Promissory Notes. 
 
 469. Promissory Notes. A written promise to pay a 
 specified sum of money on demand or at a specified time is 
 called a promissory note, or simply a note. 
 
 A note may be made payable to bearer, to a person named in the 
 note, or to the person named or his order. 
 
 The maker or drawer is the person who signs the note. 
 
 The payee or drawee is the person to whom the note is payable. 
 
 The holder of a note is the person who has lawful possession of it. 
 
 The face of a note is the sum of money named in it. 
 
 A negotiable note is a note that can be sold and transferred to any 
 one else by the holder. A note is not negotiable unless made payable 
 to bearer, or to the order of the person named in the note. 
 
 470. Forms of Notes. 
 
 1. $875.25. Boston, Mass., Nov. 1, 1897. 
 Four months after date, I promise to pay Benjamin Parker 
 
 Three Hundred Seventy-five and T 2 ^ Dollars, with interest at 
 5%, for value received. James Overton. 
 
 2. $375.25. Boston, Mass., Nov. 1, 1897. 
 Four months after date, I promise to pay Benjamin Parker, 
 
 or bearer, Three Hundred Seventy-five and t 2 q 5 ^ Dollars, with 
 interest at 5°J ,for value received. James Overton. 
 
 3. $375.25. Boston, Mass., Nov. 1, 1897. 
 Four months after date, I promise to pay Benjamin Parker, 
 
 or order, Three Hundred Seventy-five and y 2 ^ Dollars, with 
 interest at 5°] ,for value received. James Overton. 
 
 4. $375.25. Boston, Mass., Nov. 1, 1897. 
 On demand, I promise to pay Benjamin Parker, or order, 
 
 Three Hundred Seventy-five and -fife Dollars, with interest 
 at 5%, for value received. James Overton. 
 
268 INTEREST AND DISCOUNT. 
 
 471. An endorser of a note is a person who writes his 
 name on the back of a note. 
 
 An endorser of a note becomes responsible for its payment, unless 
 he writes above his signature the words without recourse. 
 
 A note that contains the words or bearer can be collected by any 
 one who has lawful possession of it. 
 
 A note that contains the words or order becomes payable to bearer 
 when the payee merely writes his name across the back. This is 
 called an Endorsement in blank. 
 
 In form 3, if Benjamin Parker sells the note to James Whitney he 
 writes on the back Pay to the order of James Whitney and signs his 
 name, or else he simply endorses in blank. 
 
 Notes that contain the words on demand are called Demand Notes, 
 and are due whenever payment is demanded; see form 4. Other 
 notes are called Time Notes. The day on which a time note is legally 
 due is called the day of its maturity. 
 
 Notes containing the words with interest bear interest from date 
 until paid. Notes not containing these words begin to bear interest 
 from the time they are due at the legal rate, if not paid when due. 
 
 Legal rate of interest is the rate established by law in the state 
 where the note is made. 
 
 Notes should contain the words value received, otherwise it may be 
 necessary to prove the note was given for a valuable consideration. 
 The face of a note, except the cents, should be expressed in words. 
 
 
 
 Exercise 123. 
 
 
 
 
 Find the day of maturity, and amount due, having given : 
 
 
 Face of Note. 
 
 Date of Note. 
 
 Time. 
 
 Rate of Int. 
 
 1. 
 
 $530.25, 
 
 Jan. 12, 1897, 
 
 60 dy., 
 
 6%. 
 
 2. 
 
 $687.45, 
 
 Mar. 22, 1897, 
 
 90 dy., 
 
 5%. 
 
 3. 
 
 $286.75, 
 
 Aug. 5, 1897, 
 
 4 mo., 
 
 4%. 
 
 4. 
 
 $944.40, 
 
 Oct. 20, 1897, 
 
 3 mo., 
 
 H%- 
 
 5. 
 
 $1262.72, 
 
 Oct. 5, 1897, 
 
 30 dy., 
 
 mo. 
 
 6. 
 
 $1875.44, 
 
 Dec. 16, 1897, 
 
 6 mo., 
 
 4%. 
 
 7. 
 
 $1521.87, 
 
 Apr. 30, 1897, 
 
 1 mo., 
 
 6%- 
 
 8. 
 
 $2849.65, 
 
 May 22, 1897, 
 
 2yr., 
 
 3£%. 
 
 9. 
 
 $1968.10, 
 
 July 10, 1897, 
 
 2 mo., 
 
 4i%- 
 
INTEREST AND DISCOUNT. 269 
 
 Find the amount due Dec. 3, 1898, on the following 
 demand notes : 
 
 10. $875.18. Concord, K H., May 10, 1897. 
 
 On demand, I promise to pay George H. Chick, or order, 
 Eight Hundred Seventy-five and -J^ Dollars, with interest 
 at 5%. Value received. Frederick D. Sibley. 
 
 11. $642.75. Lakewood, N. J., Oct. 25, 1897. 
 
 On demand, I promise to pay Harry Jones, or order, Six 
 Hundred Forty-two and T y^ Dollars, with interest at 4-J-%. 
 Value received. - George B. Atkins. 
 
 12. $1286.50. Atlanta, Ga., Apr. 22, 1897. 
 
 On demand, I promise to pay Clarence E. Garland, or 
 order, Twelve Hundred Eighty-six and T 5 ^ Dollars, with 
 interest at h\°f . Value received. Kobert Page. 
 
 13. $2548.25. St. Paul, Minn., June 17, 1897. 
 On demand, I promise to pay Fred Lacey, or order, 
 
 Twenty-five Hundred Forty-eight and j% 5 ^ Dollars, with 
 interest at 7%. Value received. William P. Wissman. 
 
 14.' $418.33. Oakland, Cal., Dec. 23, 1897. 
 
 On demand, I promise to pay Albert J. Farnham, or 
 order, Four Hundred Eighteen and T 3 ^ Dollars, with 
 interest at 4^%. Value received. Austin C. Wiggin. 
 
 15. $7486.45. Watertown, Ia., Apr. 16, 1898. 
 
 On demand, I promise to pay Harry D. Smith, or order, 
 Seven Thousand, Four Hundred Eighty-six and fify Dollars, 
 with interest at 5%, Frank j. Leavitt, 
 
270 INTEREST AND DISCOUNT. 
 
 Bank Discount. 
 
 472. Bank discount is a deduction from the amount of 
 a note or draft for cashing it before it is due. 
 
 473. The term of discount is the time from the day of 
 discounting the note to and including the day of maturity 
 of the note. 
 
 474. Days of Grace. Three days are allowed in addi- 
 tion to the time stated in the note "before the note is legally 
 due, unless the note contains the words without grace, or is 
 made in states that have abolished days of grace by statute. 
 
 475. A protest is a notice in writing by a notary public 
 to the endorsers that a note has not been paid. If a note 
 is not protested at the end of the day of its maturity the 
 endorsers are released from their obligation. 
 
 476. Bank discount is the interest for the term of dis- 
 count at. a specified rate on the amount of the note. 
 
 477. If a note draws interest, the amount of the note is 
 the face and interest. 
 
 478. Exchange. A charge is sometimes made for col- 
 lecting, if the place of payment of the note or draft is not 
 the place of discount. This charge is called exchange. 
 
 The rate of exchange is a number of cents on $1000 for 
 large amounts ; and for small amounts from \ to £ of 1% 
 of the face value of the note or draft, a fraction of $100 
 being reckoned as $100. 
 
 Note. The rate of exchange is never higher than the cost of send- 
 ing the money by express. 
 
 479. A bank draft is a written order from a bank 
 requesting another bank to pay a specified sum of money 
 to the order of the person named in the draft. Thus, 
 
INTEREST AND DISCOUNT. 271 
 
 No. 27861 
 
 <§'\x*t (Uationae (ganH of (guffafo* 
 
 Buffalo, N.Y. &e&. /5, 1897. 
 Pay to the order of fo-hn, /?. RUkcvuU $/ 72.1*5. 
 Q.n& Jvzi'vigUl&cI Qs&v-&nZu-tw~o oavcL ^ Dollars. 
 
 TO THE PARK NATIONAL BANK, \ g, &. l^t&iwmA, 
 
 NEW YORK CITY. ! Cashier , 
 
 480. A check is a draft upon a bank where the maker 
 (or drawer) has money deposited. Thus, 
 
 No. 89 
 
 Boston, 
 
 Mass., 
 
 Jbee,. /5, 1897. 
 
 
 ^First National &ank of Sosfort. 
 
 Pay to the order of fahyv Jba& — 
 
 
 $8 6. £5. 
 
 (olg/itu- 
 
 &u?o cwul> 
 
 
 jf Dollars. 
 
 
 
 a. 
 
 cf. ^WAJbhe*. 
 
 481. Certificate of Deposit. A receipt for money depos- 
 ited jn a bank is called a certificate of deposit. Thus, 
 
 No. 28642 Certificate of Deposit. 
 
 Boston, Mass., h&&. /5, 1897, 
 yaAru&fa @s. jC&CLwJX/ has deposited r in this Bank 
 
 3\/u& hwyicOb&cl Dollars 
 
 payable to rw& order in current funds on the return of this cer- 
 tificate properly endorsed. 
 
 $600 . RL&kcVuL Ro&, Cashier. 
 
272 INTEREST AND DISCOUNT. 
 
 482. A certified check is a check upon the face of which 
 the cashier of the bank has stamped the word Certified, 
 with the date, the name of the bank, and has written his 
 signature as cashier. The bank is then responsible for its 
 payment. 
 
 483. A commercial draft is in effect a letter from one 
 person to another requesting him to pay a stated sum of 
 money to the bank named in the draft. The name of the 
 person on whom the draft is drawn is written at the lower 
 left-hand corner of the draft. These drafts are sent 
 through banks instead of through the mail ; and are pay- 
 able at sight, or at a specified time after sight. 
 
 Commercial drafts are employed by creditors to demand 
 payments and collect debts through banks. 
 
 No. 93 Springfield, Mass., Jbz&. /5, i%97. 
 
 At sight pay to the order of 
 
 €be gjcconti National 3Sanfe of ^prinfffielu f/OO.™ 
 
 €n& kwncU&cL ■ Dollars. 
 
 To 1&. /if. &owiojynve,, *\ <? /j- is 
 
 &ZO-. Jr. /TCvmML. 
 
 /q6 £ve,wi tit., cfi&w- 
 
 . 
 
 484. When the person on whom a time draft is drawn 
 accepts the draft, he writes in red ink the word Accepted 
 with the date across the face of the draft and signs his 
 name. The draft is then called an acceptance, and the 
 acceptor is responsible for its payment. 
 
 Note. An acceptance is like a promissory note, the acceptor and 
 the maker of the draft taking the place respectively of the maker and 
 the endorser of the note. 
 
INTEREST AND DISCOUNT. 273 
 
 485. The proceeds or avails of a note or draft is the 
 amount of the note or draft less the discount and exchange. 
 
 486. Examples. Find the day of maturity, the time to 
 run, the discount, and the proceeds of the following notes : 
 
 1. $680. Manchester, N. H., Oct. 5, 1897. 
 Sixty days after date, without grace, I promise to pay 
 
 F. R. Thompson, or order, Six Hundred Eighty Dollars, 
 value received. 
 
 Payable at the Amoskeag National Bank. 
 
 Discounted at 6%, Oct. 14. B. F. Shaw. 
 
 Solution. Counting 60 days from Oct. 5, we have 26 days in Oct. , 
 30 days in Nov., and 4 days in Dec. Therefore, the note is due Dec. 4. 
 
 The time to run (from Oct. 14, the day of discount) is 17 days in 
 Oct., 30 days in Nov., and 4 days in Dec, or 51 days. 
 
 The discount is the interest on $680 for 51 days at 6% ; or 8£ X 
 $0,680 = $5.78 (§ 463). 
 
 The proceeds is $680 — $5.78 = $674.22. 
 
 2. $840. Manchester, N. H., Oct. 5, 1897. 
 Three months from date, I promise to pay E. A. Jackson, 
 
 or order, Eight Hundred Forty Dollars, value received, with 
 interest at six per cent. 
 
 Payable at the Amoskeag National Bank. 
 
 Discounted at 6%, Nov. 2, 1897. G-. A. Batchelder. 
 
 SSlution. The interest on the note for 3 mo. 3 dy. is 840 X $0.0156 
 = $13.02 ; and the amount of the note is $840 + $13.02 = $853.02. 
 
 The day of maturity is Jan. 8, 1898. 
 
 The time to run is 28 days in Nov., 31 days in Dec, and 8 days in 
 Jan., or 67 days. 
 
 The discount on $853.02 for 67 days at 6% is 11| X $0.85302, or 
 $9.53. 
 
 The proceeds is $853.02 — $9.53 = $843.49. 
 
 Note. In Boston, and in many other places, when the time a 
 note has to run is expressed in months, the term of discount is com- 
 puted for this number of months, and not for the exact number of 
 days contained in the months. 
 
274 INTEREST AND DISCOUNT. 
 
 Exercise 124. 
 
 Find the day of maturity, the time to run, the discount, 
 and the proceeds of the following notes, without grace : 
 
 1. $750. New York, Jan. 1, 1897. 
 Four months from date, I promise to pay to the order of 
 
 James Fay Seven Hundred Fifty Dollars, value received. 
 Payable at the National Bank of the Republic. 
 Discounted at 5%, Jan. 12. John Pray. 
 
 2. $4325.50. Boston, Mar. 4, 1897. 
 Sixty days from date, I promise to pay to James Finn, or 
 
 order, Four Thousand Three Hundred Twenty-five and T %°<y 
 Dollars, value received. 
 
 Payable at the Merchants National Bank. 
 
 Discounted at 5£%, Mar. 8. George Bellows. 
 
 3. $1300. Richmond, Va., July 14, 1897. 
 Ninety days from date, I promise to pay to the order of 
 
 Peter Bright Thirteen Hundred Dollars, value received. 
 Payable at the First National Bank. 
 Discounted at 4%, Aug. 3. George Wright. 
 
 4. $1456.30. Charleston, S. C, Aug. 27, 1897. 
 Three months after date, I promise to pay to the order of 
 
 John George Fourteen Hundred Fifty-six and T %% Dollars, 
 value received. 
 
 Payable at the Second National Bank. 
 
 Discounted at 5%, Sept. 10. John Waldorf. 
 
 5. $4550.36. Baltimore, Md., Nov. 10, 1897. 
 Four months after date, I promise to pay to the order of 
 
 John Callender Four Thousand Five Hundred Fifty and 
 T 3 ^ Dollars, value received. 
 
 Payable at the National Mechanics Bank. 
 
 Discounted at 5£%, Nov. 24. James Barton. 
 
INTEREST AND DISCOUNT. 275 
 
 6. $5000. Chicago, III., Dec. 23, 1897. 
 Six months after date, we jointly and severally promise 
 
 to pay to John Adams, or order, Five Thousand Dollars, 
 value received, with interest at five per cent. 
 
 Payable at the Metropolitan National Bank. 
 
 Discounted at 4%, Jan. 21, 1898. William Dunn, 
 
 F. E. Crockett. 
 
 Find the day of maturity, the time to run, the discount, 
 and the proceeds of the following notes, with grace : 
 
 7. $4760. Milwaukee, Wis., Jan. 1, 1897. 
 Ninety days after date, I promise to pay to the order of 
 
 James Pike Four Thousand Seven Hundred Sixty Dollars, 
 
 value received. 
 
 , Payable at the Wisconsin National Bank. 
 
 Discounted at 4^-%, Feb. 15. William Clement. 
 
 8. $2017.85. St. Paul, Minn., Jan. 14, 1897. 
 Three months after date, I promise to pay to the order 
 
 of John Brown Two Thousand Seventeen and T 8 n 5 ^ Dollars, 
 value received. 
 
 Payable at the German- American National Bank. 
 
 Discounted at 7%, Mar. 1. Timothy Bruce. 
 
 9. $9040. Galveston, Tex., Jan. 19, 1897. 
 Sixty days from date, I promise to pay to the order 
 
 of Charles Carroll Nine Thousand Forty Dollars, value 
 received. 
 
 Payable at the First National Bank. 
 
 Discounted at 5£%, Feb. 16. James Monroe. 
 
 10. $215. Augusta, Me., Jan. 28, 1897. 
 Thirty days after date, I promise to pay to the order of 
 
 James Fogg Two Hundred Fifteen Dollars, value received. 
 Payable at the Maine National Bank. 
 Discounted at 6%, Feb. 3. John Moses. 
 
276 INTEREST AND DISCOUNT. 
 
 11. $2216.85. Omaha, Neb., Dec. 15, 1897. 
 Ninety days after date, I promise to pay to the order of 
 
 F. C. Green Two Thousand Two Hundred Sixteen and T 8 ^ 
 Dollars, value received. 
 
 Payable at the Omaha National Bank. 
 
 Discounted at 7%, Jan. 8, 1898. 
 
 W. C. Colburn. 
 
 Find the proceeds of the following drafts, with grace : 
 
 12. Draft for $620 at 60 days; rate of discount 6% ; 
 exchange £%. 
 
 Solution. The discount for 63 days is 10£ x §0.620, or $6.51 ; and 
 the exchange is \% of $700, or $0.88 (every fraction of $100 being 
 reckoned as $100). The total discount is, therefore, $6.51 + $0.88 
 = $7.39, and the proceeds is $620 — $7.39 = $612.61. 
 
 13. Draft for $890 at 90 days ; rate of discount 4£% ; 
 exchange £%. 
 
 14. Draft for $12,500 at 60 days ; rate of discount 5% ; 
 exchange 15 cents on $1000. 
 
 15. Draft for $1260 at 30 days ; rate of discount 5£% ; 
 exchange \cf . 
 
 16. Draft for $1430 at 3 months ; rate of discount 6% ; 
 exchange \o/ . 
 
 17. Draft for $1875 at 4 months ; rate of discount 5% ; 
 exchange \°f . 
 
 18. Draft for $22,843 at 60 days ; rate of discount 4£% ; 
 exchange 25 cents on $1000. 
 
 19. Draft for $18,000 at 2 months; rate of discount 
 5% ; exchange •£%. 
 
 20. Draft for $3437.50 at 90 days ; rate of discount 
 5% ; exchange ±%. 
 
 21. Draft for $1287.50 at 60 days; rate of discount 
 4£% ; exchange f %. 
 
 22. Draft for $866.65 at 3 months ; rate of discount 
 5% ; exchange £%. 
 
INTEREST AND DISCOUNT. 277 
 
 Present Worth and True Discount. 
 
 487. The present worth of a debt is the sum which, if 
 put at interest, will amount to the debt when due. 
 
 488. The true discount is the difference between a sum 
 due at some future time and the present worth of that sum. 
 
 489. Example. Find the present worth and true dis- 
 count of $515 due in 7 mo. 6 dy., if money is worth 5%. 
 
 Solution. The amount of $1 at 6% for 7 mo. 6 dy. is $1.03. 
 
 As $1.03 is the amount of $1, $515 is the amount of $j*.V|, or $500. 
 
 The true discount is $515 — $500 = $15. Hence, 
 
 490. To Find the Present Worth of a Given Sum of 
 Money Due at a Stated Future Time, 
 
 Divide the given sum by the amount of $1 for the given 
 rate and time. 
 
 Exercise 125. 
 
 1. Find the present worth of $500 due in 11 mo., if 
 money is worth 5°/ . 
 
 2. Find the present worth and discount of $3334.62 due 
 in 2 yr., if money is worth 4-J-%. 
 
 3. Find the present worth and discount of $4261.33 due 
 in l„yr. 6 mo., if money is worth 6%. 
 
 4. Find the present worth and discount of $2416.50 due 
 in 7 mo., if money is worth 5%. 
 
 5. Find the present worth of $678.40 due in 16 mo., if 
 money is worth k\°j ' 
 
 6. Find the present worth and discount of $574.17 due 
 in 2 yr. 3 mo., if money is worth h\°f . 
 
 7. Find the present worth and discount of $625.13 due 
 in 8 mo., if money is worth 4%. 
 
 8. Find the present worth and discount of $715.20 due 
 in 1 yr. 4 mo., if money is worth ?>\°J . 
 
278 INTEREST AND DISCOUNT. 
 
 Exact Interest. 
 
 491. By the ordinary method interest is computed on a 
 basis of 30 days for a month ; that is, 360 days for a year. 
 
 A year has 365 days. The interest for a year counted 
 by days would thus be § §#, or \\, of the true interest. 
 
 The exact interest for any number of days is, therefore, 
 found by diminishing interest found by the ordinary method 
 by t 1 ^ of itself. 
 
 492. To Find One Seventy-third of a Number, 
 Move the decimal point two places to the left, add a third 
 
 three times, each time one place further to the right. Carry 
 only to three decimal places. 
 
 493. Example. Find the exact interest on $12,762.44 
 from May 6 to Sept. 15, at 6%. 
 
 Solution. 26 $12.76244 $280.77 
 
 30 
 
 22 
 
 $2.8077 
 
 31 
 
 2562488 
 
 0.9369 
 
 31 
 
 2562488 
 
 0.0935 
 
 15 
 
 1280.77368 
 
 0.0093 
 
 6|T32 
 
 
 $3.8464 
 
 22 
 
 
 $276.92 
 
 494. Exact interest is used by trust companies, and in 
 all calculations of national and state governments. 
 
 Note. This rule applies only to periods less than one year. For 
 periods greater than one year, we find the interest for the years, then 
 for the months and days by the above rule, and add the results. 
 
 Exercise 126. 
 Find the exact interest at 6% : 
 
 1. On $692.74 for 250 days. 
 
 2. On $1472.38 from Jan. 7, 1897 to Oct. 4, 1897. 
 8. On $1247.75 from Mar. 4, 1897 to Dec. 22, 1897. 
 4. On $1898.48 from Feb. 26, 1897 to Aug. 12, 1899. 
 
INTEREST AND DISCOUNT. 279 
 
 Annual Interest. 
 
 495. Annual interest is simple interest on the principal 
 and on each year's interest from the time each interest is 
 due until settlement. 
 
 496. Example. Find the interest on $400 for 4 yr. 
 7 mo. 20 dy., at 5%, payable annually. 
 
 Solution. Simple interest on $400 at 5% for 4 yr. 7 mo. 20 dy. is 
 
 $92.78. 
 
 Interest due the 1st year, $20, draws interest 3 yr. 7 mo. 20 dy. 
 Interest due the 2d year, $20, draws interest 2 yr. 7 mo. 20 dy. 
 Interest due the 3d year, $20, draws interest 1 yr. 7 mo. 20 dy. 
 Interest due the 4th year, $20, draws interest 7 mo. 20 dy. 
 
 Interest upon the interest = interest on $20 for 8 yr. 6 mo. 20 dy. 
 Interest on $20 for 8 yr. 6 mo. 20 dy. at 5% = $8.56. 
 The annual interest = $92.78 + $8.56 = $101.34. 
 
 Exercise 127. 
 
 1. Find the amount at annual interest of $1247.75 for 
 3 yr. 5 mo. 10 dy. at 6%. 
 
 2. Find the interest due on $987.25 in 4 yr. 9 mo. 6 dy., 
 interest at 4%, payable annually. 
 
 3. Find the interest due on $742.60 in 5 yr. 11 mo. 27 
 dy., interest at 4£%, payable annually. 
 
 4. Find the interest due May 19, 1898, on a note dated 
 Dec. 26, 1894, for $1224.60, with interest payable annually, 
 at 5%, if no interest has been paid. 
 
 5. Find the amount due May 27, 1898, on a note dated 
 Jan. 4, 1896, for $215.50, with interest payable annually 
 at 5£%, if no interest has been paid. 
 
 6. Find the amount due Jan. 16, 1897, on a note dated 
 Jan. 8, 1895, for $3115.20, with interest payable annually 
 at 5%, if no interest has been paid. 
 
280 INTEREST AND DISCOUNT. 
 
 Compound Interest. 
 
 497. Interest is compounded when it is added to, and 
 becomes a part of, the principal at specified intervals. 
 
 Interest is compounded annually, semi-annually, or quar- 
 terly, according to agreement. Interest is understood to be 
 compounded annually unless otherwise stated. 
 
 498. Example. Find the amount of $500 at compound 
 interest for 3 years at 4%. Find the compound interest. 
 
 Interest for 1st year is 4% of §500 = $20. Amount is $520. 
 
 Interest for 2d year is 4% of $520 = $20.80. Amount is $540.80. 
 Interest for 3d year is 4% of $540.80 = $21.63. Amount is $562.43. 
 The compound interest is the compound amount less the principal ; 
 that is, $562.43 — $500 = $62.43. 
 
 Note. If the time is not an integral number of years, the com- 
 pound amount is found for the number of entire years, and the 
 amount of this sum at simple interest for the portion of a year. 
 
 Exercise 128. 
 
 1. Find the amount of $356.25 for 4 yr., at 5% com- 
 pound interest. 
 
 2. Find the amount of $637.50 for 2 yr. 6 mo., at 4% 
 compound interest. 
 
 3. Find the compound interest on $800 for 3 yr. 9 mo., 
 at 6%. 
 
 4. Find the compound interest on $39.35 for 4 yr. 9 mo., 
 at 5%. 
 
 5. Find the compound interest on $300 for 2 yr., at 4%, 
 interest being compounded semi-annually. 
 
 Hint. The interest is 2% every six months. 
 
 6. Find the compound interest on $525 for 1 yr. 6 mo., 
 at 5%, interest being compounded quarterly. 
 
 7. Find the compound interest on $10,000 for 6 mo., at 
 6%, interest being compounded monthly. 
 
INTEREST AND DISCOUNT. 281 
 
 Partial Payments. 
 
 499. Partial payments, as the term implies, are pay- 
 ments of a part of a note. 
 
 There are several methods of computing interest upon such notes, 
 all recognizing the general principle that interest should cease upon 
 payments made. 
 
 500. Merchants' Rule. If a note that bears interest 
 runs one year or less, and partial payments have been made, 
 the interest is computed by the following rule : 
 
 1. Find the amount of the note at date of settlement with- 
 out regarding payments. 
 
 2. Find the amount of each payment with interest from 
 date of payment to date of settlement. 
 
 3. Subtract the sum of the payment amounts from the total 
 amount. 
 
 501. Example. A man holds a note of $460, dated 
 Jan. 20, 1897, on which the following payments are en- 
 dorsed : $140, Mar. 26, 1897; $100, June 16, 1897; 
 $160, Oct. 14, 1897. 
 
 Settlement is made Dec. 22, 1897. Find the balance 
 due, reckoning interest at 6%. 
 
 Solution. 
 
 Time from Jan. 20 to Dec. 22 is 11 mo. 2 dy. Int. on $1 = $0.055£. 
 
 Time from Mar. 26 to Dec. 22 is 8 mo. 26 dy. Int. on $1 = 0.04% 
 
 Time from June 16 to Dec. 22 is 6 mo. 6 dy. Int. on $1 — 0.031. 
 
 Time from Oct. 14 to Dec. 22 is 2 mo. 8 dy. Int. on $1 == 0.011£. 
 
 Int. on $460 = 460 X $0.055| = $25.45. Amount is $485.45 
 
 Int. on 140 = 140 X 0.044^ = 6.21. Amount is $146.21 
 Int. on 100= 100 X 0.031 = 3.10. Amount is 103.10 
 Int. on 160 = 160 X 0.011| = 1-81. Amount is 161.81 
 
 Total payment amounts $411.12 
 
 Balance due Dec. 22, 1897 $74.33 
 
282 INTEREST AND DISCOUNT. 
 
 Exercise 129. 
 
 1. A note of $618.75, dated Apr. 17, 1897, payable on 
 demand, bears the following endorsements : June 5, 
 $126.50; Aug. 20, $137.25; Nov. 17, $210. What is 
 due Jan. 1, 1898, reckoning interest at 6% ? 
 
 2. A note of $1000, dated Apr. 1, 1897, payable on 
 demand, with interest at 5%, bears the following endorse- 
 ments : May 6, $200; July 5, $225.37; Oct. 18, $322. 
 What is due Jan. 1, 1898 ? 
 
 3. A note of $835.25, dated July 1, 1897, payable on 
 demand, with interest at 4£%, bears the following endorse- 
 ments : Aug. 20, $157.50 ; Sept. 21, $180.25 ; Oct. 5, 
 $200 ; Dec. 1, $80. What is due Jan. 1, 1898 ? 
 
 4. A note of $1247.50, dated Mar. 10, 1897, payable on 
 demand, with interest at 5%, has the following endorse- 
 ments : $350.40, Apr. 14, 1897 ; $212.85, June 16, 1897 ; 
 $316.45, Aug. 25, 1897. What is due Oct. 18, 1897 ? 
 
 5. A note of $1648.25, dated Jan. 22, 1897, payable on 
 demand, with interest at 5%, has the following endorse- 
 ments : $212.60, Mar. 1, 1897 ; $168.40, May 26, 1897 ; 
 $244.40, Aug. 4, 1897 ; $744.80, Oct. 1, 1897. What is 
 due Jan. 22, 1898 ? 
 
 502. United States Rule. If a note that bears interest 
 runs more than one year, and partial payments have been 
 made, the interest is computed by the following rule : 
 
 1. Find the amount of the principal to the time when the 
 paymenb> or sum of the payments, is equal to or greater than 
 the interest. 
 
 2. From this amount deduct the payment, or sum of the 
 payments. 
 
 3. Consider the remainder as a new principal, and pro- 
 ceed as before. 
 
INTEREST AND DISCOUNT. 288 
 
 503. Example. A note of $1520, dated May 20, 1896, 
 bearing interest at 6%, had payments endorsed upon it as 
 follows : Oct. 2, 1896, $300 ; Feb. 26, 1897, $25 ; Apr. 2, 
 1897, $570 ; Aug. 9, 1897, $600. Find the amount due 
 Dec. 6, 1897. 
 
 yr. mo. dy. 
 1896 10 2 $1520 1st principal. 
 
 1896 5 20 0.022 
 
 4 12 0.022 $33.44 1st interest. 
 
 1520.00 
 $1553.44 
 
 300.00 1st payment. 
 
 1897 2 26 $1253.44 2d principal. 
 1896 10 2 0.024 
 
 4 24 0.024 $30.08 2d interest. 
 
 Since the interest is greater than the payment, we find the interest 
 on the same principal from Feb. 26, 1897, to Apr. 2, 1897. 
 
 $1253.44 2d principal. 
 0.006 
 $7.52 3d interest. 
 
 1897 4 2 30.08 2d interest. 
 
 1897 2 26 . 1253.44 
 
 
 1 
 
 6 
 
 0.006 
 
 $1291.04 
 
 
 
 
 
 $25 + $570 = 595.00 2d & 3d payments. 
 $696.04 3d principal. 
 0.02H 
 
 1897 
 
 8 
 
 9 
 
 
 $14.73 4th interest. 
 
 1897 
 
 4 
 
 2 
 
 
 696.04 
 
 0.021£ $710.77 
 
 600.00 
 $110.77 4th principal. 
 0.019^ 
 1897 12 6 $2.16 5th interest. 
 
 1897 8 9 110.77 
 
 3 27 0.019| $112.93 due Dec. 6, 1897. 
 
284 INTEREST AND DISCOUNT. 
 
 Exercise 130. 
 
 1. A note of $2000, dated Jan. 22, 1896, and drawing 
 interest at 6%, had the following endorsements : May 20, 
 1896, $100 ; July 20, 1896, $325 ; Nov. 2, 1896, $20 ; 
 Dec. 23, 1896, $125. Find the balance due Mar. 1, 1897. 
 
 2. A note of $1662.50, dated Jan. 15, 1896, and drawing 
 interest at 5£%, had the following endorsements : Apr. 30, 
 
 1896, $25 ; June 24, 1896, $25 ; Sept. 2, 1896, $625 ; Jan. 
 30, 1897, $700. Find the balance due May 12, 1897. 
 
 3. A note of $4560, dated Jan. 22, 1896, and drawing 
 interest at 5%, had the following endorsements : Jan. 11, 
 
 1897, $2000 ; Aug. 31, 1897, $500 ; Jan. 15, 1898, $1200; 
 Mar. 4, 1898, $860. Find the balance due June 15, 1898. 
 
 4. A note of $785.50, dated Jan. 30, 1896, and drawing 
 interest at 5%, had the following endorsements : July 17, 
 1896, $100; Jan. 29, 1897, $100; Dec 31, 1897, $20; 
 Mar. 16, 1898, $300 ; June 18, 1898, $50. Find the bal- 
 ance due July 23, 1898. 
 
 5. A note of $300.25, dated Aug. 4, 1896, and drawing 
 interest at 4£%, had the following endorsements : Oct. 14, 
 1896, $100; July 21, 1897, $100; Oct. 11, 1897, $50; 
 Jan. 19, 1898, $50. Find the amount due July 22, 1898. 
 
 6. A note of $1475.40, dated Feb. 12, 1896, and bearing 
 interest at 5%, had the following endorsements : July 22, 
 1896, $370; Dec. 26, 1896, $426.50; Aug. 24, 1897, 
 $112.40 ; Oct. 6, 1897, $163.25 ; Apr. 14, 1898, $185.85. 
 Find the balance due June 16, 1899. 
 
