UC-NRLF (OMPLETE RITHMETIG t Ktwnmrr^ tn mm ii i uv i i.j'rv i f ii .r'i i i i ii /^ Digitized by the Internet Archive ii^ 2008 with funding from Microsoft Corporation http://www.archive.org/details/completearithmetOOpeckrich ^rofesao-T' of Mathematics and A at-ronomy in Columbia College, and of Mechanics in the School of Mines. -^r^TOflWMW NEWYORK,CHiaftG[0 AND New ORIiEIMS. ?FBLiSHERS' NOTICE. PECK'S MATHEMATICAL SERIES. CONCISE, CONSECUTIVE, AND COMPLETE. I.— First Lessons in Numbers. II.— Manual of Practical Arithmetic. III.— Complete Arithmetic. VI.— Manual of Algebra. V. — Manual of Geometry and Conic Sections. VI.— Treatise on Analytical Geometry. VII.— Differential and Integral Calculus. VIII.— Elementary Mechanics ^without the Calculus). IX.— Elements of Mechanics (with the Calculus). 'NoTE.^ Teachers and others discovering errors in any of the above works wiU confer a famr ly communicating tTiem to us. Copyright, 1877, by William G. Peck. PREFACE. rriHE object of the following work is to present, in -^ logical order and within moderate limits, all the fundamental principles of arithmetic, together with their most important applications to the wants of the student, the artisan, and the man of business. It commences with the simplest elements, and pro- gresses by natural steps to the highest and most complex operations. All superfluous matter has been omitted, and great care has been exercised to avoid needless mul- tiplicity of cases and rules; but in no instance has any essential principle been omitted or unnecessarily ab- breviated. It is believed that the definitions are plain and concise ; that the principles are stated clearly and accurately ; that the demonstrations are full and complete ; that the rules are perspicuous and comprehensive ; and finally, that every branch of the subject is amply illustrated by well-graded examples and problems. The order of logical development is thought to be simple and practical, the meth(^ of treating successive IV PREFACE. subjects being uniform and essentially as follows : 1°. All necessary definitions are given ; 2°. A few mental exer- cises are then introduced, being so worded and so ar- ranged as to lead the pupil to a knowledge of the fundamental principles of the subject under considera- tion; 3°. The principles thus developed are used in demonstrating the required rule ; and 4°, The rule is then illustrated and enforced by a sufficient number of graded examples and problems. The Author takes this opportunity to thank the many teachers who have aided him by valuable suggestions and criticisms. Columbia College, Sept. U, 1877. CONTENTS NOTATION AND NUMERA- TION, r^g^ Formation of Numbers 1 Classification of Numbers 8 Places of Fij?ures 9 Orders of Units 9 General Principles 10 Periods of Figures 12 Roman Notation 15 FUNDAMENTAL OPERA- TIONS. I Addition H Explanation of Signs 18 Principles of Addition 20 Operation of Addition 20 II. Subtraction 28 Explanation of Signs 29 Principles of Subtraction 29 Operation of Subtraction 30 III. MtlXTIFLICATION 37 Sign of Multiplication 38 Eflfect of annexing Ciphers 41 Principles of Multiplication 41 Additional Definitions 44 IV. Division 48 Methods of indicating Division 50 Object of Division 51 Principles of Division 51 Short Division 52 Long Division 55 Contractions In Division 58 V. Factoring AND Cancelling... 65 Principles of Factoring 66 Operation of Factoring 66 Cancellation 68 Object and Principles of Can- cellation 68 PAGE VI. Greatest Common Divisor AND Least Common Multi- ple 72 Methods and Principles 73 Method by Factors 73 Additional Principle 74 Method by Continued Division. 75 Least Common Multiple 76 Definitions 76 Operation of Least Common Multiple 77 FRACTIONS. I. Common Fractions 80 Reduction of Fractions 84 Addition of Fractions 92 Subtraction of Fractions 96 Multiplication of Fractions .... 99 Division of Fractions 104 Contractions in Multiplication and Division 108 II. Decimal Fractions. Decimals, and the Decimal Point Ill Notation of Decimals Ill Numeration of Decimals 113 Decimal Currency 114 ReductionofCommonFractions to Decimals 115 Approximate Results 116 Addition of Decimals 117 Subtraction of Decimals 120 Multiplication of Decimals 124 Division of Decimals 128 in. Contractions and Business Operations 134 Aliquot Parts 134 Bills and Accounts 137 Balancing Accounts 139 71 OONTEITTS. COMPOUND NUMBERS. I. Definitions and Tables. page Scales of Compound Numbers . 141 Tables of Currency 142 Tables of Weight 143 Tables of Time 145 Measures of Length 146 Measures of Surface 147 Measures of Volume and Ca- pacity 149 Angular Measure & Longitude. 151 Metric System 153 II. Reduction. Reduction Descending 157 Reduction Ascending 160 ni. Addition of Compound Num- bers 167 IV. Subtraction op Compound Numbers , — 171 V. Multiplication of Compound Numbers 177 VI. Division of Compound Num- bers 182 Longitude and Time 188 PERCENTAGE AND ITS APPLICATIONS. 1. Percentage 190 Principles of 192 IT. Commission 199 III. Insurance 203 IV. Profit AND Loss 206 V. Taxes 209 On Property and Polls 209 Method of laying a Tax 210 VI. Simple Interest 211 Annual Interest 222 Notes 223 Partial Payments 223 Methods of Settlement 224 Supreme Court Rule 224 Mercantile Rule 226 VII. Compound Interest 228 VIII. Discount Commercial Discount 231 Present Value and True Dis- count 232 Banks and Bank Discount 234 Method of Discounting a Note. 235 IX. Stocks and Bonds 238 United States Bonds 239 PAGX X. Exchange 242 Drafts 242 Acceptances 243 Domestic Exchange 244 Foreign Exchange 246 XI. Equation of Payments 248 Equation of Accounts 253 Cash and Interest Balance 255 XII. Custom House Business — 256 RATIO AND PROPORTION. L Ratio 259 Methods of Expressing 260 II. Proportion 261 Solution. Principles used 262 Rule of Three 263 Distributive Proportion and Partnership 268 Analysis 269 POWERS, ROOTS, AND PROGRESSIONS. L Powers 275 Involution 275 n. Roots 276 Square Root 276 Cube Root 280 ni. Progressions 282 Arithmetical Progression 282 Geometrical Progression 285 MENSURATION. Polygons Triangles Property of Right-angled Tri- angles Length of a Circumference Area of a Triangle Area of a Parallelogram Area of a Trapezoid Areaofa Circle Surface of a Sphere Volume, or Content, etc Content of a Pyramid Content of a Sphere Board Measure Timber Measure ■ Method of Duodecimals 291 293 293 294 294 295 296 296 297 301 MISCELLANEOUS EX A 31- PLES 306 DEFINITIONS. 1, A Unit is a single thing ; as, one pound, one foot, one day. 2. A Number is a unit, or a collection of units; as, one pound, three days, five feet. 3. Arithmetic is the science of numbers. It treats of the properties and relations of numbers, and of the methods of computation by means of numbers. Note. — In what follows, the expressions 1°, 2°, 3°, etc., are read first, second, thirds etc. FORMATION OF NUMBERS. 4. — 1°. Numbers from one to ten are formed by collect- ing simple units, or ones. A single unit is called one ; one and one more are two ; two and one more are three ; three and one more are four ; and so on, to ten. 2°. Numbers between ten and one hundred are formed by collecting te?is and ones. One ten and one are eleven ; one te7i and two are twelve ; one ten and three are thirteen ; and so on to two tens, or twenty. Two tens and 07ie are twenty-one ; two tens and ttvo are twenty-two ; and so on to three 8 KOTA^TIOK AKD NUMERATION'. , ^ Jeiis qx^thxTty, „ Four te7is are forty ; five tens are fifty ; "c'r-'araTd^© (M ikf ■fe'/i^'tenSf or one hundred. The interme- diate numbers between thirty and forty, forty and fifty, and so on, are formed in the same manner as those be- tween twenty and thirty. 3°. Numbers between one hundred and one thousand are formed by collecting hundreds, tens, and ones; numbers between one thousand and te7i thousand are formed by collecting thousands, hundreds, tens, and ones ; and so on, indefinitely. Numbers formed by collecting ones, in the manner just explained, are called integers ; they are also called integral, or whole num- bers. CLASSIFICATION OF NUMBERS. 5. Numbers are divided into two classes, abstract and deno7ni7iate. An Abstract Number is a number whose unit is not named; as, five, seven, eleve7i, A Denominate Number is a number whose unit is named ; as, three pounds, six 7niles, seve7i 7nonths, Denominate numbei's are sometimes called concrete numbers. A denominate number may be either si7nple or compound. It is a Simple Number when all the units of the col- lection are of the same name or denomination; as, eight yards, eleven ounces, five feet. It is a Compound Number when all the units of the collection are not of the same denomination ; ns, tlweefeet mid six inches, four hows and twenty mimites, ttvo pounds and eleven ounces. Note. — All integers are simple numbers. KOTATIOK AND NUMEKATION. 9 NOTATION AND NUMERATION, 6. Notation is the method of writing numbers by means of figures, or of letters. Numeration is the method of reading written numbers. Fl GU RES. 7. The following figures are used in the common, or decimal system of notation : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. naught, one, two, three, four, five, bIx, seven, eight, nine. These figures, taken separately, are called digits. The first one, named naught, is also called a cipher, or zero ; it stands for no number. The remaining ones are called significant figures ; they stand for the numbers written below them. Figures are not numbers, but it will often be convenient to speak of them as such ; in these cases, it is to be understood that we refer to the numbers which the figures represent. PLACES OF FIGU RES. 8. If several figures are written in a line, the one on the right is said to stand in the first place, the one next to the right stands in the second place, the one next to it in the third place, and so on. Thus, in the expression 3784, 4 stands in the first place, 8 in the second place > 7 in the third place, and 3 in ihQ fourth place, ORDERS OF UN ITS. 9. The number one is called a unit of the first order ; the number ten, regarded as a collectictn of ones, is called a unit of the second order ; one hundred, regarded as a collection of tens, is called a unit of the third order ; 10 NOTATIOIT AKD NUMERATION. and so on indefinitely, the unit of each succeeding order being ten times that of the next lower one. The unit of the fourth order is one thousand ; the unit of the fifth order is ten thousand ; the unit of the sixth order is one hundred thousand ; the unit of the seventh order is one million ; and so on. Units whose order is not named are supposed to be units of the first order, or ones. GENERAL PRINCIPLES OF NOTATION AND NUMERATION. 10. Numbers are written and read in accordance with the following principles : 1°. The same digit always represents the same number of units. 2°. The order of units represented is denoted hy the place in tohich the digit stands. Z°. A cipher standing in any place shows that the num- ber contains no units of that order. Thus, the expression 777 denotes 7 hundreds, 7 tens, and 7 units, that is, it stands for seve7i hundred and seventy- seven. In like manner, the expression 507 denotes 5 hun- dreds, tens, and 7 units ^ that is, it stands iorfive hundred and seven. The expression 240 stands for two hundred and forty. Note. — Places of figures and orders of units are counted from right to left, but numbers are written and read from left to right. EXAMPLES IN NOTATION AND NUMERATION. 11. Any number less than one thousand may be written by the following RULE. Begin at the left and write the figures that denote the hundreds, tens, and units, in their proper order. ITOTATIOK A^D KUMERATIOK. 11 E55:amples. Write the following numbers : 1. Fifty seven. ' Ans. 57. 2. Ninety four. Ans. 94. 3. One hundred and sixty nine. Ans, 169. 4. Nine hundred and fourteen. A7is. 914. 5. Three hundred and sixty. 15. Two hundred and nine. 6. Two hundred and seven. 16. Five hundred and fifty. 7. Nine hundred and eight. 17. Six hundred and nine. 8. Seven hundred and seven. 18. Ninety seven. 9. Nine hundred and ninety. 19. One hundred and ten. 10. One hundred and twelve. 20. Two hundred and six. 11. Three hundred and four. 21. Six hundred and sixty. 12. Eight hundred and sixty. 22. Four hundred and five. 13. Eight hundred and four. 23. Seven hundred and six. 14. Three hundred and eight. 24. Six hundred and seven. Any written number less than one thouspnd may be read by the following RULE. Begin at the left and read the hundreds, tens, and units, in their order, translating figures into words. EXAMPLES. Read the following numbers : 1. 29. Ans. Twenty nine. 2. 107. Ans. One hundred and seven. 3. 118. Ans. One hundred and eighteen. 4. 506. 8. 270. 12. 186. 16. 999. 5. 670. 9. 809. 13. 204. 17. 400. 6. 977. 10. 422. 14. 309. 18. 207. 7. 835. 11. 109. 15. 470. 19. 554. 12 KOTATION AKD NUMERATION. Note. — Before reading a number, let the pupil name the unit of each order of figures, beginning at the right ; thus, uniiSy tens, hun- dreds. ♦ From what precedes, we see that notation is the operation of translating numbers from words into figures, and that numeration is the operation of translating numbers from figures into words. The digits of a number indicate natural parts into which it may be separated. Thus, 986 may be separated into the three parts, 900, 80, and 6, each of which has its own unit. PERIODS OF FIGURES. 13. Written numbers, containing more than three fig- ures, are separated into periods of three figures each, be- ginning at the right hand; the left-hand period may contain less than three figures. The first period, counting from the right, is called the period of units; the second is called the period of thousands ; the third is called the period of millions ; and so on, as shown in the following table, called THE NUMERATION TABLE. ^Hods\ Trillions, Billions, Millions, Thousands, Units. «M «1H «M <♦-( o o o o 02 QQ CQ CC DQ •13 ti.i'O t»_ind ci_iT3 54_('^ g«H*S g'HO g^O g-gO 2 'g®5 'gS^ '§"■2 % ^ ^ a«^^ Periods. . . 3 i 7? 8 3 2, 4 i 5, 8 i 6, 7 8 3 The number written above is read, 317 trillions, 832 billions, 415 millions, 816 tlioiisands, 783. The unit of the first period is the simple unit one ; the unit of the second period is one thousand ; the unit of the third period is a thousand times one thousand, or one million ; and so on, as indicated in the table, the unit of NOTATION AKD KUMERATION. 13 each period being equal to a thousand times that of the next lower one. The table may be continued to any desired extent ; the units of the next succeeding periods are quadrillions, quintillions, sextillions, septillions, octillions, etc. Every period, except tlie left-hand one, must contain three figures, but they may all be ciphers. Periods that contain three figures are said to be complete. Periods are written and read as explained in the last article. In writing, we make them all complete, except the one on the left ; in reading, we name the unit of each, except the one on the right. ADDITIONAL EXAMPLES. 13. Any number whatever may be written by the fol- lowing RULE. Begin at the left and write each period in order, separating it from the following one by a comma. Write the following numbers : 1. Ten thousand, two hundred and six. Ans. 10,206. 2. One hundred and fourteen thousand, eight hundred and seventy nine. Ans. 114,879. 3. Seven hundred and fifty thousand, three hundred and eighty nine. Ans. 750,389. 4. Nine hundred thousand, three hundred and fifty. 5. Six million, one hundred and sixty nine thousand, four hundred and thirty seven. 6. Seventy six million, four hundred thousand, one hundred. 7. 22 billion, 103 million, 576 thousand, 102. 8. 102 triUion, 125 milUon, 403. 9. 8 trillion, 7 billion, and 76. 14 NOTATlOIf AND KUMERATION. 10. 41 quadrillion, 817 trillion, 217 billion. ^ 11. 107 quintillion, 200 million, 757 thousand, 365. 12. 14 billion, 74 million, 231 thousand, and 5. Any written number may be read by the following RULE. I. Begin at the right and point it off into periods of three figures each ; the left-hand period may con- tain less than three figures. II. Begin at the left and read the periods in their order, naming the unit of each, except that on the right. Note. — After the number is pointed oflf, the pupil should name each period, beginning at the right ; thus, units, thousands, miUionSy UUions, etc. EXAM PLES. Eead the following numbers : 1. 104217. Arts. One hundred and four thousand, two hundred and seventeen. 2. 2304516. Ans. Two million, three hundred and four thousand, five hundred and sixteen. 3. 1001010. Ans, One million, one thousand, and ten. 4. 825314715. Ans. 825 million, 314 thousand, 715. 5. 7416. 13. 6003021715. 6. 23562. 14. 4785003298. 7. 475437. 15. 12303492816. 8. 284871. 16. 117723326419. 9. 1284576. 17. 8843412956. 10. 4534218. 18. 543521798612. 11. 88334172. 19. 254321496. 12. 24137652. 20. 1546973200849. NOTATION^ AND NUMERATION. 15 ROMAN N OTATION. 14. Roman Notation is the method of expressing numbers by letters. The letters used and the values they express are shown below : Letters ... I, V, X, L, C, D, M. Values, ... 1, 5, 10, 50, 100, 500, 1000. Other numbers are expressed by combining these letters according to the following principles : 1°. If a letter is repeated, the number that it denotes is repeated. 2°. If a letter that denotes a less number is written after 07ie that denotes a greater nuinber, the value of the latter is increased by that of the former. 3°. If a letter that deiiotes a less number is written before one that denotes a greater number, the value of the latter is diminished by that of the former. If a letter that denotes a less number is written between two that denote greater numbers, it diminishes the latter, but does not affect the former. The method of applying these principles is shown in the follomng TABLE I denotes 1 XI denotes 11 XXX denotes 30 CCCC denotes 400 II " 2 XII • 13 XL " 40 D " 500 III " 3 XIII ' 13 L " . 50 DC " 600 I^ u 4 XIV ' 14 LX -' 60 DCC " 700 V «' 5 XV ' 15 LXX " 70 DCCC « 800 VI " 6 XVT 16 LXXX " 80 DCCCC " 900 VII " 7 XVII ' 17 XC " 90 M " 1000 VIII " 8 XVIII • 18 C " 100 MM " 2000 IX •' 9 XIX ' 19 CC " 200 MDQCCLXXV de- X - 10 XX ' 20 CCC " 300 notes 1875. 16 NOTATION AKD NUMERATION. EXAM PLES. Read the following numbers : 1. XXXIX. 5. MMDXXXIL 2. XCLVIII. 6. DCOXLIII. 3. MDOXIX. 7. DCCOCXC. 4. DCCLIX. 8. CCCLXXXIII. Write the following numbers by the Koman method : 9. 42. 13. 2,940. 10. 84. 14. 3,317. 11. 119. 15. 2,150. 12. 1,214. 16. 1,555. Note. — A dash over a number written in Roman numerals in- creases the number 1,000 times. Thus, XXX stands for 30,000. revie>a;' questions. (1.) What is a unit? Example ? (2.) What is a number? Ex- ample? (3.) What is arithmetic ? What does it treat of ? (4.) Ex- plain the formation of numbers from one to ten; from ten to one hundred, etc. What is an integer, or whole number ? (5.) How are numbers classified ? What is an abstract number? A denominate number ? When is a denominate number simple ? When compound ? (6.) Define notation. Numeration. (7.) Name the ten digits. What other names has the figure naught ? Which are significant figures ? (8.) Explain what is meant by the place of a figure ? (9.) Explain what is meant by orders of units, and give the names of the orders up to the seventh. (lO.) What are the three principles of decimal notation ? How are places and orders counted, and how are numbers written and read ? (11.) Give the rules for writing and reading any number less than 1000. '(l^*) What are periods of figures? Name the units of the first six periods. (13. ) Give the general rules for notation and numeration. (14.) What is Roman notation? Explain fche method of writing numbers in this system. I. ADDITION. DEFINITIONS. 15. Addition is the operation of find- ing the sum of two or more numbers. 16. The Sum of two or more numbers is a number that contains as many units as the given num- bers taken together. Thus, 5 days is the sum of 3 days and 2 days. The numbers added, and their sum, must be similar. 17. Similar Numbers are those that have the same unit. Thus, 3 yards and 7 yards are similar, but 3 yards and 7 days are not similar. Abstract numbers are always similar, because they have the same unit. MENTAL EXERCISES. 1. John has 4 apples and James has 5 apples; how many have both ? How many apples are 4 apples and 5 apples f How many are 4 and 5 ? 5 and 4 ? 2. Frank had 8 marUes and Peter gave him 7 more ; how many had he then ? What is the sum of 8 marUes and 7 marbles? What is the sum of 8 and 7? of 7 and 8 ? 3. An arithmetic class consists of 5 loys and 7^*Vk; 18 SIMPLE NUMBERS. how many pupils are there in the class ? How many are 5 and 7? 7 and 5 ? Explanation. — Because both "boys and girls are pupils, the num- bers to be added are similar, although they appear to be dissimilar. 4. What is the sum of 5 dollars, 6 dollars, and 9 dollars? What is the sum of 5, 6, and 9 ? of 6, 9, and 5 ? of 9, 6, and 5 ? 5. A farmer has 6 oxen, 9 cows, and 8 calves; how many cattle has he ? How many are 6, 9, and 8 ? 6. What is the sum of 10, 12, 8, and 4 ? of 4, 8, 10, and 12? EXPLANATION OF SIGNS. 18. The sign of addition, +? is called plus ; when placed between two numbers, it shows that the second is to be added to the first. Thus, the expression 4 + 5 shows that 5 is to be added to 4. The sign of equality, =, indicates that the expres- sions between which it is placed are equal to each other. Thus, 4 + 5 = 9, indicates that the sum of 4 and 5 is equal to 9. An expression of equality between numbers is called an Equation ; the part on the left of the sign of equality is the first member and the part on the right is the second member. Thus, in the equation 9 + 8 = 17 the part 9 + 8 is the first member and the part 17 is the seco7id member, MENTAL EXERCISBS. 1. 4 + 8 + 7 = how many? Note. — Let the pupil supply the second member and then read the equation. ADDITION. 19 2. 4 + 5 + 7 + ^ + 1 = how many ? 3. 7 7nen + 4 men + 3 men + 9 wew = how many men? 4. 3 dollars + 2 dollars + 9 dollars = how many t?o^ 5. 6 oxen + 12 cozi;^ + 7 ca^ve5 = how many cattle ? 6. 6 + 9 + 4 + 3 + 9 + 2 + 1 = how many ? 7. 4 + 9 + 5 + 6 + 3 + 4 + 8=? 8. The sum of 4, 9, 3, 2, 7, 6, and 5 equals how many ? 9. 4 ?/«r^s + 9 yards + 11 ^/o^r^s + 8 yards = ? Note. — The signs of interrogation in examples 7 and 9 indicate that the second members are to be supplied by the pupil. 10. 14. ft. + dft. + 7/^. + 9/^. + 10/^. = ? 11. 9 + 9 + 7 + 3 + 6 + 8 + 7 + 4=? Let the pupil add the following columns: (12.) (13.) (14.) {Ih.) (16.) (17.) (18.) (19.) (20.) (21.) 6 7 3 9 5 4 7 6 7 3 7 4 8 2 1 2 8 4 2 6 8 3 2 8 8 2 7 4 4 7 5 9 8 3 2 4 6 5 3 3 4 8 3 9 9 4 2 5 5 4 2 4 5 6 3 5 3 7 7 9 3 7 4 5 7 5 4 8 2 1 8 5 9 4 6 6 1 9 9 8 1 2 6 7 4 9 5 2 6 2 Note. — The operation of adding a column of figures should be abbreviated by simply naming the result of each step. Thus, in example 12, the pupil should say 1, 9, 13, 14, 18, 23, 31, 38, 44. The exercise may be varied by adding each column from top to bottom; also by adding the lines horizontally both forward and backward. 20 SIMPLE NUMBERS. PRINCIPLES OF ADDITION. 19. The operation of addition depends on the follow- ing principles : 1°. Any number is equal to the sum of all its parts. 2°. The sum of two or more nu7nbers is equal to the sum of all their parts. OPERATION OF ADDITION. 20. Let it be required to find the sum of 564, 783, and 688. Explanation. — Having written the numbers so that Operation. units of the same order stand in the same column, we 564 begin at the right and add each column separately. 733 The sum of 8, 3, and 4, is 15 units, that is, 1 ten and 5 nnn units ; we write the 5 in the column of units and carry ■ forward the 1 and add it to the column of tens. The 2035 sum of the tens, thus increased, 1 + 8 + 8 + 6, is 23 tens, that is, 2 hundreds and 3 tens ; we write the 3 in the column of tens and carryforward the 2 and add it to the column of hundreds. The sum of the hundreds, thus increased, 3 + 6 + 7 + 5, is 20 hun- dreds, that is, 2 thousands, and hundreds; as this is the last col- umn, we set down the entire sum. The resulting number, 2,035, is the required sum, because it is the sum of all the units, tens, and hundreds of the given numbers (Art. 1.9). In like manner other numbers may be added ; hence, we have the following RULE. /. Write the nuwibers so that units of the same order shall stand in the same column. II. Add the column of units ; set down the sim- ple units of the sum, and if there are any tens, cannj them forward and add them to the next column, III. Add the column, of tens / set down th^ simply ADDITION. 21 tens, and if there are any hundreds, cam^y them forward and add them to the next column. IV. Continue this operation till all the columns have been added. Set down the entire sum of the last column. EXAMPLES. Perform the following additions : (1.) (2.) (3.) (4.) 315 29 215 8261 423 814 27 3042 719 302 891 171 Sum, 1457 1145 1133 11474 Tlie rule holds good for all simple numbers, whether abstract or denominate. (5.) (6.) (7.) (8.) 451 feet. 365 days, 187 pounds. 124 things. 817 feet. 821 days. 203 pounds. 287 things. 302 feet. 900 days. 866 pounds. 59 things. 917 feet. 76 days. 771 pounds. 803 things. 2487 feet. 2162 days. 2027 pounds. 1273 things. Proof of Addition". — Perform, the opei^ation hi/ commencing at the top of each column, and adding downward. The sum should- he the same as hefore. Note. — Every operation in addition should be proved. (9.) (10.) (11.) (12.) (13.) 9,102 8,760 25,678 87 feet. 62,743 479 325 3,002 236 feet. 4,321 73 512 21,001 1,443 feet. 78,731 810 786 715 2,010 feet. 1,239 4,312 1,420 1,630 7,818 feet. 4,241 23 SIMPLE NUMBERS. (14.) (15.) (16.) (17.) (18.) 27 yards. 7,478 days. 117,064 2,571 2,476 135 yards. 423 days. 92,973 1,701 7,884 7,271 yards. 79 ^fl^«/s. 827,569 973 3,349 185 ?/fl!rcfe. 8,102 tZ««/s. 1,351 2,045 5,876 19. Add 7,384; 326; 6,780; and 57. Ans. 14,547. 20. Add 6,740; 9,745; 5,769; 8,031; 6,543; 2,052; and 9,999. Ans. 48,879. 21. Add 89; 4,500; 423; 2,024; 5,408; 6'0,546; 9,401. 22. Add 83,746 2/«r^5 ; ^,V(^ yards ; Q92,o7'7 yards ; 456 yards ; and 7 yards. 23. Add 935,473 ^o?Z«rs; 2G2 dollars ; 13, S9'^ dollars ; 598,453 dollars ; 25 dollars ; 3,734 dollars ; and 72,405 dollars. The sign $ written before a number signifies dollars ; thus, the expression $120 is read 120 dollars. 24. Find the sum of $93,180; $279; $8,711; $371,800; $65 ; and $212,818. 25. Add 3,415 ; 17,382 ; 81,845 ; 162,345 ; and 8,342. 26. Add 8,492 /ee^; U,692 feet j 112,897 feet ; and 117,712 feet. 27. Add $8,842; $31,887; $113,214; and $887,319. 28. Add 385,842; 112,817; 32,413; and 33,335. 29. Add $88,141; $32,314; $141,003; and $89,947. 30. Add 114,312; 87,808; 3,214; 896; and 87. 31. Add 8,730 ; 3,021 ; 785 ; 879 ; and 92. 32. Add $87; $78; $114; $289; $176; and $95. 33. ^2,3Uyds.; 119,3^2 yds.; S,9Q2 yds. ; 8,962 yds. 34. Add 17,439; 410,864; 842,317; 345,876; 79,884; and 18,719. ADDITION. 23 35. Add 714,312; 182,416; 312,867; 382,843; 79,816 and 43,115. 36. Find the sum of 3,345,816; 2,882,314; 387,892 4,381,500 ; 2,874,316 ; and 887,342. 37. Add 188,841; 362,817; 411,217; 336,425; 814,316 and 45,554. 38. Add 214,333; 286,329; 851,426; 303,249; 12,456 17,324; and 22,404. 39. Add 3,329,941 ; 187,693 ; 821,436 ; 2,227,438 132,314 ; and 283,304. 40. Add 193,391; 4,180,280; 7,814,312; 88,430; 92,872 and 64,428. 41. Add 112,847; 186,320; 662,641; 3,400,300 2,810,000; and 749,209. 42. Add 682,817; 336,336; 4,150,209; 2,390,374; and 86,810,304. Dollars and cents may be written together, the cents being sep- arated from the dollars by a point. Thus, the expression $17.84 is read 17 dollars and 84 cents. Dollars and cents may be added like simple numbers. In writing them down, the separating points must stand in the same column. (43.) (44.) (45.) (46.) (47.) $18.73 15.83 1186.40 1413.30 $2,234.75 23.47 10.19 75.75 325.15 3,821.62 15.62 27.03 37.18 414.82 911.94 7.91 11.94 201.92 97.45 89.69 112.13 203.07 184.42 111.32 10,312.41 648.21 211.46 36.35 202.16 9,102.70 73.19 305.24 41.15 113.27 25,444.33 19.06 802.41 72.27 814.42 42,829.77 35.62 111.37 94.79 316.81 11,312.48 24 SIMPLE NUMBERS. 48. What is the sum of $8,311.35, 127,494.62, $143,596.22, $155,463.79, $292,986.48, $382,811.67, $482,884.20, and $919,902.20 ? 49. Add $2,863,747.25, $3,894,511.82, $8,818,416.20, $215,714,381.46, $747,719.87, and $59,107,411.28. 50. Find the sum of $53.42, $881,16, $416.49, $1,381.40, $88.88, $210.29, $6.49, $511.11, $16.84, and $2,256.00. PRACTICAL PROBLEMS. 21. A Problem is a question proposed that requires an answer. The operation of finding the answer is called the solution of the problem. Solve the following problems : 1. A farmer sold a span of horses for $318, two pairs of oxen for $420, and six cows for $290 ; how much did he receive ? A 7is. $1,028. 2. A man bought a house for $24,500, paid $1,675 for repairs, $3,140 for furniture, $375 for taxes, and then sold the whole for $2,155 more than the cost; what did he receive? Ans. $31,845. Abbreviations. — In the following problems, lbs. stands for pounds ; ft. for feet ; yds. for yards ; and hu. for hunhels. Other abbreviations will be explained in their proper places. 3. A wagon is loaded with 5 boxes of goods ; the first weighs 473 Ihs., the second 392 Ihs., the third 479 Ihs., the fourth 1,217 Ihs., and the fifth 376 lbs. ; what is the weight of the entire load ? Ans. 2,937 Ihs. 4. The first car of a freight train contains 8,117 lbs. the second 11,819 lbs., the third 9,156 lbs., the fourth 8,884 /i.9., the fifth 10,398 lbs., and the sixth 9,982 lbs. ; how many pounds are there in all ? Ans. 58,356 lbs. ADDITION. 25 5. A farm contains 79 aci es of woodland, 63 of pasture land, 50 of meadow land, and 73 of arable land; how many acres in the farm ? Ans. 265 acres. 6. A factory turned out 702 yds. of cloth on Monday, 1,023 yds. on Tuesday, 1,107 yds. on Wednesday, 997 yds. on Thursday, 910 yds. on Friday, and 1,045 yds. on Satur- day ; how many yards did it turn out in the week ? Ans. 5, '7S4: yds. 7. A merchant owes A. $2,160, B. $3,879, C. $813, D. $955, and E. $1,796 ; how much does he owe in all ? Ans. $9,603. 8. A farmer has 12 horses, 16 more oxen than horses, 42 more cows than horses and oxen together, and 22 more calves than oxen and cows together ; how many in all ? 9. A gentleman built a house ; the carpenter work cost him $4,285, the masonry $3,950, the plumbing and grates $2,783, the painting $1,975, and miscellaneous work $3,992 ; what was the entire cost ? 10. A merchant buys 56,250 bu. of corn, 30,211 bu. of oats, 18,312 bu. of barley, 2,197 bu. of wheat, and 713 bti. of rye ; how many bushels did he buy altogether ? 11. The distance from Albany to New York is 144 miles, from New York to Philadelphia 90 miles, from Philadelphia to Baltimore 98 miles, from Baltimore to AVashington 38 7niles, and from Washington to Norfolk 217 7niles ; how far is it from Albany to Norfolk by this route ? 12. In a lumber yard there are 3 7,41 2 /if. of spruce, 15,102//^. of pine, 9,187//. of oak, 171,812//f. of hemlock, 7,413/if. of ash, and 18,002/if. of chestnut; how many feet are there of all kinds ? 26 SIMPLE NUMBERS. 13. A work consists of 6 yolumes; the first volume contains 611 pages, the second 539, the third 687, the fourth 599, the fifth 580, and the sixth 679; how many pages in the entire work ? 14. A man bequeaths $15,750 to his daughter, 122,850 to each of two sons, and twice as much to his wife as to his daughter; what is the amount of his bequests ? 15. The population of Maine is 627,413, of New Hamp- shire 301,471, of Vermont 300,187, of Massachusetts 1,240,499, of Connecticut 410,749, and of Ehode Island 192,815; what is the aggregate population of these States? 16. In 1876 the number of miles of railroad in the United States was as follows : in New England 5,694, in the Middle States 15,085, in the Western States 37,055, in the Southern States 16,676, and in the Pacific States 2,960 ; how many miles in all ? 17. In 1876 the popula^tion of the several divisions of the United States was as follows : New England 3,806,850, Middle States 11,105,000, Western States 15,835,000, Southern States 12,410,000, Pacific States 1,280,395; what was the population of the entire country ? 18. A merchant bought parcels of cloth containing respectively 3,912, 1,856, 2,011, 4,550, 937, 6,303, 1,856, 2,024, 4,228, 1,345, 6,138, 607, 960, 2,445, and 8,982 yards; how many yards did he buy in all ? 19. The first of four numbers is 3,125, the second is greater than the first by 5,108, the third is equal to the sum of the first and second, and the fourth is equal to the sum of the third and first ; what is the sum of the four numbers ? ADDITIO:Nr. 27 20. The ship Orient sailed from Marseilles to Buenos Ayres, distant 6,375 miles, thence to Valparaiso 2,764 miles, thence to San Francisco 6,346 miles, thence to the Sand- which Islands 2,152 miles, thence to Melbourne 5,588 miles, thence to Yokohama b,^%^ miles, thence to Calcutta 5,115 miles, thence to Bombay 2,257 miles, thence to Suez 2,006 miles, and thence back to Marseilles 1,^1^ miles; what was the entire distance sailed ? 21. A merchant commenced business with the following capital: Cash $18,471.25, goods worth 121,419.52, bank stock $7,418.00, and other property worth 14,314.17 ; he gained $12,315.42 the first year, and $11,124.86 the second year ; how much was he worth at the end of the second year? 22. An agent collected from different individuals: 127.18, $32.52, $41.70, 83.49, $8.17, $91.94, $127.86, $14.54, $87-.78, $411.10, and $79.62 : how much did he collect in all ? 23. A man has real estate worth $20,114.50, bank stock worth $15,779.82, United States bonds worth $17,772.89, and other property worth $6,317.27 ; what is the value of his entire property ? 24. Find the sum of the following items of account : $21.27, $49.18, $412.25, $44.74, $86.92, $311.10, $8.14, $118.45, $32.41, and $52.52. REVIEW QUESTIONS. (15.) What is addition ? (16.) What is the sum of two or more numbers? Wliat numbers can be added? (17.) When are num- bers similar? Illustrate. (18.) Explain the use of the signs of addition and of equality, (19.) What are the principles of ad- dition ? (20.) Give the rule for addition. The method of proving addition. (21.) What is a problem ? The solution of a problem ? 28 SIMPLE NUMBERS. II. SUBTRACTION. DEFINITIONS. 22. Subtraction is the operation of finding the dif- ference between two numbers. 23. The Difference between two numbers is a num- ber which, added to the less, will produce the greater. Thus, 6 is the difference between 10 and 4, because 4 + 6 = 10. The greater number is called the Minuend ; the less number is called the Subtrahend ; and their difference is called the Remainder. The minuend, the subtrahend, and the remainder must be similar. MENTAL EXERCISES. 1. James has 9 marUes and Samuel has 4 marbles ; how many more marbles has James than Samuel ? If 4 mar- bles are taken from 9 marbles, how many will be left ? 4 from 9 leaves how many ? 2. John had 9 chestnuts and ate 6; how many had he left? 6 chestnuts from 9 chestnuts leayes how many chestnuts 9 6 from 9 leaves how many ? 3. Henry had 15 ce7its, but spent 9 cents ; how many cents had he left ? 9 from 15 leaves how many? What is the difference between 15 and 9 ? 16 and 9 ? 18 and 9 ? 4. William had 14 apples, of which he ate 3 and gave away 5 ; how many had he then ? What is the difference between 14 apples and the sum of 3 apples and 5 apples 9 8 from 14 leaves how many? 8 from 17 ? 8 from 19? What is the difference between 14 and 3 -f 5 ? SUBTRACTION. 29 EXPLANATION OF SIGNS. 34. The sign of subtraction, — , is called minus ; when placed between two numbers, it shows that the second is to be subtracted from the first. Thus, 5 — 3 shows that 3 is to be subtracted from 5. A parenthesis, ( ), inclosing two or more numbers, shows that the inclosed expression is to be treated as a single number. Thus, 8 — (5 — 3) shows that the differ- ence between 5 and 3 is to be subtracted from 8. MENTAL EXERCISES. 1. What is the difference between 16 — 4 and 10 ? 2. What is the difference between 16 and 10 + 4 ? 3. 20 sheep + 2 sheep — (4 sheep -f 7 sheep) = how many sheep f 4. 19 lolls — (12 halls — 6 halls) = how many halM 5. 1 7 — 9 1= how many ? 6. (17 -f 10) - (9 + 10) = how many ? 7. 15 — 7 = how many ? 8. (15 + 10) - (7 + 10) = how many? 9. $15 — $9 = how many dollars? 10. $15 H- $9 — (13 + $4) = ? 11. 24 lbs, - (8 Tbs, - 3 lbs>i = ? 12. (24 lbs. + 10 lbs.) — (5 lbs. -f 10 lbs.) = ? 13. (20 + 16) - (10 + 10 - 9) = ? 14. (30 -f 17) — (10 + 10 - 8) = ? 15. (40 + 15) - (30 + 7) = ? PRINCIPLES OF SUBTRACTION. 25. The operation of subtraction depends on the fol- lowing principles ; 30 SIMPLE NUMBERS. 1°. If all the parts of the subtrahend are taken from corresponding parts of the minuend, the sum of the par- tial remainders is equal to the required remainder. 2°. If the same number is added to both minuend and subtrahend, their difference is not changed. OPERATION OF SUBTRACTION. 26. Let it be required to find the difference between 565 and 393. Explanation.