 7. A note of $5762.45 dated Jan. 2, 1896, and drawing 
 interest at 5%, had the following endorsements : May 17, 
 1896, $500 ; Oct. 12, 1896, $750 ; Feb. 4, 1897, $1000 ; 
 Aug. 25, 1897, $1250; Mar. 1, 1898, $1500; June 15, 
 
 1898, $1050. Find the balance due Oct. 2, 1898. 
 
INTEREST AND DISCOUNT. 285 
 
 Special Rules for Partial Payments. 
 The New Hampshire Rule. 
 
 504. When a note is written with interest annually or 
 with annual interest the following is the New Hampshire 
 Rule: 
 
 If in any year, reckoning from the time the annual interest 
 began to accrue, payments have been made, compute interest 
 upon them to the end of the year in which they are made. 
 
 The amount of the payments is to be then applied, first, 
 to cancel interest upon annual interest; secondly, to cancel 
 annual interest ; and thirdly, to the extinguishment of the 
 principal. 
 
 If, however, at the date of any payment there is no interest 
 except the accruing annual interest, and the payment, or pay- 
 ments, do not exceed the annual interest at the end of the year, 
 deduct the payment, or payments, without interest on the same. 
 
 The Vermont Rule. 
 
 505. If we omit the last paragraph of the New Hamp- 
 shire Rule, we have the Vermont Rule. 
 
 The Connecticut Rule. 
 
 506. The Connecticut Rule for partial payments is : 
 Follow the United States Mule if a year's interest or more 
 
 is due at the time of a payment, and in case of the last pay- 
 ment. 
 
 If less than a year's interest is due at the time of a pay- 
 ment except the last, find for a new principal the difference 
 between the amount of the principal for an entire year and 
 the amount of the payment for the remainder of that year. 
 
 If the interest due at the time of a payment exceeds the 
 payment, find the interest on the principal only. 
 
286 INTEREST AND DISCOUNT. 
 
 507. Examples. 1. Find, by the New Hampshire Rule, 
 the amount due Sept. 22, 1896, on a note for $500, dated 
 July 16, 1893, with interest annually at 6%, on which the 
 following payments have been made : Oct. 16, 1894, $200 ; 
 Dec. 16, 1895, $20. 
 
 Principal, $500.00 
 
 1st annual interest, $30.00 
 
 Int. on 1st annual int. for 1 yr., $1.80 
 
 2d annual interest, 30.00 
 
 $600.00 $60.00 $1.80 
 Payment Oct. 16, 1894, |200 
 
 Int. on payment July 16, 1895, 9 
 
 Amt. of payment July 16, 1895, $209 = 147.20 + 60.00 + 1.80 
 
 Principal July 16, 1895, $352.80 
 
 3d annual interest, $21.17 
 
 Payment Dec. 16, 1895, 20.00 
 
 $352.80 + $1.17 
 
 Principal July 16, 1896, $353.97 
 
 4th annual interest, 3.89 
 
 Amt. due Sept. 22, 1896, $357.86 
 
 2. Find, by the Vermont Rule, the amount due Sept. 22, 
 1896, of the note in Example 1. 
 
 From Example 1, the principal July 16, 1895, is $352.80. 
 
 Principal July 16, 1895, $352.80 
 
 3d annual interest, $21.17 
 
 Payment Dec. 16, 1895, 
 
 $20.00 
 
 
 Int. on payment July 16, 1896, 
 
 0.70 
 
 
 Amt. of payment July 16, 1896, 
 
 $20.70 = 
 
 20.70 
 
 
 $352.80+ $0.47 
 
 Principal July 16, 1896, 
 
 
 $353.27 
 
 4th annual interest, 
 
 
 3.89 
 
 Amt. due Sept. 22, 1896, 
 
 
 $357.16 
 
 3. Find, by the Connecticut Rule, the amount due Sept. 
 22, 1896, on a note for $500 given Apr. 10, 1892, bearing 
 interest at 6%, which has the following endorsements : 
 
INTEREST AND DISCOUNT. 28T 
 
 June 16, 1893, $100 ; Dec. 28, 1893, $100s May 10, 1895, 
 $ 15 ; Mar. 4, 1896, $200. 
 
 Principal, $500.00 
 
 Int. on principal to June 16, 1893, 35.50 
 
 Amt. of principal June 16, 1893, $535.50 
 
 Payment June 16, 1893, 100.00 
 
 New principal June 16, 1893, $435.50 
 
 Int. on principal to June 16, 1894, 26.13 
 
 Amt. of principal June 16, 1894, $461.63 
 
 Payment Dec. 28, 1893, $100.00 
 
 Int. on payment to June 16, 1894, 2.80 
 
 Amt. of payment June 16, 1894, $102.80 
 
 New principal June 16, 1894, $358.83 
 
 Int. on principal to June 16, 1895, 21.53 
 
 Amt. of principal June 16, 1895, $380.36 
 
 Payment May 10, 1895 (less than interest), 15.00 
 
 New principal June 16, 1895, $365.36 
 
 Int. on principal to Mar. 4, 1896, 15.71 
 
 Amt. of principal Mar. 4, 1896, $381.07 
 
 Payment Mar. 4, 1896, 200.00 
 
 New principal Mar. 4, 1896, $181.07 
 
 Int. on principal to Sept. 22, 1896, 5.98 
 
 Amt. due Sept. 22, 1896, $187.05 
 
 Exercise 131. 
 
 t- Find, by the New Hampshire Eule and also by the 
 Vermont Rule, the amount due Sept. 22, 1896, on a note 
 for $1750, dated June 6, 1892, with interest annually at 
 6%, which has the following endorsements : Aug. 12, 1893, 
 $300 ; Dec. 23, 1893, $200 ; Jan. 15, 1895, $50 ; Apr. 23, 
 1896, $800. 
 
 2. Find, by the Connecticut Eule, the amount due Sept. 
 22, 1896, on a note for $1500, dated Aug. 9, 1892, with 
 interest annually at 6%, which has the following endorse- 
 ments : Mar.. 17, 1893, $250; Apr. 19, 1894, $50; Sept. 
 21, 1895, $500 ; June 26, 1896, $600. 
 
CHAPTER XIV. 
 
 STOCKS AND BONDS, EXCHANGE, ACCOUNTS. 
 Stocks and Bonds. 
 
 508. Stock Companies. A stock company is an associa- 
 tion of persons under the laws of the state for the purpose 
 of carrying on a specified business. 
 
 509. Stock. The stock represents the capital invested 
 in the business, and is issued in the form of certip'mfrs, 
 each certifying that the person named in the certificate 
 owns a stated number of shares of stock. 
 
 Note. One of the special advantages of a number of persons doing 
 business in the form of a company is that the liability of each stock- 
 holder is usually limited to the amount of his stock. 
 
 510. Bonds are written obligations under seal given by 
 a company ; or by a municipal or state government; or by 
 the national government ; in which an agreement is made 
 to pay a specified amount on or before a specified date, with 
 interest payable annually, semi-annually, or quarterly. 
 
 511. Mortgage bonds are bonds secured by mortgage, 
 but debenture bonds are simply notes under seal. 
 
 512. Registered bonds are bonds recorded by their num- 
 bers in a book called a register. The register contains the 
 names of the owners of the bonds. 
 
 Note. Registered bonds are transferred from one person to 
 another by giving proper notice to the registrar of the company. 
 
STOCKS AND BONDS. 289 
 
 513. Coupon bonds are bonds that have coupons or cer- 
 tificates of interest attached. These coupons are cut off 
 as they become due, and are given up on receipt of the 
 interest represented by them. 
 
 514. Bonds are named by giving the name of the cor- 
 poration issuing them, and the rate of interest they bear. 
 The date on which they become due is usually given, and 
 the kind of bond, registered or coupon. 
 
 Thus, U. S. coupon 4's, 1907 means United States coupon bonds 
 bearing 4 per cent interest, the principal due in 1907. U. S. registered 
 4's, 1925 means United States registered bonds bearing 4% interest, the 
 principal due in 1925. 
 
 515. Persons who buy and sell stocks and bonds are 
 called stock brokers, and their commission is called broker- 
 age. On lots of 100 shares or more, brokerage is -J% per 
 share, reckoned at $100; and on lots of less than 100 
 shares brokerage is \°] per share. Brokerage on bonds is 
 reckoned at the same rate per $100 as on stocks. 
 
 516. Stocks and bonds are said to be at par when they 
 sell for their face value ; 
 
 At a premium, or above par, when they sell for more 
 than their face value ; 
 
 At a discount, or below par, when they sell for less than 
 their face value. 
 
 The price for which stocks and bonds may be sold is 
 called their market value. 
 
 517. A company uses its receipts first to pay its ex- 
 penses ; second, to pay the interest on its bonds. If there 
 is a surplus, it is usually divided among the stockholders, 
 according to the number of shares held by each. The 
 amount allotted to each share is called the dividend per 
 share. 
 
290 STOCKS AND BONDS. 
 
 Exercise 132. 
 
 1. What is the cost of 25 shares of Boston and Maine 
 R.R. stock at 167, brokerage J ? 
 
 Solution. Each share of stock costs §167, with $0.25 for bro- 
 kerage, or $167.26 in all. Hence, 26 shares cost 26 X $167.25, or 
 $4181.25. 
 
 2. How many shares of Illinois Central R.R. stock at 
 101J can be bought for $20,400, brokerage i ? 
 
 Solution. Since the total cost of each share is $101$ + $$-, or 
 $102, $20,400 will buy *£#§ fi shares, or 200 shares. 
 
 3. What is the annual income from 150 shares of Lake 
 Shore and Michigan Ry. stock that pays an annual divi- 
 dend of 6% ? 
 
 Solution. 1 share at 6% yields $6. Hence, 160 shares at 6% yield 
 160 X $6 = $900. 
 
 4. How much must be invested in 6% stock at 107 to 
 yield an annual income of $240, brokerage £ ? 
 
 Solution. $6 is the dividend from 1 share. 
 
 $240 dividend requires 2 J ft , or 40 shares. 
 Price of 40 shares is 40 X $107 = $4280 
 
 Brokerage on 40 shares is 40 X $0.25 = 10 
 
 Total cost = $4290 
 
 5. What per cent does the investment yield, if Lake 
 Shore and Michigan Southern Ry. stock is bought at 170 ? 
 The stock pays 6% dividend ; no brokerage reckoned. 
 
 Solution. Each $170 invested yields $6 dividend. Hence, the 
 income on the investment is T f ^ of 100%, or 3^ 7 %. 
 
 6. Find the cost of 350 shares of Chicago, Milwaukee 
 and St. Paul Ry. stock at 93f , brokerage £. 
 
 7. Find the cost of 165 shares of Michigan Central R.R. 
 stock at 105f , brokerage £. 
 
 8. Find the cost of 35 shares of Reading R.R. stock at 
 23f , brokerage £. 
 
STOCKS AND BONDS. 291 
 
 9. What is the cost of 25 U. S. 4% registered 1925 
 bonds of $1000 each, at 127-J, brokerage £? 
 
 10. What is the cost of 40 Northern Pacific R.R. 1st 
 mortgage 6% registered bonds of $1000 each, at 119 J, bro- 
 kerage -J ? 
 
 11. What per cent income does the investment of Ex- 
 ample 10 yield ? 
 
 12. What is the annual income received from the invest- 
 ment of Example 10 ? 
 
 13. What is the annual income from 200 shares of 
 Chicago and Northwestern Ry. stock that pays an annual 
 dividend of 5% ? 
 
 14. What is the cost of the investment of Example 13 
 at 122J, brokerage £ ? 
 
 15. What per cent income does the investment of Ex- 
 ample 13 yield ? 
 
 16. How many shares of New York Central stock can 
 be bought for $4757.50 at 107J-, brokerage £ ? 
 
 17. How many Chicago, Burlington and Quincy 7% 
 bonds, $500 each, will $6885 buy at 114£, brokerage J ? 
 
 18. What is the annual income from the investment of 
 Example 17 ? 
 
 19. What sum of money must be invested in Northern 
 Pacific R.R. 1st mortgage 6's at 119£ to produce an annual 
 income of $2400, brokerage -J- ? 
 
 20. What sum of money must be invested in Wabash 
 R.R. 1st gold 5% bonds at 107J to produce an annual 
 income of $2000, brokerage -J- ? 
 
 21. What sum of money must be invested in Louisville 
 and Nashville R.R. unified gold 4% bonds at 84£ to produce 
 an annual income of $320, brokerage \ ? 
 
 22. What sum of money must be invested in St. Louis 
 and San Francisco Ry. general mortgage 5 °J bonds at 100 \ 
 to produce an annual income of $600, brokerage -J ? 
 
292 STOCKS AND BONDS. 
 
 23. How many shares of Chicago and Northwestern Ry. 
 stock can be. bought for $14,670 at 122£, brokerage £ ? 
 What is the brokerage? If 5% dividends are paid, what 
 per cent on his investment does the purchaser receive ? 
 
 24. How many shares of Michigan Central R.R. stock 
 can be bought for $16,940 at 105f, brokerage i ? What is 
 the brokerage ? If 4% dividends are paid, what per cent 
 on his investment does the purchaser receive ? 
 
 25. What is the cost of 40 shares of Central R.R. of New 
 Jersey stock at 92J, brokerage J ? What is the brokerage ? 
 If 6% dividends are paid, what per cent on his investment 
 does the purchaser receive ? 
 
 26. What is the cost of 250 shares of Pullman Palace 
 Car Co. stock at 171J, brokerage £ ? What is the broker- 
 age ? If 8% dividends are paid, what per cent on his 
 investment does the purchaser receive ? 
 
 27. What per cent on his investment does a purchaser 
 receive who buys New York, New Haven and Hartford 
 R.R. stock at 180£, if annual dividends of 8% are 
 declared ? 
 
 28. When West End Co. 4£% bonds are selling at 107£, 
 how much must be invested to produce an annual income 
 of $900, brokerage £ ? What per cent on his investment 
 does a purchaser receive ? 
 
 29. When Mexican Central Ry. 1st mortgage 4% bonds 
 are selling at 62£, how much must be invested to produce 
 an annual income of $200, brokerage £ ? What per cent 
 on his investment does a purchaser receive ? 
 
 30. When West Shore R.R. 1st mortgage 4% bonds are 
 selling at 108£, how much must be invested to produce an 
 annual income of $800, brokerage £ ? 
 
 31. When New England Tel. and Tel. Co. 6% bonds are 
 selling at 101£, how much must be invested to produce an 
 annual income of $900, brokerage -J ? 
 
STOCKS AND BONDS. 293 
 
 32. If a man buys a 6% bond at 120, what rate of 
 interest does he receive on the money invested ? 
 
 33. If 3% bonds are at 88-J-, what rate per cent interest 
 will a purchaser receive on his money ? 
 
 34. If an 8% stock is at 150, what rate per cent interest 
 will a purchaser receive on his money ? 
 
 35. If a 10% stock is at 175, what rate per cent interest 
 will an investor receive on his money ? 
 
 36. If a 4£% stock is at 85, what rate per cent interest 
 will a purchaser receive on his money ? 
 
 37. If 7% bonds are at 114, what rate per cent interest 
 will a purchaser receive on his money ? 
 
 38. If 6% bonds are at 130, what rate per cent interest 
 will a purchaser receive on his money ? 
 
 39. If $8000 5% stocks are sold at 90, and the proceeds 
 invested in 3^-% stocks at 60, find the increase or decrease 
 in income. 
 
 40. If $10,000 3£% bonds are sold at 65, and the pro- 
 ceeds invested in 8% bonds at 130, find the increase or 
 decrease in income. 
 
 41. If $8000 4£% stocks are sold at 70, and the pro- 
 ceeds invested in 10% stocks at 160, find the increase or 
 decrease in income. 
 
 42. If $6000 6% bonds are sold at 90, and the pro- 
 ceeds invested in 10% bonds at 135, find the increase or 
 decrease in income. 
 
 43. Find the rate of interest obtained by investing in a 
 5% bond at 124. 
 
 44. What is the price of stock if $7000 stock can be 
 bought for $5880 ? 
 
 45. Find the amount received for 100 mining shares 
 issued at $15 a share and sold at 2\°] discount. 
 
 46. How much 3£% stock must be sold at 75 \ to buy 
 $5000 4% stock at 94|, brokerage \ on each transaction? 
 
294 STOCKS AND BONDS. 
 
 47. How much stock must be sold at 76^ to raise a sum 
 sufficient to discount a note for $1075, due in 53 days, with 
 grace, and discounted at 5-J-% ? 
 
 48. A broker bought five $1000 bonds at 88|. At what 
 price must he sell them to gain $100, brokerage £ on each 
 transaction ? 
 
 49. If a broker buys bonds at 87 J, at what price must 
 he sell them to make 12£% profit, brokerage £ on each 
 transaction ? 
 
 50. Which is the more profitable stock for investment, 
 a 4% at 85 or a 3% at 63 ? a 3J% at 67J or a 4% at 81£ ? 
 
 51. Find the price of a 4£% bond to be as profitable an 
 investment as a 3£% bond at 88£. 
 
 52. Find the price of a 5% bond to be as profitable an 
 investment as a 3% bond at 89 J. 
 
 53. Find the price of a 3£% bond to be as profitable an 
 investment as a 6% bond at par. 
 
 54. Find the loss in buying $80,000 worth of bonds at 
 91f and selling at 90, brokerage £ on each transaction. 
 
 55. Which is the better investment, a 5% stock at 137 J- 
 or a 3£% stock at 91£? What rate of interest will be 
 received from each investment ? 
 
 56. A person invests $7370 in the purchase of a stock 
 at 92. What will be his loss if he sells at 90, brokerage £ 
 on each transaction ? 
 
 57. How much stock must be sold at 90f so that when 
 the seller invests the proceeds in a mortgage at 6% he 
 will receive $543.75 annual income ? 
 
 58. A person invests } of his money at 6%, § at 4£%, 
 and the rest at 3^%. What per cent does he receive on 
 the whole amount ? 
 
 59. How many shares of stock must a man sell at 107|, 
 that when he invests the proceeds in 3% stock at 7l£ he 
 may receive an annual income of $900 ? 
 
EXCHANGE. 295 
 
 Exchange. 
 
 518. Exchange. The system of paying debts due per- 
 sons living at a distance by transmitting drafts instead of 
 money is called exchange. 
 
 519. Exchange between cities of the same country is 
 called Domestic Exchange. Exchange between cities of 
 different countries is called Foreign Exchange. 
 
 520. Examples. 1 . Find the cost of the following draft 
 with grace, interest 6%, exchange %°/ premium : 
 
 $800. Cincinnati, O., Nov. 1, 1897. 
 
 Thirty days after sight, pay to the order of S. G. Clark 
 Eight Hundred Dollars, and charge to the account of 
 
 To H. S. Wright, Philadelphia. P. M. Clement. 
 
 Solution. The discount on $800 at 6% for 33 days is $4.40. 
 The proceeds of the draft is $800 — $4.40 = $795.60. 
 The exchange is \% of $800 = $4. 
 Hence, the cost of the draft is $795.60 + $4 = $799.60. 
 
 2. Find the cost of a draft of $400, payable 60 days after 
 sight with grace, interest 7%, exchange \°f discount. 
 
 Solution. The discount on $400 at 7% for 63 days is $4.90. 
 
 The exchange is ±% of $400 = $1. 
 
 The total discount is $4.90 + $1 = $5.90. 
 
 Hence, the cost of the draft is $400 — $5.90 = $394.10. 
 
 3. Find the face of a draft, payable 30 days after sight 
 with grace, that can be bought for $1000, interest 6%, 
 exchange \°] premium. 
 
 Solution. The discount on $1 for 33 days at 6% is $0.0055 ; and 
 the proceeds of $1 is $1 — $0.0055 = $0.9945. 
 
 The exchange on $1 is $0.0025 ; and the cost of $1 is $0.9945 + 
 $0.0025 = $0,997. 
 
 Hence, the face of the draft is $1000 -i- 0.997 = $1003.01. 
 
296 EXCHANGE. 
 
 Exercise 133. 
 
 1. Find the cost of a sight draft on New York of $1100, 
 exchange ±% premium. 
 
 2. Find the cost of a sight draft on New Orleans of 
 $1350, exchange £% discount. 
 
 3. Find the cost of a draft on Boston of $1600, payable 
 30 days after sight with grace, interest 6%, exchange \of 
 premium. 
 
 4. Find the cost of a draft of $500, payable 60 days 
 after sight with grace, interest 7%, exchange £% discount. 
 
 5. Find the cost of a draft of $1200, payable 90 days 
 after sight with grace, interest 7%, exchange \°J premium. 
 
 6. Find the cost of a draft of $950, payable in 30 days 
 with grace, interest 4£%, exchange at par. 
 
 7. Find the cost of a draft of $725, payable in 60 days 
 with grace, interest 5%, exchange \°J discount. 
 
 8. Find the cost of a draft of $810, payable in 90 days 
 with grace, interest 5£%, exchange J% premium. 
 
 9. Find the face of a draft, payable 30 days after sight 
 with grace, that can be bought for $274, interest 6%, 
 exchange at par. 
 
 10. Find the face of a draft, payable 60 days after sight 
 with grace, that can be bought for $1250, interest 7%, 
 exchange \°f premium. 
 
 11. Find the face of a draft, payable 60 days after date 
 witli grace, that can be bought for $1125, interest 5£%, 
 exchange \°/ discount. 
 
 12. Find the face of a draft, payable 30 days after date 
 with grace, that can be bought for $520, interest 4%, 
 exchange \°j premium. 
 
 13. Find the face of a draft, payable 90 days after date 
 with grace, that can be bought for $10,000, interest 4£%, 
 exchange at par. 
 
EXCHANGE. 297 
 
 521. Foreign Exchange. Foreign Bills of Exchange 
 are usually drawn in sets of three, of the same tenor and 
 date, called First, Second, and Third of Exchange. These 
 are sent by different mails to avoid loss or delay. When 
 one is accepted or paid the others are void. Thus, 
 £600. New York, Nov. 1, 1897. 
 
 At Sight of this First of Exchange (Second and Third of 
 the same tenor and date unpaid), pay to the order of James 
 Patterson Five Hundred Pounds, value received, and charge 
 same to account of 
 
 To Baring Blathers, ^1 _,, _ - 
 
 T _ , _ > Bliss and Morton. 
 
 London, England. J 
 
 522. Exchange for sight drafts Nov. 13, 1897, was 
 quoted in New York as follows : 
 
 On London at $4,865 for 1 pound sterling. 
 
 On Paris at 5.18£ francs for $1. 
 
 On cities in Germany at 4 reichsmarks for $0,955. 
 
 On Amsterdam at 1 guilder for $0.40£. 
 
 523. Examples. 1. Find the cost in New York of a 
 sight draft on London for £502 12s. 
 
 Solution. £502 12s. = £502.6. 
 
 502.6 X $4,865 = $2445.15. 
 
 2. Find the face of a sight draft on London that can be 
 bought for $2433.47. 
 
 Solution. $2433.47 -f $4,865 = 500.2. 
 
 £500.2 = £500 4s. 
 
 3. Find the cost in New York of a sight draft on Paris 
 
 for 2400 francs. 
 
 1 
 
 Solution. 1 franc = $ 
 
 6.18* 
 
 2400 francs = 2400 X $ -^— = $463.21. 
 5.18$ 
 
298 EXCHANGE. 
 
 Exercise 134. 
 Find the cost of a sight draft on : 
 
 1. London for £320 10s. 6d. 
 
 2. Paris for 8000 francs. 
 
 3. Hamburg for 2876 reichsmarks. 
 
 4. Amsterdam for 6486 guilders. 
 
 5. Glasgow for £5876 10s. 
 
 6. Paris for 12,842 francs. 
 
 7. Berlin for 4885 reichsmarks. 
 
 8. Kotterdam for 8282 guilders. 
 
 9. Liverpool for £1242 12s. 6d. 
 
 10. Paris for 2685 francs. 
 
 1 1 . Find the face of a sight draft on Glasgow that can 
 be bought for $2000. 
 
 12. Find the face of a sight draft on London that can 
 be bought for $4000. 
 
 13. Find the cost of a sixty-day draft on London for 
 £150, when sixty-day bills are quoted at 4.81£, and the 
 broker's commission is -J% of the cost of the draft. 
 
 14. How large a sight draft on Paris can be bought for 
 $2840 ? 
 
 15. How large a sixty-day draft on Paris can be bought 
 for $1500, when sixty-day bills are quoted at 5.17J ? 
 
 16. How large a sight draft on Berlin can be bought for 
 $8000 ? 
 
 17. How large a sixty-day draft on Hamburg can be 
 bought for $2500, when German sixty-day drafts are 
 quoted at 0.95 ? 
 
 18. How large a sight draft on Amsterdam can be bought 
 for $2200 ? 
 
 19. How large a sixty-day draft on Kotterdam can be 
 bought for $1200, when a sixty-day draft on Holland is 
 quoted at 0.40| ? 
 
AVERAGE OF PAYMENTS. 299 
 
 Average of Payments. 
 
 524. Example. John Smith has given to William 
 Jones notes as follows : $150, due May 15 ; $200, due 
 June 30 ; $500, due July 21 ; $400 due July 29. He 
 wishes to pay them all at one time. When shall they be 
 considered due ? 
 
 Solution. If all the notes were paid May 15, Smith would lose the 
 use of $200 for 46 days, of $500 for 67 days, and of $400 for 75 days. 
 
 The use of $200 for 46 days = the use of $200 X 46 for 1 day ; the 
 use of $500 for 67 days = the use of $500 X 67 for 1 day ; and the 
 use of $400 for 75 days = the use of $400 X 75 for 1 day. Smith 
 would, therefore, lose the equivalent of $9200 + $33,500 + $30,000 
 = $72,700 for 1 day, and is entitled to keep the $150 + $200 + $500 
 + $400 — $1250 as many days after May 15 as are required for the 
 use of $1250 to equal the use of $72,700 for 1 day; that is, VzV<r 
 days = 58. 2 days. 
 
 Hence, the equated time for paying the four notes is 58 days after 
 May 15 ; that is, July 12. 
 
 The work may be arranged as follows : 
 
 $150 X 00 : 
 
 
 $200 X 46 : 
 
 = $9,200 
 
 $500 X 67 = 
 
 = 33,500 
 
 $400 X 75 = 
 
 = 30,000 
 
 $1250 
 
 )$72,700 
 
 
 58.2. 
 
 Hence, 
 
 525. To Find the Equated Time for the Payment of 
 Several Debts Due at Different Dates, 
 
 Select the earliest date that any debt becomes due as the 
 standard date. 
 
 Multiply each of the debts by the number of days from 
 the standard date to the date that it becomes due, and divide 
 the sum, of the products by the sum of the debts. 
 
 The quotient is the number of days that must be added to 
 the standard date to find the average time of the payments. 
 
300 AVERAGE OF PAYMENTS. 
 
 Exercise 135. 
 
 1 . Find the equated time for the payment of $250 due 
 in 3 mo., $ 400 due in 6 mo., $700 due in 8 mo. 
 
 2. Find the equated time for the payment of $300 due 
 in 30 days, $500 due in 60 days, and $200 due in 90 days. 
 
 3. Find the equated time for the payment of $325 due 
 now, $200 due in 30 days, $460 due in 60 days, and $150 
 due in 90 days. 
 
 4. Find the equated time for the payment of $240 due 
 May 10, $420 due July 2, $310 due Sept. 14, and $600 due 
 Oct. 1. 
 
 5. Find the equated time for the payment of $275 due 
 June 21, $175 due July 16, $200 due Aug. 6, and $150 
 due Sept. 3. 
 
 6. Find the equated time for the payment of $112.30 
 due July 6, $115.25 due July 30, $232.15 due Sept. 4, and 
 $102.36 due Oct. 1. 
 
 7. A owes B $200 due in 10 mo. If he pays $120 in 
 4 mo., when should he pay the balance ? 
 
 Note. By paying 3120 in 4 mo. A loses the use of $120 for 6 mo., 
 which is equal to the use of $720 for 1 mo. Therefore, he is entitled 
 to keep the balance ($80) \\°- mo. = 9 mo. after its maturity. 
 
 8. A owed B $2000 payable in 4 mo., but at the end of 
 1 mo. he paid him $500, at the end of 2 mo. $500, and 
 at the end of 3 mo. $500. In how many months is the 
 balance due ? 
 
 9. A man, Feb. 11, 1898, gave a note for $1700 pay- 
 able in 4 mo.; but he paid Mar. 22, $400, Apr. 20, $220, 
 May 10, $300. When was the balance due ? 
 
 1 0. A man, Jan. 4, 1898, gave a note for $2500 payable 
 in 6 mo.; but he paid Feb. 4, $200, Mar. 4, $400, Apr. 4, 
 $600, May 4, $500, and June 4, $300. When was the 
 balance due ? 
 
SETTLEMENT OF ACCOUNTS. 
 
 301 
 
 Settlement of Accounts. 
 
 526. Examples. 1. Find the time for the payment of 
 the balance of the following account : 
 
 Adams & Co. in account with Bacon & Co. 
 
 Dr. 
 
 Cr. 
 
 1898. 
 
 
 
 
 
 
 Jan. 3. 
 
 To mdse. 90 dy. 
 
 $250 
 
 Apr. 11. 
 
 By cash, 
 
 $200 
 
 Mar. 7. 
 
 60 " 
 
 150 
 
 Apr. 30. 
 
 ft 
 
 100 
 
 May 3. 
 
 60 " 
 
 325 
 
 May 30. 
 
 M 
 
 125 
 
 June 7. 
 
 " 30 " 
 
 175 
 
 July 2. 
 
 u 
 
 400 
 
 Solution. We first find the equated time of the items on the 
 Debit side to be May 29, 1898 ; and the equated time of the items on 
 the Credit side to be May 30, 1898. 
 
 The total of the Debit side is $900, and the total of the Credit side 
 is $825. Hence, the balance is $900 — $825 = $75. 
 
 The difference between the equated times is 1 day. 
 
 If the account were settled at the later date, May 30, the $900 on 
 the Debit side would have been on interest 1 day, and this is equiva- 
 lent to having the balance, $75, on interest - 9 7 ° 5 °- of 1 day = 12 days. 
 , Hence, the balance should begin to draw interest 12 days before May 30; 
 that is, May 18, 1898. 
 
 2. Find the time for the payment of the balance of an 
 account if the debit and credit sides, when equated, stand 
 as follows : 
 
 Dr. 
 
 Due Feb. 18, 1898, $950 
 
 Due Jan. 20, 
 
 Cr. 
 
 1898, $850 
 
 Solution. The difference between the equated times is 29 days. 
 The balance of account is $950 — $850 = $100. 
 
 If the account were settled at the later date, Feb. 18, the $850 
 would have been on interest 29 days, which is equivalent to having the 
 balance, $100, on interest f •>-§ of 29 days = 246 \ days. Hence, to 
 increase the Debit side by an equal amount of interest, the balance 
 should remain unpaid 247 days ; that is, the balance is due Oct. 23, 
 1898. Hence, 
 
302 
 
 SETTLEMENT OF ACCOUNTS. 
 
 527. To Find the Time the Balance of an Account 
 Falls Due, 
 
 Find the equated time for each side of the account. 
 
 Multiply the side of the account that falls due first by the 
 number of days between the dates of the equated times of the 
 two sides, and divide the product by the balance of the account. 
 
 The quotient will give the number of days to the maturity 
 of the balance, to be counted forward from the later date when 
 the smaller side falls due first, and backward when the larger 
 side falls due first. 
 
 Note 1. In finding the equated time of accounts it is customary to 
 neglect cents if less than 50, and if 50 or more to consider them as $1. 
 A fraction of a day in the result is rejected if less than \ ; if $ or more 
 it is called a day. 
 
 Note 2. When an account is settled by cash at any other date 
 than that on which the balance becomes due, the interest is found on 
 the balance for the interval between the day of settlement and the day 
 the balance is due, and is added to or deducted from the balance, 
 according as the settlement is made after or be/ore the balance is due. 
 
 Exercise 136. 
 
 Find the time for paying the balance in the following 
 equated bills : 
 
 Average due. Dr. Average due. Cr. 
 
 1. May 17, 1897 . . . $950 Apr. 12, 1897 .... $1000 
 
 2. Apr. 12, 1897 . . . $950 May 17, 1897 .... $1000 
 
 3. May 30, 1898 . . $1000 June 23, 1898 .... $920 
 
 4. July 6, 1897 . . . $500 Apr. 14, 1897 .... $480 
 
 5. Aug. 13, 1897 . . $875 Sept. 13, 1897 .... $600 
 
 6. May 28, 1898 . . $500 June 4, 1898 .... $550 
 
 7. Apr. 4, 1898 . . . $400 June 6, 1898 .... $300 
 
 8. Mar. 12, 1898 . . $750 Feb. 4, 1898 $500 
 
 9. Feb. 4, 1898 . . . $750 Mar. 12, 1898 .... $500 
 
SETTLEMENT OP ACCOUNTS. 
 
 303 
 
 528. The common method of finding the balance of an 
 account is to compute the interest on each item, from its 
 date to the day of settlement, reckoning the time in days. 
 
 Dr. Int. 
 
 Apr. 8. To cash, $250 
 May 31. " 380 
 
 July 20. " ■ 420 
 
 Oct. 10. To bal. acct. 210 
 
 "10. " int. _ 
 
 $1260 
 
 $7.71 
 
 8.36 
 
 5.74 
 
 8.54 
 
 $30.36 
 
 1897. 
 
 Cr. 
 
 Apr. 4. Bymdse.,$300 
 May 19. " 350 
 
 June 9. " 610 
 
 Settled Oct. 10, 1897. 
 
 $1260 
 
 Int. 
 
 $9.45 
 
 8.40 
 
 12.50 
 
 $30.35 
 
 Hence, the cash balance = $210 + $8.54, or $218.54. 
 
 Note. When the balance of account and the balance of interest 
 fall on opposite sides, the cash balance is their difference. 
 
 Exercise 137. 
 
 Find the cash balance of the following accounts, reckon- 
 ing interest at 6% : 
 
 Apr. 5. 
 
 " 27. 
 
 June 1. 
 
 To mdse., 
 
 Dr. 
 
 $250.00 
 
 610.00 
 
 200.00 
 
 1897. CR. 
 
 Apr. 20. By cash, $200.00 
 
 " 30. " 500.00 
 
 June 4. " 400.00 
 
 Settled June 19, 1897. 
 
 1897. Dr. 
 
 Jan. 15. To mdse. 3 mo., $250.00 
 
 Feb. 25. " " 98.50 
 
 Mar. 8. " " 300.00 
 
 3. 
 
 1897. Dr. 
 
 Jan. 2. To mdse. 60 dy., $100.00 
 
 Mar. 8. 
 May 10. 
 June 2. 
 
 200.00 
 
 30 dy., 150.00 
 
 95.00 
 
 1897. 
 
 
 Cr. 
 
 Apr. 26. 
 
 By cash, 
 
 $150.00 
 
 May 17. 
 
 u 
 
 150.00 
 
 July 7. 
 
 t< 
 
 200.00 
 
 
 Settled Oct. 
 
 11, 1897. 
 
 1897. 
 
 
 Cr. 
 
 Feb. 25. 
 
 By cash, 
 
 $100.00 
 
 Mar. 22. 
 
 (i 
 
 150.00 
 
 June 21. 
 
 u 
 
 200.00 
 
 Settled Aug. 2, 1897. 
 
304 
 
 SAVINGS BANKS ACCOUNTS. 
 
 Savings Banks Accounts. 
 
 529. Savings banks receive money on deposit, and pay 
 depositors compound interest, adding the interest to the 
 principal every three months, six months, or twelve months. 
 
 530. The interval between the dates at which interest 
 is computed is called an Interest Term. 
 
 Interest is added at the end of every interest term, com- 
 puted on the smallest balance on deposit at any time during 
 the whole interest term. 
 
 Each depositor lias a bank book, in which is recorded 
 every sum deposited, every sum withdrawn, and the interest 
 due at the end of each interest term. 
 
 531. Example. Find the balance on deposit Oct. 1,1897, 
 on the following account, interest 4%, reckoned quarterly : 
 
 Deposited Jan. 1, 1897, $50 ; Feb. 4, 1897, 340 ; May 
 6, 1897, $60 ; Aug. 3, 1897, $40. Withdrawn Mar. 3, 1897, 
 $20 ; Apr. 22, 1897, $30 ; June 19, 1897, $25 ; Sept. 20, 
 1897, $40. 
 
 Statement. 
 
 Date. 
 
 Deposited. 
 
 Withdrawn. 
 
 Interest. 
 
 Bajlance. 
 
 1897. 
 
 
 
 
 
 
 
 
 
 Jan. 1, 
 
 $50 
 
 00 
 
 
 
 
 
 $50 
 
 00 
 
 Feb. 4, 
 
 40 
 
 00 
 
 
 
 
 
 90 
 
 00 
 
 Mar. 3. 
 
 
 
 $20 
 
 00 
 
 
 
 70 
 
 00 
 
 Apr. 1, 
 
 
 
 
 
 $0 
 
 50 
 
 70 
 
 50 
 
 Apr. 22, 
 
 
 
 30 
 
 00 
 
 
 
 40 
 
 50 
 
 May 6, 
 
 60 
 
 00 
 
 
 
 
 
 100 
 
 50 
 
 June 19, 
 
 
 
 25 
 
 00 
 
 
 
 75 
 
 50 
 
 July 1, 
 
 
 
 
 
 
 
 40 
 
 75 
 
 90 
 
 Aug.3, 
 
 40 
 
 00 
 
 
 
 
 
 115 
 
 90 
 
 Sept. 20, 
 
 
 
 40 
 
 00 
 
 
 
 75 
 
 90 
 
 Oct. 1, 
 
 
 
 
 
 
 
 76 
 
 76 
 
 66 
 
SAVINGS BANKS ACCOUNTS. 305 
 
 The smallest sum on deposit during the first interest term was $50. 
 The interest on $50 for 3 mo. at 4% is $0.50, which, added to the 
 balance on deposit, makes $70.50. 
 
 The smallest sum on deposit during the second interest term was 
 $40.50. The interest on $40.50 for 3 mo. at 4% is $0.40, which, added 
 to the balance on deposit, makes $75.90. 
 
 The smallest sum on deposit during the third interest term was 
 $75.90. The interest on $75.90 for 3 mo. at 4% is $0.76, which, added 
 to the balance on deposit Oct. 1, 1897, makes $76.66. 
 
 Exercise 138. 
 
 Find the balance on deposit Jan. 1, 1898, on the follow- 
 ing account : 
 
 1. Interest being 4%, computed quarterly. Deposited 
 Jan. 1, 1897, $125 ; Mar. 22, 1897, $40 ; June 8, 1897, ' 
 $35 ; July 30, 1897, $85 ; Sept. 24, 1897, $65. With- 
 drawn Apr. 2, 1897, $110 ; June 30, 1897, $40 ; Oct. 22, 
 1897, $10 ; Dec. 17, 1897, $25. 
 
 2. Interest being 3%, computed quarterly. Deposited 
 Jan. 1, 1897, $200 ; Feb. 14, 1897, $125 ; Mar. 10, 1897, 
 $75 ; May 31, 1897, $50 ; Aug. 2, 1897, $100. With- 
 drawn May 7, 1897, $25 ; June 22, 1897, $40 ; Oct. 2, 
 1897, $50 ; Nov. 4, 1897, $65 ; Dec. 14, 1897, $75. 
 
 3t Interest being 3 °f , computed semi-annually. De- 
 posited Jan. 1, 1897, $425 ; May 10, 1897, $15 ; Sept. 24, 
 1897, $200; Oct. 5, 1897, $25; Nov. 15, 1897, $65. 
 Withdrawn Feb. 1, 1897, $25 ; Mar. 20, 1897, $45 ; Aug. 
 2, 1897, $50 ; Aug. 28, 1897, $125 ; Dec. 10, 1897, $100. 
 
 4. Interest being 3%, computed annually. Deposited 
 Jan. 1, 1897, $266.50 ; May 3, 1897, $122.50 ; Aug. 2, 
 1897, $57 ; Aug. 9, 1897, $108 ; Sept. 4, 1897, $64.50. 
 Withdrawn June 15, 1897, $40 ; Oct. 8, 1897, $75 ; Nov. 
 1, 1897, $60 ; Dec. 4, 1897, $85 ; Dec. 20, 1897, $142. 
 
CHAPTER XV. 
 POWERS AND SOOTS. 
 
 532. The square of a number is the product of two 
 factors, each equal to the number (§ 69). 
 
 Thus, the squares of 1, 2, 3, 4, 6, 6, 7, 8, 9, 10, 
 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 
 
 533. The square root of a number is one of the two 
 equal factors of the number (§ 69). 
 
 Thus, the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 
 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 
 
 534. The square root of a number is indicated by the 
 radical sign ^/, or by the fractional exponent £. 
 
 Thus, V27, or 27*, means the square root of 27. 
 
 535. Since 35 = 30 + 5, the square of 35 may be 
 obtained as follows : 
 
 30+5 
 
 
 30+5 
 
 30 2 = 900 
 
 30 2 + (30X5) 
 
 2 (30 X 5) = 300 
 
 + (30X5) + 5* 
 
 5 2 = 25 
 
 30 2 + 2(30X5) + 5* 
 
 35 2 = 1225 
 
 536. Since every number of two or more figures may 
 be regarded as composed of tens and units, if we represent 
 the number of tens by a and the number of units by b, 
 (a + bf = a 2 + 2ab + b*. Hence, 
 
 The square of a member « equal to the square of the tens, 
 plus twice the tens multiplied by the units, plus the square of 
 the units. 
 
POWERS AND ROOTS. 307 
 
 537. The first step in extracting the square root of a 
 number is to separate the figures of the number into groups. 
 
 Since 1 = l 2 , 100 = 10 2 , 10,000 = 100 2 , and so on, it is evident that 
 the square root of any number between 1 and 100 lies between 1 and 
 10 ; of any number between 100 and 10,000 lies between 10 and 100. 
 In other words, the square root of any integral number expressed by 
 one or two figures is a number of one figure ; expressed by three or 
 four figures is a number of two figures, and so on. 
 
 If, therefore, an integral number is divided into groups of two 
 figures each, from the right to the left, the number of figures in the 
 root will be equal to the number of groups of figures. The last group 
 to the left may have one or two figures. 
 
 Example. Find the square root of 1225. 
 
 Solution. The first group, 12, contains the square of the tens' 
 number of the root. 
 
 The greatest square in 12 is 9, and the square root of 9 is 3. Hence, 
 
 3 is the tens' figure of the root. 
 12 25(35 The square of the tens is subtracted, and the 
 
 _9 remainder contains twice the tens X the units + the 
 
 65 ) 3 25 square of the units. Twice the 3 tens is 6 tens, and 
 
 3 25 6 tens is contained in the 32 tens of the remainder 
 
 5 times. Hence, 5 is the units' figure of the root. 
 Since twice the tens X the units + the square of the units is equal to 
 (twice the tens + the units) X the units, the five units are annexed to 
 the 6 tens, and the result, 65, is multiplied by 5. 
 
 538. The same method will apply to numbers of more 
 than two groups of figures, by considering the part of the 
 root already found as so many tens with respect to the next 
 figure of the root. 
 
 Example. Extract the square root of 7,890,481. 
 
 7 89 04 81 ( 2809 Solution. When the third group, 04, is 
 
 _4 brought down, and the divisor, 56, formed, the 
 
 48 ) 3 89 next figure of the root is 0, because 56 is not 
 
 3 84 contained in 50. Therefore, is placed both 
 
 5609 ) 5 04 81 in the root and the divisor, and the next group, 
 
 5 04 81 81, is brought down. 
 
308 POWERS AND ROOTS. 
 
 539. If the square root of a number has decimal places, 
 the number itself will have twice as many. 
 
 Thus, if 0.11 is the square root of some number, the number will 
 be (0.11 ) 2 = 0.11 X 0.11 = 0.0121. Hence, if a given number contains 
 a decimal, we divide it into groups of two figures each, beginning 
 at the decimal point and marking toward the left for the integral 
 number, and toward the right for the decimal. The last group of the 
 decimal must have two figures, a cipher being annexed if necessary. 
 
 Example. Extract the square root of 52.2729. 
 
 62.27 29(7.23 
 
 49 
 142 )fl 27 Solution. It will be seen from the groups of 
 
 2 q. figures that the root will have one integral and two 
 
 1443 ) 43 29 decimal places. 
 
 43 29 
 
 540. If a number is not a perfect square, ciphers may 
 be annexed, and an approximate value of the root found. 
 
 Example. Extract the square root of 17 to six places. 
 
 17.00 00 00(4.123106 
 16 
 81 ) 1 00 Solution. In this example, after finding 
 
 81 four figures of the root, the other three are 
 
 822 ) 19 00 found by common division. 
 
 16 44 The rule in such cases is that one less than 
 
 8243 ) 2 56 00 the number of figures already obtained may be 
 
 2 47 29 found without error by division, the divisor to 
 
 8246 ) 8 710 be employed being twice the part of the root 
 
 8 246 already found. 
 
 46400 
 
 541. The square root of a common fraction is found by 
 extracting the square root of the numerator and of the 
 denominator. If the denominator is not a perfect square, 
 multiply both terms of the fraction by a number that will 
 make the denominator a perfect square, or reduce the frac- 
 tion to a decimal and extract the root of the decimal. 
 
POWERS AND ROOTS. 309 
 
 542. Rule for Square Root. Separate the number 
 into groups of two figures each, beginning at the units. 
 
 Find the greatest square in the left-hand group and write 
 its root for the first figure of the required root. 
 
 Square this root, subtract the result from the left-hand group, 
 and to the remainder annex the next group for a dividend. 
 
 For a partial divisor, double the root already found, con- 
 sidered as tens, and divide the dividend by it. The quotient 
 (or the quotient diminished) will be the next figure of the 
 root. 
 
 To this partial divisor add the last figure of the root for 
 a complete divisor. Multiply this complete divisor by the last 
 figure of the root, subtract the product from the dividend, and 
 to the remainder annex the next group for a new dividend. 
 
 Proceed in this manner until all the groups have been thus 
 annexed. The result will be the square root required. 
 
 Note 1. If the number is not a perfect square, annex groups of 
 zeros and continue the process. 
 
 Note 2. If the given number contains a decimal, divide it into 
 groups of two figures each, beginning at the decimal point and mark- 
 ing toward the left for the integral number and toward the right for 
 the decimal number. The last group on the right of the decimal must 
 contain two figures, a zero being annexed if necessary. 
 
 nd 
 
 1 the square ] 
 
 Exercise 139. 
 root of : 
 
 
 
 1. 
 
 2916. 
 
 9. 53.7289. 
 
 17. 
 
 8£ 
 
 2. 
 
 7921. 
 
 10. 883.2784. 
 
 18. 
 
 0.9. 
 
 3. 
 
 494,209. 
 
 11. 1.97262025. 
 
 19. 
 
 1- 
 
 4. 
 
 20,164. 
 
 12. 0.0002090916. 
 
 20. 
 
 f- 
 
 5. 
 
 3,345,241. 
 
 13. 2. 
 
 21. 
 
 h 
 
 6. 
 
 125,457.64. 
 
 14. 5. 
 
 22. 
 
 !• 
 
 7. 
 
 47,320,641. 
 
 15. 0.3. 
 
 23. 
 
 3 
 4* 
 
 8. 
 
 21,609. 
 
 16. 3i. 
 
 24. 
 
 §• 
 
310 POWERS AND ROOTS. 
 
 Cube Root. 
 
 543. The cube of a number is the product of three 
 factors, each equal to the number (§ 69). 
 
 Thus, the cubes of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
 are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. 
 
 544. The cube root of a number is one of the three 
 equal factors of the number (§ 69). 
 
 Thus, the cube roots of 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 
 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 
 
 545. The cube root of a number is indicated by y/, a 
 small figure 3 being written above the radical sign, or by 
 the fractional exponent -J. 
 
 Thus, V343, or 343*, means the cube root of 343. 
 
 546. Since 35 = 30 + 5, the cube of 35 may be obtained 
 as follows : 
 
 30 +5 
 30 +5 
 
 30 2 -f- (30X5) 30 s = 27,000 
 
 + (30X5) + 5 a 3 (30 s X 5) = 13,500 
 
 30 2 + 2(30X5) + 5 a 3(30X5*)= 2,250 
 30 +5 5 3 = 125 
 
 30 3 + 2(30 2 X5) + (30 X5 2 ) 35 s = 42,875 
 -f (30 2 X5) + 2(30X5 2 ) + 5 8 
 
 30 3 + 3 (30 2 X 5) -f 3 (30 X 5 2 ) + 5 8 
 
 If we represent the number of tens by a and of units by b 
 
 (a + b) s = a 3 + 3 a 2 b + 3 ab 2 + b\ Hence, 
 
 The cube of a number is equal to the cube of the tens, plus 
 three times the product of the square of the tens by the units, 
 plus three times the product of the tens by the square of the 
 units, plus the cube of the units. 
 
POWERS AND ROOTS. 311 
 
 547. In extracting the cube root of a number, the first 
 step is to separate the figures of the number into groups. 
 
 Since 1 = l 3 , 1000 = 10 3 , 1,000,000 = 100 3 , and so on, it follows 
 that the cube root of any integral number between 1 and 1000, that 
 is, of any integral number that has one, two, or three figures, is a 
 number of one figure ; that the cube root of any integral number 
 between 1000 and 1,000,000, that is, of any integral -number that has 
 four, five, or six figures, is a number of two figures, and so on. 
 
 If, therefore, an integral number is divided into groups of three 
 figures each, from right to left, the number of figures in the root will 
 be equal to the number of groups. The last group to the left may 
 consist of one, two, or three figures. 
 
 Example. Extract the cube root of 42,875. 
 
 Solution. Since 42,875 consists of two groups, the cube root will 
 consist of two figures. 
 
 The first group, 42, contains the cube of the tens' number of the 
 root. 
 
 The greatest cube in 42 is 27, and the cube root of 27 is 3. Hence, 
 
 3 is the tens 1 figure of the root. 
 42 875 ( 35 Tlie remamder » 15,875, result- 
 
 27 ing from subtracting the cube of 
 
 3 X 30 2 = 2700 
 
 3 X (30 X 5) = 450 
 
 5 2 = 25. 
 
 3175 
 
 15 875 
 
 the tens, will contain three times 
 
 the product of the square of the 
 
 tens by the units + three times 
 
 15 gyg the product of the tens by the 
 
 square of the units 4- the cube 
 
 of the units. 
 
 Each of these three parts contains the units' number as a 
 
 factor. 
 
 Hence, the 15,875 consists of two factors, one of. which is the units' 
 number of the root ; and the other factor is three times the square 
 of the tens +• three times the product of the tens by the units + the 
 square of the units. The largest part of this second factor is three 
 times the square of the tens. 
 
 If the 158 hundreds of the remainder is divided by the 3 X 30 2 = 27 
 hundreds, the quotient will be the units' number of the root. 
 
 The second factor can now be completed by adding to the 2700 
 3 X (30 X 5) = 450 and 5 2 = 25. 
 
312 
 
 POWERS AND ROOTS. 
 
 548. The same method will apply to numbers of more 
 than two groups of figures, by considering the part of the 
 root already found as so many tens with respect to the next 
 figure of the root. 
 
 Example. Extract the cube root of 57,512,456. 
 
 57 512 456(386 
 
 3 X 30 2 = 2700 
 
 30 512 
 
 3 X (30 X 8) = 720 
 
 
 8 2 = 64 
 3484 
 
 27 872 
 
 
 2 640 456 
 
 3 X 3802 = 433200 
 
 
 [ X (380 X 6) = 6840 
 
 
 62= 36 
 
 
 440076 
 
 2 640 -156 
 
 549. If the cube root of a number has decimal places, 
 the number itself will have three times as many. 
 
 Thus, if 0.11 is the cube root of a number, the number is 0.11 X 
 0.11 X 0.11 = 0.001331. Hence, if a given number contains a decimal, 
 we divide the figures of the number into groups of three figures each, 
 by beginning at the decimal point and marking toward the left for 
 the integral number, and toward the right for the decimal. We must 
 be careful to have the last group on the right of the decimal point 
 contain three figures, annexing ciphers when necessary. 
 
 Example. Extract the cube root of 187.149248. 
 
 187.140 248(5.72 
 125 
 
 3 X 50 2 = 7500 
 
 62 149 
 
 3 X (50 X 7) = 1050 
 
 
 7 2 = 49 
 
 
 8599 
 
 60 193 
 
 
 
 1 956 248 
 
 3 X 570 2 = 974700 
 
 
 3 X (570 X 2) = 3420 
 
 
 22= 4 
 
 
 97815 
 
 it 
 
 1 956 248 
 
POWERS AND ROOTS. 
 
 313 
 
 It will be seen from the groups of figures that the root will have 
 one integral and two decimal places, and therefore the decimal point 
 must he placed in the root as soon as one figure of the root is obtained. 
 
 550. If the given number is not a perfect cube, ciphers 
 may be annexed, and a value of the root may be found as 
 near to the true value as we please. 
 
 Example. Extract the cube root of 5 to five places. 
 
 5.000(1.70997 
 
 1 
 
 3 x 
 
 3 x 10 2 = 300 
 
 (10 X 7) = 210 
 
 7 2 = _49 
 
 559 
 
 259 
 
 J2] 
 
 559 > 
 259 J 
 
 4 000 
 
 3 913 
 
 3 X 1700 2 = 8670000 
 3 X (1700 X 9) = 45900 
 
 9-2= 81 
 
 8715981 
 
 45981 
 
 3 X 1709 2 = 8762043 
 
 } 
 
 87 000 000 
 
 78 443 829 
 
 8 556 1710 
 
 7 885 8387 
 
 670 33230 
 613 34301 
 
 After the first two figures of the root are found, the next trial divi- 
 sor is obtained by bringing down the sum of the 210 and 49 obtained 
 in completing the preceding divisor, then adding the three numbers 
 connected by the brace, and annexing two ciphers to the result. 
 
 It is seen at a glance that, when the trial divisor is increased by 
 3 times the 17 tens of the root, it will be greater than 87,000 ; so that 
 is placed in the root, and 3 X 1700 2 is obtained by annexing two 
 ciphers to the 86,700. Again : the last divisor is obtained by bringing 
 down the sum of the 45,900 and 81, which were obtained in completing 
 the preceding divisor, then adding the three numbers connected by 
 the brace. 
 
 The last two figures of the root are found by division. 
 
 The rule in such cases is that two less than the number of figures 
 already obtained may be found without error by division, the divisor to 
 be employed being three times the square of the part of the root already 
 found. 
 
314 POWERS AND ROOTS. 
 
 551. The cube root of a common fraction is found by- 
 taking the cube root of the numerator and of the denomi- 
 nator. If the denominator is not a perfect cube, multiply 
 both terms of the fraction by a number that will make the 
 denominator a perfect cube, or reduce the fraction to a deci- 
 mal, and then extract the cube root of the decimal. 
 
 552. Rule for Cube Root. Separate the number into 
 groups of three figures each, beginning at the units. 
 
 Find the greatest cube in the left-hand group and write its 
 root for the first figure of the required root. 
 
 Cube this root, subtract the result from the left-hand 
 group, and to the remainder annex the next group for a 
 dividend. 
 
 For a partial divisor, take three times the square of the 
 root already found, considered as tens, and divide the divi- 
 dend by it. The quotient (or the quotient diminished) will be 
 the second figure of the root. 
 
 To this partial divisor add three times the product of the 
 first figure of the root considered as tens by the second figure, 
 and also the square of the second figure. This sum will be 
 the complete divisor. 
 
 Multiply the complete divisor by the second figure of the 
 root, subtract the product from the dividend, and to the 
 remainder annex the next group for a new dividend. 
 
 Proceed in this manner until all the groups have been 
 annexed. The result will be the cube root required. 
 
 Note 1. If the number is not a perfect cube, annex groups of zeros 
 and continue the process. 
 
 Note 2. If the given number contains a decimal, divide it into 
 groups of three figures each, beginning at the decimal point and mark- 
 ing toward the left for the integral number and toward the right for 
 the decimal number. The last group on the right of the decimal 
 must contain three figures, zeros being annexed if necessary. 
 
POWERS AND ROOTS. 
 
 315 
 
 Exercise 140. 
 Find the cube root of : 
 
 304,957.115891. 
 
 0.007821346625. 
 
 104.600290750613. 
 
 17,183,498,535,125. 
 
 122,615.327232. 
 
 116,400. 
 
 22,406,807. 
 
 1. 
 
 1331. 
 
 8. 
 
 2. 
 
 1728. 
 
 9. 
 
 3. 
 
 12.167. 
 
 10. 
 
 4. 
 
 300.763. 
 
 11. 
 
 5. 
 
 148,877. 
 
 12. 
 
 6. 
 
 2,048,383. 
 
 13. 
 
 7. 
 
 59.776471. 
 
 14. 
 
 15. 
 
 10 
 
 16. 
 
 3* 
 
 17. 
 
 H 
 
 18. 
 
 5. 
 
 19. 
 
 f- 
 
 20. 
 
 n 
 
 21. 
 
 3 
 4* 
 
 Geometrical Representation of Square and Cube Roots. 
 553. We will illustrate square root by giving a Geo- 
 
 metrical representation of the square root of 1225. 
 
 (§ 537) 
 
 The square root of 1225 is 35. 
 The square of (30 + 5) = 30 2 + 2 (30 X 5) + 5 2 . (§ 536) 
 
 The 30 2 may be represented by a square (Fig. 1) 30 in. on a side. 
 The 2 (30 X 5) may be represented by two strips 30 in. long and 
 5 in. wide of Fig. 2, which are added to two adjacent sides of Fig. 1. 
 
 Fig. l. 
 
 Fig. 2. 
 
 Fig. 3. 
 
 The 5 2 may be represented by the small square of Fig. 3 required 
 to make Fig. 2 a complete square. 
 
 In extracting the square root of 1225, the large square, which is 
 30 in. on a side, is first removed, and a surface of 325 sq. in. remains. 
 
 This surface consists of two equal rectangles, each 30 in. long, and 
 a small square whose side is equal to the width of the rectangles. 
 
 The width of the rectangles is found by dividing the 325 sq. in. by 
 the sum of their lengths ; that is, by 60 in., which gives 5 in. 
 
 Hence, the entire length of the surfaces added is 30 in. + 30 in. 
 + 6 in. = 65 in., and the width is 5 in. 
 
 Therefore, the total area is (65 X 5) sq. in. = 325 sq. in. 
 
316 
 
 POWERS AND ROOTS. 
 
 554. We will illustrate cube root by giving a Geomet- 
 rical representation of the cube root of 42,875. 
 
 The cube root of 42,875 is 35. (§ 547) 
 
 The cube of (30 + 5) = 30» + 3 ( 30 2 x 5) + 3 (30 X 52) + 5 8 . (§ 546) 
 The 30 8 may be represented by a cube whose edge is 30 in. (Fig. 1). 
 The 3 (30 2 X 5) may be represented by three rectangular solids, each 
 
 30 in. long, 30 in. wide, and 6 in. thick, to be added to three adjacent 
 
 faces of Fig. 1. 
 
 The 3(30 X 5 2 ) may be represented by three equal rectangular 
 
 solids, 30 in. long, 5 in. wide, and 5 in. thick, to be added to Fig. 2. 
 The 5 8 may be represented by the small cube required to complete 
 
 the cube of Fig. 3. 
 
 S^&\t 
 
 ^L 
 
 m 
 
 Fio. 1. 
 
 Fig. 2. 
 
 Fio. 3. 
 
 M 
 
 Fio. 4. 
 
 In extracting the cube root of 42,875, the large cube (Fig. 1), whose 
 edge is 30 in., is first removed. 
 
 There remain (42,875 — 27,000) cu. in. = 15,875 cu. in. 
 
 The greatest part of this is contained in the three rectangular solids 
 which are added to Fig. 1, and are each 30 in. long and 30 in. wide. 
 
 The thickness of these solids is found by dividing the 15,875 cu. in. 
 by the sum of the three faces, each of which is 30 in. square ; that is, 
 by 2700 sq. in. The result is 5 in. 
 
 There are also the three rectangular solids which are added to 
 Fig. 2, and which are 30 in. long and 5 in. wide ; and a cube which is 
 added to Fig. 3, and which is 5 in. long and 5 in. wide. 
 
 Hence, the sum of the products of two dimensions of all these 
 solids is 
 
 For the larger rectangular solids, 3 (30 X 30) sq. in. = 2700 sq. in. 
 
 For the smaller rectangular solids, 3 (30 X 5) sq. in. = 450 sq. in. 
 
 For the small cube, (5 x 5) sq. in. = 25 sq. in. 
 
 3175 sq. in. 
 
 This number multiplied by the third dimension gives (5 X 3175) 
 cu. in. = 15,875 cu. in. 
 
CHAPTER XVI. 
 
 MENSUBATION. 
 
 555. We have already considered Areas of Rectangles 
 and Circles ; and Volumes and Surfaces of Rectangular 
 Solids, Spheres, and Right Cylinders. 
 
 556. A polygon is a plane figure bounded by straight lines. 
 A polygon of three sides is a triangle ; of four sides, a 
 
 quadrilateral ; of five sides, a pentagon ; of six sides, a 
 hexagon; of eight sides, an octagon; of ten sides, a deca- 
 gon ; of twelve sides, a dodecagon ; and so on. 
 
 557. The area of any polygon may be found by divid- 
 ing it into triangles and finding the sum of their areas. 
 
 558. A vertex of a polygon is the point of intersection 
 of two adjacent sides. 
 
 559. A diagonal of a polygon is a straight line joining 
 any two vertices not adjacent. 
 
 Polygon. Regular Polygon. 
 
 560. A regular polygon is a polygon with all its sides 
 equal and all its angles equal. The centre of a regular 
 polygon is a point equidistant from the vertices and also 
 equidistant from the sides. The radius of a regular poly- 
 gon is the distance from the centre to any vertex. 
 
318 MENSURATION. 
 
 The radii of a regular polygon divide the polygon into 
 equal isosceles triangles ; that is, into triangles having two 
 sides equal. The apothem of a regular polygon is the dis- 
 tance from the centre to any side. 
 
 561. The area of a regular polygon = £ (perimeter X 
 apothem). 
 
 562. The apothem of a regular polygon bears a constant 
 ratio to one side. 
 
 The following table shows the ratio of the apothem to 
 one side in the most common regular polygons : 
 
 Triangle 0.2887:1. Heptagon .... 1.0382:1. 
 
 Quadrilateral . . 0.5000 : 1. Octagon 1.2071 : 1. 
 
 Pentagon .... 0.6882 : 1. Decagon .... 1.5388 : 1. 
 
 Hexagon .... 0.8660:1. Dodecagon . . . 1.8660:1. 
 
 Quadrilaterals. 
 
 LA 
 
 Trapezium. Trapezoid. Parallelogram. 
 
 563. A trapezium is a quadrilateral with no two of its 
 sides parallel. 
 
 Note. Two lines are parallel if all points of one are equally dis- 
 tant from the other. 
 
 564. A trapezoid is a quadrilateral with two of its sides 
 parallel, but the other two sides not parallel. 
 
 565. A parallelogram is a quadrilateral with its oppo- 
 site sides parallel. 
 
 566. A rhomboid is a parallelogram with its angles not 
 right angles. 
 
 567. A rhombus is a parallelogram with its angles not 
 right angles, but with all its sides equal. 
 
MENSURATION. 
 
 319 
 
 Parallelograms. 
 
 Rhomboid. 
 
 Rhombus. 
 
 Square. 
 
 Rectangle. 
 
 568. The altitude of a parallelogram or of a trapezoid 
 is the shortest distance between its parallel sides regarded 
 as bases. 
 
 569. The area of any parallelogram = base X altitude. 
 
 570. The area of a rhombus also = half the product of 
 its diagonals. 
 
 571. The area of a trapezoid = £ (sum of bases X alti- 
 tude). 
 
 Triangles. 
 
 Right. 
 
 Isosceles. 
 
 Equilateral. 
 
 Scalene. 
 
 572. A right triangle is a triangle one of whose angles 
 is a right angle. The hypotenuse is the side opposite the 
 right angle, and the other two sides, called legs, are the 
 base and the perpendicular. 
 
 573. Other kinds of triangles are, isosceles, with two 
 sides equal ; equilateral, with three sides equal ; scalene, 
 with no two sides equal. The altitude of a triangle is the 
 shortest distance from the vertex to the base or the base 
 produced. 
 
 574. When the base and altitude are given, 
 
 The area of the triangle = \ (base X altitude). 
 
320 
 
 MENSURATION. 
 
 575. When the sides of a triangle are given, 
 
 The area of the triangle is the square root of the product 
 of half the sum of the sides multiplied in succession by the 
 three remainders obtained by subtracting each side separately 
 from the half sum of the sides. 
 
 576. In a right triangle, the square of the hypote- 
 nuse is equal to the sum of the 
 squares of the other two sides. 
 
 The hypotenuse is, therefore, 
 equal to the square root of the 
 sum of the squares of the other 
 two sides ; and either leg is equal 
 to the square root of the differ- 
 ence of the squares of the hypote- 
 nuse and the other leg. 
 
 577. Examples. Find the area of : 
 
 1 . A regular hexagon, each side of which is 3 in. 
 
 Solution. The apothem = 0.8660 X 3 in. = 2.598 in.; 
 and the perimeter = 6 X 3 in. = 18 in. 
 
 Therefore, the area = i(18 X 2.598) sq. in. = 23.382 sq. in. 
 