— We write the subtra- opebatiok. hend under the minuend, so that units k^k of the same order shall stand in the same ' column. Then, beginning at the right Subtrahend, ^S hand, we see that 3 units from 5 units Remainder 1'<'2 leaves 2 units ; we therefore write 2 in the line below. Because 9 tens cannot be taken from 6 tens, we increase the latter by 10 tens, making it 16 tens ; now 9 tens from 16 tens leaves 7 tens ; we therefore write 7 in the line below. To compen- sate for the 10 tens, or 1 hundred added to the minuend, we may diminish the 5 hundreds of the minuend by 1 hundred, or what is the same thing, we may increase the 3 hundreds of the suhtrahend by 1 hundred (Principle 2°), which gives 4 hundreds ; taking 4 hun- dreds from 5 hundreds, we have 1 hnndred, which we write in the line below. The resulting number, 172, is the sum of the partial remainders obtained by subtracting the parts of the subtrahend from corresponding parts of the minuend; it is, therefore, from Principle 1°, the required remainder. In like manner we may find the difference between any two num- bers ; hence, we have the following RULE. I. Write the less number under the greater, so that units of the same order shall stand in the same column. II. Beginning at the right, subtract each figure in the lower line from the one dbove it, and write the difference in the line below. SUBTRACTION. ^1 ///. If any figure in the lower line exceeds the one above it, increase the latter hy 10, perforin the subtraction, and then add 1 to the next figure in the lower line. The operation described in the last clause of the preceding rule is called carrying. This operation, and that of adding 10, when re- quired, are performed mentally. EXAM PLES. (1.) (2.) (3.) (4.) From 663 976 Ihs, 704 /if. 1,806 yds- Subtract 580 531 Ihs. 483/^. 720 yds. Remainder, 83 445 lis. 221ft. 1,086 yds. (5.) (6.) (7.) (8.) Prom 4,236 80,502 $46,095 1555,555 Subtract 3,089 38,672 $28,736 $123,456 Remainder, 1,147 41,830 $17,359 $432,099 Proof. — Add the remainder to the subtrahend ; if the sum is equal to the minuend, the work is correct. ILLUSTRATIONS. From 75,625 376,781 lbs. $367,045 $84.16 Subtract 24,319 95,845 l bs. $106,253 $29.18 Remainder, 51,306 280,936 Ibs. $260,792 $54.98 Proof, 75,625 376,781 lbs. $367,045 $84.16 What is the difference between 13. 30,811 and 13,240? 18. 892,201 and 300,998? 14. 27,880 and 9,226 ? 19. 900,000 and 233,333 ? 15. 35,846 and 12,829 ? 20. 880,002 and 801,998 ? 16. 75,901 and 17,980 ? 21. 900,892 and 395 ? 17. 37,229 and 17,991 ? 22. 516,315 and 211,209? 32 SIMPLE NUMBERS. 23. 100,304 and 62,818 ? 25. 758,901 and 349,806 ? 24. 900,302 and 788,772 ? 26. 561,915,435 and 9,435 ? 27. The sum of two numbers is 7,817,412, and one of the numbers is 7,212,494 ; what is the other number ? 28. The greater of two numbers is 230,011 and the less is 210,299 ; what is their difference ? 29. The sum of two numbers is 485,752, and the less number is 82,992 ; what is the greater ? 30. What is the difference between 40,690,080 and 699,090 ? Perform the following indicated subtractions : 39. $57,846,203 - $7,756. 40. 14,396,802 — $83,846. 41. 3,718,412 - 807,306. 42. 887,892 — 709,378. 35. $814,316 - $91,320. 43. 68,893 - 29,394. 36. $620,306 — $413,314. 44. 4,924,863 — 43,989. 37. $813,864 — $11,899. 45. 2,814,316 — 999,007. 38. 41,336^6^5.— 7,814 «/flf5. 46. 8,904,306 — 304,216. 47. From 2,816,214/«f. subtract 1,856,394 /if. 48. How much does 3,816,204 exceed 3,334,599 ? 49. What is the difference between 740,817 and 220,198 : 50. From the sum of 862,141 and 32,843 subtract 884,109. 51. How much does the sum of 39,418 and 27,362 exceed the sum of 19,823 and 29,819 ? 52. Find the sum of 18,814 and 32,315, and subtract from it 17,794. 53. From the sum of $8,833, $141,209, and $11,362, subtract the sum of $2,843, and $10,906. 31. 81,423 - 20,120. 32. 80,200 — 1,875. 33. 18,714 — 13,392. 34. 123,387 — 94,816. SUBTRACTION. 33 54. From the sum of 88,303 feet, and 61,112 feet sub- tract the sum of 74,395 /ee^, and 0,202 /ee^. 55. From 105,242 + 522,801 subtract 131,1444-211,746. 56. From $21.56 + $42.87 + $11.72 subtract $48.99 -f- $2.65. 57. From $2,117.24 subtract $214.29 + $119.94 + 11.88. 58. From $38,140.20 subtract $16,884.49 + $22.27 + $46.71. 59. $4,547.18 + $1,620.29 - ($459.94 + $100.87 + $1,- 257.00) =z ? 60. $88,641 + $316.45 — ($19,384.22 — $6,211.88) = ? PRACTICAL PROBLEMS. 1. A. borrowed of B. $9,780 and paid $2,176 ; how much remained due ? Afis, $7,604. 2. A. purchased a farm for $10,000 and paid thereon $4,790 ; how much remained due ? Ans. $5,210. 3. B. bought merchandise, which he sold for $11,275, and made thereby $2,114 ; what was the cost price ? Atis. $9,161. 4. In 1860 the population of Maine was 627,413, and in 1870 it was 913,279 ; what was the gain in 10 years ? Ans. 285,866. 5. The sum of two numbers is 9,427, and the greater is 5,825 ; what is the less number ? 6. In 1790 the population of Connecticut was 238,141, and in 1840 it was 309,978 ; what was the gain in that period ? 7. In 1840 the population of Arkansas was 97,574, which was a gain of 07,186 in 10 years; what was the population of that State in 1830 ? 34 • SIMPLE NUMBERS. 8. How much does 57,182 exceed 18,394 ? 9. A merchant commenced business with a capital of $21,308, and retired with $74,114; how much did he make? 10. A., B., and C, commence business; A. puts in $35,000, B. $41,700, and C. $36,150 ; at the end of a year they have together $149,711 : how much did they gain ? 11. A merchant bought 500 yards of linen for $276, 3,400 yards of muslin for $325, and 75 yards of broad- cloth for $318, and sold the whole for $1,316 ; how much did he gain ? 12. A. has a yearly income of $12,000; of this he spends for rent $2,750, for taxes, repairs, and insurance, $814, for clothing $1,342, for household expenses 16,211, and the remainder he distributes in charity : how much does he distribute ? 14. B. has $12,311, and after paying his debts and giving away $2,108, he has remaining $8,199 ; what is the amount of his debts ? 14. A merchant bought cloth for $1,592, silk for $1,274, laces for $818, and sold the cloth for $2,102, the silk for $1,190, and the laces for $969 ; how much did he gain ? 15. A landholder owned 1,875 acres in Illinois, 2,396 acres in Indiana, and 13,742 acres in Michigan ; of this lie sold 813 acres in Illinois, 372 acres in Indiana, and 7,411 acres in Michigan : how many acres has he remain- ing ? 16. A., B,, and C, are in trade ; A. gains $7,055, B. gains $813 less than A., and C. gains as much as A. .and B. together, lacking $994 : what do they all gain ? SUBTRACTION. 35 17. A. bought a farm for 18,192, expended $2,815 for improvements, paid $387 for taxes, and then sold ifc so as to lose $2,282 ; for what did he sell it ? 18. A man worth $18,000 left $4,287 to his elder son, $3,751 to his younger son, $3,219 to his daughter, and the remainder to his wife ; what was the wife's portion ? 19. A man was 21 years old in 1843; in what year will he be 75 years old ? 20. A merchant bought 4 cargoes of grain ; the first contained 6,705 J?^., the second contained 842^w. less than the first, the tJiird contained 911 Jm. more than the second, and the fourth contained 3,092^2^. less than the second and third together : how many bushels were there in the four cargoes ? 21. A man bought three estates ; for the first he gave $5,260, for the second he gave $3,585, and for the third he gave as much as for the first two together ; he after- ward sold them all for $15,280 : did he gain or lose, and how much ? 22. A. travels due east at the rate of 19 miles an hour ; B. starts from the same place 1 hour later and travels in the same direction at the rate of 13 miles an hour ; how far apart are they 3 hours after A. starts ? 23. A. travels due north at the rate of 17 miles an hour; B. starts from the same place an hour earlier, and travels due south at the rate of 11 miles an hour ; how far apart are they 4 hours after A. starts ? 24. A merchant commenced business, having in cash $4,152,17, in goods $11,443.12, and in other property $5,794.22; at the end of a year he had in cash $2,158.23, 36 SIMPLE KUMBERS. in goods 117,411.98, and in other property $6,239.14: how much did he gain in the year ? 35. A gentleman purchased a house for $12,873.75, a carriage for 1720.50, a span of horses for $591.45, and a saddle horse for 1212 ;25 ; he paid for them at one time $4,374.16, at another time $3,495.17, and at a third time $2,675.14 : liow much remained unpaid ? 26. The areas of the New England States are as follows : Maine has 30,408 square miles, New Hampshire has 9,386, Vermont 9,420, Massachusetts 7,845, Connecticut 4,693, and Rhode Island 1,395 ; how many fewer square miles has Maine than all the rest together ? 27. A drover bought 24 oxen for $1,214.26, 42 cows for $2,111.79, and 40 calves for $397.11 ; he sold the oxen for $1,519.45, the cows for $2,237.18, and the calves for $318.27 ; what did he gain by the transaction ? 28. America was discovered in 1492, which was 128 years before the settlement of New England ; in what year was New England settled ? 29. A man having a sum of money, earned $8,211, and afterward lost $2,114, when he found that he had $11,415 ; how much had he at first ? 30. In a division there were 11,376 men, of whom 696 were killed in battle ; how many remained ? REVIE^A/' QUESTIONS. (22.) What is subtraction ? (23.) What is the difference between two numbers ? Illustrate. What is the minuend ? The subtrahend 1 The remainder? (24.) What is the name and use of the sign of subtraction? What is the use of the parenthesis? (25.) What are the principles of subtraction ? (20.) Give the rule for subtrac- tion. How is subtraction proved ? MULTIPLICATIO]Sr. 37 III. MULTIPLICATION. DEFINITIONS. 27. Multiplication is the operation of taking one number as many times as there are units in another. The first number, or the number to be repeated, is called the Multiplicand ; the second number is called the Multiplier ; and the result is called the Product. Thus, 4 imiUiplied by 3 is equal to 4 + 4 -f 4, or to 12. Here 4 is the multiplicand, 3 is the multiplier, and 12 is their product. Both multiplicand and multiplier are called Factors of the product. Thus, 4 and 3 slyb factors of 12. MENTAL EXERCISES. 1. What is the cost of 3 oranges at 6 cents apiece ? Explanation. — Because 1 orange costs Gets., 3 oranges will cost 6 cts. + 6 els. + Qcts., or 18 ds. From this we see that multiplication is a short method of performing the operation of addition when the numbers to be added are equal to each other. 2. A farmer sells 6 calves at $9 each ; how much does he receive ? $9 + $9 + $9 + $9 + $9 + $9 = ? How much is 6 times 19 ? 6 times 9 ? 3. If a man earns $5 a day, how much will he earn in 9 days ? What is 9 times $5 ? What is the product of 9 and 5 ? 4. How many are 4 times 5 ? How many are 5 times 4 ? Explanation. — The product does not depend on * * # * the order of the factors, as may be seen in the diagram. # * * # If we take the stars by columns we have 4 times * * * * 5 stars ; if we take them by horizontal rows, we have * # * # 5 times 4 stars ; in either case, we have 20 stars. * # * # 5. How many are 9 times 8 ? 8 times 9 ? 4 times 10 ? 7 times 12? 12 times 12 ? 38 SIMPLE NUMBERS. SIGN OF MULTIPLICATION. 28. The Sign of Multiplication, x, when placed between two numbers, indicates that their product is to be taken. Thus, the expression 5x7 shows that 5 is to be multiplied by 7, or that 7 is to be 7nultiplied ly 5. CONDENSED 1 VIULT IPLICATION TABLE. 1 2 3|4 5 6 7 8 9 |10 11 il2 2 4 6 1 8 10 12 14 16 18 20 22 I24 3 6 9 i 12 15 18 21 24 27 30 ZZ 36 4 8 12 1 16 20 24 28 32 36 40 44 48 5 10 15 1 20 25 30 35 40 48 _45, 54 i5o i 60 _5S„ 66 60 6 12 18I24 30 36 42 72 H 14 21 28 35 42 49 56 64 63 72 70 77 84 8 i6 24I32 40 48 56 1 80 88 i 96 9 i8 27 I36 45 54 63 72 81 i9o 99 |io8 10 20 30 1 40 50 60 70 80 90 100 no |l20 11 22 33 44 55 66 1 77 88 99 |iio 121 1x32 12 24 36I48 60 72 1 84 96 108 |l20 132 1 1144 13 26 39IS2 65 78 1 91 104 117 130 143 II56 14 28 42 1 56 70 84 1 98 112 126 I14O 154 |i68 15 30 45 60 75 ^ 1 90 1 105 120 135 I15O 165 ii8o 16 32 48 64 80 96 1X2 128 144 |i6o 176 I192 17 34 51 68 85 102 119 136 153 !i7o 187 204 18 36 54 72 90 108 i 126 144 162 |i8o 198 I216 19 38 57r76| 95 114 nz 152 1 171 190 209 !228 20 40 60 80 100 120 140 r6o ! 180 200 1 220 {240 Use of the Table.— Find and the multiplier in the first the nuiltiplicand in the upper line column ; their product will then be MULTIPLICATION. 39 found in the same column with the multiplicand and in the same line with the multiplier, By reversing the process, any number given in the table may be separated into factors. Thus, 187=17 x 11. MENTAL EXERCISES. 1. Add by 2's from 2 to 40. By 3's from 3 to 60. By 4's from 4 to 80. By 5's from 5 to 100. 2. Subtract by 2's from 40 to 2. By 3's from 60 to 3. By 4's from 80 to 4. By 5's from 100 to 5. Note. — Let these exercises be continued to the limit of the table. 3. What is the cost of 12 lbs. of sugar at 15 cts. a pound ? 12 times 15 cts, are how many cents ? What is the product of 15 ds. by 12 ? Of 15 by 12 ? 4. A man earns $18 a week, and it costs him $11 a week to hve ; how much can^he save in 13 weeks ? in 15 weeks? in 9 weeks? (118— $11) xl3 = ? (18-11) x (14— 6)=: ? Note. — The multiplier must always be abstract ; the multipli- cand may be either abstract, or denominate ; the product is always similar to the multiplicand. In practice, we multiply as though both factors were abstract, and then determine the unit of the product from the nature of the question. 5. (26-9) X 8 = ? 9. 8 X 10-14 = ? 6. (24-13) X (15-7) = ? 10. 4 X 9—13 x 2 = ? 7. (20-7) X (19-8) = ? 11. (4 + 9)-^(18-7) = ? 8. (3 + 9) X (11 + 6) = ? 12. (7 + 10) X (12-3) V MULTIPLICATION BY ONE FIGURE. 29. Lfet it be required to multiply 946 by 8. Explanation. — Multiplying 6 units by operation. 8, we have 48 units; that is, 4 tens and Multiplicand, 946 8 units ; we set down 8 in the units' place, Multiplier 8 and carry the 4 tens to the next column. 7" Multiplying 4 tens by 8 we have 32 tens, Product, 7,568 which increased by the 4 tens brought forward give 36 tens, or 3 hundreds and 6 tens; we set down 6 in the tens' place, and carry 40 SIMPLE NUMBERS. the 3 hundreds to the next column. Multiplying 9 hundreds by 8 and adding the 3 hundreds brought forward, we have 75 hundreds, which we set down. The resulting number 7,568 is the required product. In like manner we may proceed in all similar cases ; hence, the following RULE. Begin at the right hand and multiply each figure of the multiplicand by the multiplier, setting down and carrying as in addition. Note. — We may multiply by any number from 10 to 20 by the same rule. EXAMPLES. Perform the following multiplications: (1.) - (2.) . (3.) (4.) Multiplicand, 357 8645 2079 84123 Multiplier, 5 8 9 6 Product, 1785 69160 18711 504738 (5.) (6.) (7.) (8.) 8842 yds. 3749 in. 13146 lbs. $81386 4 7 9 8 85368 yds. 26243 in. 118314 lbs. 1651088 (9.) (10.) (11.) (12.) 5G432 13596 /i^. 14382 lbs. $87645 12 15 17 19 13. 43875x9 = ? 14. 14876//. X 11 = ? 15. $87653 X 14 = ? 16. 79792 ZZ^.v. X 12 = ? 17. 16749 X (24-7) =? 18. 86639 X 12 = ? 19. ^39864 X 18 = ? 20. $222794 x 19 = ? 21. 637489 x 9 = ? 22. 333333 x 16 = ? MULTIPLICATION. 41 EFFECT OF ANNEXING CIPHERS. 30. Every cipher that we annex to a number moves each of its digits one place to the left, that is, it converts units into teyis, tens into hundreds, and so on ; but this is the same as multiplying the number by 10 ; hence, To inultiply a numher hy 10, we annex one cipher ; to multiply it hy 100, lue annex two ciphers ; to mul- tiply it hy 1000, we annex three ciphers ; and so on. Thus, 75 X 10 = 750; 75 x 100 =z 7,500; 75 x 1000 = 75,000 ; 75 x 10,000 = 750,000 ; and so on. To multiply by any numher of tens, we first multiply by the given numher and then annex one cipher to the product ; to multiply by any nvmher 'of hundreds, we multiply by the given numher and annex ^?C6> ciphers ; to multiply by any number of thousands, we mul- tiply by the given number and annex three ciphers ; and so on. Thus, 8x4 tens — 32 tens = 320 ; 6x4 hundreds - 24 hundreds = 2,400 : 7 X 3,000 = 21,000 ; and so on. PRINCIPLES OF MULTIPLICATION. 31. The operation of multiplication depends on princi- ples already explained and also on the following : If all the imrts of the multiplicand are multiplied hy ea^h part of the multiplier, the sum of the p^artial products is equal to the required product. MULTIPLICATION BY ANY NUMBER. 32. Let it be required to find the product of 458 and 346. Explanation. — Ha\ing written the numbers so that units of the same order stand in the same column, we begin at the right and multiply all the parts of the multiplicand by 6, as explained in Art. 29 ; this gives 2748 for the first partial product. We next multiply all the parts of the multiplicand by 4 tens, or 40. Product, 158468 OPERATION. Multiplicand, 458 Multiplier, 346 ' 2748 Partial products, 1833 1374 42 SIMPLE NUMBERS. Multiplying 8 units by 40 (Art. 30), we have 320, that is, 3 hun- dreds and 2 tens ; we omit the cipher, write 2 ^in the column of tens, and carry 3 to the column of hundreds, and so on ; this gives the second partial product. We next multiply all the parts of the multiplicand by 3 hundreds. Multiplying 8 units by 300 we have 2400, or 2 thousands and 4 hun- dreds ; we omit the two ciphers, write 4 in the column of hundreds, and carry 2 to the column of thousands, and so on ; this gives the third partial product. The sum of the products thus obtained is 158,468 ; but this is the sum of the partial products found hy multiplying all the parts of the multiplicand hy each part of the multiplier ; it is therefore the required product (Art. 31)- In like manner we may find the product of any two numbers ; hence, the following RULE. I. Write the jnultipUer under the multiplicand, so that units of the same order shall stand in the same column. II. Beginning at the Hght, multiply all the parts of the multiplicand hy each figure of the multiplier, writing the first figure of each partial product under the corresponding multiplier. III. Find the sum of the partial products. EXAMPLES (1.) (2.) (3.) (4.) (5.) 843 1817 7287 325 9372 27 69 75 503 98 5901 16353 36435 975 74976 1686 10902 51009 1625 84348 22761 125373 546525 163475 918456 Proof. — Multiply the multiplier by the multipli- cand ; if the product is the same as before, the ivorJc is correct. MULTIPLICATIOlf. 43 (6.) Proof, (7.) Proof, 345 572 835 794 572 345 794 835 690 2860 3340 3970 2415 2288 7515 2382 1725 1716 5845 6352 197340 197340 662990 662990 Multiply 8. 875 by 349. 18. 46,137 by 841. 9. 11,843 by 216. 19. 50,246 by 322. 10. 1,781 by 74. 20. 61,532 by 742. 11. 999 by 77. 21. 184,387 by 994. 12. 1,754 by 306. 22. 97,418 by 887. 13. 7,506 by 45. 23. 107,309 by 1,206. 14. 2,016 by 1,008. 24. 320,009 by 344. 15. 8,435 by 371. 25. 99,897 by 284. 16. 4,572 by 614. 26. 44,479 by 3,227. 17. 32,183 by 179. 27. 56,263 by 7,777. When the multiplier terminates in ciphers, multiply the sig- nificant part, and annex the ciphers as explained in Article 30. (28.) (29.) (30.) (31.) (32.) 31 875 42 87 3194 290 4300 31000 1500 127000 8990 3762500 1302000 130500 405638000 33. 87 by 78,000. 36. 4,968 by 3100. 34. 314 by 87,000. 37. 19,872 by 26000. 35. 414 by 82,000. 38. 346,843 by 4500. If both multiplicand and multiplier terminate in ciphers, multi- ply the significant parts, and annex as many ciphers as there are in both factors. 44 SIMPLE NUMBERS. 39. 8,840 by 7,250. 45. 32,400 by 32,400. 40. 2,040 by 8,060. 46. 18,750 by 16,000. 41. 10,800 by 870. 47. 37,590 by 92,000. 42. 37,300 by 8,170. 48. 33,330 by 27,100. 43. 88,320 by 36,000. 49. 877,000 by 209,000. 44. 45,100 by 8,190. 50. 337,800 by 99,000. 51. 3,876 X 879 - 2,799 x 74 zir ? 52. 12,483 X 4,520 — 38,795 x 89 = ? 