 2. A parallelogram, base 12 in., altitude 7 in. 
 Solution. The area = (12 x 7) sq. in. = 84 sq. in. 
 
 3. A trapezoid, if its altitude is 10 in., and its parallel 
 sides are 16 in. and 12 in., respectively. 
 
 Solution. The sum of the bases is 16 in. + 12 in. = 28 in. 
 Therefore, the area = \ (28 X 10) sq. in. = 140 sq. in. 
 
 4. A triangle, base 12 in., altitude 8 in. 
 Solution. The area = i (12 x 8) sq. in. = 48 sq. in. 
 
 5. A triangle, sides 5 in., 6 in., 7 in. 
 
 Solution. The half sum of the sides is I (6 + 6 + 7) in., or 9 in. 
 Hence, the area = V9X4X8X2 sq. in. = V216 sq. in. = 14.696 sq. in. 
 
MENSURATION. 321 
 
 6. The base of a right triangle is 20 ft., and the perpen- 
 dicular is 15 ft. Find the hypotenuse. 
 
 Solution. V20 2 + 15 2 = V400 + 225 = V625 = 25. 
 Therefore, the length of the hypotenuse is 25 ft. 
 
 7. The base of a right triangle is 16 ft., and the hypote- 
 nuse is 20 ft. Find the perpendicular. 
 
 Solution. V20 2 — 16 2 = V400 - 256 = Vl44 = 12. 
 Therefore, the length of the perpendicular is 12 ft. 
 
 Exercise 141. 
 Find the area of : 
 
 1. A parallelogram, base 18 in., altitude 11 in. 
 
 2. A triangle, base 16 in., altitude 12 in. 
 
 3. A rectangle, base 24 in., altitude 18 in. 
 
 4. A square, side 18 in. 
 
 5. A rhombus, diagonals 8 in. and 10 in. 
 
 6. A triangle, sides 12 in., 11 in., and 10 in., respectively. 
 
 7. A regular hexagon, side 4 in. 
 
 8. A regular octagon, side 2 in. 
 
 9. A triangle, base 185 yd., altitude 154 yd. 
 
 10. A square, side 212 yd. 
 
 11. A rectangle, base 106 yd., altitude 66 yd. 
 12,. A parallelogram, base 24 ft., altitude 18 ft. 
 
 13. An equilateral triangle, side 132 yd. 
 
 14. A right triangle, base 164 ft., perpendicular 150 ft. 
 
 15. A regular pentagon, side 5 J in. 
 
 16. A parallelogram, base 122 yd., altitude 76 yd. 
 
 17. A regular decagon, side 2\ in. 
 
 18. A triangle, base 82 cm , altitude 51 cm . 
 
 19. A rhombus, diagonals 16 ft. and 12 ft. 
 
 20. A circle, diameter 72 ft. 
 
 21. A trapezoid, parallel sides 106 ft. and 56 ft., respec- 
 tively, altitude 48 ft. 
 
322 MENSURATION. 
 
 22. Find the number of hektars in a triangular field, 
 one side of which is 82.1 m , and the distance to this side 
 from the opposite corner 47.3 m . 
 
 23. Find the number of acres in a triangular field, one 
 side of which is 343.6 ft., and the distance to this side 
 from the opposite cornel- 163.2 ft. 
 
 24. Find the area of a circle that has a radius of 10 in. ; 
 of a circle that has a diameter of 10 ft.; of a circle that 
 has a circumference of 30 in. 
 
 25. A horse is tied by a rope 27. 8 m long; over what 
 part of a hektar can he graze ? 
 
 26. How many square feet in a circle that has a diam- 
 eter of 17# yd.? 
 
 27. How many square feet in a circle that has a circum- 
 ference of 117 yd.? 
 
 28. Find the area of a triangle whose sides are 73 ft., 
 57 ft., and 48 ft. 
 
 29. Find the number of hektars in a triangular field 
 whose sides are 37.5 m , 91. 7 m , and 78.9 m . 
 
 30. Find the number of hektars in a triangular field 
 whose sides are 67.5 m , 81.2 m , and 102.7 m . 
 
 31. Find the number of acres in a triangular field whose 
 sides are 227 ft., 342 ft., and 416 ft. 
 
 32. Find the number of acres in a triangular field whose 
 sides are 79.08 ch., 57.03 ch., and 102.19 ch. 
 
 33. Find the number of square rods in a triangle whose 
 sides are 7 rd. 2 yd. ; 6 rd. 5 yd. ; and 9 rd. 4£ ft. 
 
 34. One diagonal of a trapezium is 10 rd., and the per- 
 pendiculars upon it from the opposite corners are 6 rd. and 
 8 rd. Find the area. 
 
 35. Find the area of a lot of land in the shape of a 
 trapezium, if one diagonal is 108 ft., and the perpendicu- 
 lars upon it from the opposite corners are 55 ft. and 
 60 ft. 
 
MENSURATION. 323 
 
 36. What is the area of the ground covered by a tent, 
 the base of which is a regular heptagon 25 ft. on a side ? 
 
 37. How many paving stones will be required to pave a 
 rectangular court 60 ft. long and 40 ft. wide, if each stone 
 is in the shape of a regular hexagon 5 in. on a side ? 
 
 38. At $225 an acre, what is the value of a field in the 
 shape of a regular pentagon 250 yd. on a side ? 
 
 39. A rectangular field 100 yd. wide contains 3£ A. 
 What is its length ? 
 
 40. The dimensions of a rectangle are 45 yd. and 28 yd. 
 What is the length of its diagonal ? 
 
 41. A field has the shape of a right triangle, and the two 
 legs are 75 yd. and 60 yd., respectively. What decimal of 
 an acre does the field contain ? 
 
 42. Compare the areas of a square and an equilateral 
 triangle, if the perimeter of each is 60 ft. 
 
 43. Find the area of a field in the shape of a trapezoid, 
 if the altitude is 240 yd., and the parallel sides are 510 yd. 
 and 725 yd., respectively. 
 
 44. The legs of a right triangle are each equal to 12 ft. 
 Find the hypotenuse. 
 
 45. A city lot in the shape of a right triangle has for its 
 base 119 ft., and for its perpendicular 120 ft. Find the 
 area and the hypotenuse of the lot. 
 
 46. Find the base and the area of a right triangle, hypot- 
 enuse 130 yd., and perpendicular 112 yd. 
 
 47. Find the perpendicular and the area of a right tri- 
 angle, hypotenuse 164 ft., and base 160 ft. 
 
 48. Find the hypotenuse and the area of a right triangle, 
 base 100 yd., and perpendicular 105 yd. 
 
 49. Find the hypotenuse and the area of a right triangle, 
 base 96 ft., and perpendicular 110 ft. 
 
 50. Find the area of a field in the shape of a right tri- 
 angle, if the hypotenuse is 709 yd., and one leg 660 yd. 
 
324 MENSURATION. 
 
 51. A rectangular field is 345 yd. long and 152 yd. wide. 
 What is the length of its diagonal ? 
 
 52. The legs of a right triangle are 44 ft. 4 in. and 13 ft. 
 9 in., respectively. Find the length of its hypotenuse. 
 
 53. The hypotenuse of a right triangle is 7 ft. 1 in., and 
 one leg is 6 ft. 5 in. Find the other leg and the area. 
 
 54. The hypotenuse of a right triangle is 3 ft. 1 in., and 
 one leg is 2 ft. 11 in. Find the other leg and the area. 
 
 55. The area of a lot in the shape of a right triangle is 
 1560 sq. yd., and the base is 80 yd. Find the perpendicu- 
 lar and the hypotenuse. 
 
 56. The area of a right triangle is 60 sq. in., and one leg 
 is 8 in. Find the hypotenuse and the other leg. 
 
 57. The length and diagonal of a rectangular field are 
 60 rd. and 65 rd., respectively. What is its area? 
 
 58. What is the length of a side of a square that con- 
 tains 390,625 sq. ft.? 
 
 59. Express to six places of decimals the length of the 
 diagonal of a square in terms of a side. 
 
 60. The hypotenuse of a right triangle is 95 ft., and the 
 two legs are as 3 to 4. Find the legs and the area. 
 
 61. St. Mark's Square in Venice has the shape of a trape- 
 zoid. The parallel sides are 61 yd. and 90 yd., respec- 
 tively, and the altitude is 192 yd. What is its area ? 
 
 62. The perimeter of a regular hexagon is 45 in. Find 
 its area. 
 
 63. A circular pond contains 12 acres. Express its 
 diameter in feet. 
 
 Note. Multiply the area in square feet by 0.31831, and take the 
 square root for the radius. 
 
 64. What is the diameter of a circle whose area is 
 1262 sq. ft.? 
 
 65. What is the diameter of a circle whose area is 
 2206 sq. ft.? 
 
MENSURATION. 
 
 325 
 
 Solids. 
 
 578. A right prism is a solid bounded by two equal 
 parallel polygons, called the bases, and by rectangles, called 
 the lateral faces. The altitude of a right prism is the 
 shortest distance between its bases. 
 
 Right Prism. 
 
 Regular Pyramid. 
 
 Right Cone. 
 
 579. A regular pyramid is a solid bounded by a regular 
 polygon, called the base, and by isosceles triangles, called 
 the lateral faces. These triangles all terminate in a point 
 called the vertex of the pyramid. The altitude of a regular 
 pyramid is the shortest distance from its vertex to its base. 
 The slant height is the altitude of the lateral faces. 
 
 580. A cone is a solid bounded by a circle, called the 
 base, and by a curved surface, called the lateral surface, 
 which terminates in a point called the vertex. 
 
 581. A right cone is a cone whose vertex is in the per- 
 pendicular erected at the centre of the base. The altitude 
 of a right cone is the shortest distance from its vertex to its 
 base. The slant height of a right cone is the distance from 
 its vertex to the circumference of its base. 
 
 582. A frustum of a regular pyramid or a frustum 
 of a right cone is the part of the pyramid or of the cone 
 left after its top has been cut off by a plane parallel to its 
 base. The lateral faces of a frustum of a regular pyramid 
 are trapezoids. 
 
326 MENSURATION. 
 
 583. The bases of a frustum of a regular pyramid or of 
 a frustum of a right cone are the base of the pyramid or 
 cone and the section made by the cutting plane. The 
 altitude of the frustum of a regular pyramid or of the 
 frustum of a right cone is the shortest distance between 
 its bases. 
 
 Frustum of a Regular Pyramid. Frustum of a Right Cone. 
 
 584. The area of the lateral surface of a right prism, 
 of a regular pyramid, or of the frustum of a regular pyra- 
 mid, is the sum of the areas of its lateral faces. 
 
 585. The lateral surface of a right prism = perimeter of 
 base X altitude. 
 
 586. The volume of a right prism = base X altitude. 
 
 587. The lateral surface of a regular pyramid = £ ( per- 
 imeter of base X slant height). 
 
 588. The volume of a regular pyramid = £ (base X alti- 
 tude). 
 
 589. The lateral surface of a right cone = £ (circumfer- 
 ence of base X slant height). 
 
 590. The volume of a right cone = J- (base X altitude). 
 
 591. The lateral surface of a frustum of a regular pyra- 
 mid or of a frustum of a right cone = £ (sum of perimeters 
 of bases X slant height). 
 
MENSURATION. 327 
 
 592. To find the volume of a frustum of a regular pyra- 
 mid or of a frustum of a right cone, we take the sum of the 
 areas of its bases and the square root of their product ; and 
 multiply this sum by one third the altitude. 
 
 593. Examples. 1. Find the lateral surface and the 
 volume of a right prism, base a square 3 ft. on a side, and 
 altitude 5 ft. 
 
 Solution. The perimeter of base = 4 X 8 ft. = 12 ft. 
 Hence, the lateral surface = (12 X 5) sq. ft. = 60 sq. ft. 
 The area of base = (3x3) sq. ft. — 9 sq. ft. 
 Hence, the volume = (9X5) cu. ft. = 45 cu. ft. 
 
 2. Find the lateral surface and the volume of a regular 
 pyramid, base a square 6 ft. on a side, and altitude 4 ft. 
 
 Solution. The peri meter o f base = 4x6 ft. = 24_ft. 
 The slant height = V3 a + 4 2 ft. = V9 + 16 ft = V25 ft. = 5 ft. 
 Hence, the lateral surface = i(24 X 5) sq. ft. = 60 sq. ft. 
 The area of base = (6x6) sq. ft. = 36 sq. ft. 
 Hence, the volume = £ (36 X 4) cu. ft. = 48 cu. ft. 
 
 3. Find the volume of a frustum of a regular pyramid, 
 bases squares, 5 ft. and 3 ft., respectively, on a side, and 
 altitude 6 ft. 
 
 Solution. The areas of the bases are 25 sq. ft. and 9 sq. ft., 
 respectively ; and the square root of their product is 15 sq. ft. 
 Hence, the volume = £ X 6 (25 + 9 + 15) cu. ft. = 98 cu. ft. 
 
 Exercise 142. 
 
 1. Find the volume of a triangular prism, height 11 in., 
 and sides of the ends 2 in., 3 in., and 4 in., respectively. 
 
 2. Find the capacity in bushels of a bin 6 ft. long, the 
 end of which is a square 3 ft. 3 in. on a side. 
 
 3. Find the lateral surface and the volume of a regular 
 pyramid, base a regular hexagon 9 in. on a side, altitude 
 40 in., and slant height 40.75 in. 
 
328 MENSURATION. 
 
 4. Find the number of cubic yards in a prism, base a 
 square 200 ft. on a side, height 40 ft. 
 
 5. How many square yards of canvas are required for 
 a conical tent 9 ft. 11 in. high, diameter of base 20 ft.? 
 
 6. Find the volume and the lateral surface of a frustum 
 of a regular pyramid, bases squares, 24 in. and 12 in. on a 
 side, respectively, altitude 17£ in., slant height 18£ in. 
 
 7. Find the volume and the lateral surface of a frustum 
 of a right cone, radii of bases 50 cm and 30 cm , respectively, 
 altitude 48 cm , and slant height 52 cm . 
 
 8. Find the volume and the surface of a sphere whose 
 diameter is 17.2 cm . 
 
 9. A right cylinder is 3 ft. 2 in. in diameter and 4 ft. 
 6 in. high. Find its volume and its lateral surface. 
 
 10. Find the length of an edge of a cubical vessel that 
 will hold a ton of water. 
 
 11. A rectangular tank 6 ft. long and 4£ ft. wide holds 
 108 cu. ft. of water. What is the height of the tank ? 
 
 12. Find the total surface of a regular pyramid, base a 
 square 5 ft. on a side, and slant height 20 ft. 
 
 1 3. The circumference of the base of a right cone is 12 ft., 
 and the height of the cone is 12 ft. Find the volume. 
 
 14. Find the surface of a megaphone in the shape of a 
 frustum of a right cone, diameters of the upper and lower 
 bases 24 in. and 3 in., respectively, slant height 30 in. 
 
 15. Find the difference between the volume of a frus- 
 tum of a regular pyramid, bases squares, 8 ft. and 6 ft., 
 respectively, on a side and altitude 9 ft., and the volume 
 of a right prism, base a square 7 ft. on a side, altitude 9 ft. 
 
 16. Find the surface and the volume of a sphere whose 
 diameter is 28 in. 
 
 17. Find the ratio of the volume of a cube of wood 15 in. 
 on an edge to the volume of the largest sphere that can be 
 turned from it. Find the ratio of their surfaces. 
 
MENSURATION. 329 
 
 18. Find the ratio of the volume of a cube of wood to 
 the volume of the largest right cylinder that can be turned 
 from it. Find the ratio of their surfaces. 
 
 19. Find the ratio of the volume of a right cylinder of 
 wood to the volume of the largest right cone that can be 
 turned from it. Find the ratio of their lateral surfaces. 
 
 20. Find the length of an edge of a cube that contains 
 100 cu. in. 
 
 21. The Great Pyramid of Egypt was originally made 
 in the form of a regular pyramid, altitude 480£ ft., and 
 base a square 764 ft. on a side. Find in acres the area of 
 the ground covered by the pyramid. Find in cubic yards 
 the volume, and in square yards the lateral surface of the 
 pyramid. 
 
 22. The mast of a ship is 80 ft. high, and the diameters 
 of its ends are 4 ft. 6 in. and 2 ft., respectively. Find its 
 value at 75 cents a cubic foot. 
 
 23. A spherical shot 6 in. in diameter is melted and cast 
 into a cylinder 3 in. in diameter. What is the height of 
 this cylinder ? 
 
 24. A cylindrical pail 14 in. High, holds 2 cu. ft. of 
 water. What is the diameter of its base ? 
 
 25. A regular pyramid 14 in. high has for its base an 
 equilateral triangle 6 in. on a side. What is its volume ? 
 
 26. A right prism 8 in. high has for its base a trapezoid 
 whose altitude is 4 in., and whose parallel sides are 5 in. 
 and 3 in., respectively. What is the volume and the total 
 surface of the prism ? 
 
 27. A rectangular room is 18 ft. long, 16 ft. wide, and 
 12 ft. high. What is the distance from the upper right- 
 hand corner to the opposite lower left-hand corner ? 
 
 28. A conical spire 40 ft. high has a base 15 ft. in diam- 
 eter. Find the cost at 5 cents a square inch of gilding the 
 spire. 
 
330 MENSURATION. 
 
 594. Similar Figures. Figures that have the same 
 shape are called similar figures. 
 
 595. The corresponding lines of similar figures are pro- 
 portional. 
 
 596. The surfaces of similar figures are to each other as 
 the squares of their corresponding dimensions; and their 
 volumes are to each other as the cubes of their cori'esponding 
 dimensions. 
 
 597. The corresponding dimensions of similar figures are 
 to each other as the square roots of their surfaces, or as the 
 cube roots of their volumes. 
 
 598. Examples. 1. A rectangle is 8 in. long and 6 in. 
 broad. Find the length and the area of a similar rectangle 
 whose breadth is 9 in. 
 
 Solution. 6 : 9 = 8 in. : required length. 
 Therefore, the required length = 12 in. 
 The area of the given rectangle = 48 sq. in. 
 Hence, 48 sq. in. : required area = 6 2 :9 2 = 4 :9. 
 Therefore, the required area = 108 sq. in. 
 
 2. The altitude of a right prism that contains 8 cu. ft. is 
 3 ft. Find the altitude of a similar right prism that con- 
 tains 27 cu. ft. 
 
 Solution. 3 ft.: required altitude = V8 : V27 = 2:3. 
 Hence, the required altitude = 4£ ft. 
 
 Exercise 143. 
 
 1. If the diameter of the moon is reckoned at 2000 mi., 
 and that of the earth at 8000' mi., find the ratio of their 
 surfaces and the ratio of their volumes. 
 
 2. If the diameters of two circles are 20 in. and 40 in., 
 find the ratio of their circumferences, and of their surfaces. 
 
 3. If the areas of two circles are 8000 sq. in. and 36,000 
 sq. in., respectively, find the ratio of their diameters. 
 
MENSURATION. 331 
 
 4. If the volumes of two spheres are 100 cu. in. and 
 1000 cu. in., respectively, find the ratio of their diameters. 
 
 5. If an ox 7 ft. in girth weighs 1500 lb., what will be 
 the girth of a similar ox that weighs 2500 lb.? 
 
 6. The surface of a pyramid is 560 sq. in. What is 
 the surface of a similar pyramid whose volume is 27 times 
 as great ? 
 
 7. The volume of a pyramid is 1331 cu. in. What is 
 the volume of a similar pyramid whose surface is 4 times 
 as great ? 
 
 8. If a well-proportioned man 5 ft. 10 in. high weighs 
 160 lb., what should a man 6 ft. high weigh, to the nearest 
 tenth of a pound ? What should be the height, to the 
 nearest tenth of an inch, of a man who weighs 210 lb.? 
 
 9. A three-gallon jug and a one-gallon jug are similar. 
 Find to three decimals the ratio of their diameters. 
 
 10. Two hills have exactly the same shape ; one is 
 900 ft. high, the other 1200 ft. Find the ratio of their 
 surfaces, and also the ratio of their volumes. 
 
 11. A ball 3 in. in diameter weighs 4 lb. ; another ball 
 of the same metal weighs 9 lb. Find the diameter of the 
 second ball to the nearest thousandth of an inch. 
 
 12. If Apollo's altar were a perfect cube 10 ft. on an 
 edge, -what would be the edge of a new cubical altar con- 
 taining twice as much stone ? 
 
 13. A man standing 40 ft. from a building 24 ft. wide 
 observed that, when he closed one eye, the width of the 
 building hid from view 90 rods of fence which was parallel 
 to the width of the building. Find the distance from the 
 eye of the observer to the fence. 
 
 14. A bushel measure and a peck measure are of the 
 same shape. Find the ratio of their heights. 
 
 15. If the height and the diameter of a cylinder are 
 both doubled, in what ratio is the volume altered ? 
 
CHAPTER XVII. 
 
 CONTINUED FRACTIONS AND SCALES OF NOTATION. 
 
 599. As in decimals we often require a result accurate 
 to a specified number of places, so in common fractions we 
 often require the most nearly accurate value of a ratio 
 that can be expressed by a fraction with a denominator 
 limited to a certain size. 
 
 600. Example. Find the most nearly accurate value of 
 the ratio of the circumference of a circle to the diameter 
 expressed by a fraction with a denominator less than 10 ; 
 less than 100 ; less than 1000. 
 
 Solution. The ratio 3.1416 is true to the nearest ten-thousandth. 
 Reducing Jfflfo to its lowest terms, we have J£fa. 
 
 Then, as in the margin, we divide the denom- 
 177)1250(7 inator by the numerator; the last divisor by 
 
 1 239 the last remainder ; and so on, as in finding 
 
 ll)l77(16 the greatest common measure. 
 
 176 If, therefore, we divide both terms of the 
 
 '" fraction Jjf v by the numerator, we have r-—r ; 
 
 11 / T77 
 
 and if we omit the fraction in the denomi- 
 nator, we have for the required ratio with a denominator less than 
 10, 3 7 , or y. 
 
 If we put the fraction -^ in the form of tttt and omit the fraction 
 
 in the denominator, the ratio becomes 
 
 3^ = 3^ = 111; 
 
 which shows that 2 7 2 - is the most nearly accurate value of the ratio 
 expressed by a fraction with a denominator less than 100, and that 
 fff is the most nearly accurate value of the ratio expressed by a frac- 
 tion with a denominator less than 1000. 
 
CONTINUED FRACTIONS. 333 
 
 601. Continued Fractions. After the quotients have 
 been found the results may be written in a fractional form 
 as follows : 
 
 1 
 
 7-f 
 
 16 +n 
 
 Such a fraction is called a continued fraction. 
 
 602. To find the successive approximate values of a con- 
 tinued fraction we begin at the top and take first one, then 
 two, then three, and so on, of its parts. Thus, 
 
 The first approximate value is 3. 
 
 The second is 3 + \ = - 2 T 2 -. 
 
 The third is 3 + — L- = 3 + T \% = f f f. 
 
 The fourth is 3 + 
 
 7 + 
 
 »+A 
 
 and this = 3 + -\ T = 3 + tf& = fffi, or 3.1416. 
 
 'T7T , 
 
 603. In reducing the part of a continued fraction 
 selected for an approximate value, we begin with the 
 last fraction. 
 
 Thus, find the value of the continued fraction 
 
 1 
 
 2 + - 1 
 
 3 + -L 
 
 -I 
 
 1= .. JL-«. J-= 6, 
 
 4i 3 " 3 A «' 2H T "- 
 
334 CONTINUED FRACTIONS. 
 
 Exercise 144. 
 
 1. Change f T , }£, f£ 7 , -\\^ to continued fractions. 
 
 2. Find the approximate values of f ^ ; |^ ; |f^. 
 
 3. Find a series of fractions approximating to 0.236 ; 
 0.2361; 1.609. 
 
 4. Find a series of fractions approximating to 0.382 ; 
 1.732; 0.6253. 
 
 5. Find approximate values of ££} ; f J^ ; $}£ ; f ^£. 
 
 6. Find the proper fraction that, when changed to a 
 continued fraction, will have 2, 3, 5, 6, 7 as quotients. 
 
 7. Find a series of fractions approximating to the ratio 
 of the pound troy (5760 gr.) to the pound avoirdupois 
 (7000 gr.). 
 
 8. Find a series of fractions approximating to the ratio 
 of the side of a square to its diagonal ; that ratio being 
 1:1.414214, nearly. 
 
 9. Find a series of fractions approximating to the ratio 
 of the ar to the square chain, from the equality 
 
 1 ar = 0.2471 sq. ch. 
 
 10. Find a series of fractions approximating to the ratio 
 of the weight of the 48-pound shot to the weight of the 
 French shot of 24 k *. 
 
 11. If the mean diameter of the Earth is reckoned at 
 7912 mi., and that of Mars 4189 mi., find a series of frac- 
 tions approximating to the ratio of the mean diameters of 
 these two planets. 
 
 12. Find a series of fractions approximating to the ratio 
 of a cubic yard to a cubic meter, from the equality 
 
 1 cu. yd. = 0.76453 cbm . 
 
 13. Find a series of fractions approximating to the ratio 
 of the kilometer to the mile, from the equality 
 
 1™ = 1.09362 yd. 
 
SCALES OF NOTATION. 335 
 
 14. Find the proper fraction that, if changed to a con- 
 tinued fraction, will have as quotients 1, 7, 5, 2. 
 
 15. Find a series of fractions approximating to 0.5236 ; 
 approximating to 0.7854. 
 
 16. Find a series of fractions approximating to the con- 
 tinued fraction that has as quotients 7, 2, 1, 2, 6, 4 ; that 
 has as quotients 1, 2, 3, 4, 5, 6. 
 
 Scales of Notation. 
 
 604. The common mode of representing numbers is 
 called the common scale of notation, and io is called its 
 radix or base. 
 
 605. In the common or decimal scale every figure placed 
 to the left of another represents ten times as much as if it 
 were in the place of that other. 
 
 606. Instead of the radix number 10, any other integral 
 number might be used as the base of a system of notation. 
 
 Thus, the number 6532 stands for : 
 
 In the scale of 10, 6 X 10 3 + 5 X 10 2 + 3 X 10 + 2. 
 In the scale of 8, 6 X 8 3 + 5 x 8 a + 3 X 8 + 2. 
 In the scale of 7, 6 X 7 3 + 5 X 7 2 + 3 x 7 + 2. 
 
 607. A given number can be changed from one scale to 
 another scale. 
 
 608. Examples. 1. Express 6532 in the scale of 6. 
 
 Solution. The quotients and remainders of the successive divi- 
 sions by 6 are as follows : 
 
 6532 
 
 1088 remainder 4. 
 
 6 1 181 remainder 2. 
 
 6 [30 remainder 1. 
 
 5 remainder 0. 
 
 Therefore, 6532 expressed in the scale of 6 is 50,124. 
 
336 SCALES OF NOTATION. 
 
 2. Change 50,124 from the scale of 6 to the scale of 8. 
 
 Solution. 8 )50124 
 
 8 )3440 remainder 4. 
 
 8 )250 remainder 0. 
 
 8 [20 remainder 6. 
 
 1 remainder 4. 
 
 Therefore, the number required is 14,604. 
 
 Since 50,124 is in the scale of 6, each figure has six times the value it 
 would have one place to the right. Hence, at the beginning we have 
 to divide 6 X 5 + by 8, and we get 3 for the quotient and 6 for the 
 remainder. The next partial dividend is 6 X 6 4- 1, or 37, and this 
 divided by 8 gives 4 for the quotient and 5 for the remainder. The 
 next partial dividend is 6 X 5 + 2, or 32, and this divided by 8 gives 
 4 for the quotient and for the remainder ; and so on. 
 
 3. Change 14,604 from the scale of 8 to the scale of 10. 
 
 Solution. 10114604 
 
 10 ] 1215 remainder 2. 
 10 [101 remainder 3. 
 6 remainder 5. 
 Therefore, the required number is 6532. 
 
 4. Add 56,432 and 15,646 (scale of 7). 
 
 Solution. The process differs from that in the decimal 
 scale only in that when a sum greater than seven is reached, 
 we divide by seven (not ten), set down the remainder, and 
 
 add the quotient with the next column. 
 
 5. Subtract 34,561 from 61,235 (scale of 8). 
 
 Solution. When the number of any order of units in 
 
 the minuend is less than the number of the corresponding 
 
 order in the subtrahend, we increase the number in the 
 24454 
 
 minuend by eight instead of ten as in the common scale. 
 
 6. Multiply 5732 by 428 (scale of 9). 
 
 5732 
 
 Solution. We divide each partial product by nine, set 
 
 down the remainder, and add the quotient to the next 
 
 partial product. 
 8o28o 
 
 2712127 
 
SCALES OF NOTATION. 337 
 
 Divide 2,712,127 by 5732 (scale of 9). 
 
 428 
 5732)2712127 
 25238 
 17722 
 12564 
 
 51477 
 51477 
 
 The operations of multiplication and subtraction involved in this 
 problem are precisely the same as in the decimal notation. The only 
 difference is that the radix number is 9 instead of 10. 
 
 Exercise 145. 
 
 Change 4852 of the common scale to : 
 
 1. The scale of 7. 5. The scale of 6. 
 
 2. The scale of 2. 6. The scale of 5. 
 
 3. The scale of 9. 7. The scale of 8. 
 
 4. The scale of 3. 8. The scale of 4. 
 
 Change : 
 
 9. 54,231 of the scale of 6 to the common scale. 
 
 10. 54,231 of the scale of 7 to the common scale. 
 
 1 1 . 54,231 of the scale of 8 to the common scale. 
 
 1 2. 54,231 of the scale of 9 to the common scale. 
 
 Perform the following arithmetical processes : 
 
 13. Add 67,814; 76,406; 88,718 (scale of 9). 
 
 14. Add 44,231 ; 13,432 ; 12,304 (scale of 5). 
 
 15. Subtract 77,614 from 114,672 (scale of 8). 
 
 16. Subtract 52,515 from 112,252 (scale of 6). 
 
 17. Multiply 14,612 by 6502 (scale of 7). 
 
 18. Multiply 72,645 by 46,723 (scale of 8). 
 
 19. Divide 162,542 by 6522 (scale of 7). 
 
 20. Divide 468,722 by 5432 (scale of 9). 
 
CHAPTER XVIII. 
 
 SERIES. 
 
 609. Series. A succession of numbers that proceed 
 according to some fixed law is called a series. The succes- 
 sive numbers are called the terms of the series. 
 
 610. A series that ends at some particular term is called 
 a finite series. A series that continues without end is 
 called an infinite series. 
 
 611. The number of different kinds of series is un- 
 limited ; in this chapter we shall consider only Arith- 
 metical Series, Geometrical Series, and Harmonical Series. 
 
 Arithmetical Progression. 
 
 612. A series of numbers that increase or decrease by a 
 common difference is called an Arithmetical Series or an 
 Arithmetical Progression. 
 
 Thus, the numbers 6, 8, 11, 14 form an arithmetical progression 
 with a common difference 3 ; and the numbers 12, 10, 8, 6 form an 
 arithmetical progression with a common difference 2. 
 
 613. In the increasing arithmetical progression 
 1st 2d 3d 4th 6th 6th 
 2, 5, 8, 11, 14, 17, 
 
 we find any term, as the 6th, by adding to the first term 
 the product of the common difference by a number one 
 less than the number of the term : 2 + (3 X 5), or 17. 
 In the decreasing arithmetical progression 
 
 1st 2d 3d 4th 5th 6th 7th 
 50, 46, 42, 38, 34, 30, 26, 
 
SERIES. 339 
 
 we find any term, as the 7th, by subtracting from the first 
 term the product of the common difference by a number 
 one less than the number of the term : 50 — (4 X 6), or 26. 
 Hence, 
 
 614. To Find Any Term of an Arithmetical Progression, 
 
 Multiply the common difference by a number one less than 
 the number of the required term. Add this product to the 
 first term if the series is an increasing series ; subtract this 
 product from the first term if the series is a decreasing series. 
 
 Exercise 146. 
 
 1. Find the seventh term of the series 3, 5, 7, etc. 
 
 2. Find the fifteenth term of the series 2, 7, 12, etc. 
 
 3. Find the sixth term of the series 2, 2|, 3f, etc. 
 
 4. Find the twentieth term of the series 2, 3 J, 4^, etc. 
 
 5. Find the seventh term of the series 21, 19, 17, etc. 
 
 6. Find the twelfth term of the series 18, 17 J, 16f, etc. 
 
 7. If the first term of a series is 5, and the common 
 difference 2j, find the thirteenth and eighteenth terms. 
 
 8. If the fourth term of a series is 18, and the common 
 difference 3, find the seventh and eleventh terms. 
 
 9. If the fifth term of a decreasing series is 52, and the 
 common difference 3^, find the twelfth and eighteenth terms. 
 
 10. If the fourth term of a series is 14, and the twelfth 
 term 38, what is the common difference ? 
 
 Hint. The difference between the fourth and twelfth terms is 
 evidently eight times the common difference. 
 
 Find the common difference in a series : 
 
 11. If the fourth term is 12 and the seventh term 27. 
 
 12. If the first term is 20 and the fourth term 40. 
 
 13. If the first term is 2 and the eleventh term 20. 
 
 14. If the third term is 7 and the eighth term 12^. 
 
 15. If the first term is 1 and the fourth term 19. 
 
340 SERIES. 
 
 615. The sum of seven terms of the series 3, 5, 7, etc., is 
 
 3+ 5 + 7+ 9 + 11 + 13 + 15, 
 in reverse order is 15 + 13 + 11+ 9+ 7+ 5+ 3. 
 
 Hence, twice the sum is 18 + 18 + 18 + 18 + 18 + 18 + 18, 
 or 7 X 18. 
 
 Therefore, the sum is £ (7 X 18), or 63. 
 
 Here 7 is the number of terms, and 18 is the sum of the 
 first and last terms. Hence, 
 
 616. To Find the Sum of the Terms of an Arith- 
 metical Progression, 
 
 Multiply one half the sum of the first and last terms by 
 the number of terms. 
 
 Thus, the sum of eight terms of the series whose first term is 3 and 
 last term 38 is 8 X £ (3 + 38) = 164. 
 
 Exercise 147. 
 
 1. Find the sum of 1, 5, 9, etc., to twenty terms. 
 
 2. Find the sum of 4, 5£, 7, etc., to eight terms. 
 
 3. Find the sum of 8, 7f , 7£, etc., to sixteen terms. 
 
 4. Find the sum of 20, 18£, 16£, etc., to seven terms. 
 
 5. Find the sum of the first twenty natural numbers. 
 
 6. Find the sum of the natural numbers from 37 to 53 
 both inclusive. 
 
 7. Find the sum of a series of thirty terms, if the first 
 term is 21 and the last 59. 
 
 8. Find the sum of the series whose first two terms are 
 3 and 9, and the last term 75. 
 
 9. Find the sum of a series of twenty terms whose 
 third and fifth terms are 10 and 15, respectively. 
 
 10. A body falls through a space of 16^ ft. in the first 
 second of its fall, and in each succeeding second 32£ ft. 
 more than in the second just before. How far will a stone 
 fall in the seventh second ? How far in seven seconds ? 
 
SERIES. 341 
 
 11. A travels 8 miles the first day, 11 miles the second, 
 14 miles the third, and so on, and overtakes in 17 days B 
 who started at the same time, and traveled at a uniform 
 rate. What is B's rate per day ? 
 
 12. In a potato race 100 potatoes are placed in a straight 
 line 3 ft. distant from each other. A boy, starting from a 
 basket 3 ft. from the first potato, is required to pick them 
 up one by one and carry them to the basket. To finish the 
 race, how far must the boy run ? 
 
 13. How many times a day does a clock strike that 
 strikes the hours only ? 
 
 14. A body falls through a space of 4.9 m in the first 
 second of its fall, and in each succeeding second 9.8 m more 
 than in the second just before. A stone dropped from a 
 balloon was 35 seconds in reaching the ground. How high 
 was the balloon ? 
 
 Geometrical Progression. 
 
 617. A series of numbers each term of which after 
 the first is obtained by multiplying the preceding term 
 by a constant multiplier is called a Geometrical Series, 
 or a Geometrical Progression. The constant multiplier 
 is called the ratio. 
 
 Thus, 3, 6, 12, 24 form a geometrical progression with a ratio 2 ; 
 and 97 3, 1, -J- form a geometrical progression with a ratio £. 
 
 618. In the geometrical progression 
 
 1st 2d 3d 4th 5th 6th 
 
 2, 6, 18, 54, 162, 486, 
 
 the second term is 2 X 3; the third term is 2 X 3 2 ; the fourth 
 term is 2 X 3 3 ; the fifth term is 2 X 3 4 ; and so on. Hence, 
 
 619. To Find Any Term of a Geometrical Progression, 
 
 Multiply the first term by that power of the ratio that is 
 one less than the number of the term required. 
 
342 SERIES. 
 
 620. If any two consecutive terms of a geometrical pro- 
 gression are known, the ratio may be found by dividing 
 the second of these terms by the first. 
 
 If two terms not consecutive are known, the ratio may 
 be found as in the following 
 
 Example. Find the ratio in a geometrical progression if 
 the second and sixth terms are 5 and 80, respectively. 
 
 Solution. 6th term = 2d term x (ratio) 4 . 
 
 Therefore, (ratio) 4 = 4£ = 16, 
 
 and ratio = Vl6 = 2. 
 
 Exercise 148. 
 
 1. Find the eighth term of the series 2, 6, 18, etc. 
 
 2. Find the fifth term of the series 8, 4, 2, etc. 
 
 3. Find the seventh term of the series 2, 3, 4^, etc. 
 
 4. Find the sixth term of the series 4, 2§, 1$, etc. 
 
 5. Find the eighth term of the series 4, 10, 25, etc. 
 
 6. Find the fifth term of the series J, ^ gij, etc. 
 
 7. Find the ninth term of the series 4, 2, 1, etc. 
 
 8. Find the sixth term of the series 6, 9, 13£, etc. 
 
 9. Write the first six terms of the geometrical series 
 whose fifth and sixth terms are 112 and 224, respectively. 
 
 10. The seventh and ninth terms of a geometrical series 
 are 100 and 144, respectively. Find the twelfth term. 
 
 11. A capital of $1000 is increased by ^ of itself each 
 year. What will it be at the beginning of the fifth year ? 
 
 12. A capital of $1000 is increased by jfo of itself each 
 year. What will it be at the beginning of the sixth year ? 
 
 621. In the geometrical progression 4, 12, 36, 108, 324, 
 etc., the sum of five terms is 4 + 12 + 36 + 108 + 324. 
 
 If we multiply this sum by the ratio 3, and from the 
 product subtract the sum of the five terms, we shall have 
 twice the sum of the terms. That is, 
 
SERIES. 343 
 
 12 + 36 + 108 + 324 + 972 
 4 + 12 + 36 + 108 + 324 
 
 Therefore, the sum = 
 
 972-4. 
 972-4 
 
 The numerator is the difference between the product of 
 the last term by the ratio and the first term ; and the 
 denominator is the ratio minus 1. Hence, 
 
 622. To Find the Sum of the Terms of a Geometrical 
 Progression, 
 
 Multiply the last term by the ratio, and subtract the first 
 term from the product. Divide the remainder by the ratio 
 minus 1. 
 
 If the ratio is less than 1, 
 
 Multiply the last term by the ratio, and subtract the product 
 from the first term. Divide the remainder by 1 minus the 
 ratio. 
 
 Exercise 149. 
 
 1. Find the sum of 2, 6, 18, etc., to six terms. 
 
 2. Find the sum of 1, 2, 4, etc., to nine terms. 
 
 3. Find the sum of 3, 9, 27, etc., to five terms. 
 
 4. Find the sum of 2, 3, 4£, etc., to eight terms. 
 
 5. Find the sum of 1, J, ^, etc., to eight terms. 
 
 6. Find the sum of 1, ^, i, etc., to ten terms. 
 
 7. Find the sum of £, £, }, etc., to eight terms. 
 
 8. Find the sum of the first six terms of the series 
 whose first term is 3 and ratio 5. 
 
 9. Find the sum of the first eight terms of the series 
 whose first term is 3 and ratio ^. 
 
 10. A man saved in one year $64, and in each succeed- 
 ing year, for 9 years more, l£ times as much as in the pre- 
 ceding year. Find the whole amount he saved. 
 
344 SERIES. 
 
 623. If we represent the first term by a, the last term 
 by I, the ratio by r, the number of terms by n, and the sum 
 of the terms by s, we have, if the ratio is less than 1 (§ 622), 
 
 a — r X I 
 
 Now, since Z= a X r" _1 , rl = ar"; and we have 
 _ a — ar* _ a X (1 — r") 
 1 — r 1 — r 
 
 Since r is less than 1, r" becomes smaller as n becomes 
 larger ; and when n is too large to be counted, r" becomes 
 too small to be considered. We then have 
 
 s = Hence, 
 
 1 — r 
 
 624. To Find the Sum of the Terms of an Infinite, 
 Decreasing Geometrical Progression, 
 
 Divide the first term by 1 minus the ratio. 
 
 Exercise 150. 
 Find the sum of the infinite series : 
 
 1. h h h • 6 - 0.212121 
 
 2. §, |, A, • 7. 0.9999 
 
 3. i,A,A> ' 8 ' 0.232323- 
 
 4. h A, t**> • 9 - 0.36848484 
 
 5. 0.171717 10. 0.15272727 
 
 625. A series is called a Harmonical Series, or a Har- 
 monical Progression, if the reciprocals of its terms form an 
 arithmetical series. 
 
 Thus, the numbers 1, i, i, £ form a harmonical progression, since 
 1, 2, 3, 4, the reciprocals of the terms, are in arithmetical progression. 
 
 626. Questions involving a harmonical progression may 
 be solved by writing the reciprocals of the terms so as to 
 form an arithmetical progression. 
 
CHAPTER XIX. 
 
 COMMON LOGARITHMS. 
 
 627. In the common system of notation the expression 
 of numbers is founded on their relation to 10. 
 
 Thus, 5434 indicates that this number consists of 10 3 five times, 
 10 2 four times, 10 three times, and four units. 
 
 628. But any number may be expressed, exactly or 
 approximately, as a power of 10. 
 
 Thus, 5434 is greater than 10 3 and less than 10 4 , and is expressed 
 approximately by 10 3 - 7351 . 
 
 629. When a number is expressed as a power of 10, the 
 exponent of 10 is called the logarithm of that number. 
 
 Thus, if 10 8 - 7861 = 5434, the logarithm of 5434 = 3.7351, a state- 
 ment that is generally written log 5434 = 3.7351. 
 
 . . . 10 2 
 
 630. By division — 2 = 1 ; and by the rule for expo- 
 
 10 2 
 nents (§ 108) — 2 == 10 2 ~ 2 = 10°. Therefore, 10° = 1. 
 
 Since 10°= 1, log 1 = 0, 
 
 10 x = 10, log 10 = 1, 
 
 10 2 = 100, log 100 = 2, 
 
 10 3 = 1000, log 1000 = 3, and so on. 
 
 631. It is often convenient to write the reciprocals of 
 powers of 10 in an integral form; — is written 10 _1 ; —* 
 
 is written 10 _2 ; — -„ is written 10 ~ 3 ; and so on. 
 10 3 
 
 Note. An exponent with the minus sign prefixed is called a nega- 
 tive exponent ; and an exponent with the plus sign prefixed, or with- 
 out any sign, is called a positive exponent. 
 
and 
 
 1, 
 
 
 
 1 and 
 
 o 
 
 
 
 2 and 
 
 3, 
 
 
 
 and ■ 
 
 -1, 
 
 
 
 1 and 
 
 -2, 
 
 
 
 2 and- 
 
 -3, 
 
 and 
 
 so on. 
 
 346 COMMON LOGARITHMS. 
 
 632. Since lO" 1 = 0.1, log 0.1 = — 1, 
 
 10-* = 0.01, log 0.01 = — 2, 
 
 10- 3 = 0.001, log 0.001 = — 3, and so on. 
 
 633. It is evident, therefore, that the logarithms of all 
 numbers between 
 
 1 and 10 will lie between 
 
 10 and 100 will lie between 
 100 and 1000 will lie between 
 
 1 and 0.1 will lie between 
 0.1 and 0.01 will lie between 
 0.01 and 0.001 will lie between 
 
 634. If a number is less than 1, its logarithm is nega- 
 tive (§ 633), but is written in such a form that its decimal 
 part is always positive. 
 
 Thus, log 0.0284 = - (1.5467) = (— 1)+ (— .5467) = - 2 + .4533. 
 
 If the integral part of a logarithm is negative, the minus 
 sign is written over the integral part. 
 Thus, log 0.0284 is written 2.4533. 
 
 635. Every logarithm, therefore, consists of two parts : 
 a positive or negative integral number called the character- 
 istic, and a positive decimal called the mantissa. 
 
 Thus, in the logarithm 2.4533, the integral number 2 is the charac- 
 teristic, and the decimal .4588 is the mantissa. In the logarithm 
 2. 1638, the negative integral number — 2 is the characteristic and the 
 positive decimal .4533 is the mantissa. 
 
 636. If a logarithm has a negative characteristic, it is 
 customary to change its form by adding 10 or a multiple 
 of 10 to the characteristic, and indicating the subtraction 
 of the same number from the result. 
 
 Thus, the logarithm 2.4533 is written 8.4533 — 10, and the loga- 
 rithm 13.4533 is written 7.4533 - 20. 
 
COMMON LOGARITHMS. 34? 
 
 637. From an inspection of § 633, we deduce the follow- 
 ing rules for writing the characteristic of a logarithm : 
 
 Rule 1. If the given number is greater than 1, make the 
 characteristic of its logarithm one less than the number of 
 figures to the left of the decimal point in the number. 
 
 Rule 2. If the given number is less than 1, make the 
 characteristic of its logarithm negative, and one more than 
 the number of zeros between the decimal point and the first 
 significant figure of the given number. 
 
 Thus, the characteristic of log 7849.27 is 3 ; the characteristic of 
 log 0.037 is - 2, or 8.0000 - 10. 
 
 638. The mantissa of the logarithm of any number 
 depends only upon the sequence of the digits of the 
 number, and is unchanged so long as the sequence of 
 the digits remains the same. 
 
 For changing the position of the decimal point in a number is 
 equivalent to multiplying or dividing the number by a power of 10. 
 Its logarithm, therefore, will be increased by or diminished by the 
 exponent of that power of 10 ; and since this exponent is integral, the 
 mantissa, or decimal part of the logarithm, will be unchanged. 
 Thus, 27,196 = 10 4 - 434 *, 2.7196 = lO - 434 *, 
 
 2719.6 = 10 3 - 434 *, 0.27196 = lO 9 - 4345 -™, 
 
 27.196 = 10 1 - 4345 , 0.0027196 = W.ms-w, 
 
 639. A Four-place Table of Logarithms. A four-place 
 table of logarithms is given on pages 350 and 351. This 
 table contains the mantissas of the logarithms of all inte- 
 gral numbers under 1000, the decimal point being omitted. 
 
 Note. Tables containing logarithms to more decimal places can 
 be procured, but this table will serve for many practical uses, and will 
 enable the student to understand the use of five-place, seven-place, and 
 ten-place logarithms in work that requires greater accuracy. 
 
 640. In working with a four-place table, numbers cor- 
 responding to logarithms will be correct to four significant 
 digits. 
 
348 COMMON LOGARITHMS. 
 
 To Find the Logarithm of a Given Number. 
 
 641. The characteristic of the logarithm is determined 
 by the rules of § 637. 
 
 642. If the given number consists of a single digit, as 4, 
 8, etc., the mantissa of its logarithm is the same as the 
 mantissa of the logarithm of 40, 80, etc. 
 
 643. If the given number contains two digits, it is in 
 the column headed N, and the mantissa of its logarithm is 
 on the same line, and in the column headed 0. 
 
 Thus, log 28 = 1.4472, log 0.086 = 8.9345 - 10, 
 
 log 40 = 1.6021, log 7 = 0.8451. 
 
 644. If the given number contains three digits, or three 
 significant digits followed by one or more zeros, the first 
 two digits of the number are in the column headed N, and 
 the third at the top of the page in the line containing the 
 figures 0, 1, 2, etc. The mantissa of its logarithm is in 
 the column headed by the third figure and on the same line 
 with the first two figures. 
 
 Thus, log 742 =2.8704, log 84,100 =4.9248, 
 
 log 6090 = 3.7846, log 0.00261 = 7.4166 - 10. 
 
 645. If the given number contains four or more digits, 
 the mantissa of its logarithm is found as in the following 
 
 Examples. 1. Find the logarithm of 2034. 
 
 Solution. The required mantissa is (§ 638) the same as the man- 
 tissa for 203.4 ; hence, it is found by adding to the mantissa for 203 
 four tenths of the difference between the mantissas for 203 and 204. 
 
 The mantissa for 203 is 3076 ; and for 204 is 3096. 
 
 The difference between the mantissas for 203 and 204 is 21, and 0.4 
 of 21 = 8. Hence, 8 must be added to 3075. 
 
 Therefore, the mantissa for 203.4 is 3075 + 8 = 3083. 
 
 Therefore, log 2034 = 3.3083. 
 
 2. Find the logarithm of 0.0015764. 
 
 Solution. The required mantissa is (§ 638) the same as the man- 
 tissa for 157.64 ; hence, we add to the mantissa for 157 sixty-four 
 
COMMON LOGARITHMS. 349 
 
 hundredths of the difference between the mantissas for 157 and 
 158. 
 
 The mantissa for 157 is 1959 j and for 158 is 1987. 
 
 The difference between the mantissas for 157 and 158 is 28, and 
 0.64 of 28 = 18. Hence, 18 must be added to 1959. 
 
 Therefore, the mantissa for 157.64 is 1959 + 18 = 1977. 
 
 Therefore, log 0.0015764 = 7.1977 - 10. 
 
 Note. When the fraction of a unit in the part to be added to the 
 mantissa for three figures is less than 0.5, it is neglected ; when it is 
 0.5 or more, it is taken as one unit. 
 
 Exercise 151. 
 Find the logarithm of : 
 
 1. 
 
 70. 
 
 6. 
 
 6897. 
 
 11. 
 
 77,860. 
 
 16. 
 
 5.0009. 
 
 2. 
 
 101. 
 
 7. 
 
 9901. 
 
 12. 
 
 30,127. 
 
 17. 
 
 0.3769. 
 
 3. 
 
 333. 
 
 8. 
 
 4389. 
 
 13. 
 
 730.84. 
 
 18. 
 
 0.070707. 
 
 4. 
 
 3491. 
 
 9. 
 
 1111. 
 
 14. 
 
 0.008765. 
 
 19. 
 
 0.03723. 
 
 5. 
 
 1866. 
 
 10. 
 
 58,343. 
 
 15. 
 
 8.0808. 
 
 20. 
 
 98.871. 
 
 646. From § 633, we deduce the following rules for writ- 
 ing the decimal point in the antilogarithm of a logarithm ; 
 that is, in the number corresponding to a logarithm : 
 
 Rule 1. If the characteristic of the given logarithm is 
 positive, make the number of figures in the integral part of 
 its antilogarithm one more than the number of units in the 
 characteristic of the logarithm. 
 
 Rule 2. If the characteristic of the given logarithm is 
 negative, make the number of zeros between the decimal point 
 and the first significant figure of its antilogarithm one less 
 than the number of units in the characteristic of the loga- 
 rithm. 
 
 Thus, the number corresponding to the logarithm 4.7542 contains 
 five figures in its integral part ; the decimal corresponding to the log- 
 arithm 7.2816 — 10, that is 3.2816, contains two zeros between the 
 decimal point and the first significant figure. 
 
350 
 
 COMMON LOGARITHMS. 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 13 
 14 
 
 0000 
 0414 
 0792 
 1139 
 1461 
 
 0043 
 0453 
 0828 
 1173 
 1492 
 
 0086 
 0492 
 0864 
 1206 
 1523 
 
 0128 
 0531 
 0899 
 1239 
 1553 
 
 0170 
 0569 
 0934 
 1271 
 1584 
 
 0212 
 0607 
 0969 
 1303 
 1014 
 
 0253 
 0045 
 1004 
 1335 
 1044 
 
 0294 
 0682 
 1038 
 1367 
 1673 
 
 0334 
 0719 
 1072 
 1399 
 1703 
 
 0374 
 0755 
 1106 
 1430 
 1732 
 
 15 
 
 16 
 17 
 18 
 19 
 
 1761 
 2041 
 2304 
 2553 
 
 2788 
 
 1790 
 2068 
 2330 
 2577 
 2810 
 
 1818 
 2095 
 2355 
 2601 
 2833 
 
 1847 
 2122 
 2380 
 2625 
 2856 
 
 1875 
 2148 
 2405 
 2648 
 2878 
 
 1903 
 2175 
 2430 
 2672 
 2900 
 
 1931 
 2201 
 2455 
 2695 
 2923 
 
 1959 
 2227 
 2480 
 2718 
 2945 
 
 1987 
 2253 
 2504 
 2742 
 2967 
 
 2014 
 2279 
 2529 
 2705 
 2989 
 
 20 
 
 21 
 22 
 23 
 24 
 
 3010 
 3222 
 3424 
 3617 
 3802 
 
 3032 
 3243 
 3444 
 3636 
 3820 
 
 3054 
 3263 
 3464 
 3655 
 3838 
 
 3075 
 3284 
 3483 
 8674 
 
 3856 
 
 3096 
 3304 
 3502 
 3692 
 3874 
 
 3118 
 3324 
 3522 
 3711 
 3892 
 
 3139 
 3345 
 3541 
 8729 
 3909 
 
 3160 
 3365 
 3560 
 3747 
 3927 
 
 3181 
 3385 
 3579 
 3766 
 3945 
 
 3201 
 3404 
 3598 
 3784 
 3962 
 
 25 
 
 26 
 
 27 
 28 
 29 
 
 3979 
 4150 
 4314 
 4472 
 4624 
 
 3997 
 4166 
 4330 
 
 4487 
 4639 
 
 4014 
 4183 
 4346 
 4502 
 4654 
 
 4031 
 4200 
 4362 
 4518 
 4669 
 
 4048 
 4216 
 4378 
 4533 
 
 4683 
 
 4005 
 4232 
 4393 
 
 4548 
 4698 
 
 4082 
 4249 
 4409 
 4564 
 4713 
 
 4099 
 4265 
 4425 
 4579 
 4728 
 
 4116 
 4281 
 4440 
 4594 
 4742 
 
 4133 
 4298 
 
 4456 
 4009 
 4757 
 
 30 
 31 
 32 
 33 
 34 
 
 4771 
 4914 
 5051 
 5185 
 5315 
 
 4786 
 4928 
 5065 
 5198 
 5328 
 
 4800 
 4942 
 5079 
 5211 
 5340 
 
 4814 
 4955 
 6092 
 6224 
 5353 
 
 4829 
 4960 
 
 6105 
 6237 
 5366 
 
 4843 
 4983 
 6119 
 5250 
 5378 
 
 4857 
 4997 
 6132 
 5263 
 5391 
 
 4871 
 5011 
 5145 
 5276 
 6403 
 
 4886 
 5024 
 6159 
 6289 
 5416 
 
 4900 
 5038 
 5172 
 5302 
 5428 
 
 35 
 36 
 37 
 38 
 39 
 
 5441 
 5563 
 5682 
 5798 
 5911 
 
 5453 
 6575 
 5694 
 5809 
 6922 
 
 5465 
 5587 
 6705 
 5821 
 6933 
 
 5478 
 5599 
 5717 
 6832 
 6944 
 
 5490 
 6611 
 5729 
 5843 
 5955 
 
 5502 
 5623 
 5740 
 6855 
 6966 
 
 5514 
 5635 
 6752 
 5866 
 6977 
 
 6527 
 6647 
 
 5703 
 
 5877 
 6988 
 
 5539 
 5658 
 5775 
 5888 
 5599 
 
 5551 
 5670 
 5786 
 5899 
 6010 
 
 40 
 41 
 42 
 43 
 44 
 
 6021 
 6128 
 6232 
 6335 
 6435 
 
 6031 
 6138 
 6243 
 5345 
 6444 
 
 6042 
 6149 
 8258 
 
 6355 
 6454 
 
 6053 
 6100 
 6203 
 6305 
 6464 
 
 0004 
 6170 
 6274 
 6375 
 6474 
 
 6075 
 6180 
 6284 
 6385 
 6484 
 
 6085 
 6191 
 6294 
 6395 
 6493 
 
 6096 
 0201 
 6304 
 6406 
 6003 
 
 6107 
 0212 
 6314 
 
 0415 
 0513 
 
 6117 
 6222 
 6325 
 6425 
 6522 
 
 45 
 
 46 
 47 
 48 
 49 
 
 6532 
 6628 
 6721 
 6812 
 6902 
 
 6542 
 6637 
 6730 
 6821 
 6911 
 
 6551 
 6646 
 6739 
 6830 
 6920 
 
 6561 
 6656 
 6749 
 6839 
 6928 
 
 6571 
 6665 
 6758 
 6848 
 6937 
 
 6580 
 6675 
 6767 
 6857 
 6946 
 
 6590 
 6684 
 6776 
 6866 
 6955 
 
 6599 
 6693 
 6785 
 6875 
 6964 
 
 6609 
 6702 
 07! »4 
 6884 
 6972 
 
 6618 
 6712 
 6803 
 6893 
 6981 
 
 50 
 51 
 52 
 53 
 54 
 
 6990 
 7076 
 7160 
 7243 
 7324 
 
 6998 
 7084 
 7168 
 7251 
 7332 
 
 7007 
 7093 
 7177 
 7259 
 7340 
 
 7016 
 7101 
 7185 
 7207 
 7348 
 
 7024 
 7110 
 7193 
 
 7275 
 7350 
 
 7033 
 7118 
 7202 
 7284 
 7364 
 
 7042 
 7126 
 7210 
 7292 
 7372 
 
 7050 
 7135 
 7218 
 7300 
 7380 
 
 7059 
 7143 
 7220 
 7308 
 7388 
 
 7067 
 7152 
 7235 
 7316 
 7396 
 
COMMON LOGARITHMS. 
 
 851 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 55 
 
 56 
 57 
 58 
 59 
 
 7404 
 
 7482 
 7559 
 7634 
 7709 
 
 7412 
 7490 
 7566 
 7642 
 7716 
 
 7419 
 7497 
 7574 
 7649 
 7723 
 
 7427 
 7505 
 7582 
 7657 
 7731 
 
 7435 
 7513 
 7589 
 7664 
 7738 
 
 7443 
 7520 
 7597 
 7672 
 7745 
 
 7451 
 
 7528 
 7604 
 
 7679 
 
 7752 
 
 7459 
 7536 
 7612 
 7686 
 7760 
 
 7466 
 7543 
 7619 
 7694 
 
 7767 
 
 7474 
 
 7551 
 7627 
 7701 
 
 7774 
 
 60 
 
 61 
 62 
 63 
 64 
 
 7782 
 7853 
 7924 
 7993 
 80G2 
 
 7789 
 7860 
 7931 
 8000 
 8069 
 
 7796 
 7808 
 7938 
 8007 
 8075 
 
 7803 
 7875 
 7945 
 8014 
 
 8082 
 
 7810 
 7882 
 7952 
 8021 
 
 8089 
 
 7818 
 7889 
 7959 
 8028 
 8096 
 
 7825 
 7896 
 7966 
 8035 
 8102 
 
 7832 
 7903 
 7973 
 8041 
 8109 
 
 7839 
 7910 
 7980 
 8048 
 8116 
 
 7846 
 7917 
 7987 
 8055 
 8122 
 
 65 
 
 66 
 67 
 68 
 69 
 
 8129 
 8195 
 8261 
 8325 
 8388 
 
 8136 
 8202 
 8267 
 8331 
 8395 
 
 8142 
 8209 
 8274 
 8338 
 8401 
 
 8149 
 8215 
 
 8280 
 8344 
 8407 
 
 8156 
 8222 
 8287 
 8351 
 8414 
 
 8162 
 8228 
 8293 
 8357 
 8420 
 
 8169 
 8235 
 8299 
 8363 
 8426 
 
 8176 
 8241 
 8306 
 8370 
 8432 
 
 8182 
 8248 
 8312 
 8376 
 8439 
 
 8189 
 8254 
 8319 
 8382 
 8445 
 
 70 
 
 71 
 
 72 
 73 
 
 74 
 
 8451 
 8513 
 8573 
 8633 
 8692 
 
 8457 
 8519 
 8579 
 8639 
 8698 
 
 8463 
 8525 
 8585 
 8645 
 8704 
 
 8470 
 8531 
 8591 
 8651 
 8710 
 
 8476 
 8537 
 8597 
 8657 
 8716 
 
 8482 
 8543 
 8603 
 8663 
 
 8722 
 
 8488 
 8549 
 8609 
 8669 
 8727 
 
 8494 
 8555 
 8615 
 8675 
 8733 
 
 8500 
 8561 
 8621 
 8681 
 8739 
 
 8506 
 8567 
 8627 
 8686 
 8745 
 
 75 
 
 76 
 
 77 
 78 
 79 
 
 8751 
 8808 
 8865 
 8921 
 8976 
 
 8756 
 8814 
 8871 
 8927 
 8982 
 
 8762 
 8820 
 8876 
 8932 
 
 8987 
 
 8768 
 8825 
 8882 
 8938 
 8993 
 
 8774 
 8831 
 8887 
 8943 
 8998 
 
 8779 
 8837 
 8893 
 8949 
 9004 
 
 8785 
 8842 
 8899 
 8954 
 9009 
 
 8791 
 
 8848 
 8904 
 8960 
 9015 
 
 8797 
 8854 
 8910 
 8965 
 9020 
 
 8802 
 8859 
 8915 
 8971 
 9025 
 
 80 
 
 81 
 
 82 
 83 
 
 84 
 
 9031 
 9085 
 9138 
 9191 
 9243 
 
 9036 
 9090 
 9143 
 9196 
 
 9248 
 
 9042 
 9096 
 9149 
 9201 
 9253 
 
 9047 
 9101 
 9154 
 9206 
 9258 
 
 9053 
 9106 
 9159 
 9212 
 9263 
 
 9058 
 9112 
 9165 
 9217 
 9269 
 
 9063 
 9117 
 9170 
 
 9222 
 9274 
 
 9069 
 9122 
 9175 
 9227 
 9279 
 
 9074 
 9128 
 9180 
 9232 
 9284 
 
 9079 
 9133 
 9186 
 9238 
 9289 
 
 85 - 
 
 86 
 87 
 88 
 89 
 
 9294 
 9345 
 9395 
 9445 
 9494 
 
 9299 
 9350 
 9400 
 9450 
 9499 
 
 9304 
 9355 
 9405 
 9455 
 9504 
 
 9309 
 9360 
 9410 
 9400 
 9509 
 
 9315 
 9365 
 9415 
 9465 
 9513 
 
 9320 
 9370 
 9420 
 9469 
 9518 
 
 9325 
 9375 
 9425 
 9474 
 9523 
 
 9330 
 9380 
 9430 
 9479 
 9528 
 
 9335 
 9385 
 9435 
 9484 
 9533 
 
 9340 
 9390 
 9440 
 9489 
 9538 
 
 90 
 91 
 92 
 93 
 94 
 
 9542 
 9590 
 9638 
 9685 
 9731 
 
 9547 
 9595 
 9643 
 9689 
 9736 
 
 9552 
 9600 
 9647 
 9694 
 9741 
 
 9557 
 9005 
 9652 
 9699 
 9745 
 
 9562 
 9609 
 9657 
 9703 
 9750 
 
 9566 
 9614 
 9661 
 9708 
 9754 
 
 9571 
 9619 
 9666 
 9713 
 9759 
 
 9576 
 9624 
 9671 
 9717 
 9763 
 
 9581 
 9628 
 9675 
 9722 
 9768 
 
 9586 
 9633 
 9680 
 9727 
 9773 
 
 95 
 
 96 
 97 
 
 98 
 99 
 
 9777 
 9823 
 9868 
 9912 
 9956 
 
 9782 
 9827 
 9872 
 9917 
 9961 
 
 9786 
 9832 
 9877 
 9921 
 9965 
 
 9791 
 
 9836 
 9881 
 9926 
 9969 
 
 9795 
 9841 
 9886 
 9930 
 9974 
 
 9800 
 9845 
 9890 
 9934 
 
 9978 
 
 9805 
 9850 
 9894 
 9939 
 9983 
 
 9809 
 9854 
 9899 
 9943 
 
 9987 
 
 9814 
 9859 
 9903 
 9948 
 9991 
 
 9818 
 9863 
 9908 
 9952 
 9996 
 
352 COMMON LOGARITHMS. 
 
 To Find the Antilogarithm of a Given Logarithm. 
 
 647. If the given mantissa can be found in the table, 
 the first two figures of the required number are in the 
 column headed N on the same line with the mantissa, and 
 the third figure is at the top of the column that contains 
 the mantissa. 
 
 The position of the decimal point is determined by the 
 characteristic (§ 646). 
 
 Thus, the number corresponding to the logarithm 
 2.9736 is 941, 0.8169 is 6.56, 
 
 6.0899 is 1,230,000, 7.8739 - 10 is 0.00748. 
 
 648. If the given mantissa cannot be found in the table, 
 find in the table the two adjacent mantissas between which 
 the given mantissa lies, and the three figures corresponding 
 to the smaller of these two mantissas are the first three 
 significant figures of the required number. The fourth 
 figure is found as in the following 
 
 Examples. 1. Find the antilogarithm of the logarithm 
 3.7936. 
 
 Solution. The two adjacent mantissas of the table between which 
 the given mantissa 7936 lies are 7931 and 7938. The corresponding 
 numbers are 621 and 622. The smaller of these, 621, contains the 
 first three significant figures of the required number. 
 
 The difference between the two adjacent mantissas is 7, and the 
 difference between the corresponding numbers is 1. 
 