63. 1,379 X 794 + 145,902 x 86 = ? 54. (14,749 - 3,892) x (12,700 - 8,309) = ? 55. (265,484 — 142,184) x (8,794 — 3,684) = ? 56. (18,943 + 37,711) x (27,385 — 7,965) =: ? 57. (7,890 + 8,901) x (3,700 + 6,400) = ? 58. (276 + 3,276) x 875 - 4,962 x 79 = ? 59. (1,846 — 199) x 79 — (i;329 - 211) x 9 = ? 60. (5,946 + 9,544) x 284 + (8,305 + 95) x 890 = ? ADDITIONAL DEFINITIONS. 33. A Continued Product is a product of more than two factors. Thus, 3 x 4 x 5 is a continued product; it indicates that the product of 3 and 4 is to be multi- pHed by 5. A continued product may contain any number of factors. Its value is independent of the order of the factors. Thus, 3x4x5 = 3x5x4 = 4x3x5. 34. A Composite Number is a number composed of two or more integral factors. Thus, 21 is a composite number, because it is equal to 3x7; the number 30 is composite, because it is equal to 2 x 3 x 5. To multiply a number by a composite number whose factors are known we have the following multiplicatio:n^. 45 RULE. Multiply the iiiultiplicand hy one factor of the multiplier, then multiply the result hy another factor, and so on, till all the factors have been used. EXAMPLES. 1. Multiply 324 by 36, that is, by 9 x 4, or by VZ x 3. riBST OPERATION. SECOND OPERATION. 324 324 9 12 2916 Partial product. 3888 Partial product. 4 3 11664 Total product. 11664 Total product. Note. — In the following examples the factors of the multiplier may be found by means of the multiplication table. If the factors are unequal, we generally begin with the greater one. 2. 873x144 = ? 7. 736x48 = ? 3. 887/y. X 84 = ? 8. 4,315 x 176 = ? 4. 13,845x63=? 9. $8,712x209 = ? 5. 38,257 yds. x 96 = ? 10, 48 x 11 x 15 x 16 = ? 6. 7,836/^5. X 132 =? 11. 234x14x12x7 = ? PRACTICAL PROBLEMS. 1. What will 455 lbs. of sugar cost at 14 cents per pound ? Ans. 6370 ds. = $63.70. 2. What will 692 pounds of beef cost at 26 cents per pound ? Ans, 17,992 cts. = 1179,92. 3. If one barrel of pork costs $15, what will 1,728 barrels cost? ^?^5. 125,920. 4. If a train travels 35 miles an hour, how far will it travel in 425 hours? Ans. 14,875 miles. 46 SIMPLE . NUMBERS. 5. An army contains 106 regiments, and each regiment contains 1,128 men ; how many men in the army ? 6. If it requires 720 barrels of provisions to feed an army for one day, how many barrels will it require for 365 days ? Ans. 262,800 barrels. 7. There are 320 rods in a mile ; how many rods are there in 50 miles ? 8. If a railway costs $42,500 per mile, and is 385 miles long, how much does it cost ? 9. A field containing 56 acres produces 29 iu. of rye to the acre ; what is the total yield '? 10. Sound travels at the rate of 1,142 feet per second ; how far will it travel 3,600 seconds, or one hour ? 11. The distance from New York to Bridgeport is 56 miles, and there are 320 rods in a mile; how many rods from New York to Bridgeport ? 12. In an orchard there are 214 rows of trees, and each row contains 241 trees ; how many trees are there in the orchard ? 13. What is the continued product of 92, 37, and 45 ? 14. A freight train consists of 21 cars; each car contains 85 barrels of flour, and each barrel of flour weighs 196 pounds : how many pounds in the entire cargo ? 15. The distance from Bridgeport to New Haven is 18 miles; each mile contains 1,760 yards, and each yard 3 feet : how many feet from Bridgeport to New Haven ? 16. In an orchard there are 14 rows of peach trees ; each row contains 27 trees, and each tree bears 108 peaches : how many peaches in the orchard ? MULTIPLICATION. 47 17. A man earns $18.75 a week, and all his expenses are $11.25 a week ; how much can he save in 37 weeks ? 18. A woman bought IS yds. of ribbon at 27 cts. a yard, and 42 yds. of muslin at 1 6 cts. a yard ; what did she pay for both ? 19. A man has a barn worth $475, a house worth 5 times as much as the barn, and land worth 3 times as much as the house and barn together; what are they all worth ? 20. A man traveled 295 miles in 6 days ; for 5 days he traveled at the rate of 53 miles a day: how far did he travel the sixth day? 21. The diameter of Mercury is 2,967 miles ; the diam- eter of Saturn is 24 times that of Mercury ; and the diam- eter of the sun is 12 times that of Saturn : what is the sun's diameter ? 22. A courier had to travel a certain distance in 13 hours ; for 5 hours he traveled at the rate of 12 miles an hour, but finding that he was behind time, he increased his speed and finished the journey at the rate of 14 miles an hour : what was the distance traveled ? 23. The distance from Chicago to Albany is 835 miles; a passenger train starts from Chicago and runs toward Albany at the rate of 38 miles an hour, and at the same time a freight train starts from Albany and runs toward Chicago at the rate of 13 miles an hour : how far apart are they at the end of 12 hours ? 24. A steamboat runs from St. Louis to Cairo in 11 hours at the rate of 17 miles an hour; from Cairo to Mem- phis in 15 Jirs., at the rate of 16 ?m. an hour; from Memphis to Vicksburgh in 23 hrs., at the rate of 18 mi. an hour; and 48 SIMPLE NUMBERS. from Vicksburgh to New Orleans in 21 hrs., at the rate of 19 mi. an hour : what was her running distance from St. Louis to New Orleans ? 25. A fox starts from a certain place and runs at the rate of 616 yds. a minute ; at tlie end of three minutes a dog starts from the same place and follows the fox at the rate of 792 yds. a minute : how far apart are they at the end of 9 minutes ? REVIE^A^ QUESTIONS. (27.) What is multiplication? Multiplicand? Multiplier? Pro- duct ? Define factors. (28.) Explain the use of the sign of multi- plication. (29.) Give the rule for multiplying by a single figure. (30.) How do you multiply a number by 10 ? By 100 ? By 1000 ? By any number of tens ? (31.) What is the fundamental principle of multiplication ? (32.) Give the rule for multiplication by any number. If the multiplicand is denominate, what will be the nature of the product? What is the method of proving multiplication? What is the rule for multiplying when both factors terminate in ciphers ? (33.) What is a continued product ? How many factors may such a product contain ? (34.) What is a composite number? How do you multiply by a composite number whose factors are known ? IV. DIVISION. DEFINITIONS. 35. Division is the operation of finding how many times one number is contained in another, or of finding one of the equal parts of a number. The number to be divided is the Dividend ; the num- ber by which it is divided is the Divisor ; the result of the division is the Quotient ; and the part of the divi- dend that remains after the operation is the Remainder. DIVISION. 49 When the remainder is 0, the division is said to be exact ; in this case both the divisor and tke quotient are Jactors of the divi- dend. MENTAL EXERCISES. 1. If 24 ajyples are divided equally among 6 boys, how many apples will each boy receive ? What is one of the 6 equal parts of 24 apples ? How many times is 6 con- tained in 24 ? Note. — To use the multiplication table or a division table, find the divisor in the left hand column, and on the same line find the divi- dend ; the quotient will be the number at the head of the corres- ponding column. Let the pupil familiarize himself with this method of using the table. 2. What is the quotient of 36 divided by 6 ? of 144 by 12 ? of 126 by 14 ? of 72 by 9 ? of 153 by 17 ? of 90 by 18? of 95 by 19? of 56 by 14? 3. How many 7's can be taken from 23, and what will remain ? What is the quotient of 23 divided by 7, and what is the remainder ? Subtract by 7's from 23 as far as possible, and find the remainder. Note. — Division may be regarded ^s a short method of continued subtraction. The number of times that the divisor can be taken from the dividend is equal to the quotient. 4. If 47 peaches are divided into 9 equal parcels, how many will there be in each parcel, and how many over ? In this case, what is the dividend 9 The divisor 9 The quotient 9 The remainder 9 5. If you divide 126 by 11, what is the quotient, and what is the remainder? 149 by 12? 74 by 6 ? 190 by 15 ? 6. If you divide $154 among 14 children, how much will each child receive ? What is the quotient of 154 divided by 14 ? How many times is 14 contained in 154. 50 SIMPLE NUMBERS. Note. — Division is performed as though both numbers were abstract and the unit is determined from tlie nature of the question. In the last example $1.54 is equal to 154: cts.; we divide 154 by 14, which gives 11 ; hence, the answer is 11 cts. If the dividend and divisor are similar the quotient is abstract ; if the dividend is denominate, the quotient is of the same denomination as the dividend. 7. How many yards of cloth, at 17 a yard, can be bought for 184 ? If 12 yds. of cloth cost $84, what is the cost of 1 yd. f How many times is 7 contained in 84 ? What is the quotient of 84 by 7 ? of 84 by 12 ? METHODS OP INDICATING DIVISION. 36.-1°. The sign of division, -^, when placed be- tween two numbers, indicates that the first is to be divided by the second. Thus, the expression 12 -^ 3 indicates that 12 is to be divided by 3. 2°. The same operation may be indicated by writing the dividend over the divisor, with a line between them. Thus, the expression -^^, which is read 12 divided by 3 is equivalent to the expression 12 -r- 3. 3°. Division may also be indicated by writing the divisor on the left of the dividend, with a curved line between them. Thus, the expression 3)12 is equivalent to the expression J^, or to 12 -^ 3. MENTAL EXERCISES. 1. What is the quotient of 144 by 12 ? of $96 ^ 12? of $96 -T- $12 ? of 84 -^ 12? How many times is $12 contained in $84 ? 2. ¥ X -V- = ? 4. 192 -^ (23 - 7) = ? 3. 190 -^ 19 - 54 -^ 6 = ? 5. (200 _ 8) ^ 16 = ? DIVISION. 51 6. (108 + 45) -^ (10 -f- 7) = ? 7. (4 + 9 + 26) -^ 13 = ? 9. (12 x 17) -^ 6 = ? 8. 156 -^ (19 - 6) = ? 10. (12 X 16) -^ (13 - 5) = ? 11. (99 -^ 11) X (154 -^ 14) = ? 12. The product of two numbers is 64 and one of the numbers is 16 ; what is the other ? 64 -f- 16 == how many ? 13. Write the following by means of signs: the sum of $12 and $10, divided by $11, is equal to the quotient of the difference between $18 and $4, divided by $7. Note. — If a thing is divided into 2 equal parts, each part is called one half; if divided into 3 equal parts, each is called one third ; if into 4 each is one fourth ; if into 5 each is one fifth ; and so on. Thus, i is one half; f is one half of 3, or 3 Iialves of 1 ; J is ojie third; f is one third of 8, or two thirds of 1 ; -|^ is one fourth; f is one fourth of 2, or 2 fourths oi 1 ; | is one fourth of 3, or 3 fourths of 1 ; and so on. Expressions of the form ^, i, |, f , etc., are called fractions. Fractions are treated more fully hereafter. 14. What is J of 4 ? J of 8 ? ^ of 16 ? i of 3 ? | of 12 ? i of 15 ? J of 9 ? J of 4 ? J of 8 ? i of 04 ? 15. Read the expressions f , f , |, ^, j\, f|, -^, ||. OBJECT OF DIVISION. 37. Division is the reverse of multiplication. In multiplication, we have two factors given, to find their product ; in division, we have the 2^'^oduct and one factor given, to find the other factor. PRINCIPLES OF DIVISION. 38. The operation of division depends on the principles obtained by reversing those of Article 30, and also on the following, obtained by reversing that of Article 31 : If WG divide all the parts of the dividend by the divisor, 52 ' SIMPLE NUMBERS. the sum of the jmrtial quotients is equal to the required quotient, 39. There are two cases: 1°. Short Division, in which the divisor contains but one figure ; and, 2°. Long Division, in which the divisor contains more than one figure. In the first case most of the operation is performed mentally ; in the second case the different steps of the operation are written out. The principles employed are the same in both cases. I°CASE. SHORT DIVISION. 40. Let it be required to divide 26,812 by 4 : Explanation. — Having written the operation. numbers as shown in the margin, we Dividend, begin at the left and divide the dif- Divisor, 4 ) 26812 ferent parts of the dividend by the divisor. Quotient, 6703 Since 3 is not divisible by 4, we divide 26 by 4 ; this gives 6 for a quotient with 2 for a remainder -, hence, there are 6 thousands in the quotient ; we write 6 in the column of thousands and to the remainder we annex the following figure of the dividend giving 28 hundreds. The quotient of S8 by 4 is 7 ; hence, there are 7 hundreds in the quotient ; we therefore write 7 in the column of hundreds. Since there is no remainder and since 1 is smaller than 4 there are no tens in the quotient ; we there- fore write in the place of tens and annex the following figure to 1 giving 12 units. The quotient of 12 by 4 is 3, which we write in the column of units. In like manner we may treat all similar cases ; hence, the follow- ing RULE. /. Write they divisor on tfie left of the dividend, and draw a line between theiiv. II. Divide the first figure of the dividend by the divisor and set the quotient underneath, or, if the Divisioif. 53 first figure is less than the divisor^ divide the first two figures and set the quotient under the second. III. To the remainder annex the following figure of the dividend, divide the result hy the divisor and set the quotient underneath, or, if the result is less than the divisor, put a cipher in the quotient, annex another figure, and proceed as he fore. IV. Continue the operation till all the figures of the quotient have been found. EXAMPLES. Perform the following divisions: (1.) (2.) (3.) (4.) (5.) 5)785 6)804 8)1624 7)392 9)1926 Ans, 157 134 203 56 214 (6.) (7.) (8.) (9.) (10.) 4)1544 3)825 8)4896 9)792 7)2415 Ans. 380 275 612 88 345 If there is a remainder after the last partial division, we write it over the divisor and annex the result to the quotient. Thus, 27 -i- 4 = 6J indicates that the quotient of 27 by 4 is 6 with a remainder 3. The expression 6f may be read G and 3 divided hy 4, or 6 and S/t^wr^As. (11.) (13.) 5)176 4)8140 (13.) (14.) 7)8146 9)4023 11634 447 Ans, 35| 2035 Proof. — Multiply the quotient hy the divisor and. to the product add. the remainder ; if the result is equal to the d^ividend, the worh is correct. Thus, in example (11.), 35 x 5 + 1 = 176; hence, the work is correct. 54 SIMPLE NUMBERS. Divide 15. 12,360 by 4. 16. 3,730 by 5. 17. 20,202 by 6. 18. 37,904 by 4. 19. 90,872 by 8; 20. 640,339 by 7. Divide 21. 1639,145 by 7. 22. 1454,396 by $8. 23. 321,314/Z>. 99 = ? 5. 1,913,578 -^ 42 = ? DIVISION. 59 6. 15,336 -^ 72 = ? 9. 1,461,870 -v- 7 x 5 x 3 = ? 7. 93,312 -^ 108 = ? 10. 26,964 -^ 11 x 5 x 2 = ? 8. 674,201 -^ 110 = ? 11. 93,696 -^ 11 x 7 x 3 = ? To divide by 10, 100, 1000, &c., we point off as many figures from the right of the dividend as there are ciphers in the divisar ; the part on the left is the quotient and the part on the right is the remainder ^ (Art. 30). 12. Divide 8,759 by 100. An8, SH^. 13. 746 by 10. 15. 4,981 by 100. 17. 3,425 by 100. 14. 1,382 by 100. 16. 8,637 by 1000. 18. 94,276 by 10. Note. — If the divisor is composed of a significant part followed by ciphers, we cut off the ciphers and also the same number of fig- ures from the right of the dividend ; we then divide the remaining part of the dividend by the significant part of the divisor ; to find the true remainder we annex to the partial remainder the figures cut off from the dividend. 19. Divide 37,843 by 2,500. Ans. 16^%\. •Explanation, — The operation operation. is equivalent to dividing first by 25 00 ) 378 43 ( 15 100 and then by 25. The first ' ' partial remainder is 43, the sec- ond partial remainder is 3, and 128 the first divisor is 100 ; hence, ^ok by the rule, we have the true remainder equal to 3 x 100 + 43, True remainder, 343 or 343. 20. Divide 98,742 by 1,700. Ans. 68^^. Perform tbe following indicated divisions : 21. 8,436 -T- 2,100. 24. 2,564,310 -=- 84,000. 22. 8,566 -^ 2,500. 25. 217,896 -j- 7,200. 23. 17,439 -^ 1,700. 26. 1,310,741 -^ 64,000. Note. — If both dividend and divisor terminate in ciphers, we strike off from the right of each as many as are common to both, and then perform the division. 60 SIMPLE NUMBERS. 27. Divide 875,000 by 2,500. OPERATION. Explanation. — Striking off two ar rvrv \ o^v-rx r.r. / n^^ . 1 » T . . 1 ^ . 25,00 ) 87o0 00 ( 350 ciphers from eacli is equivalent to ^ dividing both by 100, which ob- 7o viously does not affect the result- 225 ing quotient. -j^p. Perform the following divisions : 28. 1,831,200 by 240. 35. 98,710 by 8,400. 29. $1,350,500 by 3,G50. 36. 66,920 by 8,800. 30. 087,500 yds. by 27,500 yds. 37. 8,623,000 by 250. 31. 201,600 by 3,600. 38. 47,890 by 8,600. 32. 41,580 by 540. 39. 35,100 by 4,800. 33. 71,820 by 87. 40. $1,400 by $270. 34. 1,749,600 by 360. 41. 368,000 by 4,200. PRACTICAL PROBLEMS. 43. The following problems afford exercises in review of the four fundamental operations, Addition, Subtrac- tion, Multiplication, and Division. 1. An estate worth $41,185 was divided equally among 5 persons ; what was the share of each ? Aiis. $8,237. 2. An estate worth $41,185 was divided equally among a certain number of heirs so that each received $8,237 ; how many heirs were there ? Ans. 5. 3. The capital of a joint-stock company is $13,125 and is divided into 175 shares ; what is the value of each share ? Ans. $75. 4. If a ship sails 5,712 miles in 48 days, how many miles does she sail per day ? Ans. 119. 5. If a ship sails 114 miles in 1 day, how many days will ittakeher to sail 2,622 miles ? Ans. 23. DIVISION. 61 6. A farmer paid $13,216 for a farm of 112 acres; how much did he pay per acre ? 7. IIow many acres of land can be bought with $26,432, at the rate of $59 per acre ? 8. In a field of corn there are 21,033 hills and each row contains 171 hills; how many rows are there ? 9. The mean diameter of the earth is 7,912 miles, and that of the sun is 854,496 miles ; how many diameters of the earth are there in the sun's diameter ? 10. A grocer bought 55,664 pounds of flour put up in barrels, each of which contained 196 pounds ; how many barrels were there in the lot ? 11. There are 4,032 yards of cloth in 96 equal pieces ; how many yards are there in each piece ? 12. A field produces 3,404 bushels of oats at the rate of 37 bushels per acre ; how many acres are there in the field? 13. Twenty pieces of cloth contain 39 yards each ; 32 pieces coutain 38 yards each ; and 17 pieces contain 43 yards each ; how many yards in all ? 14> A merchant bought 1 75 yards of cloth at 7 dollars per yard and afterwards sold 72 yards at 9 dollars per yard and the remainder at 8 dollars per yard ; how much did he gain ? 15. A dealer bought 27 barrels of flour at $14 per barrel and gave in exchange 32 cords of wood at $8 per cord and paid the balance in cash ; how much cash did he pay ? 16. A man's income is $3,150 per year and his expenses are $2,817 per year ; how much can he save in 6 years ? 17. A farmer bought 32 acres of land at $95 per acre, 6'i SIMPLE NUMBEKS. 71 acres at $47 per acre, 38 acres at $62 per acre, and 19 acres at $88 per acre ; what did he pay for the whole ? 