 The difference between the smaller of the two adjacent mantissas, 
 7931, and the given mantissa, 7936, is 6. Therefore, the number to be 
 annexed to 621 is \ of 1 = 0.71, and the fourth significant figure of 
 the required number is 7. 
 
 Hence, the required number is 6217. 
 
 2. Find the antilogarithm of the logarithm 7.3884 — 10. 
 
 Solution. The two adjacent mantissas of the table between which 
 the given mantissa 3884 lies are 3874 and 3892. The corresponding 
 numbers are 244 and 246. The smaller of these, 244, contains the 
 first three significant figures of the required number. 
 
COMMON LOGARITHMS. 353 
 
 The difference between the two adjacent mantissas is 18, and the 
 difference between the corresponding numbers is 1. 
 
 The difference between the smaller of the two adjacent mantissas, 
 3874, and the given mantissa, 3884, is 10. Therefore, the number to 
 be annexed to 244 is f § of 1 = 0.55, and the fourth significant figure 
 of the required number is 6. 
 
 Hence, the required number is 0.002446. 
 
 Exercise 152. 
 Find the antilogarithms of the following logarithms : 
 
 1. 
 
 3.9017. 
 
 7. 
 
 2.9850. 
 
 13. 
 
 8.7324-10. 
 
 2. 
 
 1.2076. 
 
 8. 
 
 4.5388. 
 
 14. 
 
 9.5555 - 10. 
 
 3. 
 
 0.4442. 
 
 9. 
 
 0.8550. 
 
 15. 
 
 6.0216 - 10. 
 
 4. 
 
 1.0090. 
 
 10. 
 
 9.9992 - 10. 
 
 16. 
 
 7.0080 - 10. 
 
 5. 
 
 4.8697. 
 
 11. 
 
 7.0016 - 10. 
 
 17. 
 
 8.2361 - 10. 
 
 6. 
 
 1.9214. 
 
 12. 
 
 9.2618 - 10. 
 
 18. 
 
 9.4513 - 10. 
 
 649. Since every factor of a product may be expressed 
 as a power of ten (§ 628), 
 
 The logarithm of a product is equal to the sum of the loga- 
 rithms of its factors (§ 69). 
 
 650. Example. Find by logarithms the product of 
 908.4 X 0.05392 X 2.117. 
 
 Solution. log 908.4 = 2.9583 
 
 log 0.05392 = 8.7318 - 10 
 
 log 2.117 = 0.3257 
 
 2.0158 = log 103.7. 
 Therefore, the required product is 103.7. 
 
 Exercise 153. 
 Find by logarithms the value of : 
 
 1. 948.22 X 0.4387. 4. 270.05 X 0.0087. 
 
 2. 1.9704 X 0.0786. 5. 11.163 X 0.3333. 
 8. 380.25 X 0.00673. 6. 777.78 X 0.0787. 
 
354 COMMON LOGARITHMS. 
 
 7. 216.21 X 0.76312. 11. 2.6537 X 0.2313. 
 
 8. 0.56127X1.2312. 12. 37.587X12.371. 
 
 9. 0.86311X56.371. 13. 89.313X2.3781. 
 10. 59.795 X 0.7955. 14. 9.1765 X 0.089. 
 
 15. 4786 X 5.4187 X 0.00218 X 0.8652. 
 
 16. 3.1416 X 7.77 X 184 X 0.01865. 
 
 17. 0.7854X129.6X63.45X0.0021. 
 
 18. 1842.65X9.876X0.843X0.0265. 
 
 19. 12.48 X 44.63 X 32.78 X 0.004587. 
 
 20. 0.9876 X 0.8765 X 0.7654 X 0.6543. 
 
 651. Any required power of a given power of a number 
 
 is found by multiplying the exponent of the given power 
 
 by the exponent of the required power. 
 
 Thus, the cube of 10 2 = 102>< 3 = 10 6 ; the fifth power of 10* = 
 104x5 = loao. Hence, 
 
 The logarithm of a power of a number is found by multi- 
 plying the logarithm of the number by the exponent of the 
 power. 
 
 652. Example. Find by logarithms the cube of 0.0497. 
 
 Solution. log 0.0497 = 8.6964 — 10 
 
 8 
 log 0.0497 3 = 6.0892 - 10 = log 0.0001228. 
 Therefore, 0.0497 8 = 0.0001228. 
 
 Notk. The real product is 26.0892 — 30. Removing 20 from the 
 26 and — 20 from the — 30, we have 6.0892 — 10. 
 
 Exercise 154. 
 Find by logarithms the value of : 
 
 1. 5.06 s . 6. 0.7685*. 11. 2.861415 4 . 
 
 2. 2.501 6 . 7. 0.9611 8 . 12. 3.79125 6 . 
 
 3. 1.716 7 . 8. 0.0231 2 . 13. 0.021875*. 
 
 4. 1.178 10 . 9. 0.8567 3 . 14. 0.87152 7 . 
 
 5. 7.6821 6 . 10. 0.5438 6 . 15. 0.95956 8 . 
 
COMMON LOGARITHMS. 355 
 
 653. Any required root of a given power of a number 
 is found by dividing the exponent of the power by the 
 index of the root. 
 
 Thus, the sixth root of 10 7 , that is, VlO 7 = 10*. Hence, 
 
 The logarithm of a root of a number is found by dividing 
 the logarithm of the number by the index of the root. 
 
 654. Examples. 1. Find the cube root of 271. 
 
 Solution. log 271 = 2.4330 
 
 Divide by 3, 3 )2.4330 
 
 0.8110 = log 6.471. 
 
 Therefore, ^271 = 6.471. 
 
 2. Find the fifth root of 0.4654. 
 
 Solution. log 0.4654 = 9.6679-10 
 
 Add, 40. - 40 
 
 Divide by 5, 5 )49.6679-50 
 
 9.9336 -10 = log 0.8582. 
 
 Therefore, ^0.4654 = 0.8582. 
 
 If the given number is less than 1, its logarithm, when written in 
 the ordinary form, will have a — 10 annexed. In this case, the form 
 of the logarithm should be changed as in Example 2, so that, when 
 the logarithm is divided by the index of the root, the negative number 
 of the quotient shall be — 10. 
 
 Exercise 155. 
 Find by logarithms the value of : 
 
 1. 13*. 8. 879™. 15. 93.73*. 
 
 2. 29*. 9. 0.609*. 16. 21.97 g . 
 
 3. 471*. 10. 0.8716*. 17. 7.935 1 . 
 
 4. 288*. 11. 0.021641*. 18. 0.815 f . 
 
 5. 1019*. 12. 0.9825*. 19. 2.81451 
 
 6. 1281*. 13. 0.42184*. 20. 0.04165 A . 
 
 7. 1862*. 14. 0.02187*. 21. 4,516 ; 298 T \ 
 
356 COMMON LOGARITHMS. 
 
 655. Since a quotient is equal to the divideud divided 
 by the divisor, 
 
 The logarithm of a quotient is equal to the logarithm of 
 the dividend minus the logarithm of the divisor (§ 108). 
 
 656. Example. Divide 905.6 by 38.45. 
 
 Solution. log 905.6 = 2.9569 
 
 log 38.45 = 1.5849 
 
 1.3720 = log 23.55. 
 Therefore, 905.6 -r 38.45 = 23.55. 
 
 657. The Cologarithm of a Number. The logarithm 
 of the reciprocal of a number is called the cologarithm of 
 the number. 
 
 Thus, colog 30 = log 3V = log 1 — log 30. 
 
 Since log 1=0, log 1 — log 30 = — log 30. 
 
 Now, in the expression — log 30, the minus sign affects 
 the entire logarithm, both the characteristic and the man- 
 tissa. To avoid a negative mantissa, we can substitute 
 for — log 30 its equivalent 
 
 (10 - log 30) - 10. 
 
 Therefore, the cologarithm of a number is found by sub- 
 tracting the logarithm of the number from 10, and annex- 
 ing — 10 to the remainder. 
 
 Note. The best way to perform the subtraction is to begin at the 
 left, subtract from 9 each figure except the last significant figure, and 
 subtract the last significant figure from 10. 
 
 658. Examples. 1. Find the cologarithm of 4007. 
 
 Solution. 10. — 10 
 
 log 4007 = 3.6028 
 colog 4007= 6.3972-10 
 
 2. Find the cologarithm of 0.004007. 
 Solution. 10. — 10 
 
 log 0.004007 = 7.6028 - 10 
 colog 0.004007 = 2.3972 
 
COMMON LOGARITHMS. 357 
 
 659. By using cologarithms, the inconvenience of sub- 
 tracting the logarithm of the divisor is avoided. 
 
 Thus, the example of § 656 is usually solved as follows : 
 
 log 905.6 = 2.9569 
 colog 38.45 = 8.4151 - 10 
 
 1.3720 as log 23.55. 
 
 Therefore, 905.6 -f 38.45 = 23.55. 
 
 660. Example. By the use of cologarithms find the 
 
 7.56 X 4667 X 567 
 value ol 899 ^ x 00337 x 23435 ' 
 
 Solution. log 7.56 = 0.8785 
 
 log 4667 = 3.6690 
 
 log 567 = 2.7536 
 
 colog 899.1 =7.0462-10 
 
 colog 0.00337 = 2.4724 
 
 colog 23,435 = 5.6301 - 10 
 
 2.4498 = log 281.7. 
 
 Note. Log 899.1 is 2.9538, and its cologarithm is (10-2.9538)— 10, 
 or 7.0462 — 10 ; log 0.00337 is 7.5276 — 10, and subtracting this from 
 10 — 10 we obtain for its cologarithm 2.4724 ; log 23,435 is 4.3699, and 
 its cologarithm is (10 — 4.3699) — 10, or 5.6301 — 10. 
 
 Exercise 156. 
 Find by logarithms the value of : 
 
 56.407 _ 75.46 X 0.0765 
 
 1. 
 
 13.045 ' 93.08 X 98.071 
 
 857.06 98 X 537 X 0.0079 
 
 * 3079.8* ' 67309 X 0.0947 
 0.9387 314 X 7.18 X 8132 
 
 ' 598.6 ' ' 519X827X3.215' 
 
 3069 212 X 2.16 X 8002 
 
 * 0.7891* ' 536X351 X 7.256* 
 
 9. (fj)« 11. (5A) 2 - 13. (fi-f) 5 . 15. (|||)3. 
 
 10. (if)«. 12. (4^)*. 14. (£3)8. 16 . (HH) 
 
358 COMMON LOGARITHMS. 
 
 19.258 X 3.1416 X 812.72 
 716.4 X 8.002 X 21.465 
 2018 X 0.00261 X 1728 
 
 1412 X 0.0965X0.08621* 
 44,816 X 17.265 X 181 
 
 28,754 X 1.2871 X 206.45* 
 216.1 X 5280 X 144.2 
 
 187.42 X 4622.6 X 156.8* 
 
 5982.55 X 0.02987 X 0.9852 
 
 42.875 X 34.62 X 28.47 
 14.718 X 48.67 X 96.542 
 
 2746.2 X 0.0467 X 2.1876* 
 
 / 83.25 X 4267 X 0.008576" 
 ' ^0.0327 X 687.5 X 0.005003* 
 
 */ 4.163 2 Xl7.74 4 X 0.7183* 
 ^3.013 2 X 34.34 X 0.08137*' 
 
 7132 X 9.245 X 0.5477 s 
 
 17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 23 
 
 25 V™ 
 
 93 X 0.000173* X 0.01 
 
 5 / 65.02 2 X 0.002753 X 97.98* 
 ' "V 7.298 X 0.04754 X 8.156 2 ' 
 
 «/ 23.79 2 X 0.00756 X 0.4648* 
 ' "V 4723* X 0.6571 X 0.8246* 
 
 6012 X 0.6012* X 0.6012* 
 5926X0.5926* X 0.5926* 
 
 / 0.03214 X 3.718 3 X 0.07824* \f 
 ' \ 0.05142 X 0.4728* X 1.239 s / 
 
 / 0.07986 X 0.7555* X 0.5557* \f 
 ' \0.06897 X 0.5777* X 0.05698V 
 
 / 0.07543 X 0.7689* X 0.8965 2 \* 
 " V 0.06987 X 0.07986* X 0.9867V 
 
CHAPTEK XX. 
 
 APPLICATIONS OF LOGARITHMS. 
 Compound Interest Problems. 
 
 661. In compound interest, if P represents the principal, 
 r the rate per cent, n the number of years, A the amount, 
 then (1 -4- r) represents the amount of $1 for 1 year. 
 
 (1 -f- r) 2 represents the amount of $1 for 2 years. 
 
 (1 + r) 8 represents the amount of $1 for 3 years. 
 
 (1 -+■ r ) n represents the amount of $1 for n years. 
 
 P X (1 + r ) n represents the amount of $P for n years. 
 
 Therefore, A = P X(l + r) n , 
 
 and log A = log P-\-n X log (1 + r). 
 
 662. Examples. 1. Find the amount of $150 for 6 
 years at 4% compound interest. 
 
 Solution. Here P = 160, r = 0.04, n = 6. 
 Therefore, log .A = log 150 + 6 X log 1.04. 
 log 150 = 2.1761 
 6 X log 1.04 = 0.1020 
 
 log A - 2.2781 = log 189. 7. 
 Hence, the required amount is $189.70. 
 
 2. What principal will amount to $500 in 5 years at 
 ^\°f compound interest ? 
 
 Solution. Here A = 500, r = 0.045, n = 5. 
 Therefore, 500 = P X (1.045) 5 ; and P = -^^ • 
 
 log 500 = 2.6990 
 5 X colog 1.045 = 9.9045 - 10 
 
 log P = 2.6035 = log 401.4. 
 
 Hence, the required principal is $401.40. 
 
360 COMPOUND INTEREST PROBLEMS. 
 
 3. At what rate of interest will $360 amount to $481.80 
 in 5 years at compound interest ? 
 
 Solution. Here A - 4S1.80, P = 300, n = 5. 
 
 Therefore, 481.80 = 360 X (1 + r) 6 , 
 
 „ , , r 481.80 _ 4 . s/481.80 
 
 log 481.80= 2.6828 
 colog360 = 7.4437 - 10 
 5 )0.1265 
 log (1 + r) = 0.0253 = log 1.06. 
 
 Hence, the required rate of interest is 6%. 
 
 4. In what time at 3£% compound interest will $540 
 amount to $619.40 ? 
 
 Solution. Here A = 619.40, P = 540, r = 0.035. 
 Therefore, log 619.40 = log 540 + n X log 1.035, 
 
 n X log 1.035 = log 619.40 + colog 540, 
 
 log 619.40 4- colog 540 
 
 and n = \ , MC " 
 
 log 1.035 
 
 _ 2.7920 + 7.2676 -10 
 
 0.0149 
 
 0.0596 
 
 0.0149 
 Hence, the required time is 4 years. 
 
 = 4. 
 
 Exercise 157. 
 
 1. Find the compound interest on $1280 for 7 years 
 at 4£%. 
 
 2. Find the compound interest on $2645 for 5 years 
 at 3£%. 
 
 3. Find the amount of $848 for 6 years at 5% com- 
 pound interest. 
 
 4. Find the amount of $3600 for 5 years at 5£% com " 
 pound interest. 
 
 5. What principal will amount to $720 in 4 years at 
 6% compound interest ? 
 
COMPOUND INTEREST PROBLEMS. 361 
 
 6. What principal will amount to $1640 in 6 years at 
 3% compound interest? 
 
 7. At what rate of interest will $648 amount to $788.20 
 in 5 years at compound interest ? 
 
 8. At what rate of interest will $2415 amount to $3237 
 in 6 years at compound interest ? 
 
 9. In what time at 4-j-% compound interest will $1265 
 amount to $1576 ? 
 
 10. In what time at 5% compound interest will $1845 
 amount to $2413 ? 
 
 663. The table on the following page has been made 
 with the aid of logarithms. It shows the amount of $1 at 
 compound interest at various per cents for from 1 to 20 
 years. The compound interest on $1 is found by subtract- 
 ing 1 from the proper number shown in the table. 
 
 664. Examples. 1. What principal will in 10 years 
 at 6% compound interest yield $1898.04 interest ? 
 
 Solution. The interest on $1 for 10 yr. at 6% is $0.79085. 
 Since $0.79085 is the interest on $1, 
 
 $1898.04 is the interest on $^^|, or $2400. 
 0.79085 
 
 2. In what time will $1600 at k\°j compound interest 
 yield $1000 interest ? 
 
 Solution. Since $1600 yields $1000, $1 will yield T ^ of $1000, 
 or $0,625, in the same time, and $1 will amount to $1,625. By the 
 table, $1 will in 11 yr. at 4£% amount to $1.62285, and in 12 yr. to 
 $1.69588. Hence, the required time is a little more than 11 yr. 
 
 3. At what rate, compound interest, will $1500 yield 
 $1201.41 interest in 15 years ? 
 
 Solution. Since $1500 yields $1201.41 interest in 15 yr., $1 in 
 15 yr. will yield T ^ of $1201.41, or $0.80094, and $1 will amount 
 in 15 yr. to $1.80094. In the table, opposite 15 yr., we find in the 4% 
 column the amount of $1 is $1.80094. Therefore, the rate is 4%. 
 
362 
 
 COMPOUND INTEREST PROBLEMS. 
 
 665. Table. 
 
 Showing the amount of $1 at compound interest for: 
 
 Vlt. 
 
 2 PER CENT. 
 
 2£ PER CENT. 
 
 3 PER CENT. 
 
 31 PER CENT. 
 
 4 PER CENT. 
 
 1 
 
 1.02000 
 
 1.02500 
 
 1.03000 
 
 1.03500 
 
 1.04000 
 
 2 
 
 1.04040 
 
 1.05063 
 
 1.06090 
 
 1.07123 
 
 1.08160 
 
 3 
 
 1.06121 
 
 1.07689 
 
 1.09273 
 
 1.10872 
 
 1.12486 
 
 4 
 
 1.08243 
 
 1.10381 
 
 1.12551 
 
 1.14752 
 
 1.16986 
 
 5 
 
 1.10408 
 
 1.1.5141 
 
 1.15927 
 
 1.18769 
 
 1.21665 
 
 6 
 
 1.12616 
 
 1.15969 
 
 1.19405 
 
 1.22926 
 
 1.26532 
 
 7 
 
 1.14869 
 
 1.18869 
 
 1.22987 
 
 1.27228 
 
 1.81608 
 
 , 8 
 
 1.17166 
 
 1.21840 
 
 1.2(5677 
 
 1.31681 
 
 1.36857 
 
 9 
 
 1.19509 
 
 1.24886 
 
 1.30477 
 
 1.36290 
 
 1.42:;:: 1 
 
 10 
 
 1.21899 
 
 1.28009 
 
 1.34392 
 
 1.41060 
 
 1.48024 
 
 11 
 
 1.24337 
 
 1.31209 
 
 1.381J:5 
 
 1.45997 
 
 1.53945 
 
 12 
 
 1.26824 
 
 1.34489 
 
 1.42576 
 
 1.51107 
 
 1.60103 
 
 13 
 
 1.29961 
 
 1.37851 
 
 1.46853 
 
 1.56396 
 
 1.(5(5507 
 
 14 
 
 1.31948 
 
 1.41297 
 
 1.51259 
 
 1.61870 
 
 1.7:5168 
 
 15 
 
 1.34587 
 
 1.44830 
 
 1.65797 
 
 1.67535 
 
 1.80094 
 
 16 
 
 1.37279 
 
 1.48451 
 
 1.60471 
 
 1.73399 
 
 1.87208 
 
 17 
 
 1.40024 
 
 1.52162 
 
 1.66286 
 
 1.794(58 
 
 1.04790 
 
 18 
 
 1.42825 
 
 1.55966 
 
 1.70248 
 
 1.85749 
 
 2.0268S 
 
 19 
 
 1.45681 
 
 1.59865 
 
 1.75351 
 
 1.92250 
 
 2.10685 
 
 20 
 
 1.48595 
 
 1.63862 
 
 1.80611 
 
 1.98979 
 
 2.19112 
 
 YB. 
 
 4J PER CENT. 
 
 5 PER CENT. 
 
 5J PER CENT. 
 
 6 PER CENT. 
 
 7 PER CENT. 
 
 1 
 
 1.04500 
 
 1.05000 
 
 1.05500 
 
 1.06000 
 
 1.07000 
 
 2 
 
 1.09203 
 
 1.10260 
 
 1.11303 
 
 1.12360 
 
 1.1 441 to 
 
 3 
 
 1.14117 
 
 1.16763 
 
 1.17424 
 
 1.19102 
 
 1.22504 
 
 4 
 
 1.19252 
 
 1.21551 
 
 1.23882 
 
 1.26248 
 
 1.31080 
 
 5 
 
 1.24618 
 
 1.27628 
 
 1.80606 
 
 1.33823 
 
 1.40255 
 
 6 
 
 1.80296 
 
 1.34010 
 
 1.37884 
 
 1.41852 
 
 1.50073 
 
 7 
 
 1.36086 
 
 1.40710 
 
 1.46468 
 
 1.60363 
 
 1.60578 
 
 8 
 
 1.42210 
 
 1.47746 
 
 1.53469 
 
 1.69385 
 
 1.71819 
 
 9 
 
 1.48610 
 
 1.55133 
 
 1.61909 
 
 1.(58948 
 
 1.83846 
 
 10 
 
 1.55297 
 
 1.62889 
 
 1.70814 
 
 1.79085 
 
 1.96715 
 
 11 
 
 1.62285 
 
 1.71034 
 
 1.80209 
 
 1.89830 
 
 2.10485 
 
 12 
 
 1.69588 
 
 1.79586 
 
 1.90121 
 
 2.01220 
 
 2.25210 
 
 13 
 
 1.77220 
 
 1.88565 
 
 2.00577 
 
 2.13293 
 
 2.40986 
 
 14 
 
 1.851H4 
 
 1.97993 
 
 2.11609 
 
 2.2(5090 
 
 2.57853 
 
 15 
 
 1.93528 
 
 2.07893 
 
 2.23248 
 
 2.80666 
 
 2.75903 
 
 16 
 
 2.02237 
 
 2.18287 
 
 2.35626 
 
 2.54035 
 
 2.95216 
 
 17 
 
 2.11338 
 
 2.29202 
 
 2.48480 
 
 2.69277 
 
 3.15882 
 
 18 
 
 2.20848 
 
 2.40662 
 
 2.62147 
 
 2.85434 
 
 3.37993 
 
 19 
 
 2.30786 
 
 2.52695 
 
 2.76565 
 
 3.02660 
 
 3.61(55:5 
 
 20 
 
 2.41171 
 
 2.65330 
 
 2.91776 
 
 3.20714 
 
 3.86968 
 
COMPOUND INTEREST PROBLEMS. 363 
 
 Exercise 158. 
 
 1. A man deposits $60 in a savings bank, and draws 
 out his whole account at the end. of 8 years, with 4% 
 compound interest. What amount does he receive ? 
 
 2. What will $100 amount to in 7 years with interest 
 at 8% per annum, compounded semi-annually ? 
 
 3. In how many years will a sum of money double 
 itself at 6%, compounded annually ? 
 
 4. In how many years will a sum of money treble 
 itself at 6%, compounded annually ? 
 
 5. In how many years will $87 amount to $99 at 3%, 
 compounded annually ? 
 
 6. In how many years will $100 amount to $175 at 
 4%, compounded annually ? 
 
 7. At what rate per cent will a sum of money double 
 itself in 12 years, compound interest ? 
 
 8. At what rate will a sum of money treble itself in 
 19 years, compound interest ? 
 
 9. At what rate will $80 at compound interest amount 
 to $110 in 8 years ? 
 
 10. What sum must be invested at 5%, compound in- 
 terest, to amount to $1200 in 7 years ? 
 
 11., What sum must be invested at 4%, compound in- 
 terest, to amount to $2000 in 10 years?. To amount to 
 $5000 in 8 years ? 
 
 12. At what rate compound interest will $462.50 yield 
 $277.98 interest in 12 years ? 
 
 13. What principal will in 10 years at 6% amount to 
 $3612.22, interest being compounded semi-annually ? 
 
 14. In what time at 5% will $1250 amount to $2000, 
 interest being compounded semi-annually ? 
 
 1 5. At what rate per annum will $500 amount to $779.83 
 in 9 years, interest being compounded semi-annually ? 
 
364 ANNUITIES. 
 
 Annuities. 
 
 666. An annuity is a sum of money to be paid at reg- 
 ular intervals of time, as years, half years, quarter years. 
 
 667. A perpetual annuity is an annuity that continues 
 forever. 
 
 668. A certain annuity is an annuity that begins at a 
 specified time and ends at a specified time. 
 
 669. A contingent annuity is an annuity that depends 
 upon some particular event, as the death of an individual. 
 
 Life insurance, dowers, and pensions are examples. 
 
 670. The final value of an annuity is the sum to which 
 all its payments at compound interest will amount at the 
 end of the annuity. 
 
 671. The present value of an annuity is the sum which 
 at compound interest will amount to its final value. 
 
 672. A sinking fund is the final value of sums of money 
 set apart at regular intervals of time, and invested at com- 
 pound interest, to pay a debt due at a stated time. 
 
 673. To Find the Final Value of an Annuity. 
 
 If S represents the amount paid each year, 72 the amount 
 of $ 1 at interest for 1 year, n the number of years, and A 
 the final value, then, 
 
 The amount at the end of the 
 1st year = S, 
 2d year = S+SR, 
 3d yea,r= S+SR+SR 2 , 
 
 nth year = S + SR + SR 2 + + /SB— 1 . 
 
 That is, A = S+SR+SR 2 + + SE*-\ (1) 
 
 Multiplying (1) by R, we have 
 
 AR = SR + SR 2 + SM S + + SB— 1 + S&. (2) 
 
ANNUITIES. 365 
 
 Subtracting (1) from (2), we have 
 
 AR-A = SR n - S, 
 or A(R-l) = S(R n -l). 
 
 Therefore, A = — ^ — * • 
 
 K — 1 
 
 Here, R — 1 is the interest on $1 for 1 year, and R n — 1 is the 
 compound interest of $1 for n years. 
 
 674. To Find the Present Worth of an Annuity. 
 
 If P represents the present worth of the annuity, then 
 the amount of P for n years = A, the final value of the 
 annuity in n years. 
 
 The amount of P for n years 
 
 = P (1 + r)» = PR% (§ 661) 
 
 and -*-*£^- (§673) 
 
 Therefore, PR* 
 
 JX — J- 
 
 R n -1 
 
 R — 
 
 1 
 
 
 
 S(R n - 
 
 "^ 
 
 
 
 R- 
 
 1 ' 
 
 
 S(R n - 
 
 "1)_ 
 
 8 
 
 
 R n (R- 
 
 p. wri t.t.p.n 
 
 -1) 
 
 s 
 
 R- 
 
 1 
 
 1 - 
 
 and P = — ^— — -f = - X 
 
 i2" 
 
 The value of P may be written — X ( 1 — — ) • 
 
 ■ R — l\ R n J 
 
 If the annuity is perpetual, n is too large to be counted ; 
 therefore, R n is too large to be counted and — = 0. 
 Hence, P== _^ = |. 
 
 675. Examples. 1. What is the value of a sinking 
 
 fund, if $6000 is set apart annually for 6 years and put at 
 
 4% compound interest ? 
 
 . S(R»-1) $6000 x (1.046-1) 
 Solution. A = V R _ 1 ; - = - ^1 * • (§ 673) 
 
 By the use of logarithms, A is found to be $39,750. 
 
366 ANNUITIES. 
 
 2. Find the present value of an annuity of $500 for 
 
 5 years, if money is worth 4%. 
 
 8 R»-l 500 1.04 5 -1 tmm - M 
 
 SoumoH. P=_x-^= — X-j^p- (§674) 
 
 log 1.04 =0.0170 
 5 
 
 log 1.04 5 = 0.0850 = log 1.216. 
 ™ r n 500 0.216 
 
 Therefore, P = 0^4* 1^16 ' 
 
 log 500 = 2.6990 
 log 0.216 = 9.3345 - 10 
 colog 0.04 = 1.3979 
 colog 1.216 = 9.9150-10 
 
 log P= 3.3464 = log 2220. 
 
 Hence, the present value of the annuity is §2220. 
 
 3. Find the present value of a perpetual annuity of 
 $500, if money is worth 4%. 
 
 Solution. P = f = O^ol = $12 > 500 - <§ 674 > 
 
 Exercise 159. 
 
 1. Find the present value of an annuity of $300 for 
 
 6 years, if money is worth 5%. 
 
 2. Find the present value of an annuity of $600 for 
 
 4 years, if money is worth 5£%. 
 
 3. Find the present value of an annuity of $800 for 
 
 5 years, if money is worth 6%. 
 
 4. Find the present value of a perpetual scholarship of 
 $900, if money is worth 3£%. 
 
 5. Find the present value of a perpetual fellowship of 
 $3200, if money is worth 4£%. 
 
 6. What is the value of a sinking fund, if $25,000 is 
 set apart yearly for 7 years at ±\°f compound interest ? 
 
 7. What is the value of a sinking fund, if $18,000 is 
 set apart yearly for 5 years at 3£% compound interest? 
 
an;n UiTIES. 
 
 367 
 
 676. The table at the bottom of the page shows the 
 average number of years persons live after the ages indi- 
 cated. This table is known as the Carlisle Table because 
 it is based upon the rate of mortality, as carefully observed 
 at Carlisle, England. Several other tables of Expectancy 
 of Life have been compiled from other data and are in 
 common use with insurance companies. 
 
 The table on the following page has been made with 
 the aid of logarithms. It shows the present value of an 
 annuity of $1 per annum at compound interest from 1 to 
 40 years at 3£% and at 4%. 
 
 677. 
 
 Carlisle Table of Expectancy of Life. 
 
 
 EX- 
 
 
 EX- 
 
 
 EX- 
 
 
 EX- 
 
 AGE. 
 
 PECTANCY. 
 
 AGE. 
 
 PECTANCY. 
 
 AGE. 
 
 PECTANCY 
 
 AGE. 
 
 PECTANCY. 
 
 
 
 38.72 
 
 26 
 
 37.14 
 
 52 
 
 19.68 
 
 78 
 
 6.12 
 
 1 
 
 44.68 
 
 27 
 
 36.41 
 
 53 
 
 18.97 
 
 79 
 
 5.80 
 
 2 
 
 47.55 
 
 28 
 
 35.69 
 
 54 
 
 18.28 
 
 80 
 
 5.51 
 
 3 
 
 49.82 
 
 29 
 
 35.00 
 
 55 
 
 17.58 
 
 81 
 
 5.21 
 
 4 
 
 50.76 
 
 30 
 
 34.34 
 
 56 
 
 16.89 
 
 82 
 
 4.93 
 
 6 
 
 51.25 
 
 31 
 
 33.68 
 
 57 
 
 16.21 
 
 83 
 
 4.65 
 
 6 
 
 51.17 
 
 32 
 
 33.03 
 
 58 
 
 15.55 
 
 84 
 
 4.39 
 
 7 
 
 50.80 
 
 33 
 
 32.36 
 
 59 
 
 14.92 
 
 85 
 
 4.12 
 
 8 
 
 50.24 
 
 34 
 
 31.68 
 
 60 
 
 14.34 
 
 86 
 
 3.90 
 
 9 
 
 49.57 
 
 35 . 
 
 31.00 
 
 61 
 
 13.82 
 
 87 
 
 3.71 
 
 10 
 
 48.82 
 
 36 
 
 30.32 
 
 62 
 
 13.31 
 
 88 
 
 3.59 
 
 11 
 
 -48.04 
 
 37 
 
 29.64 
 
 63 
 
 12.81 
 
 89 
 
 3.47 
 
 12 
 
 47.27 
 
 38 
 
 28.96 
 
 64 
 
 12.30 
 
 90 
 
 3.28 
 
 13 
 
 46.51 
 
 39 
 
 28.28 
 
 65 
 
 11.79 
 
 91 
 
 3.26 
 
 14 
 
 45.75 
 
 40 
 
 27.61 
 
 66 
 
 11.27 
 
 92 
 
 3.37 
 
 15 
 
 45.00 
 
 41 
 
 26.97 
 
 67 
 
 10.75 
 
 93 
 
 3.48 
 
 16 
 
 44.27 
 
 42 
 
 26.34 
 
 68 
 
 10.23 
 
 94 
 
 3.53 
 
 17 
 
 43.57 
 
 43 
 
 25.71 
 
 69 
 
 9.70 
 
 95 
 
 3.53 
 
 18 
 
 42.87 
 
 44 
 
 25.09 
 
 70 
 
 9.18 
 
 
 3.46 
 
 19 
 
 42.17 
 
 45 
 
 24.46 
 
 71 
 
 8.65 
 
 97 
 
 3.28 
 
 20 
 
 41.46 
 
 46 
 
 23.82 
 
 72 
 
 8.16 
 
 98 
 
 3.07 
 
 21 
 
 40.75 
 
 47 
 
 23.17 
 
 73 
 
 7.72 
 
 99 
 
 2.77 
 
 22 
 
 40.04 
 
 48 
 
 22.50 
 
 74 
 
 7.33 
 
 100 
 
 2.28 
 
 23 
 
 39.31 
 
 49 
 
 21.81 
 
 75 
 
 7.01 
 
 101 
 
 1.79 
 
 24 
 
 38.59 
 
 50 
 
 21.11 
 
 76 
 
 6.69 
 
 102 
 
 1.30 
 
 25 
 
 37.86 
 
 51 
 
 20.39 
 
 77 
 
 6.40 
 
 103 
 
 0.83 
 
368 
 
 ANNUITIES. 
 
 678. 
 
 Table. 
 
 Showing the present value of an annuity of $1 per annum, 
 at compound interest from 1 to Jfi years at 3^°f and at 4°/o- 
 
 YR. 
 
 3J PER CENT. 
 
 4 PER CENT. 
 
 YR. 
 
 3£ PER CENT. 
 
 4 PER CENT. 
 
 1 
 
 0.96018 
 
 0.90154 
 
 21 
 
 14.09797 
 
 14.02910 
 
 2 
 
 1.89909 
 
 1.88010 
 
 22 
 
 15.10713 
 
 14 45112 
 
 3 
 
 2.80164 
 
 2.77609 
 
 23 
 
 15.02041 
 
 14.85084 
 
 4 
 
 3.03708 
 
 8.62980 
 
 24 
 
 16.06887 
 
 16.24606 
 
 6 
 
 4.51505 
 
 4.45182 
 
 ■25 
 
 10.48152 
 
 15.02208 
 
 6 
 
 6.32855 
 
 5.24214 
 
 26 
 
 10.89035 
 
 15.98277 
 
 7 
 
 0.11454 
 
 c. 00206 
 
 27 
 
 17.28537 
 
 16.82060 
 
 8 
 
 0.87390 
 
 0.73275 
 
 28 
 
 17.00702 
 
 10.00300 
 
 9 
 
 7.00709 
 
 7.43533 
 
 29 
 
 18.03577 
 
 10.98372 
 
 10 
 
 8.31661 
 
 8.11090 
 
 80 
 
 18.39205 
 
 17.29203 
 
 11 
 
 9.00155 
 
 8.7004H 
 
 81 
 
 18.73028 
 
 17.58849 
 
 12 
 
 9.00333 
 
 8.88607 
 
 32 
 
 19.00887 
 
 17.87355 
 
 13 
 
 10.30274 
 
 0.06666 
 
 88 
 
 19.39021 
 
 18.14766 
 
 14 
 
 10.92052 
 
 10.50312 
 
 34 
 
 19.70008 
 
 18.41120 
 
 15 
 
 11.51741 
 
 11.11839 
 
 35 
 
 20.00000 
 
 18.00401 
 
 16 
 
 12.09412 
 
 11.05230 
 
 30 
 
 20.29049 
 
 18.90828 
 
 17 
 
 12.66188 
 
 12.10507 
 
 37 
 
 20.57053 
 
 19.14258 
 
 18 
 
 13.18908 
 
 12.05930 
 
 38 
 
 20.84109 
 
 19.30780 
 
 19 
 
 13.70984 
 
 13.13394 
 
 39 
 
 21.10250 
 
 19.58449 
 
 20 
 
 14.21240 
 
 13.69033 
 
 40 
 
 21.35507 
 
 19.79277 
 
 679. Examples. 1. Find the present value of an 
 annuity for $500 for 5 yr. at 4%. 
 
 Solution. The present value of 81 for 5 yr. at 4% by the table is 
 $4.45182 ; and of $500 is 500 X $4.45182, or $2225.01. 
 
 2. A person 41 years of age pays $9797.75 for a life 
 annuity. If interest is reckoned at 4%, find the amount of 
 the annuity. 
 
 Solution. According to the table on page 367, the expectancy of 
 
 life for a person 41 years of age is about 27 years. 
 
 The present value of an annuity of $1 for 27 yr. at 4% is $16.32959. 
 
 9797 75 
 Hence, the amount of the annuity is $ ' t or $600. 
 
 10. 32959 
 
ANNUITIES. 369 
 
 Exercise 160. 
 
 1. Find the present value of an annuity of $900 for 
 15 years at 4%. 
 
 2. Find the present value of an annuity of $1500 for 
 12 years at 4%. 
 
 3. Find the present value of an annual pension of $144 
 for 10 years at 3J%. 
 
 4. Find the present value of a scholarship of $200 for 
 25 years at 2>\°f . 
 
 5. Find the present value of an annuity of $2500 for 
 30 years at 4%. 
 
 6. Find the present value of an annuity of $250 for 
 12 years at 3£%. 
 
 7. A person 22 years old has a life annuity of $750. 
 Find its present value at 4%. 
 
 8. A person 35 years old has a life annuity of $1800. 
 Find its present value at 4%. 
 
 9. A person 53 years old has a life annuity of $500. 
 Find its present value at 4%. 
 
 10. A person 75 years old has a life annuity of $2400. 
 Find its present value at 3£%. 
 
 11. A boy 15 years old has a life annuity of $3250. 
 Find jts present value at 4%. 
 
 12. A person 22 years old pays $4948.19 for a life 
 annuity. If interest is 4%, find the amount of the annuity. 
 
 13. A person 29 years old pays $7465.84 for a life 
 annuity. If interest is 4 °f , find the amount of the annuity. 
 
 14. A person 35 years old pays $9368.14 for a life 
 annuity. If interest is 3£%, find the amount of the 
 annuity. 
 
 15. A person 44 years old pays $5933.35 for a life 
 annuity. If interest is 3£%, find the amount of the 
 annuity. 
 
370 COOPERATIVE BANKS. 
 
 Cooperative Banks. 
 
 680. A cooperative bank is a mutual corporation with 
 the object of the accumulation of a capital to be loaned to 
 its members, especially for the purchase of homes. 
 
 681. Shares. The capital stock is usually divided into 
 shares of final value $200 each, that are paid for in monthly 
 instalments of $1 each. The number of shares that any 
 member may purchase is limited, usually to twenty-five. 
 Each shareholder pays $1 a month per share until his 
 shares are worth $200 each. The shares are then said to 
 be matured. At maturity the shareholder receives $200 
 in money for each share he holds. 
 
 If no profits were added to the value of the shares, it 
 would take 200 months, that is, 16 years 8 months to 
 mature the shares ; but the profits generally reduce this 
 time to between 10 and 12 years. 
 
 682. Loans. Any shareholder may borrow $200 on 
 each share he holds, provided he furnishes the security 
 required by law. Security may be by mortgage upon real 
 estate or upon the shares themselves. If the shares are 
 offered as security, no shareholder is allowed to borrow 
 more than the present value of his shares. 
 
 The amount of a loan is usually limited to $2000. 
 
 683. When the accumulation of the various payments 
 has reached a certain sum, these funds are offered at auction 
 and loaned to the shareholder who offers proper security 
 and bids the highest premium, in addition to interest at 
 the rate of 6% per annum. This interest and premium is 
 added to the general fund, and at stated times is credited 
 equally among the various shares. 
 
 684. Fines. To prevent payments falling into arrears, 
 a fine, usually 2 cents a month on every dollar not paid 
 
COOPERATIVE BANKS. 371 
 
 when due, is imposed on delinquent shareholders, whether 
 the delinquency is the monthly instalment, interest, or 
 premium. 
 
 685. Examples. 1. Find the cost at compound interest 
 of a cooperative bank share, if the share matured in 11 
 years, and money was worth 4%. 
 
 Solution. 11 yr. = 132 mo. $1 was paid monthly for 132 mo. 
 The rate of interest was 4% yearly, or \°j monthly. 
 
 g(g»-l) _ *lx(1.003j*»-1) 
 A ~ R-\ ~ 1.003* -1 (§673) 
 
 By the four-place table of logarithms, the value of this fraction is 
 found to be $163.80 ; by a seven-place table, the value is $165.46. 
 The seven-place table gives a very close approximation. 
 
 2. Find the cost at compound interest of a loan of $200 
 from a cooperative bank, if the borrower pays $1 per month 
 interest. The shares are worth $60, and mature in 7 years, 
 and money is worth 4%. 
 
 Solution. The borrower pays monthly $2 for 7 yr., that is, 84 mo. 
 The final value of $2 deposited monthly for 84 mo. at 4% is found by 
 § 673 to be $191.40, and the compound amount of $60 at 4% for 7 yr. 
 is $78.96. Hence, the cost of the loan is $191.40 + $78.96 = $270.36. 
 
 Exercise 161. 
 
 Find the cost at compound interest of a : 
 
 1. Cooperative bank share that matured in 10 years, 
 when money was worth ty°/ . 
 
 2. Cooperative bank share that matured in llj years, 
 when money was worth 6%. 
 
 3. How much more does it cost to borrow $2000 from 
 a cooperative bank, monthly interest being $12, and the 
 shares maturing in 10 years, than to borrow $2000 at com- 
 pound interest for 10 years, if money is worth 5°/ in both 
 cases ? 
 
CHAPTER XXI. 
 
 MISCELLANEOUS PEOBLEMS. 
 
 In solving these problems, logarithms should be used whenever 
 they can be used with advantage. 
 
 1. Make six different numbers with the digits 1, 2, 3, 
 and find their sum. 
 
 2. Make six different numbers with the digits 2, 3, 5, 
 and find, by logarithms, their continued product. 
 
 3. Make six different numbers with the digits 8, 7, 3, 
 and find, by logarithms, their continued product. 
 
 4. Find, by logarithms, the inissing term in each of the 
 following proportions : 
 
 (i) 7.13 : 3.57 : : 4.18 : ?. (iii) 7.37 : ? : : 86.1 : 43.7. 
 (ii) 5.89 : 76.3 : : ? : 38.7. (iv) ? : 69.7 : : 3.79 : 29.4. 
 
 5. Find, by logarithms, the value of 0.08* ; 2734* ; 
 21.97*j ? 3 - 6 ; 9.71* ; 7.936*. 
 
 _ _. , ., . „ J>/ 4.79*X 3.1416X12.72 
 
 6. Fmd the value of V— 0^36^14^8 
 
 7. If the air-line distance between two points is 1534 ft., 
 and the difference of level is 34 ft., what is the horizontal 
 distance between the two points ? 
 
 8. If the road distance is 1 mi., and the rise 347 ft., 
 find the horizontal distance. 
 
 9. If the road distance is half a mile, and the hori- 
 zontal distance 2513 ft., find the difference of level. 
 
 10. The diagonal of a rectangular floor is 34.6 ft., and 
 the width is 17.8 ft. Find the length of the floor. 
 
 11. The height of a tower on the bank of a river is 55 ft., 
 and the length of a line from the top of the tower to the 
 opposite bank is 78 ft. Find the breadth of the river. 
 
MISCELLANEOUS PKOBLEMS. 873 
 
 12. The number of seamen at Portsmouth is 800, at 
 Charlestown 404, and at Brooklyn 756. A ship is com- 
 missioned whose complement is 490 seamen. Determine 
 the number to be drafted from each place to obtain a pro- 
 portionate number from each. 
 
 13. Show, without division, that 36,432 contains 8, 9, 
 11 as factors. 
 
 14. Find the smallest multiplier that will make 47,250 
 a perfect cube. 
 
 1 5. Find the proper fraction that, when reduced to a con- 
 tinued fraction, has for quotients 1, 3, 5, 7, 2, 4. 
 
 16. If the meter is equal to 1.09362 yd., find a series of 
 four fractions that will express more and more nearly the 
 true ratio of the meter to the yard. 
 
 17. Find the square factors contained in 33,075. 
 
 18. The height of St. Peter's, Rome, is yf^ of a mile, 
 and that of St. Paul's, London, is ^y ¥ of a mile. How 
 many feet higher is St. Peter's than St. Paul's ? 
 
 19. How many days elapsed between the annular eclipse 
 of May 15, 1836, and that of March 15, 1858 ? 
 
 20. In a gale, a flagstaff 60 ft. high snaps 28.8 ft. from 
 the bottom ; and, not being wholly broken off, the top 
 touches the ground. If the ground is level, how far is the 
 top frjnn the bottom ? 
 
 21. Seventeen trees are standing in a straight line, 
 20 yd. apart ; a man walks from the first to the second 
 and back, then to the third and back, and so on. How far 
 does he walk ? 
 
 22. A canal is 14f mi. long and 48 ft. wide. At one 
 end is a lock 80 ft. by 24 ft., with a fall of 8 ft. 6 in. 
 How many barges can pass through the lock before the 
 water in the canal is lowered 1 in.? 
 
 23. Find the capacity, in liters and in bushels, of a box 
 1.7 m long, 87 cm wide, and 31 cm deep. 
 
374 MISCELLANEOUS PROBLEMS. 
 
 24. Find the number of kilograms of olive oil, specific 
 gravity 0.915, required to fill a rectangular vessel 2.3 m long, 
 1.8 m wide, and 74 cm deep. 
 
 25. How many tons in a block of marble 4 ft. long, 
 34 in. wide, 17.3 in. thick, specific gravity 2.73 ? 
 
 26. Find the surface of a sphere 18.3 in. in diameter. 
 
 27. Find the number of acres in a circular field 213 yd. 
 2 ft. in diameter. 
 
 28. How many cubic inches in a 10-inch globe ? in a 20- 
 inch globe ? What is the ratio of their volumes ? 
 
 29. How many balls 3 in. in diameter can be cast from 
 a pig of iron 7 ft. long, 6.7 in. wide, 3.8 in. thick, if the 
 waste in melting and casting is reckoned at 3J% ? 
 
 30. Find the difference in length, at 80° F., of a glass 
 rod and a steel rod, each 3 ft. long at 0° C, if the expansion 
 at 100° C. is 0.00085 for glass and 0.0012 for steel. 
 
 31. A grain of gold is beaten into leaf to cover 56 sq. in. 
 What weight will be required to gild the faces of a cube 
 whose edge is 3£ ft.? 
 
 32. What premium must be paid, at the rate of 4J%, for 
 insuring a vessel worth $100,000, in order that in the event 
 of loss the owner may receive both the value of the ship 
 and the premium ? 
 
 33. By selling goods at 60 cents a pound, 8% is lost. 
 What advance must be made in the price to gain 15% ? 
 
 34. The sharpest grade on Mt. Washington liy. is 
 1980 ft. to the mile. What fraction of a foot is the rise 
 for each foot ? What is the per cent of grade ? 
 
 35. Find the square root, to four decimal places, of the 
 reciprocal of 0.0043. 
 
 36. The population of a city in 1890 was 12,298, show- 
 ing a decrease of 8£% on its population in 1880 ; in 1880 
 there was an increase of 7£% on the census of 1870. What 
 was its population in 1870 ? 
 
MISCELLANEOUS PROBLEMS. 375 
 
 37. Find the increase of income obtained by transferring 
 25 shares of 3% stock at 94f to 4% stock at 104f, broker- 
 age \ on each transaction. 
 
 38. Each person in breathing spoils the air of a closed 
 room at the rate of about 8 cu. ft. a minute. An audience 
 of 400 persons enter a closed hall 70 ft. by 40 ft., and 20 ft. 
 high. How long will it take them to spoil the air ? 
 
 39. How long can the windows and doors of a school- 
 room be safely kept closed when occupied by 50 children, 
 if the room is 25 ft. by 20 ft. and 10 ft. high ? 
 
 40. A pays B $230 as the present value of $300 due in 
 5 years. Which gains by the payment, and how much, if 
 interest is reckoned at 5 °f compound interest ? 
 
 41. Find the quantity of coal required by a steamer for 
 a voyage of 4043 mi., if her rate per hour is 14.04 knots, 
 and her consumption of coal 87 long tons per day. 
 
 42. Find the area of a circular ring whose inner and 
 outer diameters are 7.36 in. and 10.64 in., respectively. 
 
 43. A and B can do a piece of work in 13-J- days ; A and 
 C in lOf days ; A, B, and C in 1\ days. In how many 
 days can A do the work alone ? 
 
 44. If 3 men working 11 hours a day can reap 20 A. 
 in 11 days, how many men working 12 hours a day can 
 reap-a field 360 yd. long and 320 yd. broad in 4 days ? 
 
 45. Find the area of a triangle whose sides are 12 in., 
 5 in., and 13 in., respectively. 
 
 46. The four sides of a field measured in succession are 
 237 ft., 253 ft., 244 ft., and 261 ft., and the diagonal meas- 
 ured from the end of the first side to the end of the third 
 side is 351 ft. Find the area of the field. 
 
 47. The four sides of a field measured in succession are 
 361 ft., 561 ft., 443 ft., and 357 ft., and the distance from 
 the beginning of the first side to the end of the second side 
 is 682 ft. Find the area of the field, 
 
376 MISCELLANEOUS PROBLEMS. 
 
 48. Find the altitude of a triangle, if each side is 1000 ft. 
 
 49. Find the three altitudes of a triangle, if its sides are 
 17.8 mm , 23.6 mm , and 31.5 mm , respectively. 
 
 50. How many square inches in the surface of a sphere 
 that has a radius of 12.37 in.? 
 
 5 1 . Find the area of the surface of the largest globe that 
 can be turned out from a joist 4 in. by 6 in. 
 
 52. How many cubic inches in a globe that has a diam- 
 eter of 10 in.? 
 
 53. If a tree is round, and its girth is 17 ft. 6 in., find 
 its diameter. Find the area of a cross section, and also the 
 number of cubic feet in the largest sphere that can be cut 
 from it. 
 
 54. Find the weight in kilograms and in pounds of an 
 iron ball 21.5 cm in diameter, specific gravity 7.47 ; of a tin 
 ball 13 cm in diameter, specific gravity 7.29 ; of a lead ball 
 17.3 cm in diameter, specific gravity 11.35 ; of a silver ball 
 1.31 cm in diameter, specific gravity 10.47. 
 
 55. A slab of cast iron 4 ft. 2£ in. long, 17 in. wide, and 
 8£ in. thick, specific gravity 7.31, is cast into 2-lb. balls. 
 If there is a loss of 5% in melting, how many balls are 
 obtained, and what is the diameter of each ? 
 
 56. How many pounds will a ball of iron 30 in. in 
 diameter weigh, if the specific gravity of the iron is 7.31 ? 
 
 57. If the specific gravity of ice is 0.930, find the weight 
 and the surface of each of three spheres of ice whose diam- 
 eters are l cm , 10 cm , and l m . 
 
 58. Find the capacity in gallons of a round cistern 13 ft. 
 in diameter and 9 ft. deep. 
 
 59. A cylinder is 10 in. in diameter and 12 in. long. 
 Find the area of each end, the lateral surface, the total sur- 
 face, and the contents in gallons. 
 
 60. What must be the diameter of a cylinder 10 in. deep 
 that it may hold 1 gallon ? 
 
MISCELLANEOUS PROBLEMS. 377 
 
 61. Find the volume of a cylinder 8 in. in diameter and 
 11 in. high. 
 
 62. Find the dimensions of three cylinders that have the 
 diameters equal to the heights, and hold 1 gallon, 1 quart, 
 and 1 liter, respectively. 
 
 63. How many cubic yards in a pyramid 123 ft. high, 
 with a square base 210 ft. on a side ? 
 
 64. Find the capacity of a cup, whose mouth is 4 in. 
 square, and whose sides are four equilateral triangles. 
 
 65. The largest of the Egyptian pyramids is 147 m high, 
 with a base 231 m square. Find its volume in cubic meters. 
 
 66. The slant depth of a conical cup is 93 mm , and the 
 diameter at the top 8 cm . What is its capacity ? 
 
 67. The volume of a cone is l cbm ; its height is equal to 
 the radius of its base. Find the dimensions of the cone. 
 
 68. Find the capacity in pints of a cylinder, diameter 
 1.9375 in., height 2.4375 in.; of a cylinder, diameter 3£ in., 
 height 3| in. ; of a cylinder, diameter 3}| in., height 5 T ^ in. 
 
 69. Find the capacity in pecks of a cylinder, diameter 
 15.865 in., height 12.5 in.; of a cylinder, diameter 9.25 in., 
 height 4.25 in.; of a cylinder, diameter 18.5 in., height 8 in. 
 
 70. What must be the diameter of a circle to contain 
 78.54 sq. ft.? to contain 314.16 sq. ft.? 
 
 71* What must be the diameter of a circle to contain 
 1 A.? to contain 9 A.? 
 
 72. What must be the diameter of a circle to contain 
 l ha ? to contain 25 ha ? 
 
 73. Divide $1270 into parts proportional to 4^, 5 J, 6f. 
 
 74. How much water will a hemispherical bowl hold 
 that is 10 in. in diameter ? 
 
 75. At 50 cents a square foot, what will it cost to gild 
 a hemispherical dome 10 ft. in diameter ? 
 
 76. If the moon is a sphere 2170 miles in diameter, how 
 many million bushels would it hold if hollow ? 
 
378 MISCELLANEOUS PROBLEMS. 
 
 77. If the earth is 7920 miles in diameter, and the 
 air is 40 miles deep, how many cubic miles of air are 
 there ? 
 
 78. What is the difference between 2 feet square and 
 2 square feet ? between a foot square and a square foot ? 
 between half a foot square and 6 in. square ? 
 
 79. Find the volume of a frustum of a right pyramid 
 whose lower base is a square 3 ft. on a side, upper base a 
 square 2 ft. on a side, and height 4 ft. 
 
 80. Find the capacity in liquid quarts of a tin pan 10 in. 
 in diameter at the top, 8 in. in diameter at the bottom, and 
 4 in. deep. 
 
 81. How many hektoliters will a circular vat hold 5 m in 
 diameter at the top, 4.57 m in diameter at the bottom, and 
 1.17 m deep ? 
 
 82. If 4 cu. in. of iron weigh 1 lb. avoirdupois, what is 
 the weight in grains of 1 cu. in. of iron ? What is the 
 specific gravity of the iron ? 
 
 83. If 4 cu. in. of iron weigh 1 lb., what is the diameter 
 of a 6-lb. ball ? of a 32-lb. ball ? 
 
 84. At £ lb. to the cubic inch, what is the weight of a 
 rectangular block of iron 17.36 in. by 8.7 in. by 1.76 in.? 
 What would be its diameter if cast into a ball, if 11% is 
 allowed for waste ? 
 
 85. At £ lb. to the cubic inch, what is the weight of a 
 rectangular block of iron 71.4 in. by 8 J in. by 3£ in.? 
 What would be its diameter if cast into a ball, if 11% is 
 allowed for waste ? 
 
 86. What is the diameter of a cylinder 11 in. long that 
 will hold 2 gallons ? 
 
 87. What is the diameter of a cylinder 9 in. long that 
 will hold 2 gallons ? 
 
 88. What is the diameter of a cylinder 30 cm long that 
 will hold 10 liters ? 
 
MISCELLANEOUS PROBLEMS. 379 
 
 89. Find the circumference of a globe, if the number of 
 square centimeters in its surface is three times the number 
 of cubic centimeters in its volume. 
 
 90. Find the diameter of a circle, if the number of inches 
 in its circumference is equal to the number of square feet 
 in its area. 
 
 9 1 . How many times does a carriage wheel 3 ft. 2 in. in 
 diameter turn in going a mile on a smooth road ? 
 
 92. A point in the tire moves, while the wheel turns 
 once, just four times the diameter of the wheel. How far 
 does a spike head in the tire travel while a wheel, 3 ft. 
 2 in. in diameter, travels 1 mi.? 
 
 93. An oil can is formed of two cylinders connected by 
 a frustum of a cone. . The upper cylinder, or neck, is 6 cm in 
 diameter, and 75 mm high ; the lower cylinder is 13 cm in 
 diameter, and 153 mm high ; the total length of the can 
 is 30 cm . Find the capacity of the can in liters. 
 
 94. A common tunnel is formed of a frustum of a cone 
 terminated with a cylinder. The height of the frustum is 
 14 cm , and the diameters of the two bases are 175 mm and 
 16 m,n , respectively. The cylinder is 8 cm long. Find the 
 capacity of the tunnel in liters. 
 
 95. A pan in the form of a frustum of a cone is 10 cm 
 deep, J.2 cm across the bottom, and 23 cm across the top. Find 
 the capacity of the pan in liters. 
 
 96. Find the number of square centimeters of sheet iron 
 in a stovepipe 4 m long, 26 cm in diameter, and l mm thick, if 
 the edges lap one centimeter. Find the weight of the pipe, 
 if the specific gravity of the sheet iron is 7.8. 
 
 97. A steam boiler is formed of a cylinder terminated 
 at each end by a hemispherical cap of the same diameter. 
 The length of the cylinder is 3.4 m , interior diameter 0.8 m . 
 Find the number of hektoliters of water required to fill 
 the boiler half full. 
 
380 MISCELLANEOUS PROBLEMS. 
 
 98. A spherical bomb is 32 cm in diameter, and the sides 
 38 mm thick. If the specific gravity of the metal is 7.2, 
 what is the weight of the bomb and its capacity ? 
 
 99. The diameters of a lamp shade are 25 cm and 7 cm , and 
 its slant height is 134 mm . Find its curved surface in square 
 centimeters. 
 
 100. A niche is formed like a half -cylinder surmounted 
 by a quarter of a sphere. The height of the cylinder is 
 1.2 m , the diameter 0.8 m . Find the volume of the niche, and 
 the area of its interior surface. 
 
 101. What is the expense, at 30 cents a square yard, of 
 painting the walls and ceiling of a room 22 ft. 6 in. long, 
 13 ft. 6 in. wide, and 10 ft. high ? 
 
 102. In what time will an empty cistern be filled by 
 three pipes whose diameters are £ in., £ in., and 1 in., if 
 the largest alone would fill it in 40 min.? The rates of 
 flow are proportional to the squares of the diameters. 
 
 103. How many gallons of water are contained in a 
 length of 50 yd. of a canal, if its width at the top is 8 yd. 
 and at the bottom 7 yd., and its depth 5 ft.? 
 
 104. A man who rows 4 miles an hour in still water 
 takes 1 hr. 12 min. to row 4 miles up a river. How long 
 will it take him to row down again ? 
 
 105. How long must a ladder be to reach a window 
 40 ft. from the ground, if the distance of the foot of the 
 ladder from the wall is 9 ft.? 
 
 106. If 3 oz. of gold 15 carats fine are mixed with 7 oz. 
 12 carats fine, what will be the fineness of the compound ? 
 What must be the fineness of 11 oz. that, when added to 
 this compound, the whole may be 14 carats fine ? 
 
 107. Find the surface of each face of a cube whose 
 volume is 14 cu. ft. 705.088 cu. in. 
 
 108. Determine the depth of conical wineglasses 2\ in. 
 across the top that 60 of them may hold a gallon. 
 
MISCELLANEOUS PROBLEMS. 381 
 
 109. What must be the length of spermaceti candles $• 
 of an inch in diameter that six of them may weigh a pound, 
 if the specific gravity of spermaceti is 0.943 ? 
 
 110. A cylinder 10 in. across and 10 in. high contains 
 0.3927 cu. ft. of water. How many shot 0.1 in. in diam- 
 eter must be poured in to raise the water to the top ? 
 
 111. How deep must a round cistern 4 ft. in diameter be 
 made to be lined with the same amount of lead as a cubical 
 cistern 4 ft. on an edge ? Compare their capacities. 
 
 112. The material for lining a cubical cistern cost $10. 
 Find the cost of the material for lining two similar cisterns 
 which shall each hold one half as much. 
 
 113. If 5 excavators sink a circular shaft 8 ft. in diam- 
 eter and 125 fathoms deep in 100 days of 10 hr. each, how 
 many nights of 7 hr. each will 4 excavators be in sinking 
 a shaft 6 ft. in diameter and 75 fathoms deep, if the diffi- 
 culty of working by night is one seventh greater than by 
 day, and the hardness of the ground in the smaller shaft is 
 to that in the larger shaft as 7 is to 5 ? 
 
 114. Find the number of dry quarts a tub will hold that 
 is 22 in. across the top, 20 in. across the bottom, and 18 in. 
 deep. 
 
 115. Find the number of dry quarts a cylinder will 
 hold that is 28 in. long and has a diameter of 18 in. 
 
 116. How high will 2 quarts of milk stand in a cylindri- 
 cal pail 7 in. in diameter ? How high will 2 quarts of oats 
 stand in the same pail ? 
 
 117. Find the capacity in gallons of a cylindrical boiler 
 1 ft. in diameter and 4 ft. 10 in. long; of a cylindrical 
 boiler 1 ft. 6 in. in diameter and 3 ft. 6 in. long; of a 
 cylindrical boiler 2 ft. 8 in. in diameter and 5 ft. 6 in. long. 
 
 118. Find the capacity of a tumbler 3J- in. across the 
 bottom, 3-J- in. across the top, and 3^- in. deep ; of a cylin- 
 drical tumbler 3£ in. in diameter and 3£ in. deep. 
 
382 MISCELLANEOUS PROBLEMS. 
 
 686. Ellipse. If a point moves continuously so that 
 the sum of its distances from 
 two fixed points, called the 
 foci, always remains the 
 same, the point traces a 
 curve called an ellipse. 
 
 Two common examples of an 
 ellipse are the shadow of a circular plate and a section of a right 
 cylinder not parallel to the bases. 
 
 687. To Find the Area of an Ellipse, 
 
 Multiply the product of its longest and shortest diameters 
 by 0.7854 (i of 8.1^16). 
 
 119. Find the area of an ellipse whose longest and 
 shortest diameters are 11 in. and 8 in., respectively. 
 
 120. The ends of a rope 100 ft. long are fastened to 
 stakes placed 80 ft. apart on level ground. A ring, to 
 which a kid is tied, plays freely on the rope. How far 
 from a straight line joining the stakes can the ring be pulled ? 
 
 121. If the stakes of Ex. J 20 are placed 25 ft. apart, by 
 how many per cent is the kid's pasturage increased, pro- 
 vided he can graze 18 in. beyond the rope when stretched ? 
 
 122. A cylindrical log, 11 in. in diameter, is sawed off 
 at such a slant that the pieces are 8 in. longer on the 
 longest than on the shortest side. Find the diameters of 
 the ellipse thus made, and its area. 
 
 123. Find the area of an ellipse, if its longest diameter 
 is 12 in. and its shortest diameter 9 in. 
 
 688. To Find the Capacity of Any Round Vessel, like 
 a Cup, Saucer, Bowl, or Tunnel, 
 
 If spherical, the vessel holds two th irds as much as a 
 cylinder of the same diameter and depth; if conical, one 
 third as much; if like a coffee cup, one half as much. 
 
MISCELLANEOUS PROBLEMS. 383 
 
 124. Find the number of quarts a conical tunnel will 
 hold if it is 9 in. across the top and 8 in. deep. 
 
 125. Find the number of pints a spherical bowl will 
 hold if it is 5 in. across the top and 2\ in. deep. 
 
 126. Find the number of pints a spherical bowl will 
 hold if it is 4 in. across the top and 3£ in. deep. 
 
 127. Find the capacity in pints of a coffee cup 3 in. 
 across the top and 3 in. deep. 
 
 128. Find the capacity in liters of a spherical wash 
 bowl 30 cm in diameter and 5 cm deep. 
 
 129. Find the capacity in liters of the basin of a foun- 
 tain 89 cm in diameter and 31 cm deep. 
 
 130. Find the capacity in quarts of a bowl 10 in. in 
 diameter and 4 in. deep. 
 
 131. Find the capacity in pints of a saucer 6 in. across 
 and \\ in. deep ; of a bowl 7 in. across and 3 in. deep. 
 
 132. How many gallons will a spherical basin 5 ft. in 
 diameter and 2 ft. deep hold ? 
 
 133. How many gallons will a spherical bowl 30 in. in 
 diameter and 1 ft. deep hold ? 
 
 134. Find the capacity in pints of a saucer 5 in. across 
 and 2 in. deep. 
 
 135. Find the capacity in gallons of a paraboloid (shaped 
 like a^coffee cup) boiler 25 in. across and 14 in. deep. 
 
 136. Find the capacity in quarts of a conical funnel 
 9 in. across and 7 in. deep. 
 
 689. To Find the Number of Gallons a Cask will Hold, 
 
 Multiply by 0.65 the difference between the Imng and head 
 diameters expressed in inches, and add the product to the 
 head diameter for the mean diameter. 
 
 Divide the product of the length of the cask expressed in 
 inches and the square of the mean diameter by 294; ^ ie 
 quotient is the number of gallons the cask will hold. 
 
384 MISCELLANEOUS PROBLEMS. 
 
 137. Find the number of gallons contained in a full cask 
 whose bung diameter is 24 inches, head diameter 22 inches, 
 and length 30 inches. 
 
 138. Find the number of gallons contained in a full cask 
 whose bung diameter is 22 inches, head diameter 20 inches, 
 and length 28 inches. 
 
 139. Find the number of gallons contained in a full cask 
 whose bung diameter is 20 inches, head diameter 18 inches, 
 and length 28 inches. 
 
 690. Sound travels in still air at 32° F. 1090 ft. a sec- 
 ond, and 1.1 ft. a second faster for every degree Fahrenheit 
 increase in temperature. 
 