18. The factors of one number are 19, 17, and 23; of ano-ther number, 31, 29, and 11 ; and of a third number, 77 and 83 : what is the sum of the numbers ? 19. Two men set out from the same point and travel in opposite directions; the first travels at the rate of 43 miles per day, and the second at the rate of 37 miles per day : how far apart are they at the end of 7 days ? 20. A farmer bought 6 oxen at $65 each, 12 cows at $42 each, and 142 sheep at 16 each ; what did he pay for the whole ? 21. In a freight car there are 6 boxes of goods, each weighing 382 pounds; 13 barrels, each weighing 218 pounds ; and 37 bags, each weighing 179 pounds : how many pounds in all ? 22. In 19 bales of cloth, each bale containing 16 pieces, and each piece containing 42 yards, how many yards ? 23. What number multiplied by 86 will give the same product as 163 multiplied by 430 ? 24. How many yards of muslin at 14 cents a yard must be given in exchange for 35 bushels of oats at 56 cents a bushel ? 25. A., B., and 0. enter into partnership; A. puts in $7,200, B. puts in 1700 more than A., and C. puts in $550 less than A. and B. together : what is the capital of the firm? 26. A.'s income is 5 times B.'s, B.'s income is 3 times C.'s, and C.'s income is $1,325 ; what is the entire income of A., B., and 0, '^ DlVISIOl^. 63 27. A farmer bought 154 acres of land at $64 per acre, and sold the whole for 111,704 ; how many dollars did he gain per acre ? 28. The distance from New York to Albany is 144 miles, and each mile contains 5,280 feet ; how many hours will it take a man to walk from New York to Albany if he walks at the rate of 352 x 60 feet an hour ? 29. The sum of two numbers is 10,370, and the second is 4 times the first ; what are the numbers ? t 30. The first of three numbers is 24, the second is 3 times the first, and the third is 4 times the sum of the first and second ; what is the difference between the second and third ? 31. Write down 4,617, multiply it by 12, divide the pro- duct by 9, add 365 to the quotient, and from the sum sub- tract 5,521 ; what is the final result ? 32. Mrs. White has 3 houses worth $12,530, $11,324, and ^9,875, also a farm worth $6,720. To her daughter she gave one third the value of the houses and one fourth the value of the farm, and then she divided the remainder equally between her two sons; how much did each re- ceive ? * 33. What IS the difference between the cost of 425 sheep at $4.75 apiece and 38 cows at $48.25 apiece ? 34. The distance from Chicago to San Francisco is 2,448 miles ; how long will it take a man to walk the whole dis- tanca at the rate of 24 miles a day ? 35. Two men had an equal interest in a herd of cattle ; one took 72 at $35 apiece and the other took the rest at $42 apiece ; how many cattle in the herd ? 64 SIMPLE NUMBERS. 36. A man bought 4 horses at $116 apiece and 2 colts at 156 apiece, and paid for them in flour at $12 a barrel ; how many barrels of flour did it require to make the pay- ment ? 37. A man travels due north for 7 days at the rate of 37 miles a day ; he then returns on his path at the rate of 29 miles a day; how far is he from the starting point at the end of 12 days trayel ? ^. A man bought 742 acres of land at 118 an acre ; he sold at one time 211 acres at $22 an acre and at another time he sold 184 acres at 125 an acre ; at what rate per acre must he sell the rest to gain 13,867 ? 39. A. bought a farm for $3,612 ; he sold half of it at 156 an acre and received for it $2,408 : how many acres did he buy and what did he give per acre ? 40. How many horses worth $112 apiece can be bought for 28 oxen woith $63 each, 52 cows worth $42 each, 175 sheep worth $6 each, and $2,394 in cash ? REVIE^A;' QUESTIONS. (35.) What is division? Define the dividend ; the divisor; the quotient ; and the remainder. When is division exact ? What are factors ? (36.) What does the sign of division indicate when placed between two numbers ? In what other ways may division be indi- cated ? (37.) What is the relation between multiplication and divi- sion? (38.) State the leading principle of division. (30.) What is short division ? What is long division ? (40.) Give the rule for short division. Method of Proof? (41.) Give the rule for long division. (42.) How do you divide by a composite number ? What is the method of determining the true remainder ? How do you divide by 10, 100, 1,000, etc. ? How do you divide by a number that ends in ciphers ? How do you divide when both dividend and divi- sor terminate in ciphers ? (43.) What are the fundamental opera- tions of arithmetic ? FACTORING AI^D CANCELLIKG. 65 V. FACTORING AND CANCELLING. DEFINITIONS. 44. A Factor of a number is one of its exact integral divisors, (Art. 35). Thus, 2, 3, and 4, are factors of 12. 45. A Composite number is a number composed of two or more Integral Factors (Art. 34). Thus, 15 is a composite number, because it is the product of 3 and 5. A Prime number is one that cannot be separated into any integral factors except 1 and the number itself. Thus, 2, 3, 5, etc., are prime numbers. 46. Factoring is the operation of separating a num- ber into integral factors. The factors of a number may be either prime; or composite. Com- posite factors may themselves be factored, and so on, till all the fac- tors are prime. Thus, 24=2 x 12=2 x 2 x 6=2 x 2 x 2 x 3 ; hence, the prime factors of 24 are 2, 2, 2, and 3. MENTAL EXERCISES. 1. What is the product of 2 and 3 ? What are the fac- tors of 6 ? of 9 ? of 15 ? of 77 ? of 121 ? of 144 ? 2. What is the continued product of 2, 3, and 7 ? of 2 and 3 ? of 2 and 7 ? of 3 and 7 ? What are the prime factors of 42 ? What are the composite factors of 42 ? In how many ways may 42 be factored ? Note. — Because every number is the product of 1 and of the number itself, these numbers are not specially considered in the operation of factoring, 3. What are the prime factors of 4 ? of 9 ? of 27 ? of 81 ? of 121 ? of 143 ? of 187 ? of 198 ? of 225 ? Note. — The product of two or more factors that are equal is called a power. The name of the power depends on the number of 8 66 SIMPLE KUMBEE8. equal factors. Thus, 3 x 3, or 9, is the second power, or the squa/re of 3 ; 3 X 3 X 3, or 27, is the third power, or the cube of 3; 3x3x3x3, or 81, is ihe fourth potcer of 3 ; and so on. Every lower power of a number is a factor of a higher power of the same number. 4. What are the factors of 81 ? of 27 ? of 9 ? How many prime factors has 81 ? How many has 27 ? How many has 9 ? What is the square of 6 ? the third power of 5 ? the fourth power of 4 ? the fifth power of 2 ? 5. What is the second power of 10 ? the third power of 10 ? the fourth power of 10 ? How many ciphers are required to write the square of 10 ? the cube of 10 ? 6. What is the fifth power of 10 ? How many ciphers in the fifth power of 10 ? What power of 10 is 100 ? 1,000 ? 10,000 ? 100,000 ? 1,000,000 ? PRINCIPLES OF FACTORING. 47. The operation of resolving a number into prime factors depends on the following principles : 1°. A nicmher is equal to the continued product of all its prirne factors. 2°. If a number is divided hy one of its prime factors, the quotient is equal to the continued product of all the others. OPERATION OF FACTORING. 48. Let 210 be separated into prime factors. Explanation.— We first divide by 2, which is a operation. prime factor ; we next divide the first quotient by 2)210 3, which is also a prime factor ; we then divide the 3)105 second quotient by 5, and find 7 for a quotient. — — The numbers 2, 3, 5, and 7 are the required factors ; ^)j^ that is, 7 210 = 2 X 3 X 4 X 5. FACTOEIN"G AKD CAKCELLIKG. 67 In like manner other composite numbers may be factored ; hence, the following RULE. Divide the given nurriber by one of its prime factors; then divide the quotient by one of its prime factors ; and so on, till a quotient is found that is a prime number ; the several divisors and the last quotient are the required factors. Note. — It will be found convenient to begin the division with the smallest prime factor. EXAMPLES. Resolve the following numbers into their prime factors : 1. 42. Ans. 2x3x7. Note. — If there are more than two factors in any indicated prod- uct, the sign of multiplication may be replaced by a simple dot ; thus, 3 . 3 . 7 is equivalent to 2 x 3 x 7. 2. 180. Ans. 2.2.3.3.5. 5. 770. Ans. 2 . 5 . 7 . 11. 3.378. ^ws. 2.3.3.3.7. 6. 1,575. ^w. H, and A. 10. f, A. and |f . 73. To reduce two or more fractions to their least common denominator. Let it be required to reduce -^, |f , and |-J to equivalent fractions having the least possible common denominator. Explanation.— Here we first opebatiok. reduce each fraction to its low- 8 2 2x8 16 est terms (Art. 70) ; we next 12 ^^ S ^^ o ^ o ^^ 94' find the least common multiple, 24, of the new denominators, 15 3 3x6 18 which will be the required de- 20 4 4 ^ ^ — 24' nominator ; we then divide this multiple by each denominator 20 5 5x2 10 separately, and multiply both 48 12 12 X 2 24' terms of the corresponding frac- tion by the quotient. Thus, we multiply both terms of | by 8, both terms of J by 6, and both terms of -^-^ by 2. In like manner we may treat all similar cases ; hence, the fol- lowing RULE. I. Reduce each fraction to its simplest terms. II. Find the least common multiple of all the denominators for a common denominator; divide this by each denominator separately and multiply the corresponding numerator by the quotient. 92 COMMON^ FRACTIONS. EXAM PLES. Eeduce each of the following groups of fractions to a common denominator : 1. h h i> and T^. Ans. J|, Jf, fi and 4|. 2. I, I, }, and ^^. Ans, yVo. ^. t¥o. and y^-. 3. f, T^, If, and A. 17. W. M. and «• 4. i A, H, and •^. 18. HJ, ^V, and ^^3. ^- A» A» yt^ and -^. 19. f, f, y^-, and f|. 6. f, tt. and If. 20. «, A. ft. and JJf . 7. if, if, and tt- 21. i, i h h and f . 8. J^, 1, and A. 22. f , ^s^, A, A, and if 9. t3^, t^, and ff . 23. A. M. W' and ^. 10. tt, A, and a- 24. W, t¥o, U, and if. 11. i^, A, and i«. 25. if, ^, i^i^, and li. 12. A. ii and «. 26. ^, ^, ii, and iff. 13. A. tIt, and ^f^. 27. f^, f f , ^VV, and -W- 14. A, ^, and A. 28. 4|, f J, if, and -,f^. 15. H, T^g, and ■^. 29. 5f, 11^, ^, and J^- 16. fi, if, and e. 30. 6f, llf, 4^, and ff. Note 1.— If there are any Integral, or any mixed numbers, reduce them to the form of simple fractions (Arts. 67 and 68). Note 2.— Complex fractions are reduced to simple ones by the rule for division of fractions (Art. 81). ADDITION OF FRACTIONS. DEFINITION. 14. Addition of Fractions is the operation of find- ing the sum of two or more fractions. MENTAL EXERCISES. 1. What is the fractional unit of ^ of $1 ? of ^6^. of $1 ? How many times is this unit taken in $^ and $r^ ? ADDITION. 93 2. What is the sum of -^ and -^ ? of ■^, y\, and -^ ? of A. A. A and A ? of A, A, H, and ^ ? 3. How many twelfths are there in f? How many twelfths in f ? What is the sum -^ and ^ ? What is the sum of f and | ? What is the sum of f and f ? Note. — The fractions must be reduced to the same fractional unit before they can be added, that is, they must be reduced to a common denominator, 4. A man bought ^ of a pound of indigo at one time and I of a pound at another time ? how mucli did he buy in all ? What is the sum of J and f ? of f and | ? 5. What is the sum of $4^ and $2^ ? of ^ and J? of ^ and ff ? What mixed number is equal to || ? 6. What is the sum of $|, 1^, and 1^? of |, \, and 3^? of IJ, JV? and ^? What mixed number is equal to If^? OPERATION OF ADDITION. 75. Let it be required to find the sum of f and -f . Explanation. — Having operation. reduced the fractions to a 4 3 28 15 43 common denominator, we 5 7 35 35 ^^ 35 ^^ * see that the first is equal to the fractional unit 3^5 taken 28 times, and the second to the same unit taken 15 times ; hence, the sum is equal to this unit taken 38 + 15, or 43 times, that is, to f f or to Ig^. In like manner we may treat all similar cases ; hence, the fol- lowing RULE. /. Reduce the fractions to simple fractions having a common denominator. II. Add their numerators for a new numerator, and write the sum over the common denominator. 94 COMMON FRACTIOiJS. MENTAL EXERCISES. 1. What is the sum of |y\, $^, and IJ| ? Ans. |1|. Add the following groups of fractions : 2. I, I and t'j. 9. |, Jf , and J#. 3. I, f , and i. 10. I, tV and -W. 4. I, f, and T^V. 11. i, ^, h and «. 5. i A. and A- 12- f, A, H. and f . 6. A, i ^, and J. 13. |, ^, i, |, and |. 7. f, 4, and ^. 14. ^\, J, |, 4^, and J. 8. J, J, i, and i. 15. +, -ft-, f, -g^, and i- . When there are mixed numbers, add the sum of the fractional parts to the sum of the whole numbers. 16. 4^, 6J, 2i, and |. Ans. 12 + ff = 13^- 17. 10|, 7i, 8|, and 16}. Ans. 42^. 18. 1-^, 6f, 18if , and 2^. Ans. 28^^. 19. 2^, 6^, and 12H- 30. 2^, 5^, and 6^. 20. 67^^, 4^, and 600f. 31. 6^, 4|, and 13|. 21. 13^-, 99|, and 512-jV 32. 6f, llt^j, and 9. 22. 14t^, d^, and 88^- 33. {i, -^, and 2^- 23. 900^, 450^%, and 6^. 34. ^, ll-fj, and 5f. 24. 21i-, 98^^, and 14^^. 35. 18-J-, 2^^, and 6J. 25. li, 4^, and 6|. 36. ^, ^, and 8^. 26. ^, 7tV, and 8^. 37. 9^, ||, and «. 27. ^, ^S, and 7^^. 38. 5f 3-^, and «. 28. 5i, i and 7^. 39. 16^ |i, and fj. 29. 4|, 3^, and 3^. 40. 8^, «, and f|. ■41. $1 + $4 + 1^ + $1 + $3i z= ? 42. 37i?J5. + 24|/^'5. + ^/5. + 4ii-ZJ5. = ? 43. 46} + 118f + 319| + 1^ = ? 44. ^^yds. + m^yds. + l^\yds. + 82tt«/«^6^ = ? ADDITION. 95 PRACTICAL PROBLEMS. 1. A farmer has 3 fields ; the first contains 31-| acres, the second 49f acres, and the third 59^ acres : how many acres has he in all ? Ans. 14:0^^ Acres, 2. A man earned $3| the first day, $4| the second day, 15^ the third day, $7i the fourth day, $4^ the fifth day, and $3f the sixth day ; how much did he earn in the six days ? Ans. $28f. 3. A. traveled 17f miles the first day, f of 17f miles the second day, 22-J iniles the third day, and 36^- 77iiles the fourth day ; how far did he travel in the four days ? 4. B. works S^Jiours on Monday, 9^hot(rs on Tuesday, S^Q liours on Wednesday, lOj hours on Thursday, 9^ hours on Friday, and 10^ hours on Saturday ; how many hours does he work during the week ? 5. A farmer sells 3 tons of hay for 147. ISf, 3 cows for $111,421, a horse for ^173.16|, and 100 hushels of oats for $62.87|^; how much does he receive for all ? 6. How many dollars will pay for a coat worth $14}, a hat worth %b^, a vest worth %Q\, a pair of pants worth $8, and a pair of boots worth $9^ ? 7. How many pounds of butter in 4 tubs weighing respectively 27^/^',^., 34|/^*\, ^^lhs., and 29f Z^>s.? 8. How many tons of coal in 5 loads weighing respec- tively li, lA, lA. H. and 11 tons 9 9. How many yards in 4 pieces of cloth, measuring respectively 27^yd-<^., 37f //r/,^., 39^ yds., and 30{^ yds.? 10. A farm contains 26^ acres of plough land, 39^ acres of wood land, 61f acres of pasture land, and 42^ acres of meadow land ; how many acres in the farm ? 96 COMMON FEACTIONS. SUBTRACTION OF FRACTIONS. DEFINITIONS. 76. Subtraction of Fractions is the operation of finding the difference between two fractions. MENTAL EXERCISES. 1. "What is the difference between f and f ? ^ and ^ ? ^and^%,? MandH? Mand|4? ii and ^ ? 2. .What is the difference between |- and -f? Explanation. — The minuend is equal to f^ and the subtrahend is equal to i| ; the difference is therefore equal to the fractional unit -^Q taken, 25 — 12, or 13 times, that is, it is to ^f. 3. A boy had -J of a dollar, but he spent -^ of a dollar for a slate ; how much had he left ? What is the differ- ence between -J and ^ ? ||- and ^ ? 4. -f- - I = ? 6. ^Ibs. -ilb. = ? 5. $1^ -$^=? 7. 2^ft. - {iff. = ? OPERATION OF SUBTRACTION. 77. Let it be required to find the difference between I and f . Explanation. — Having reduced the given fractions to the common unit, -^^, we see that the minuend contains this unit 35 times and the subtrahend 16 times ; hence, the re- mainder contains it 35 — 16, or 19 times ; the remainder is there- fore H. Since all similar examples may be treated in the same manner, we have the following RULE. /. Reduce the fractions to simple fractions having a common denominator. r. 5 8" 7 35 "56 16 56^ 19 "56* SUBTRACTION. 9*? //. Subtract the nuinerator of the subtrahend from that of the minuend for a ne^w numerator, and write the difference over the common denomi- nator. EXAMPLES. Find the difference between, 1. «andtt. 17. $18i| and $13^. 2. A and A. 18. 14^ ft. and l^ft. 3. 37^ and 33^. 19. lOm and 4^^. 4. ^ and 4J. 20. 206^^6/5. and 194f ^^^5. 6. 13^^^ and 9^. 21. 4ni\yds. and 21^ yds. 6. SO-jij and 47^. 22. $11811- and |24|4. 7. 42 and 30t^. 23. $22-3!^ and $9-3%. 8. 90^ and 25^. 24. 2463^5- and 194f. 9. 46-1 and 15J. 25. l,476ff and 894^. 10. T^ and f off. 26. 177f^'^^.and66H^i*. 11. 98|- and 45f. 27. 163|??2f. andl08|wi. 12. 150^ and 65^1,^. 28. $864iV and $648|. 13. ^ and f 29. 146|^owsand97^^ow5. 14. ^^ and If. 30. %%1^ acres Q^ndiUll acres. 15. 134 and 7^. 31. 1,884t^w. and l,801^m. 16. 843ff and 94^. 32. ^^O^rods and 199^. ro^5. 33. $180^ H- $27^ - - ($431 + I39|) = ? 34. 146J rods + 73 J rods — (24^ rods — 6^ rods) = ? 35. nShu. + 375|^.«^. - (46 5w. ~ 7i Jw.) = ? PRACTICAL PROBLEMS. 1. A. has $4^, and B. has $d^ ; how much more has A. than B. ? Ans. I J. 2. A. bought bQ^ pounds of butter, and sold ^ of it to 4 98 COMMON FRACTIONS. one person and 13^ pounds to another ; how much had he left? Ans, 14i\l^ pounds, 3. A grocer bought 2 hogsheads of sugar, each weigh- ing 1,302 2J02inds ; he sold -J of one hogshead and 455^ pounds from the other : how many pounds had he remain- ing ? Ans. 1,714^ ?5s. 4. A merchant bought 2 pieces of cloth ; the first con- tained 3^ yards, and the second 414^ yards; he then sold 69^ yards : how many yards had he left ? 5. A man bought a farm of 211| acres, and sold llY^Sj. acres of it ; how much had he remaining ? 6. From a cask containing 60 1 gallons of cider there were drawn off Yt\^ gallons ; how much was there left ? 7. A sloop has on board 406| tons of coal, of which 311f tons is anthracite, and the remainder cannel ; how much cannel does she contain ? 8. A merchant had a piece of silk containing 4:2^ yards, from which he sold 17|- yards j how much had he remain- ing? 9. A person having $49f, spent I4f in getting to Bos- ton, and $5^ in getting to Portland ; how much had he left on reaching Portland ? 10. If a man has $11^ and spends I8f, how much will he have left ? 11. A tailor had a piece of cloth containing 2^ydsj he cut off 4J- yds. to make a coat and If yds. to make a pair of pants : how many yards were left ? 12. A man had to walk dill miles; ^® walked 30^ miles the first day, 33| miles the second day, and finished the journey the third day : how far did he walk the third day ? MULTIPLICATION. 99 13. A cask of wine contained 42 1^ gallo7is ; of this 13^ gallons were drawn off, and 12^ gallons leaked out: how much remained in the cask ? 14. A grocer bought Sd^lbs. of tea; of this he sokl 13|- lbs. to one customer, 9^ lbs. to a second customer, and VZ^lbs. to a third customer : how many pounds had he left ? 15. A laborer earned $18| and received as a gift $14^; he then bought a barrel of flour for $12, and groceries to the amount of $8|^ : how much had he remaining ? 16. A drover bought 4 cows for $168J, and after paying $34^ for pasturage, he sold them for I203J-; did he gain or lose, and how much ? 17. A man traveled in a certain direction 34^ miles the first day, and 37f miles the second day ; then, retracing his path, he traveled 28^ 7niles on the third day, and 34^ miles on the fourth day : how far was he then from the starting point ? 18. What is the difference between 57J yds. + 72^^ yds. and 211 yds. — 94^ yds. 9 19. A. bought a house for $11,320^, and after expending $1,31 If for repairs, sold it for $12,500 ; did he gain or lose, and how much ? MULTIPLICATION OF FRACTIONS. DEFINITION. 78. Multiplication of Fractions is the operation of finding the product of two or more fractions. MENTAL EXERCISES. 1. What is -J of 5 ? I of 5 ? I of 5 ? What is 5 taken ^ of 1 time ? 5 taken f of one time ? 5 taken | of 1 time ? What is 5 x i? 5 x | ? 5 x |? 5 x -f? 5 x H? 100 COMMOIT FRACTIONS. 2. If ^ of an orange is divided into 5 equal parts, how much of a whole orange is 1 of these parts ? how much is 2 of the parts ? 3 of the parts ? What is | of ^ ? | of ^ ? fofi? WhatisJ xi? ix I? ix|? }xi? Explanation. — The expression ^ of | is equivalent to the ex- ession I x I; for 4 of -^ is the sar therefore equal to |^ x |^ (Art. 27). 3. What is 4- of i ? f of ■^? f of ^ ? What is ^ of f ? f of I ? What is 4- X i? f X i ? f X I ? I X f ? Is there any difference between f of f , and f of ■§ ? What isf xl? What is f x|? A xf ? OPERATION OF MULTIPLICATION. 79. Let it be required to multiply f by -f . Explanation. — We first multi- opbbation. ply f by 5, which, according to Prin- 3 5 3x5 15 ciple 1° (Art. 65) gives ?-^; but ^ '^ 4 X 7 "" 28 this result is 7 times the required product^ because the multiplier used is 7 times the given multiplier ; hence, to find the true product, we must divide it by 7, which, according to Principle 3° (Art. 65), 3 X 5 15 gives -; =, or —r . ^ 4x7 28 In like manner we may treat all similar cases ; hence, the fol- lowing RULE. Reduce the factors to the form of simple frac- tions ; then multiply the num^erators together for a new numerator, and the denominators for a new denominator. E X AM PLES. 3 1. Multiply 7i by |. Ans. ^ x ^ = | = 4i. 2. Multiply 2i by ^ of f Ans. ^ x ^ x ^ = g- MULTIPLICATION". > ' '^ ' V , foi Note.— After indicating the operation, m^wcJ^'oVciy^* factor* tKiit'it? common to any numerator and any denominator. If the final result is an improper fraction, reduce it to a whole, or to a mixed number ; if it is a proper fraction, reduce it to its lowest terms. The rule may be simplified in the following cases : 1°. To multiply a whole number by a simple fraction : Multiply it by the numerator of the fraction and divide the result by the denominator. 3. Multiply 928 by |. Ans. ?|? x | =^^^2^ = 348. 1 o o 2°. To multiply a whole number by a mixed number : Multiply it first by the fractional part of the mixed niunber, then by the integral part, and find the sum of the results. (4.) 928 (5.) 3)1143 6i n 348 I of 928. 381 i of 1143. 5568 6 times 928. 8001 7 times 1143. 5916 Product. 8382 Product. Perform the following indicated operations : 6. f Xt%. 15. f of7ix}of90. 7. V X f 16. 345 X 4|. 8. 1 X 12|. 17. 3f X 2f. 9. 7i X 8i. 18. I X I X A- 10. ^ X 61|. 19. 3i X 1 X llf 11. I of 7 X f. 20. A X A X ^. 12. 154 x^. ^1- 3ix WVx^. 13. 6f x -W- 2^- 3i X 7i X H- 14. ^ off xi|. 23. ^x2i X Hi i02f '' '■ ' ' 'COMMON FRACTIONS. ^ ' '^^i^^D^rui^'k^^. 29. A of 3i X H. 25. 2| X 5| X iff. 30. tt X f X 3f . 26. 114m X 81^. 31. 2174- X 112|. 27. f ofA xH. 32. «x-Jx A- 28. 2i X If X |. 33. 8i X 8J X 8J. 34. (56| + 24i) X (13i + 9f ) = ? 35. (Ill -f- 302i) X (107^ - 30^) = ? 36. (207| -f 39i) X (lOO^ij: - 66^V) = ? 37. (4451 - 36i) x (36f - 21^) = ? 38. (999f - ^^) X m^ - 72|) = ? 39. (256 - 7i) X (394 + ^) =? 40. (224 + 3A) X (88| -4^)=? PRACTICAL PROBLEMS. 1. If a man earns 133^ per week, how much will he earn in a year of 52 weeks ? A^is. 11,733^. 2. A farmer bought 43 acres of land at |104f per acre, 16 cows at I28f each, and 2 plows at II 1^^ each ; what did they all cost him ? Ans. $4,985-^. 3. What must be paid for 600 barrels of flour at $5.37^ per barrel ? Ans. $3,225. 4. What is the cost of 33^ lbs. of tea at 93f cents a pound? Ans. $31.25. 5. If a man can travel 7f miles in 1 hour, how many miles can he travel in 6J hours ? 6. If it takes If bushels of wheat to sow an acre, how many bushels will it take to sow 7^^ acres ? 7. A grocer bought 100 barrels of flour at $6-J per bar- rel ; he sold 49 barrels at 17^ per barrel, and the rest at $7^ per barrel : how much did he gain ? MULTIPLICATION. 103 8. A. bought 319f acres of land at $200 per acre ; he then sold 250f acres at $250 per acre, and the remainder at $26 6f per acre : how much did he gain ? 9. A drover bought 64 sheep at $7f a piece ; he then sold 30 of them at $6|^ a piece, and the remainder at f 8 J a piece : did he gain or lose, and how much ? 10. A. starts from Cincinnati and travels at the rate of 5f miles an hour ; at the end of 3|- hours B. starts from the same place and travels in pursuit at the rate of 6 J miles an hour : how far apart are they at the end of 5;^ hours ? 11. What is the cost of f of a piece of cloth containing 13J yards at $2J a yard ? 12. A., B., and C. own a tract of land; A's share is 62|- acres, B's share is 1^ times as much as A's, and C's share is lOJ acres greater than A's and B's together : how many acres in the whole tract ? 13. A man traveled 112J miles in 3 days ; the first day he traveled f of the whole distance, and the second day he traveled ^ of the distance he did the first day : how far did he travel the third day ? 14. A woman is 24f years old, and her husband lacks 7f years of being twice as old ; what are the united ages of the two ? 15. What will f of -J of a yard of cloth cost at the rate of A of $3| per yard? 16. How many yards in 8 pieces of cloth, each contain- ing 37| yards ? 17. If a train of cars runs 22j^ mi, an hour, how far will it run in 8 J hrs 9 104 COMMON" FE ACTIONS. DIVISION OF FRACTIONS. DEFINITION. 80. Division of Fractions is the operation of find- ing the quotient of one fraction by another. MENTAL EXERCISES. 1. In 1 orange, how many ffths of an orange 9 How many times is \ contained in 1 ? What is the quotient of Ibyi? oflby^? of 1 by | ? 2. What is the quotient of 1 by I ? of 2 by | ? of 3 by \ ? How do you divide an integral number by a fractional unit? Explanation.— The quotient of 1 by | is 7 ; but the quotient of 3 by \ is 3 times as great as tlie quotient of 1 by | ; bence it is 3 x 7, that is, we multijply the given nwriber ly the denominator of the frac- tional unit. 3. What IS the quotient of 3 by ^ ? of | by ^^ ? of } by \ ? How do you divide a simple fraction by a fractional unit ? Explanation. — The quotient of 3 by | is 3 x 7 ; but the quotient of f by I is only \ as great as the quotient of 3 by \ ; hence it is 3x7 — 2— , that is, we multiply the numerator of the given fraction by the denominator of the fractional unit. 4. What is the quotient of f by 1 ? of | by 2 ? of f by 5 ? How do you divide a simple fraction by a whole num- ber ? Explanation.— The quotient of f by lis | ; but the quotient of f by 5 is only \ as great as the quotient of f by 1 ; hence it is SI S ; X = or - — -, that is, we multi'ply the denominator of the given 4 5 4x5 fraction by the whole number. DIVISION. 105 OPERATION OF DIVISION. 81. Let it be required to divide -f- by ^ : Explanation. — Here the divi- operation. sor is equal to i taken 4 times, that 3 4 3/1 \ is, to 1^ X 4 ; hence, to find the quo- -;z -^ -^ = -^ -r- (^ X 41 tient we divide f by ^ and that ^ ' result by 4. To divide f by i 3x5 3 5 15 we multiply its numerator by 5 = = - x - = — ■» 3x5 7x474 28 (Art. 80, Ex. 3), which gives -y- ; to divide the result by 4 we multiply its denominator by 4, (Art. 80, Ex. 3), which gives = — ^, and this is the same thing as = x 2, or — . Here we have inverted the divisor, that is, we have made its terms change places, and then we have proceeded as in multiplication. In like manner we may treat all similar cases ; hence, the RULE. Reduce both dividend and divisor to simple frac- tions ; then invert the divisor and proceed as in multiplication. Note. — Before performing the multiplication cancel and reduce as explained in Art. 79. EXAMPLES. 1. Divide Z\ by |. Ans. ^ x | = y = 5^. To divide a whole number by a simple fraction we may Multiply it hy the denom^inator of the fraction and divide the result hy the numerator. 2. Divide 27 by f . Ans. y x | = ^^- = 331- Perform the following indicated operations : 106 C0MM02S' rRACTIONS. 5. *^«. 21. fofJI -AofH. 6. tt - A- 22. 1 of 7|- 4 of 2,15. 7. iM-«- • 23. (8i + 3i) - 7|. 8. 241 -r- ^. 24. (7f + 8i)H-iof6^. 9. 1275 -J- ff. 25. iof4| --(2^+31). 10. f - t X A- 26. 7f X 84-T-3i X ^. 11. V-^lof-A. 27. ^ofSJ-^aiJ X 7i. 12. iofif-v-if XtV- 28. (3+ + 15f ) - 27A- 13. |of«-HofJ. 29. 25^ -^ - m + 5i). 14. W- -fof*. 30. llT»r - - (^ + ^T^r)- 15. W - -¥• 81. 14A - - f of 15. 16. ^^- r21. • 32. 214} - hiof25ii. 17. m- -i^. 33. i of} of 5 -=-21^. 18- 54 -=- Sf. 34. 4 X 7i -=- 8 X 19I-. 19. 611^ H- 20A- 35. i of 15| -=- 9i X f 20. 10015 \ - 66f. 36. (32i + 7})H-f of-A. 37. 2J X 3f + 3 -^ 4^ - 7|: = ? 38. (3^ + 7f ) -- 2A + 71- ^ 5f = ? 39. (Hi + 17}) -- (33i _ 4J) + 8^ zz: ? • 40. (i of 6} - 24) -^ (25 + 1^ of 3|) = ? Note. — The following problems afford exercises on all the opera, tions that can be performed upon fractions : PRACTICAL PROBLEMS. 1. If Hi yds. of silk cost $13, what is the cost of 1 yard? of 5 yds. ? Ans. UH; and I8|. 2. If 37| ounces of silver cost $31J, what is the cost of 1 ounce? of 150 ounces ? Ans. If; and $125. 3. If 3| hu. of buckwheat cost $2|, what does 1 bu. cost ? What is the cost of 30 bu. ? A71S. %\^\ and 119. 4. A. divides $3,000^ into 7 equal shares and gives ij DIVISIOIT. 107 of these shares to a benevolent society ; how much does he give to the society? Ans. $1,928||. 5. A. can build a wall in 10 days, B. can do it in 12 days, and 0. in 15 days ; what part of the wall can they all build in 1 day ? A^is. xd" + tV + tt — i- 6. How long will it take them all to build the wall ? Explanation. — Because it takes 1 day to build \ of the wall, it will take 4 days to build the whole. 7. What number multiplied by 1| will give 14} ? 8. The difference of two numbers is 15-^ and the greater number is 20}^ ; what is the less number ? 9. A man inherits f of an estate and gives his son -J- of his share ; what part of the estate does the son receive ? 10. If f of a ton of coal costs $13, what will 7 tons cost ? 11. If coffee costs 13 J cents a pound, how much can be bought for $10 ? for $16 ? 12. How many pounds of coffee can be bought for 1784 at $^ per pound ? at $| ? 13. A. can do a piece of work in 3 days and B. can do it in 2 days; liow long will it take them both to do it? 14. A. bought 24:^ yds. of cloth at $4^ a yard, and sold the whole for $1 28| ; what did he gain per yard ? 15. How many pounds of sugar at 12J cts. a pound must be given for 16-| lbs. of butter at 22^ cts. a pound ? 16. If 6 men can do a piece of work in 7^ days, how long will it take one man to do it ? 17. If a man can walk 10^ miles in IJ hours, how far can he walk in 1 hour ? in 5|- hours ? 18. A merchant owning ^^ of a vessel, sold | of his share for $1,640 ; what was the vessel worth at that rate ? 108 COMMON FRACTIONS. 19. A. can mow a piece of grass in 4 days, and B. can do it in 2 days ; how long will it take both to do it ? 20. A man made a journey in 6J days, traveling at the rate of 22| miles a day ; on his return he traveled at the rate of 24|- miles a day : how many days did it take him to return ? 21. A merchant bought a piece of cloth containing SQ^ycls. for $G5J ; at what rate must he sell it per yard so as to gain 125^ ? 22. A. sets out from Detroit and travels towards Buffalo at the rate of 6f miles an hour; at the end of 2 J hours B. sets out from Detroit and follows at the rate of 8;^ miles an hour: how far apart are they at the end of 5f hours? 23. A farmer sold to a grocer 32^ bu. of com at ^^ a bushel, and 86 lbs. of butter at If a pound. He received in pay 200 lbs. of sugar at $^ a pound, and the remainder in money ; how much money did he receive ? 24. A regiment lost 220 men in battle, which was 4 men more than f of the whole regiment ; how many men were there in the regiment ? 25. A. owned J of a ship and sold f of his share to B. ; B. then sold f of what he bought to C. for $3000; what was the whole ship worth at that rate ? CONTRACTIONS IN MULTIPLICATION AND DIVISION. 83. The rules for multiplication and division of frac- tions lead to certain contractions in multiplication and division of whole numbers, of wliich the following are some of the most important. 1°. The fraction J-f^ is equal to 25 ; hence, to multiply a number by 25, we may DIVISION. 109 Annex 2 ciphers and divide the result by 4- To divide a number by 25, we may Multiply it by 4 cu^^^ divide the result by 100. EXAM PLES. 1. Multiply 3,416 by 25. Ans. ?i^ = 85400. 2. Divide 5,875 by 25. Ans. ^^^^^ ^ = 235. 3. 394 X 25 = ? 8. 9,850 ^ 25 = ? 4. 3,724 X 25 = ? 9. 93,100 ^ 25 = ? 5. 8,123 X 25 = ? 10. 87,525 -^ 25 = ? 6. 10,201 X 25 = ? 11. 46,350 -^ 25 = ? 7. 4,386 X 25 = ? 12. 174,025 ^ 25 = ? Let the pupil deduce rules for multiplying and dividing by 12J, by33i, andbyl25: 13. 81 X 12i = ? 21. 10,125 -^12^=? 14. 914 X 124 = ? 22. $11,425 -^ $12^ = ? 15. $4,834 X 12i = ? 23. 9,125 yds. -M2i = ? 16. 1375 X 331 ^ ? 24. 13,500 ^33-^=? 17. 28,452/^5. X 33^ = ? 25. $29,100 -^334=? 18. l,%tQyds. X 125 = ? 26. 8,700 «/^5.-^33i?/^5. = ? 19. $4,365 X 125 = ? 27. $2,250 -^ 125 = ? 20. 34,115 ijds. X 124 = ? 28. 10,000/5. -j- 125 = ? REVIEW QUESTIONS. (60.) What is a fractional unit? What is a half, a third, a fourth, &c. ? What is the reciprocal of a number ? (61.) What is a fraction ? How do you write a common fraction ? What is the denominator? the numerator? What are terms? (63.) In how many ways may we regard a fraction ? Illustrate. (64.) What is a proper fraction? Illustrate. An improper fraction? Illustrate. 110 DECIMAL FRACTIONS. A mixed number ? Illustrate. A simple fraction ? Illustrate. A complex fraction ? Illustrate. (65.) State the fundamental prin- ciples of fractions. (66.) What is reduction? (67.) Give the rule for reducing a whole number to a fraction with a given unit. (68.) Give the rule for reducing a mixed number to a fraction. (69.) Give the rule for reducing an improper fraction to a mixed number. (70.) Give the rule for reducing a fraction to its lowest terms. (7ii.) Give the rule for reducing fractions to a common denominator. (73.) Give the rule for reducing fractions to their least common denominator. (74.) What is addition of fractions? (75.) Give the rule for addition of fractions. (76.) What is sub- traction of fractions ? (77.) Give the rule for subtraction of frac- tions. (78.) What is multiplication of fractions ? (71).) Give the rule for the multiplication of fractions. (80.) What is division of fractions? (81.) Give the rule for division of fractions. (82.) Give a rule for multiplying by 25. Give a rule for dividing by 25. II. DECIMAL FRACTIONS. DEFINITIONS. 83. A Decimal Fraction is a fraction whose denom- inator is 10, 100, 1,000, or some higher power of 10, (Art. 46), Thus, ^, -jig^, yfoir? ^^c, are decimal fractions. MENTAL EXERCISES. 1. If 1 is divided into 10 equal parts, what is one of the parts called ? 2 of the parts ? 5 of the parts ? 2. If i^g- is divided into 10 equal parts, what is one of the parts called ? 2 of the parts ? 17 of the parts ? 8. If yJ^ is divided into 10 equal parts, what is 1 of the parts called ? 3 of the parts ? 27 of the parts ? 4. WhatisT^oofTV? Aof-rb? A of 1,000 ? -^ of 10,000? What power of 10 is 100? 1,000? 10,000? 100,000? 1,000,000? REDUCTION. Ill DECIMALS, AND THE DECIMAL POINT. 84. Decimal fractions may be written in two ways: their denominators may be expressed, as in ordinary fractions; or their denominators may be indicated by means of a point followed by one or more figures. In the latter case they are called Decimals, and the point (.) used in writing them is called the Decimal Point. NOTATION OF DECIMALS. S5. Decimals are written in the same manner as whole numbers, and both may be written together, decimals on the right and whole numbers on the left, as shown in the following NUMERATION TABLE. ^ o I ^ 's I ^ . ^. 4 I I Oil— irta53?HcoJ2 rSS?5 ^ Si 3963042.5749826.. v_ Whole Numbers. Decimals. Note. — In whole numbers, places of figures and orders of units are counted from the decimal point toward the left ; in decimals, they are counted from the decimal point toward the right. A figure in the first place of decimals denotes tenths ; in the second place it denotes hundredths ; in the third place it denotes thousandths ; and so on, as indicated in the table. Hence, to write a decimal we have the following 113 DECIMAL FKACTIONS. RULE, Write the number of tenths in the first decimat place, the number of hundredths in the second place, the number of thousandths in the third place, and so on. EXAM PLES. 1. Write three fte7iths, as a decimal. Ans. .3. 2. Write twenty seven/ hu7idredths. Ans. .27. 3. ^YvitG forty eight /thousandths. Ans. .048. Note. — In Example 2, because 27 hundredths is the same as 2 tenths and 7 hundredths, we write 2 in tlie first place of decimals and 7 in the second place. In Example 3, because 48 thousandths is the same as tenths, 4 hundredths, and 8 thousandths, we write in the first place, 4 in the second place, and 8 in the third place. Let the student in like manner explain each of the following examples : 4. Two hundred mid thirteen /thousandths. 5. One thousand and six/ten thousandths. 6. Four thousand two hundred and seven/millionths. 7. Two hundred and seventy four thousand three hun- dred and forty three /millionths. 8. Twenty three million, two hundred and four thousand, five hundred and seventy seven/hundred milUonths. A mixed decimal is a mixed number whose fractional part is a decimal. Thus, six, and three/ tenths is a mixed decimal; it may be written 6.3. In all such cases the integral part is written on the left of the decimal point. 9. Twenty, 2Lndi forty four /hundredths. Ans. 20.44. 10. Thirty seven, and seventy two /thousandths. 11. Forty seven, and two hundred and nine /milUonths. Note. — In the preceding examples, decimals are in italics. The sign / separates the numerator from the denominator. REDUCTION". 113 From what precedes, we see that a decimal fraction may Be expressed decimally by writing its numerator, and then placing a decimal point so that the number of figures following it shall be equal to the number of ciphers in the denominator. If the number of figures in the numerator is less than the number of ciphers in the denominator, a sufficient number of ciphers must be prefixed, that is, tvritten before the numerator. EXAMPLES. 1. A = -3. 4. 9tWV ::= 9.313. 2. Tff^ = .045. 5. 4r«o = 4.079. 3. tMt^ = .0017. 6. 256TnjW^ = 256.00117. NUMERATION OF DECIMALS. 86. From what has been explained, we see that a deci- mal may be read by the following RULE. Bead the significant part as a whole number, and add the name of the lowest unit of the decimal. Note. — Before reading a decimal the pupil should numerate it, that is, he should begin at the left hand and name the units of each place : thus, tenths, hundredths, thovsandths, etc. , according to the table. Read the following decimals : 1. .087. Ans. Eighty seven j thousandths. 2. .000317. Ans. Three hundred and seventeen I millionths. 3. .0027. 6. .52346. 9. .11122. 4. .10364. 7. .50067. 10. .224785. 5. .00201. 8. .320315. 11. .0067412. 114 DECIMAL FRACTIONS. Note. — In mixed decimals we read the integral and the decimal parts separately. 12. 120.009. Ans. One hundred and twenty, and nine/thousandths. 13. 19.00015. 15. 150.15632. 17. 45.36251. 14. 212.1236. 16. 34.001725. 18. 111.009265. DECIMAL CU RRENCY. 87. The currency of the United States is purely deci- mal, the primary unit being 1 Dollar. In it Dimes are te?iths of a dollar, Cents are hundredths of a dollar, and Mills are thousandths of a dollar. Dollars, cents, and mills are generally written in the form of a mixed decimal, the decimal point being placed after dollars. Thus, the expression 174.853, denotes 74 dollars, 8 dimesy 5 cents, and 3 mills ; it is read 75 dollars 85-]^ cents. Note. — An eagle is equal to $10. In business transactions the terms eagle, dime, and mill are but little used, sums of money being expressed in dollars and cents. FUNDAMENTAL PRINCIPLES. 88. Moving the decimal point one place to the right changes tenths to units, hundredths to tenths, and so on ; but this is equivalent to multiplying the decimal by 10 : hence, the following principle : 1°. Moving the decimal point one place to the right is equivalent to multiplying the decimal hy 10. In like manner we have the following principle : 2°. Moving the decimal point one place to the left is equivalent to dividing the decimal by 10. Annexing a cipher to a decimal multiplies both numera- tor and denominator by 10 ; but this does not alter the REDUCTIOJS^. 115 value of the fraction (Art. 65); hence, the following- principle : 3°. Annexing one or more ciphers to a decimal does not change its value. In like manner we have the following principle : 4°. Striking out one or more terminal ciphers does not change the value of a decimal, REDUCTION OF COMMON FRACTIONS TO DECIMALS. 89. Let it be required to reduce, that is, to change I to the form of a decimal. Explanation. — The value of | is equal to opbratiox. 5 -J- 8 (Art. &2) ; to find this quotient we annex 8"\'=i00n three ciphers to 5, which is equivalent to mul- tiplying it by 1,000, and then perform the .625 division ; but this result is 1,000 times as great as the true value of the fraction ; we therefore divide it by 1,000, which is done by pointing oflf three decimal figures (Priii. 2, Art. 88). In like manner we may treat all similar cases ; hence, the RULE. Annex ciphers to the numerator and divide the result by the denominator ; then point off from the right of the quotient a number of decimal figures equal to the number of ciphers annexed. Note. — If the number of figures in the quotient in less than the number of ciphers annexed, prefix the requisite number of ciphers. EXAMPLES. Reduce the following fractions to decimals : 3. fj. 6. ^^. 9. r^. 116 DECIMAL FEACTIONS. Note. — To reduce a mixed number to a decimal form, we reduce the fractional part to a decimal and annex the result to the inte- gral part. 10. 19J. 13. 11-^. 14- ai^Ar- 11. 2m. 13. 110^17. 15-*A%- APPROXIMATE RESULTS. 90, Ifc may happen tliat the division described in the last article will not terminate, no matter how many ciphers we annex. In this case the decimal found by stopping at any particular step of the division is called an approx- imate value of the given fraction. Thus .1904 is an approximate value of ^. 'In this case ^ is greater than .1904 and less than .1905 ; hence, it differs from either by less than they differ from each other, that is, by less than .0001. In like manner the approximate value of a frac- tion found ,by stopping at any decimal figure differs from the true value of the fraction by less than the correspond- ing decimal unit. If we stop at any decimal figure and increase it by 1 when the next figure is equal to, or greater than 5, the error can never exceed ^ the corresponding decimal unit. Thus, f = .667, and -J- = .333, each to within less than -J of .001. This is the practical method of finding approximate values of decimals. Note. — In applying the principles of decimals to practical cases we shall habitually follow the method of approximation just ex- plained, and, except in special cases, we shall limit the approximation either to three or to four decimal places. In United States money we shall habitually limit the approxima- tion to three decimal places ; each result will then be true to within Iialf a mill, or the twentieth of a cent. ADDITIOI^". 117 EXAM PLES. Reduce the following fractions to decimals, carrying the approximation to the fourth place : 1. foff 6. 4f f 11. 21|i. 2. «. 7. iofSJ. 12.1^^. 3. lA. 8. Aof^i 13. 14^. 4. A. 9. f of I of 44. 14. IS^Vt- 5. dii. 10. ^ X 4f. 15. 4fi|. ADDITION OF DECIMALS. DEFINITION. 91. Addition of Decimals is the operation of find- ing the sum of two or more decimals. MENTAL EXERCISES. 1. AVhat is the sum of 4 tenths and 5 tenths? of .3 and .6 ? How many units and tefiths of a iinvt in the sum of .8 and .9 ? What is the sum of .3, .5, and .9 ? of .5, .7, .9, and .6 ? 2. How would you read 49 hundredths in tenths and hundredths 9 How many hutidredfhs are there in 6 te?iths? What is the sum of .5 and .49 ? of .59 and .4 ? How would you read three hundred and seventeen/thousandths in tenths, hmidredfhs, and thousandths ? 3. What is the sum of 13 cts. and 22 cts. ? of $.13 and 1.22 ? Is there any difference between 13 cts. and $.13 ? What is the sum of 25 cts. and $.45 ? of $.4, 1.25, and 1.7 ? of .4, .25, and .7 ? Note. — Addition of decimals depends on tlie same principles as addition of integers. 118 DECIMAL FRACTIOlfS. OPERATION OF ADDITION. 93. Let it be required to find the sum of 4.035, 76.19, and 114.0305. Explanation. — We write the decimals operation. so that units of the same order shall stand in 4.035 the same column ; this will bring all the lyo iq decimal points in one column: then begin- /o.ly ning at the right, we add each column sep- 114.0305 arately, setting doicn and carrying as in sim- ~ pie numbers. Hence, the Sum. 194.2555 RULE. Write the decijnals so that units of the same order shall stand in the same column, and add as in sim- ple numbers. Note. — The decimal points of the numbers to be added, and of their sum, must stand in the same column. EXAMPLES. (1.) (2.) (3.) (4.) (5.) 3.057 5.6000 5.43 0.105 $3.97 14.086 17.0032 12.998 0.0012 $4,295 209.3154 35.9070 317.0971 0.25 $11,464 226.4584 58.5102 335.5251 0.3562 $19,729 6. Find the sum of .632, .718, 3.202, and 111.1. 7. Of .0049, 47.0426, 37.041, and 360.0039. 8. Of $81,053, $67,412, $93,172, and $14.38. 9. Of $59,317, $69,565, $8,213, and $7,775. 10. Of 3.25 Ihs., 47.348 Ihs., 748.4 lbs., and 29.32 lbs. 11. Of 672.5 yds., 4.923 yds., 80 yds., and .0764 yds. 12. Of 72.5 + 140 + 340.03 + 21.5715 + 4.0008. 13. Of 2.8146 + .0938 + 8.875 + 231.2788 + 4.0087. ADDITIOlf. 119 14 Of 54.3 fU + 7.29 ft + 180.0046 ft + 187 ft + 3.024 /if. 15. Of 57.038 + 95.00487 + 53.4690 + 107.00003. 16. Of $62.70 + $2.03 + $4,009 + $78.15 + $114. 17. Of .0009 + 3.0021 + .128 + 8.0469 + 59. 18. Of 3.0102 lu, + 11.5008 hu, + 73.07 lu, + 2.92 lu. H- 9.5 hu. 19. 2.005 + 110.301 + .069 + 7.375 + 2.25 = ? 20. 17.215 + 3.0567 + 2.072 + 4.009 + 54.75 = ? 21. 29.157 ft + 8.0016 ft + 77.29 ft + 32.004/^. + 8.848 /f. = ? 22. 14.2351 + 651.012 + 2.219 + 3.157 + 13.614 = ? 23. $861.55 + $378.25 + $461.37 + $683.57 + $1,205.47 = ? 24. 213.7 U. + 2.913 lu. + 14.769 lu, + .0078 lu. = ? 25. 15.753^^5. + 2.069yds. + ItQUdyds. + 10.27yds. + 3.2107 yds. = ? PRACTICAL PROBLEMS. 1. A boy paid 28 cts. for a slate, 75 cts. for paper, and 94 cts. for an Arithmetic ; what did he pay for all ? Ans. . $1.97. 2. One field contains 5.3 acres, a second contains 11.43 acres, a third contains 17.59 acres, and a fourth contains 3.175 acres ; how many acres in all of them ? 3. A. bought 16 hams for $31.87|^, a bag of coffee for $17.92, a chest of tea for $12.75, and a firkin of butter for $21.37-|- ; what did they all cost ? 4. A man bought candles for $6.89, flour for $25.56, raisins for $1.12J- (^. e. for $1,125), cheese for $8.37^-, and sugar for $5.44 ; what was the cost of the whole ? 1 20 DECIMAL FRACTIOI^S. 5. A merchant sold 4- pieces of muslin ; the first con- tains 34.25 yds.f the second 38.056 yds., the third 40.2 yds., and the fourth 37,225 yds. ; how many yards in all ? 6. A farmer has 4 bins of wheat ; in the first there are 86.35 hu., in the second 73.125 hu., in the third 96.5 hi , and in the fourth 74.3 bu. ; how many bushels in all ? 7. In 5 piles of wood there are respectively 4.316 cords, 8.23 cords, 11.25 cords, 7.364 cords, and 13.819 cords ; how many cords are there in all the piles ? 8. B. bought a house for 15,000, a store for $6,290, mer- chandise for 123,654.12, a horse for 1278.53, a farm for 19,371.60, bank stock for $11,500, and a watch for 892.72^; Avhat did the whole cost him ? SUBTRACTION OF DECIMALS. DEFINITION. 93. Subtraction of Decimals is the operation of finding the difference between two decimals. MENTAL EXERCISES. If 6 teyiths are taken from 9 tenths, how much will remain ? What is the difference between 23 tenths and 14 tenths ? 23 tenths is equal to how many units, and how mmy tenths? 2. What is the difference between 35 hundredths and 2 tenths 9 How many tenths in 35 hundredths 9 What is the difference between .45 and .25 ? 4.5 and .6 ? .7 and .15 ? 4.5 and 2.2 ? 3.5 and 2.7 ? 3.2 and 1.9 ? Note. — Subtraction of decimals depends on the same principles as subtraction of integers. SUBTRACTION. 121 OPERATION OF SUBTRACTION. 94. Let it be required to subtract 4.079 from 11.362. Explanation. — The subtrahend is opbration. written under the minuend, so that units Minuend 11 362 Subtrahend, 4.079 of the same order shall stand in the same column ; this will bring the decimal points into the same column: the operation is Remainder, 7.283 then performed as in the subtraction of simple numbers. Hence, the following RULE. Write the subtrahend under the minuend, so that units of the same order shall stand in the saine column ; then subtract as in simple numbers. Note. — The decimal points of the minuend, of the subtrahend, and of the remainder must stand in the same column. EXAMPLES. (1-) (2.) (3.) (4.) Minuend, 5.316 17.0091 1075.0567 $312,475 Subtrahend, 2.013 11.9902 287.9374 $214,268 Remainder, 3.303 5.0189 787.1193 198.207 Note. — If the subtrahend contains more decimal figures than the minuend, annex the requisite number of ciphers to the minuend, or conceive them to be annexed (Principle 3°, Art. 88). (5.) (6.) (7.) (8.) (9.) 13.700 13.7 884.1300 884.13 $8. 8.299 8.299 33.7865 33.786 5 $4^5 5.401 5.401 8503435 850.3435 $3^5 10. From 298.789 subtract 196.493. Ans. 102.296. 11. From 2684.11 subtract 199.8637. Ans. 2484.2463. 122 DECIMAL FRACTI0K8. Find the difference between, 12. 127.334 and 55.827. 21. ^bft. and 25.0003//?. 13. 94.8607 and 27.861. 22. 6 yds. and .0006 yds. 14. 986.444 and 98.6438. 23. $14,003 and 19.875. 15. $17,025 and $7,255. 24. 13.4072 and 9.1875. 16. 2.867/if. and .9965/i?. 25. 18.65 and 12.0734. 17. $661.40 and $95,472. 26. 17.314 and 12.9921. 18. $25,000 and $1,077. 27. 13.3125 and 8.4139. 19. 100 yds. and 99.001 yds, 28. $34,883 and $9.43. 20. 41.02 and 40.021. 29. 87.007/^5. and 10.895/Z>.^. 30. $1.87 + $3.945 + $27— ($6.42 + $15.07 + $.25)=? 31. 125.6 Ihs. + 27.42 Z2>s.- (4.3 Ihs. + 12.11 Ihs. + 9 Ihs.) =? 32. $5,000 + $325,175 — ($2,710.75 - $147.56) = ? 33. ($794.26-$215.875)-($456.375-$211.12)=:? PRACTICAL PROBLEMS, 1. Mr. Holmes bought a cow for $45,125 and sold her for $49.18; what did he gain? 2. From a piece of cloth containing 42.37 yds., 16.89 yds. were cut off; how many yards remained in the piece? 3. What is the difference between $875,043 and $704.91 ? 4. How much must I add to $617.37^ to make $922.75 ? 5. A.'s income is $6,250 per year, of which he spends $3,142.75 and lays up the rest ; what does he lay up ? 6. From $981.43 + $456.81 subtract $498.75. 7. From $10,000 subtract $4,367.18 + $3,587.47. 8. From $965 + $341.60 subtract $433.33 + $89.47. 9. A man received the following sums: $27.40, $68.75, $810.47, $386.59, and $2.20; he paid out the following sums: $78.67, $129.72, $119.46, and $3.88; how much had he left? SUBTRACTION". 133 10. A. had, at the beginning of the year, goods worth $10,500 ; during the year he bought goods to the amount of $9,345.75, and sold to the amount of $13,450.95; at the close of the year he had goods worth $11,122.37 ; how much did he make during the year ? 11. A lady bought a dress for $42.18, a bonnet for $17.65, and a pair of gloves for $1.87|- ; she gave for them a $100 bill ; how much change ought she to receive ? 12. A. bought 37.41 cords of wood, of which he sold 8.3 61 and burned 13.426 C. ; how many cords had he left? 13. A flagstaff is made up of two parts, the upper part being 27. 84= ft. long, and the lower part 57.86/^. long: now if the lower part is set 11.31/if. in the ground, how many feet of the whole staff is above the ground ? 14. A man had $137.26, of which he spent $17. 87 J for coal, $22.12i for flour, $7.42 for soap, and $32.79 for a suit of clothes ; how much had he left ? 15. From a hogshead of sugar containing 397.25 lbs., a grocer sold parcels as follows : 110.25 lbs., 64.5 Z^s., 14.25 lbs,, 29.375 lbs., 39.23 lbs., and 16.33 lbs. ; how much was left ? 16. A. is to travel bdHi^niles in 3 days ; the first day he travels 196.4:miles, and the second day he travels 201.25 miles : how many miles must he travel the third day ? 17. A farmer had a colt worth $147^, which he traded for a cow worth $42,375, 4 calves worth $22|, and the balance in cash ; how much cash did he receive ? 18. A merchant bought a piece of cloth for $75f, a box of ribbons for $25|, and a quantity of thread for $27.87 ; he sold the cloth for $87,125, the ribbons for $22.16, and the thread for $21f : did he gain or lose, and how much ? 124 DECIMAL FRACTION^. MULTIPLICATION OF DECIMALS. DEFINITION. 95. Multiplication of Decimals is the operation of iinding the product of two decimals. MENTAL. EXERCISES. 1. If a copy-book costs 2 tenths of a dollar, what will 4 copy-books cost? What is 4 times 2 tenths of a dollar? What is 6 times 2 tenth's? How many units and how many tenths in the product? How many tenths are 6 times 9 tenths 9 How many units and how many tenths in the product ? 2. If a melon costs 3 tenths of a dollar, what does 1 tenth of a melon cost? What is 1 tenth oi 3 tenths? What is the product of .3 by .1 ? of .3 by .2 ? of .3 by ,5 ? of . 7 by . 9 ? of . 8 by . 8 ? What is the decimal unit of the product of tenths by tenths 9 3. What is 1 tenth of 4 hundredths 9 What is the pro- duct of 4 hundredths by 1 ^ewjf^ 9 of .04 by .2 ? of .04 by .4 ? of .05 by .7 ? of .09 by .9 ? What is the decimal unit of the product OiX hundredths by tenths9 4. What is the product of .005 by .2 ? of .007 by .09 ? of .006 by .008 ? of .7 by .6 ? of .7 by .04 ? of .03 by .07 ? 5. What is one hundredth of one hundredtli ? What is the product of 3 hundredths by 7 hundredths 9 What is the decimal unit of the product of hundredths by hun- dredths 9 What is the product of .09 by .06 ? of .07 by .08 ? Note. — Multiplication of decimals depends on the same principles as multiplication of integers, and also on the rule for the multiplica- tion of comm,(m fractions., (Art. 79). MULTIPLICATION. 125 OPERATION OF MULTIPLICATION. 96. Let it be required to find the product of 7.8 and .82. Explanation. — Here the decimals opebation. are first changed, to equivalent common 78 82 fractions and multiplied together by '-o X .o/C = -- x jx^ the rule of Article 7^) ; the resulting fraction is then reduced to the decimal __ __ (3,395^ form ; in doing this, we have actually 1000 multiplied the given decimals together, without reference to their decimal points, and in the result we have pointed off as many decimal figures as there are in both factors. Since all similar cases may be treated in the same manner, we have the following RULE. Multiply as in simple numbers, and point off, frojn the right of the product, as many decUnal figures as there are in both factors. Note. — If the number of places in the product is less than the number of decimal places in both factors, prefix as many ciphers as may be necessary. EXAM PLES. 1. What is the product of 3.05 by 4.102 ? Ans. 12.5111. 2. What is the product of .003 by .042 ? Ans. .000126. Perform the following multiplications : 3. $38.4 by 16.7. 9. .0463 /^>5. by .0081. 4. -$14.25 by .375. 10. 701.005 x 60.06. 5. 1,500 ^>?^ by .00014. 11. 456.05 x 3.825. 6. $1,009 by .0012. 12. 308.25 x .0775. 7. 146.05 ?/^s. by 128.6. 13. 27.032 x 14.3. 8. Mhft. by .16. 14. $380.06 x 22. 126 DECIMAL FRACTION'S. 15. $24.07 X .125. 25. 0.0156 rods x 6.75. 16. t75 X .33. 26. .2897 x 3020. 17. $456.87 X .066. 27. $37.55 x 45.64. 18. $798,007 X .08. 28. 3.005 x 21.82 x 14.71. 19. $.034 X .08. 29. 8.013 x 11.7 x 0.774. 20. 1lA6lbs. X 2.7504. 30. 12.12 x 300.7 x 8.004. 21. 42.2 X 2.004. 31. 0.713 x 2.346 x 2.005. 22. 79.004 X .00473. 32. $12.5 x 7.2 x 16.5. 23. 412.5384 x 1.00003. 33. 4:.2 lbs. x 8.1 x 2.4. 24. ^OMyds, x .00293. 34. 1.7 yds. x 11.4 x 82.3. Note. — To multiply a decimal or a mixed decimal by 10, 100, 1,000, etc., move tlie decimal point as many places to tlie right as there are ciphers in the multiplier, annexing ciphers to the multi- plicand if necessary. 35. What is the product of 77.56 by 10? Ans. 775.6. 36. What is the product of .0075 by 100 ? Ans. .75. 37. What is the product of 6.6 by 1000 ? Ans. 6600. 38. ($31.45 + $18.2) x 7.2 — $240.15 = ? 39. {ISO.Qlbs. - SQAlbs.) x (67.2 - 3.47) = ? 40. $150.75 x 16.3 + $211.5 x 16 — $114.25 x 9 = ? 41. (dS.4:yds. + 5Q.4:yds.) x 7.2 - 18.36^^^/5. x 8.1 = ? 42. (463.45 + 31.4 - 2.175) x (18.2 — 11.07) = ? . PRACTICAL PROBLEMS. 1. What is the cost of 17 barrels of flour at $6.37^ a barrel ? 2. Of 85^ lbs. of tea at $1.37i a pound ? 3. Of 311 yds. of linen at Q4^cts. a yard? 4. Of 41^ gallons of wine at $3.12| a gallon ? 5. Of 278 cords of wood at $9.62| a cord ? . 6. Of 17 lbs. of tea at $.75 a pound ? MULTIPLICATION. 127 7. Of 7.5 reams of paper at $3.62|- a ream ? 8. Of 2,754 sheep at $5,121 apiece ? 9. Of 47.75 bu, of corn at $.875 a bushel ? Note. — Let the student apply the rule for approximate results explained in Art. 90, finding values to the nearest mill. To secure uniformity the rule should be applied at each step of the operation. 10. A grocer sold 25.5 lbs. of sugar at 12^ cts. a pound, and 18.6 lbs. of lard at Id^cts. a pound ; how much did he receive for both ? 11. A farmer sold Sl^bu. of oats at 4:2^ cts. a bushel, and 35:^ bu. of potatoes at 37-J cts. a bushel ; he received for the same 4:3^ yds. of muslin at 12^ cts. a yard, and the balance in cash : how much cash did he receive ? 12. A. sold 75 bu. of wheat at I1.12J- a bushel, 36.2 bu. of beans at $2,374 ^ bushel, and 97^ lbs. of butter at 22^ cts. a pound ; what did he receive for the whole ? 13. A man's wages are $18,874 ^ week, and his expenses are $13.25 a week ; how much can he save in 14^ weeks ? 14. A man was to walk 245| in 7 days : for the first 3 days he walked at the rate of 34.36 miles a day, and for the next three days he walked 36.75 miles a day; how far had he to walk the seventh day ? 15. The distance from St. Louis to New Orleans is 1332 miles ; tAvo boats start at the same time, one from St. Louis , and the other from ISTew Orleans, are run towards each other ; the boat from St. Louis makes 230f miles a day, and the one from New Orleans 196| miles a day: how far apart are they at the end of 2^ days ? 16. A man starts from a certain point and travels in a certain direction at the rate of 7.25 miles an hour ; at tlie 128 DECIMAL FRACTIOiq"S. end of 2J hours a second man starts from the same point and travels in an opposite direction at the rate of 6.29 miles an hour : how far apart are they at the end of the sixth hour ? 17. A carpenter earned $12.87^ a week for 3 weeks; the first week he spent 18.333, the second week he spent $9.18, the third week he spent $7|, and the rest he saved ; how much did he save ? 18. A gardener sold his cabbages for 1212.87^, and his turnips for $118.33 ; the cost of raising the cabbages was 1119.75, and the cost of raising the turnips was $99.87^: what was his profit on the two crops ? 19. A man bought 43 sheep at the rate of 4 dollars and 67i cents a piece, and sold the lot for 215 dollars and 42^ cents ; did he gain or lose, and how much ? 20. A man made a journey as follows: he traveled 7j hours by rail at the rate of 22.75 miles an hour, 9-| hours by stage at the rate of 6.75 miles an hour, and 11.75 hours on foot at the rate of 4.62 miles an hour; what was the length of the journey ? DIVISION OF DECIMALLS. DEFINITION. 97. Division of Decimals is the operation of find- ing the quotient of one decimal by another. MENTAL EXERCISES. 1. What is the product of .3 by .5 ? What then is the quotient of .15 by .5 ? How many decimal places in the dividend? in the divisor? in the quotient? 2. What is the product of .12 by .13 ? What then is DIVISION^. 129 the quotient of .0156 by .12 ? by .13 ? How many decimal places in the dividend? in the divisor? in the quotient ? 3. What is the product of .003 by .9 ? What then is the quotient of .0027 by .9 ? How does the number of decimal places in the quotient compare with the number in the dividend and in the divisor. Note. — Division of decimals depends on the same principles as division of integers, and also on the rule for the division of frac- tions (Art. 81). OPERATION OF DIVISION. 98. Let it be required to divide 7.8 by .125. Explanation. — Here we opebation. have reduced the given ded- •j'^g 'j'g ]^25 mals to common fractions A A' 'A Ay. ,x. 1 * -125 10 • 1000 and divided by the rule of Art. 81 ; we have then re- _^ 78 1000 _ 78000 duced the result to the deci- — 10 ^ 105 — 105 — '~ mal form ; in doing this, we have actually annexed three = 624 -7- 10 =: 62.4 decimal ciphers to the divi dend (Art. 88), and divided the result by the divisor, without ref- erence to the decimal point ; then from the quotient we have pointed off as many decimal figures as the number in the reduced dividend exceeds that in the divisor. All similar cases may be treated in the same manner ; hence, the following RULE. Annex decimal ciphers to the dividend if neces- sary ; then divide as in simple numbers and^ point off from the right of the quotient as many deci7)^al figures as the number of decimal places in the dividend exceeds that in the divisor. 5 130 DECIMAL FRACTIONS. Notes. — 1. The dividend must contain as many decimal figures as the divisor, but it may contain more. If the number of decimal figures is the same in both, the quotient is a whole number. 2, If the number of figures in the quotient is less than that re- quired by the rule, a sufficient number of ciphers must be prefixed. EXAMPLES. 1. Divide 40.05 by 45. Ans. 8.9. 2. Divide .0141 by .00047. Ans, 30. 3. Divide 2.3 by 1437.5. Ans. .0016. Perform the following indicated divisions, limiting ap- proximate values to the fourth decimal place : 4. .00125 -^ .5. 15. 15.875 -f- 35.25. 5. $34.75 -^ 25. 16. 480 — 3.12. 6. 46.103 -j- 2.14. 17. $1.8 -^ 28.8. 7. 7.8125 ^ 31.25. 18. 17.1031 yds, ^ .63. 8. $2756.25 -^ 31.5. 19. .09925 -^ .37. 9. $68,875 -f- 14.5. 20. 3.72812 ^ 4.07. 10. 3414.52 -^ 30.25. 21. $18.1771 -^ 27.13. 11. 16.025 -r- .045. 22. 101.6688 -~ 43.08. 12. .9375 /«J. -T- .075. 23. 1.51088 -^ .019. 13. 112.1184 -^ 9.16. 24. 187.12264 ^ 1.52. 14. 9322.15 -^ 6.275. 25. $71.1022 -^ $9.43. Note.— To divide a decimal or a mixed decimal by 10, 100, 1000, &c., move the decimal point as many places to the left as there are ciphers in the divisor, prefixing ciphers to the dividend if necessary. 26. What is the quotient of 77.56 by 10 ? Ans. 7.756. 27. What is the quotient of .0075 by 100 ? Ans. .000075. 28. What is the quotient of 6.6 by 1000? Ans. .0066. 29. ($28 + $11.75) -r- 1.25 -f $38.75 = ? 30. $50 -^ 5.75 -f ($10 - $3.75) x 1.2 = ? 31. ($13.75 - $1.87i) -^ (12.75 - 4.5) = ? DIVISION. 131 (63.5/if. - 24.25/0 -r- (17.25 - 11.75) = ? uu. 4^7/ds. X 2.2 + lliyds. -^ 1.25 + IS.Sl 6 yds. = 34. (47.3 Z^5. + 6.7 lbs.) -^ (34.18 - 16.78) = ? 32 33. 4^2/^ PRACTICAL PROBLEMS. 1. If a man can earn $519.75 in 13.5 weeks, how much can he earn in 1 week ? A?is. $38.50. 2. If 20.5 bic. of buckwheat cost $12.71, what is the cost of 1 bu. ? of 7i bii. ? Atis. 62 cts. ; $4.65. 3. If a barrel of flour costs 15.75, how many barrels can be bought for $1,035 ? A7is. 180 bbls, 4. If 1,000 acres of land cost $17,586, what is the cost of 1 ^. ? of 75i A. ? Ans. $17.58^^ ; $1,32.743. 5. If 1 acre costs $25.62, how many acres can be bought for $1,242.57? 6. K 75.3 cords of wood cost $640.05, how much will 1(7. cost? 6.3 a? 7. If 1 cord of wood costs $8.25, how much wood can be bought for $156. 75 ? For $30.52^ ? 8. At $4.28 a yard, how much cloth can be bought for $74.90 ? How much for $152.52^ ? 9. A farmer sold 27.5 lbs. of butter at 20 cts. a pound, and 17.5 bu. of oats at 75 cts. a bushel ; he took in payment in sugar at 12^ cts. a pound: how many pounds did he receive ? 10. There are 31.5 gallons in a barrel; how many bar- rels are there in 2756.25 gallons ? 11. There are l,"^ 60 yds. in 1 mile; how many miles are there in 23,760 yds.? In 26,840 yds.? 12. A merchant buys a piece of cloth containing 35 yds. 132 DECIMAL FRACTIONS. for $87.50; he wishes to sell it so as to gain 117.50: at what price must he sell it per yard ? 13. A man can travel 4:4:1.5 miles in 7.5 days; how far can he travel in 1 day ? in 9 J days ? 14. A man travels 7.25 days at the rate of 211.5 miles a day; on his return he makes the whole journey in 5 days: how many miles is that per day ? 15. A. and B. start at the same time from points 147.16 miles apart, and travel toward each other till they meet ; if A. travels at the rate of 7.75 mi. an hour, and B. at 6.4 mi. an hour, how long before they meet ? 16. A speculator bought 78.25 acres of land for 19,781.25, and sold it so as to gain 13.50 an acre ; what did he get per acre ? 17. If 621! bu. of wheat cost 1592.87^, what is the cost o^lbu.f of 6.6 bu. 9 18. If 115 lbs. of beef cost $19.89^, what will 93 lbs. cost at the same rate ? 19. If I pay $39.48 for 28 bu. of wheat, what must I pay for 48 bushels at the same rate? 20. A grocer bought 114 gallons of vinegar at 22^ cts. a gallon, and sold it so as to gain $7.98 ; at what rate per gallon did he sell it ? 21. A. starts from a certain place and travels along a road at the rate of 4.66 miles an hour ; B. starts 13.75 miles behind him and travels in the same direction at the rate of 5.91 miles an hour: how long before B. will over- take A. ? 22. A farmer sold 22.5 bu. of wheat at $1.18 a bushel, and a certain number of bushels of oats at 68 cts. a bushel ; DIVISION. 133 he received for his oats $22. 41 more than he did for his wheat : how many bushels of oats did he seU ? 23. If mibs. of coffee cost $2.07, what will lib. cost? What will 31 J /^5. cost? 24. How many bushels of oats at 62^ cts. a bushel will pay for 4J thousand of lumber at 17.50 a thousand ? 25. A farmer exchanged 70 Ui. of rye at $0.92 a bushel, for 40 hi. of wheat at 11.371- a bushel, and the balance in oats at 10.40 a bushel ; how many bushels of oats did he receive ? 26. If a man can travel 32.48 miles in .8 of a day, how far can he travel in 5.3 days ? 27. What is the sum of the quotients of 24 by 9.6, of 42.75 by 11.4, and of 17.85 by 4.2 ? 28. A. and B. start together and travel in the same direction around an island whose circuit is 4.2 miles ; A. travels at the rate of 4.6 mi. an hour, and B. at the rate of 5.2 mi, an hour: how many hours before they are together again ? 29. A man bought a farm containing 64.5 acres for $1,773.75 ; what was that per acre ? 30. How many cords of wood at $8 a cord must be paid for 24 yards of cloth at $3.50 a yard ? 31. If 60 bushels of turnips cost $18.60, how much will 19 bushels cost ? 32. If 10 tons of coal cost $57.50, how many tons can be bought for $235.75 ? 33. A tailor cuts from 31.25 yds. of cloth 6 coats, each taking 3.75 pds., and makes the rest into vests, each taking 1.25 yds. ; how many vests does he make ? 134 BUSINESS OPERATIONS. III. CONTRACTIONS AND BUSINESS OPERATIONS. ALIQUOT PARTS. 99. An Aliquot Part of a number is one of the equal parts, whether integral or fractional, into which the num- ber can be divided. The principal aliquot parts of a dollar are shown in the following TABLE. 50 cts., equal to J of II. 12|^ ds., equal to \ of $1. 33i ds., '' '' i '' '' 25 ds., '' " \ '' '' 20 ds., '' '' \ '' '' 10 ds.. a "tV" " 6 J ds.. a "t^" " 5 ds,, (( "^•' " 100. To find the cost of any number of things when 1 thing costs an aliquot part of a dollar, we have the following RULE. Divide the number of things by the number of times the price of one thing is contained in $1; the quotient will be the required number of dollars. EXAM PLES. 1. What is the cost of 64 bushels of oats at 50 cents a bushel? Ans. 1-^ = $32. 2. What will 116 pounds of beef cost at 20 cents a pound ? Ans. $J^ = -^23.20. 3. Of 250 melons at 25 ds. each ? 4. Of 144 pencils at VZ^cts. each ? 5. Of 75 oranges at 5 ds. each ? ALIQUOT PARTS. 135 6. Of 69 yds. of sheeting at 33i ds. a yard ? 7. Of 50 dozen marbles at 6J cts. a dozen ? 8. Of 73 lbs. sugar at 12J cts. a pound ? 9. 17 ^o;2e;i eggs at 25 cts. a dozen ? 10. Of 117 ^w.«r^5 berries at 20 cts. a quart ? 11. Of 47 lbs. coffee at ^?>^cts. a pound ? ' 12. Of 145 lbs. rice at 6J cts. a pound ? 13. Of 87.3 Ihs. coffee at 33^ cts. a pound ? 14. Of 315 lbs. sugar at 12^ cts. a pound ? * 15. Of 70 lbs, of butter at 33^ cts. a pound ? 16. Of 35 doz, eggs at 20 cts. a dozen ? 101. To find the cost of things sold by the hundred, or by the thousand, we have the following RULE. Multiply the cost of 100, or 1000 things, hy the mnnher of things, and move the decimal point 2, or 3 places to the left. EXAMPLES. 1. What is the cost of 460 oranges at $3.50 joer hundred ? . $3.50x460 ._,_ Ans. j^r = $16.10. 2. Of 1,726/j?. of boards at $3 per 1,000 /if. ? 3. Of 47,555 bricks at $7.50 per 1,000 ? 4. Of freight on 8,714 lbs. at 62^ cts, a hundred? 5. Of 83,750/i^. of stone at $60 a thousand 9 6. Of 763 lbs, of pork at $4.50 a hundred 9 7. Of 511 lbs. of beef at $7 a hundred 9 8. Of 1,432 lbs. of pork at $8.25 per hundred 9 136 BUSINESS OPERATIONS. 9. Of 8,741 /if. of plank at $30 per thousand f 10. Of 4,875//^. of boards at $17 per thousand f 11. Of 7,320 papers tacks at $30 a thousand 9 12. Of 756/if. of stone at $5 a hundred? 13. Of 3,450 oysters at $2 a hundred? 14. Of 7,846 lis. of hay at dOcts, a hundred? 102. To find the cost of things sold by the ton, that is, by the two thousand pounds, we have the following RULE. Multiply half the cost of a ton hy the number of pounds, and move the decimal point in the product three places to the left. EXAMPLES. I. What is the cost of 3,475 lis. of plaster at $7.50 per . o A ^3.75x3475 .^_„^ ton ? Ans. j^^^ = $13.03|. 2. Of transporting 6,742 lbs. at $7 a ton ? 3. Of 6,527 Ihs. of oats at $30J a ton ? 4. Of 18,747 lbs. of coal at $8| per ton ? 5. Of 8,142 lbs. of iron at $100 a ton ? 6. Of 3,120 lbs. of wool at $660 a ton ? 7. Of 1,620 lbs. of hay at $16 per ton ? 8. Of 5,782 lbs. of pig iron at $22 a ton ? 9. Of 7,711 lbs. of straw at $15 per ton ? 10. Of 8,824 lbs. of hay at $15 per ton ? II. Of 3,509 lbs. of wheat at $40 a ton ? 12. Of 2,250 lbs. of coal at $7.40 a ton ? 13. Of 14,710 lbs. of crushed stone at $3 a ton? 14. Of 16,318 lbs. of meal at $37 a ton ? BILLS AND ACCOUNTS. 137 BILLS AND ACCOU NTS. 103. A bill is a written statement of goods sold, ser- vices rendered, or money paid, with the date of each item. The ordinary form of a bill of items is shown below, in which @ stands for at. i^/c/e^ S^ '^o. y/ ^■a'C^.4'i€id. ix^ ^-e-ci . . . . @ ^0. fS ^^fS So 7^ /4 ^■^& ^■044.4'l^d -O^ dU'O.lZ^ . . . @ .-/^ ^s Ss O^^^^^fiS^f^-^ /(^y // 'C^pt/C, d cJ^ciJc/elf< I, DEFINITIONS AND TABLES. DEFINITIONS. 105. A Denominate Number is one whose unit is named; as, ^ feet, b pounds, 16 pennyweights (Art. 5). Numbers that have the same unit are of the same denomination ; those that have different units are of different denominations. Thus, ^feet, and 1 feet, are of the same denomination ; dfeet, and 7 yards, are of different denominations. 106. A Compound Number is a denominate num- ber whose units are of .the sams hind but of different denominations; as, ^pounds 6 ounces. Denominate numbers are of the same Mud, when they can be expressed in terms of a common unit. Thus, ^pounds, and 6 ounces, are of the same kind because both can be expressed in ounces. If two denominate numbers are of the same kind, that which has the greater unit is said to be of the higher denomination. Thus, 3 pounds is of a higher denomination than 6 ounces. SCALES OF COMPOUND NUMBERS. 107. The Scale of a compound number is a succes- sion of numbers showing how many times the unit of each denomination is contained in the unit of the next 142 COMPOUND NUMBERS. higher denomination. Thus, in English cnrrency, i far- things make 1 penny, 12 pe7ice make 1 shilUng, and 20 shillings make 1 pound; hence, the scale of Enghsh cur- rency is 4, 12, 20. In this scale, 4 connects farthings and pence, 12 connects pence and shillings, and 20 connects shillings and pounds. In United States currency the scale is 10, 10, 10, 10. The scale of English currency is varying ; that of the United States is uniform. The scale of United States currency is called the scale of tens, or the decimal scale. The scales of the most important compound numbers are indicated in the following tables TABLES OF CUERENCY. 1°. UNITED STATES CURRENCY. 108. The United States currency was established by act of Congress in 1792 ; its primary unit is 1 dollar, TABLE. 10 mills (m.) make 1 cent ct 10 cents ** 1 dime d, 10 dimes '' 1 dollar $, 10 dollars " 1 eagle B, In business transactions the terms eagle and dime are nearly obsolete ; the term mill is seldom used except in official reports and in laying taxes. Note. — The currency of the Dominion of Canada is decimal, and like that of the United States, it is reckoned in dollars and cents. TABLES. 143 2\ ENGLISHCURRENCY. 109. This is the national currency of Great Britain. TABLE. 4 farthings {far. or qr.) make 1 penny d, 12 pence '^ 1 shilling s. 20 shillings " 1 pound £. 21 shillings " 1 guinea G» The primary unit of this currency is 1 pound sterling, and the corresponding coin is called a sovereign. The value of the sovereign is $4.8665. Note. — The sign £, like the sign $, is written before the number to which it refers ; thus, £25. 3°. FRENCH CURRENCY. 110. This is the national currency of France. TABLE. 10 centimes {cent.) make 1 decime d. 10 decimes " 1 franc fr. The primary unit of this currency is 1 franc; its value is 1%.^ cents, that is, $1 is equal to about 5.18 francs. French money is usually expressed in francs and decimals of a franc. TABLES OF WEIGHT. 4°. TROY WEIGHT. 111. This is used in weighing gold, silver, and some kinds of precious stones. TABLE. 24 grains {gr.) make 1 pennyweight. . .dwt. 20 pennyweights " 1 ounce oz. 12 ounces " 1 pound lb. Tr, 144 COMPOUND J^UMBERS. Explanation. — The pound Troy is the primary unit of weight in Great Britain and the United States. It was declared by an act of Parliament, which took effect in 1826, that the brass weight of one pound Troy, then in the custody of the Clerk of the House of Commons should be the standard, and that all other weights should be derived from it. It was enacted that the pound Troy should con- tain 5, 7Q0 grains, and that 7,000 such grains should make a pound avoirdupois. Note. — The English system of weights and measures has been adopted by act of Congress, and is the legal system of the United States. 2\ APOTHECARIES' WEIGHT. 112. This differs from Troy weight in the mode of subdividing the ounce ; it is used in weighing medicines. TABLE. 20 grains (gr.) make 1 scruple ^ 3 scruples " 1 dram 3 8 drams " 1 ounce | 12 ounces " 1 pound S) 3'. AVOIRDUPOIS WEIGHT. 113. This weight is used in weighing the ordinary articles of trade and commerce. TABLE. 16 ounces (oz.) make 1 pound lb. 25 pounds " 1 quarter .... . . . . qr, 4 quarters " 1 hundredweight . . . cwt 20 hundredweight " 1 ton T, The primary unit is 1 pound, equal to 7,000 grains Troy. In weighing coarse articles liable to wastage, as coal at the mines, and the like, it is customary to call 112 Ws. a hundredweight, and 38 £b8. a qumrter. TABLES. 146 TABLES OF TIME. DEFINITION AND EXPL AN A T I N . 114. Time is a measured portion of duration. Jts primary unit is 1 mean, or average, solar day. Explanation. — An astronomical year is the time required for the earth to revolve about the sun ; but this period does not contain an exact number of days ; hence, for civil purposes, an artificial year is adopted. The length of the civil year is sometimes 365 days, and sometimes 366 days, and these are so distributed that after a long period the a'derage length of the civil year is equal to that of the astronomical year. Every year divisible by 4 (except centennial years not divisible by 400) are leap years and contain 366 days each ; all other years are common yea/rs and contain 365 days each. TABLE. 60 seconds (sec.) make 1 minute min, 1 hour hr. 1 day da, 1 week wh. 1 common year — yr. 1 leap year yr. 1 