 Sound travels in still air at 0° C. 332 m a second, and 
 60.9 cm a second faster for every degree Centigrade increase 
 in temperature. . 
 
 140. The flash of a gun is seen 7£ sec. before the report 
 of the gun is heard ; there is no wind, and the temperature 
 is 73° F. How far off is the gun ? 
 
 141. A meteor was seen to burst; the report followed 
 in 4 min. 17 sec. What was its distance, if the average 
 temperature of the intervening air was 50° F.? 
 
 142. How long will it take for an explosion at the equa- 
 tor to be heard at the antipodes of the place, if the circum- 
 ference of the earth at the equator is reckoned at 40,000 km , 
 and the average temperature at the equator at 23° C? 
 
 143. If an explosion at the equator occurs at sunset and 
 the average temperature east of the spot is 22° C, and that 
 to the west 24° C, how far from the antipodes will the 
 sound waves meet ? 
 
 144. How far off is the lightning when the thunder fol- 
 lows in 13 sec, the temperature being 76° F.? 
 
 145. How long would it take sound to go through a 
 whispering tube 3 mi. long, temperature 61° F.? 
 
MISCELLANEOUS PROBLEMS. 385 
 
 146. Sound travels in iron about 10£ times as fast as in 
 air. How long, then, after seeing the blow of a sledge 
 hammer given on the other end of an iron pipe \\ mi. long, 
 may I expect to hear the sound by the iron ; and how long 
 after, to hear the sound through the air in the pipe j ther- 
 mometer 63° F. ? 
 
 147. Two gunners fire at each other simultaneously from 
 forts \\ mi. apart; the wind, at 70° F., blows steadily from 
 one fort to the other, at 11 mi. an hour. How soon will 
 each hear the report of the other's gun ? Suppose one ball 
 flies on the average 987 ft. a second, the other 818 ft. a 
 second ; when will each receive the other's shot ? 
 
 148. Sound travels in water about 4.26 times as fast as 
 in air. How many seconds sooner would the sound of a 
 torpedo exploded under water 2 mi. off reach you by water 
 than by air, at 68° F. ? 
 
 691. In the average state of the atmosphere, the distance 
 at which an object is visible at sea may be found by the 
 following relation : 
 
 The square of the distance in miles is seven fourths the 
 height of the object in feet. 
 
 The square of the distance in kilometers is fifteen times 
 the height in meters. 
 
 Hence, log miles = 0.1215 + \ log feet. 
 
 log kilometers = 0.5880 + i log meters. 
 
 149. A hill 482 ft. high is 8 mi. from the shore. How 
 many miles out at sea is it visible ? 
 
 150. A sailor at the topmast 80 ft. above the sea can 
 just see a sailor at the topmast of a similar ship. How 
 many miles apart are the vessels ? 
 
 151. How far is a mountain 1000 m high visible ? a 
 mountain 2000 m high? 
 
386 MISCELLANEOUS PROBLEMS. 
 
 152. If a man stands on a bluff that raises his eyes ll in 
 above the sea, how far can he see from the shore ? 
 
 153. A sailor at sea is at a distance of 171 km from a 
 mountain when the top of the mountain is just visible. 
 How high is the mountain ? 
 
 154. A vessel approaching Valparaiso at daybreak just 
 makes out the peak of Aconcagua, 22,427 ft. high and 140 
 mi. back from the coast. How far is the vessel from land 
 if the eye of the observer is 30 ft. above the water ? 
 
 155. If Mount Washington is 6293 ft. high and 76 mi. 
 in an air line from Cape Elizabeth, how far out from the 
 Cape will its peak be visible in the ordinary state of the 
 atmosphere ? 
 
 156. How many acres of water can a man see if he 
 Stands on a raft with his eyes just 6 ft. above the water, 
 and no land is in sight ? 
 
 157. How far would a mountain 29,000 ft. high be 
 visible ? one 5000 ft. high ? one 1000 ft. high ? 
 
 158. How high must a mountain be in order to be visible 
 at sea level 50 miles ? 100 miles ? 150 miles ? 
 
 159. What distance can be seen from the top of a 
 mountain 4 miles high? 
 
 692. Pendulum. A body suspended by a straight line 
 from a fixed point so as to swing freely is called a pendulum. 
 
 693. The number of vibrations that pendulums make in 
 a given time is inversely as the square root of their lengths. 
 
 A pendulum that passes its central point of rest once every mean 
 solar second is 39.138 in. long. 
 
 160. Find the length of a pendulum that beats half- 
 seconds ; of a pendulum that beats quarter-seconds. 
 
 161. How many centimeters long is a pendulum that 
 swings 80 times a minute ? a pendulum that swings 30 
 times a minute ? 
 
MISCELLANEOUS PROBLEMS. 387 
 
 162. If a cannon ball is suspended by a line wire 176 ft. 
 long in the central well of the Bunker Hill Monument, 
 how many times a minute will it swing ? 
 
 163. How long is a pendulum that swings three times 
 in two seconds ? that swings five times in two seconds ? 
 
 694. A body falling from rest in a vacuum falls 16^ ft. 
 or 4.903 m in the first second ; it then has acquired a veloc- 
 ity of 32£ ft. or 9.806 m . 
 
 A falling body increases its velocity in proportion to the 
 time it is falling ; and the distance fallen is in proportion to 
 the square of the number of seconds of time it is falling. 
 
 Thus, a body falling from rest in a vacuum in half a second has 
 fallen 4^-g- ft. or 1.2257 m , and has acquired a velocity of 16 r ^ ft. or 
 4.903 m per second ; in 3 sec. it has fallen 144| ft. or 44.127 m , and has 
 acquired a velocity of 96^ ft. or 29.418 m per second. 
 
 The velocity of heavy bodies falling short distances in air will not 
 be much less than in a vacuum. 
 
 164. What velocity in meters a second will a cannon 
 ball acquire in falling three quarters of a second ? in fall- 
 ing three and a quarter seconds ? 
 
 165. How long will it take a leaden ball, rolling off a 
 table 29 in. high, to reach the floor ? 
 
 16€. What velocity will a crowbar attain in falling end- 
 wise from a balloon 2000 m high ? How long will it be in 
 coming down ? 
 
 167. What velocity will a crowbar attain in falling end- 
 wise from a balloon one mile and a quarter high ? How 
 long will it be coming down ? 
 
 168. How long will it take a ball, rolling off a table, to 
 drop l cm ? 1 in. ? 10 cm ? 6 in ? 
 
 169. If Carisbrook Well is 210 ft. deep, how long after 
 a pebble is dropped will it be heard to strike the bottom, 
 if the velocity of sound is 1120 ft. a second ? 
 
588 MISCELLANEOUS PROBLEMS. 
 
 170. How long after a pebble is dropped will it be heard 
 to strike the bottom of a ventilating shaft 1600 ft. deep, 
 if the temperature is 68° F.? 
 
 171. If a rock dropped over a precipice strikes the 
 bottom in 7£ sec, how high is the precipice ? 
 
 172. How long after a pebble dropped down a shaft 
 133 ft. deep will it be heard to strike the bottom, if the 
 temperature is 59° F.? 
 
 695. If a plunger fits tightly in a small cylinder, and by 
 it water is forced into a large cylinder, a plunger in the large 
 cylinder is lifted with a force nearly equal to the product of 
 the force with which the little plunger is driven in multiplied 
 by the square of the ratio of the diameters of the two cylinders. 
 
 173. Find the lifting power of a hydraulic press, the 
 plunger being l cm in diameter and driven with a force of 
 100 kg , if the lifting piston is l m in diameter. 
 
 174. If the plunger is £ in. in diameter, and is driven 
 with a force of 1000 lb., how much can it lift with a lifting 
 piston 4 ft. in diameter ? 
 
 175. If the plunger is 2 in. in diameter, and is driven 
 with a force of 1000 lb., how much can it lift with a lifting 
 piston 2 ft. in diameter ? 
 
 176. The water stands in a fissure in a rock 10 m high 
 and 12 m long. What pressure is exerted to split the rock 
 on the lowest meter's width ? on the highest meter's 
 width ? in the whole fissure ? 
 
 N«te. This pressure is found by multiplying the surface upon one 
 md% by the height of water above the centre line, counting the product 
 as Tolume of water, and then finding the weight of this volume of 
 water. The principle is precisely the same as in the hydraulic press. 
 
 177. A dam is 100 ft. long and 10 ft. deep, and the 
 water is just flowing over it. What pressure is exerted 
 •n the lowest two feet of the dam? 
 
MISCELLANEOUS PROBLEMS. 
 
 178. Water is running 2 ft. over a dam that is 180 ft. 
 long and 12 ft. deep. Find the pressure on the dam. 
 
 179. Water is running 9 in. deep over a dam that is 
 78 ft. long and 8 ft. deep. Find the pressure on the dam. 
 
 696. The velocity with which water will flow out of a hole 
 in the side of a reservoir is nearly proportional to the square 
 root of the depth of the hole below the surface of the water ; 
 and is about 82 ft. a second at the depth of 16 ft. 
 
 180. With what velocity will water flow through a hole 
 9 ft. below the surface ? 
 
 181. With what velocity will water leave a fountain 
 having free play, and a head of 25 ft.? a head of 100 ft.? 
 
 182. If a hole in the side of a cistern 4 ft. below the 
 surface of the water is delivering 10 gal. an hour, how 
 many gallons would it deliver with 5 ft. more head ? 
 
 183. If a pipe 2 in. in diameter, and 1 ft. long, inserted 
 in a dam, the head of water being kept constant, delivers 
 4 gallons of water a minute, how many gallons a minute 
 may be expected when another pipe of the same length, but 
 2\ in. in diameter, is substituted for the two-inch pipe ? 
 
 184. If a one-inch pipe, 20 in. long, is substituted for 
 the two-inch pipe, 1 ft. long, in Example 183, and the flow 
 is fotmd to be 5 pints a minute, what part of the decrease of 
 flow is due to the smaller area of the orifice, and what part 
 to the increased friction on the sides of the longer pipe ? 
 
 697. The quantity of water issuing from a hole is in pro- 
 portion to the square root of the head ; and the velocity is in 
 proportion to the square root of the head. 
 
 The work which the water can do is in proportion to the 
 quantity multiplied by the square of the velocity ; that is, 
 
 The work is in proportion to the square root of the cube of 
 the head. 
 
390 MISCELLANEOUS PROBLEMS. 
 
 185. A miller is using water flowing through the gate- 
 way under 4 ft. head. How much more work could he do 
 if the head was raised to 9 ft.? how much more if the head 
 was raised to 25 ft.? 
 
 698. Work is the act of changing the position of a body 
 by overcoming the resistance to the change. 
 
 699. Units of Work. The unit of work is the work 
 done in raising a weight of one pound through a distance 
 of one foot. This unit is called a foot-pound. 
 
 The corresponding metric unit is the kiloyram-meter. 
 
 700. Horse Power. The rate at which work is done is 
 called power. A horse power is the power to do 33,000 foot- 
 pounds of work per minute, or 550 foot-pounds per second. 
 
 186. A cross section of a stream of water is a rectangle 
 6 ft. by 2£ ft. ; the velocity is 40 ft. per minute. There is 
 a fall of 10 ft. where a water wheel is erected that utilizes 
 70% of the work. Find the horse power of the wheel. 
 
 187. Find the horse power of the wheel of Example 186, 
 if the fall of the water is 14 ft. 
 
 188. A cross section of a stream of water is a rectangle 
 5 ft. by 4 ft.; the velocity is 50 ft. per minute. There is 
 a fall of 12 ft. where a water wheel is erected that utilizes 
 65% of the work. Find the horse power of the wheel. 
 
 189. Find the horse power of the wheel of Example 188, 
 if the fall of the water is 16 ft. 
 
 190. A cross section of a stream of water is a trapezoid 
 whose altitude is 3£ ft., and parallel sides 6 ft. and 5 ft., 
 respectively ; the velocity is 150 ft. per minute. There is 
 a fall of 9 ft. where a water wheel is erected that utilizes 
 75% of the work. Find the horse power of the wheel. 
 
MISCELLANEOUS PROBLEMS. 391 
 
 701. When a body is moving in a circle, the centrifugal 
 force is about 1.227 of the continued product of the weight of 
 the body, the number of feet in the radius of the circle, and 
 the square of the number of revolutions in a second. 
 
 Thus, a body going round a circle of 5 ft. radius once a minute 
 
 presses away from the centre with a force equal to 1.227 X 5 X — 
 
 of the weight of the body. 
 
 Note. When the radius is measured in meters, the multiplier 
 4.025 must be used in place of 1.227. 
 
 191. If a top 3 in. in diameter is making 200 revolu- 
 tions a second, with what force does the outer layer pull 
 away from the centre ? 
 
 192. If a sling 30 in. long contains a stone that weighs 
 -J- lb., and is whirled round 80 times a minute, what is the 
 force pulling on the string ? 
 
 193. With what force does a locomotive that weighs 60 
 tons running 30 mi. an hour, on a curve of 800 ft. radius, 
 bear against the outer rail ? If the locomotive is running 
 60 mi. an hour, with what force does it bear on the outer rail ? 
 
 194. If washed wool is put wet into a wire basket 1.2 m 
 in diameter, and the basket is set to spinning at the rate of 
 180 revolutions a second, with what force is water wrung 
 out of the wool ? 
 
 195. If steel pens are revolved in a basket 32 cm in diam- 
 eter, 17 revolutions a second, with what force is the oil 
 drained from them ? 
 
 196. The top of a wheel is at each instant moving with 
 twice the velocity of the carriage, and is moving in a curve 
 whose centre, at the instant, is as far below ground as the 
 point is above ground. What, then, is the force exerted 
 to separate the mud from the top of a wheel 3 ft. 2 in. in 
 diameter, when the carriage is moving at the rate of 10 
 miles an hour? 
 
392 
 
 MISCELLANEOUS PROBLEMS. 
 
 702. When a chain of uniform thickness hangs from 
 two points not in the same vertical line, it hangs in a curve 
 called a common catenary. 
 
 The length of the chain from the' lowest point to any point selected 
 may be called half-chain. The height of the point selected above the 
 lowest point is called sag. The horizontal distance of the point 
 selected from the lowest point is called half-span. The horizontal 
 force with which the point selected is drawn inward is called tension. 
 The radius of the circle which will fit the curve at its lowest point is 
 called radius. The straight line which touches the curve at the point 
 selected is called tangent. 
 
 703. 1. The tension is equal to the weight of a piece of 
 the same chain as long as the radius. 
 
 2. The radius is equal to the sum of the half-chain and 
 sag multiplied by the difference of the half-chain and sag, 
 and divided by twice the sag. 
 
 3. The log of half-span is equal to the log of sum of half- 
 chain and sag plus log of their difference, plus log of the dif- 
 ference of these two logarithms plus colog of sag plus 0.0612. 
 
 4. Radius divided by half-chain measures the "batter" at 
 the point selected ; that is, measures the horizontal falling 
 back for every unit of vertical ascent in a straight line 
 tangent at that point. 
 
 197. How strong a horizontal pull on a chain, weighing 
 half a pound to the yard, is required to make the lowest 
 part curve with an 18-in. radius ? with a 6-ft. radius ? 
 
MISCELLANEOUS PROBLEMS. 393 
 
 198. A f-in. rope, weighing £ lb. to the yard, is fastened 
 at one end to a staple, and near the other end, on the same 
 level, runs over a pulley, and has a 25-lb. weight hung to it. 
 What is the radius of its curvature at the middle ? 
 
 199. A shower wets the rope of Example 198, and in- 
 creases its weight 40 °f ; what does its radius now become ? 
 
 200. A steam tug, in attempting to move a ship, straight- 
 ened the hawser until the radius of the lowest point was 
 1980 ft. The rope was wet, and weighed 3£ lb. to the 
 yard. With what force was it stretched ? 
 
 201. A chain 31 ft. long hangs between points on a 
 level, and sags 4 ft. What is the radius at the lowest 
 point ? 
 
 202. The whole chain, in Example 201, weighs 18 lb. 
 What is the horizontal tension ? What is the distance 
 between the points ? What is the slant, or batter, of the 
 end of the chain ? 
 
 203. A chain weighing l kg to the meter is suspended 
 from points on a level ; the length of chain is 31 m , and it 
 sags 1.3 m . Find all the conditions, and find how much it 
 falls below a level at 10 cra from each end. 
 
 204. A chain 100 m long, weighing 14 oz. to the foot, 
 is suspended from points on a level 80 m apart, and sags 
 26.5£>. m . What is the radius at the middle, the batter at 
 the ends, and the horizontal tension ? 
 
 205. If the chain of Example 204 is shortened 5 m , the 
 sag is decreased 4 m . What is the radius and tension ? 
 
 704. A lever is a rigid bar that can be moved about a 
 fixed axis called the fulcrum. 
 
 The perpendicular from the fulcrum to the line in which 
 the power acts is the power arm of a lever. 
 
 The perpendicular from the fulcrum to the line in which 
 the weight acts is the weight arm of a lever. 
 
394 MISCELLANEOUS PROBLEMS. 
 
 705. The ratio of the power to the weight raised by a 
 lever is equal to the ratio of the weight arm to the power arm. 
 
 206. How heavy a rock placed 6 in. from the fulcrum 
 can a man, who weighs 180 lb., raise with a crowbar 5 ft. 
 6 in. long ? 
 
 207. Two weights of 30 lb. and 20 lb., respectively, at 
 the ends of a horizontal lever 5 ft. long balance. Find 
 how far and in which direction the fulcrum must be moved 
 for the weights to balance when each is increased by 5 lb. 
 
 208. A man who weighs 160 lb., wishing to raise a rock, 
 leans with his whole weight on a horizontal crowbar 5 ft. 
 long, which is propped at the distance of 4 in. from the 
 end in contact with the rock. Find the force he exerts on 
 the rock, and the pressure the prop has to sustain, if the 
 weight of the crowbar is not reckoned. 
 
 209. A child weighing 56 lb. is seated at one end of a 
 plank 16 ft. long, and a child weighing 72 lb. is at the 
 other end. Find the distance of each child from the ful- 
 crum when the plank is used for a seesaw. 
 
 210. In a pair of nutcrackers if the nut is placed at a 
 distance of 1 in. from the hinge, and the hand presses at 
 a distance of 8 in. from the hinge, find the pressure upon 
 the nut for every ounce of pressure exerted by the hand. 
 
 211. A body is weighed in both arms of a false balance, 
 and its apparent weights are 2.56 lb. and 2.25 lb. Find 
 its true weight. 
 
 212. In a steelyard the weight of the beam is 15 lb., 
 and the distance of its centre of gravity from the fulcrum 
 is 3 in. Find the distance from the fulcrum a weight of 
 6 lb. must be placed to balance the beam. 
 
 706. With the Wheel and Axle, the ratio of the power to 
 the weight to be raised is equal to the ratio of the radius of 
 the axle to the radius of the wheel. 
 
MISCELLANEOUS PROBLEMS. 395 
 
 213. A cask weighing 160 kg is attached to a rope wound 
 on an axle 19 cm in diameter ; at one end of the axle is a 
 wheel 175 cm in diameter. With what force must a man pull 
 down on a rope passing over the wheel to raise the cask? 
 
 214. A. rope passes over a single pulley. How much 
 force is required to raise 180 lb. attached to one end of a 
 rope if 1 % of the force is required to overcome friction ? 
 
 215. If the radius of the wheel is four times that of the 
 axle, and the string round the wheel can support a weight 
 of 50 lb. only, find the greatest weight that can be lifted. 
 
 216. Find the ratio of the radii of a wheel and axle that 
 a force of 100 lb. may just support a weight of 1 ton. 
 
 217. The radius of a wheel is 80 cm and the radius of the 
 axle is 12 cm . What weight can be supported by a force of 
 30 kg ? Find the work done if the weight is raised 60 cm . 
 
 707. With the Screw, the ratio of the power to the weight 
 is equal to the ratio of the distance between two consecutive 
 threads to the circumference described by the end of the 
 power arm. 
 
 218. The power arm of a screw is 16 in. long, and by 
 one turn of the screw the head advances one eighth of an 
 inch. If the power is 3 lb., find the weight lifted. 
 
 219. In a screw used to raise a load of 10 tons, the 
 power is 50 lb., acting by an arm 4 ft. long. Find the 
 distance between two consecutive threads. 
 
 220. The lever of a screw is 1 ft. 9 in. long, and the 
 power applied at the end is 100 lb. What must be the 
 distance between the threads that a pressure of 5000 lb. 
 may act on the press board ? 
 
 221. The lever of a screw is 3 ft. 6 in. long, and the 
 distance between the threads is \ in. What power must 
 be applied at the end of the lever to produce a pressure of 
 10 tons on the press board ? 
 
396 MISCELLANEOUS PROBLEMS. 
 
 708. Chemists employ initial letters to signify fixed 
 proportional quantities. Multiples of these are indicated 
 by small figures placed after them ; and multiples of com- 
 pounds are indicated by figures placed before them. 
 
 Thus, if the gram is the unit, H signifies Is of hydrogen, and O 
 signifies 16s of oxygen; then H 2 means 18s of water, and 2 ll 2 o 
 means 36s of water ; while H 2 + O would mean 2s of hydrogen and 
 16s of oxygen mixed, but not chemically united. An electric spark 
 passed through H 2 + O would produce H 2 0, with an explosion. 
 
 709. The chemical symbols and numerical proportions 
 of a few common elements are as follows : 
 
 Element. Symbol. Numerical Equivalent. 
 
 Hydrogen H 1 
 
 Oxygen O 16 
 
 Sulphur S 32 
 
 Iron (ferrum) . . . . Fe 66 
 
 Calcium Ca 40 
 
 Sodium (natron) . . . Na 23 
 
 Carbon C 12 
 
 Boron B 11 
 
 222. What per cent of water is oxygen ? what per cent 
 hydrogen ? 
 
 223. What per cent of quicklime, CaO, is oxygen ? 
 
 224. What per cent of water in slacked lime, Ca0 2 H 2 ? 
 
 225. What per cent of pure marble, CaC0 3 , is oxygen ? 
 
 226. What per cent of gypsum, called plaster of Paris, 
 CaS0 4 + 2 H 2 0, is sulphur ? 
 
 227. What per cent of washing soda, Na-jCC^ + 10 H 2 0, 
 is carbon ? 
 
 228. In 118 lb. of Glauber salts, Na^SO, + 10 H 2 0, how 
 many ounces of sulphur ? 
 
 229. How many ounces of soda, Na 2 + H 2 0, in 7 lb. 
 of borax, Na^O, + 10 H 2 ? 
 
 230. What per cent of pure alcohol, C 2 H 6 0, is carbon? 
 What per cent of pure white marble, CaC0 8 , is carbon ? 
 
MISCELLANEOUS PROBLEMS. 397 
 
 231. What per cent of pure acetic acid (the acid of 
 vinegar) is carbon, the formula being C 2 H 4 2 ? 
 
 232. How much acetic acid can be obtained from 12 lb. 
 of alcohol, C 2 H 6 0, if there is no waste ? 
 
 233. How many grains of carbon in 1 oz. avoirdupois of 
 oxalic acid, C 2 H 2 4 + 2 H 2 ? 
 
 234. How many milligrams of carbon in 3 g of tartaric 
 acid, C 4 H 6 6 ? 
 
 235. How many kilograms of carbon in 95 kg of white 
 sugar, C 12 H 2 20ii ? 
 
 236. The formula of camphor is Ci H 16 O. How many 
 grams of carbon in 14 kg of camphor ? 
 
 237. In 20 kg of oil of vitriol, H 2 S0 4 , how many grams of 
 sulphur ? 
 
 238. What per cent of oil of vitriol is water ? what per 
 cent sulphuric acid, S0 3 ? 
 
 239. In 3.5 g of black oxide of iron, FeO, how many mil- 
 ligrams of jron ? 
 
 240. Eed iron-rust consists of 70% iron and 30% 
 oxygen. Find its formula. 
 
 241. The choking vapor of burning sulphur is sulphur 
 and oxygen in equal parts. Find its formula. 
 
 242. Copperas is 28.9% sulphuric acid, 25.7% oxide of 
 iron,j45.4% water. Find its formula. 
 
 Solution. Water being 18, oxide of iron 72, and sulphuric acid 
 80, first seek multiples of 72 and 80, in the ratio of 25.7 to 28.9 ; that 
 is, of 0.8892 to 1. But 72 and 80 are in almost exactly that ratio. 
 This gives FeS0 4 + water ; and it remains to find a multiple of 18 
 which is to 152 as 45.4 is to 54.6; that is, which is 0.8315 of 152, or 
 126.4. But 7 X 18 = 126; and the addition of seveh parts of water 
 gives FeS0 4 + 7 H 2 0. 
 
 243. Spirits of turpentine is 11.76% hydrogen and 
 88.24% carbon. Find its formula. What per cent of 
 oxygen combined with spirits of turpentine are required 
 to make camphor, C 10 H 16 O? 
 
398 MISCELLANEOUS PROBLEMS. 
 
 710. Ohm and Ampere. The unit of the resistance of 
 the conductor to an electrical current is an ohm. The unit 
 of the strength of an electrical current through a conductor 
 is an ampere. These units have been arbitrarily chosen. 
 
 Note. The strength of the electrical current for an ordinary arc 
 lamp is about 10 amperes. 
 
 711. A volt is the force required to send an electrical 
 current of one ampere through a conductor that offers a 
 resistance of one ohm. 
 
 712. The resistance of the conductor to an electrical cur- 
 rent is directly proportional to the length of the conductor, 
 and inversely proportional to the area of its cross section. 
 
 713. The strength of an electrical current in amperes is 
 equal to the force in volts divided by the resistance in oh /us. 
 
 244. If the resistance of 1 mile of wire 2 mm in diameter 
 is 4.72 ohms, what is the resistance of 3 miles of wire of 
 the same material 3 mm in diameter ? 
 
 245. What length of copper wire l mm in diameter has 
 the same resistance as 720 m of copper wire 4 rani in diameter ? 
 
 246. The conductivity of iron is | that of copper. If 
 the resistance of a copper wire 1 mile long and £ in. in 
 diameter is 6.8 ohms, what is the resistance of an iron 
 wire ^ in. in diameter and 5 miles long? 
 
 247. If 50 volts force 54.8 amperes of electrical current 
 through a lamp, what is the resistance? 
 
 248. If the resistance of an electric lamp is 2.8 ohms 
 when a current of 10 amperes is passing through it, what 
 is the voltage ? 
 
 249. Five arc lamps on a circuit have each a resistance 
 of 2.35 ohms. The resistance of the wires is 1.2 ohms and 
 of the dynamo is 0.75 ohm. What voltage is required to 
 send a current of 15 amperes through the circuit ? 
 
TABLES. 
 
 399 
 
 Table of Specific Gravities. 
 
 NAME OF SUBSTANCE. 
 
 SPECIFIC GRAVITY 
 
 WEIGHT OF CUBIC 
 FOOT IN POUNDS. 
 
 Acid, acetic, strongest . 
 
 11 nitric, " 
 
 11 sulphuric, " 
 
 Air 
 
 Alcohol, pure .... 
 11 of commerce . 
 Aluminum, lightest . . 
 Brass, average .... 
 Brick, common . . . 
 
 " pressed .... 
 
 Cedar, dry 
 
 Cherry, dry, average . . 
 Coal, bituminous, average 
 
 " anthracite, . . . 
 Copper, cast .... 
 
 Cork 
 
 Glass, average .... 
 Gold, pure, cast . . . 
 Granite, average . . . 
 Gypsum, heaviest . . . 
 
 Hydrogen 
 
 Ice 
 
 Iron, cast, average . . 
 
 " wrought, average . 
 Lead, cast 
 
 " white 
 
 Lignum vitse .... 
 
 Lime 
 
 Marble, average . . . 
 Mercury ...... 
 
 Milk^ 
 
 Nitrogen 
 
 Oil, linseed ..... 
 Oak, dry white . . . 
 
 Oxygen 
 
 Pine, dry white . . . 
 Platinum, hammered 
 
 Salt 
 
 Sand, average .... 
 Silver, hammered . . . 
 Slate, lightest .... 
 
 Steel 
 
 Tin, cast 
 
 Water, sea 
 
 Zinc 
 
 1.062 
 
 1.583 
 
 1.841 
 
 0.001292 
 
 0.792 
 
 0.834 
 
 2.560 
 
 7.611 
 
 2.000 
 
 2.400 
 .350 to 0.600 
 
 0.715 
 
 1.250 
 
 1.500 
 
 8.788 
 
 0.240 
 
 2.760 
 19.258 
 
 2.720 
 
 2.288 
 
 0.0000893 
 
 0.930 
 
 7.150 
 
 7.770 
 11.350 
 
 7.235 
 
 1.333 
 
 0.804 
 
 2.720 
 13.580 
 
 1.032 
 
 0.001250 
 
 0.940 
 
 0.830 
 
 0.001429 
 
 0.400 
 21.841 
 
 2.130 
 
 1.650 
 10.500 
 
 2.110 
 
 7.816 
 
 7.291 
 
 1.026 
 
 7.190 
 
 66.4 
 ' 98.94 
 115. 
 0.0808 
 
 49.43 
 
 52.1 
 160. 
 475.7 
 125. 
 150. 
 22 to 37.5 
 
 44.7 
 
 78.1 
 
 93.75 
 549.3 
 
 15. 
 
 172.5 
 
 1204. 
 
 170. 
 
 143. 
 
 0.00558 
 
 68.1 
 447. 
 486. 
 709.4 
 452. 
 
 83.3 
 
 50.25 
 170. 
 849. 
 
 64.5 
 0.0781 
 
 58.8 
 
 51.9 
 0.0893 
 
 25. 
 1365. 
 133. 
 103. 
 656.2 
 132. 
 488.6 
 456. 
 
 64.13 
 449.4 
 
400 
 
 TABLES. 
 
 Table of Selected Constants. 
 
 Diagonal of square in terms of side 
 Side in terms of diagonal . . . 
 Cube root of 2 ...... . 
 
 Cube root of £ 
 
 Square root of 20 
 
 Square root of 3 
 
 Square root of £ 
 
 Square root of 30 
 
 Square root of 5 
 
 Extreme and mean ratio . . . 
 Extreme and mean ratio . . . 
 
 NUMBER. LOGARITHM. 
 
 1.4142136 
 0.7071068 
 1.2599210 
 0.7937002 
 4.4721359 
 1.7320508 
 0.5773503 
 5.4772256 
 2.2360680 
 0.3819660 
 0.6180340 
 
 0.1505150 
 9.8494850 
 0.1003433 
 9.8996567 
 0.6505150 
 0.2385606 
 9.7614394 
 0.7385606 
 0.3994850 
 9.5820247 
 9.7910124 
 
 Circumference in terms of diameter 
 Diameter in terms of circumference 
 Circle in terms of diameter square 
 Square in terms of the circle . . 
 Sphere in terms of diameter cube 
 Cube in terms of sphere .... 
 
 3.1415927 
 0.3183099 
 0.7853982 
 1.2732395 
 0.5235988 
 1.9091406 
 
 0.4971499 
 9.5028501 
 9.8950899 
 0.1049101 
 9.7189986 
 0.2810014 
 
 One meter in feet . . . 
 One foot in meters . . 
 One meter in yards . . 
 One yard in meters . . 
 One centimeter in inches 
 One inch in centimeters . 
 One mile in kilometers . 
 One kilometer in miles . 
 
 3.2808693 
 0.3047973 
 1.0936231 
 0.9143918 
 0.3937043 
 2.5399772 
 1.6093295 
 0.6213768 
 
 0.5159889 
 9.4840111 
 0.0388677 
 9.9611323 
 9.5951702 
 0.4048298 
 0.2066450 
 9.7933550 
 
 One square foot in square meters 
 One square meter in square feet . 
 One sq. inch in square centimeters 
 One square centimeter in sq. inches 
 
 One acre in hektars 
 
 One hektar in acres 
 
 0.0929014 
 10.7641036 
 
 6.4514847 
 0.15500308 
 
 0.4046784 
 
 2.4710982 
 
 8.9680221 
 1.0319779 
 0.8096596 
 9.1903404 
 9.6071100 
 0.3928900 
 
 One cubic foot in cubic centimeters 
 One cubic meter in cubic feet . . 
 One U.S. bushel in cubic inches . 
 One U.S. bushel in cu. centimeters 
 One cubic foot in U.S. bushels 
 One U.S. gallon in liters . . . 
 One liter in U.S. gallons . . . 
 
 28,316.085 
 
 35.315017 
 
 2150.42 
 
 35,238.117 
 
 0.8035640 
 
 3.7853103 
 
 0.2641791 
 
 4.4520332 
 1.5479668 
 3.3325232 
 4.5470127 
 9.9050204 
 0.5781015 
 9.4218985 
 
 One gram in grains . 
 One grain in grams . 
 One pound in kilograms 
 One kilogram in pounds 
 
 15.4323487 
 0.0647990 
 0.4535927 
 2.2046212 
 
 1.1884320 
 8.8115680 
 9.6566660 
 0.3433340 
 
VB 17443 
 
 M306073 
 
 W4 £3 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY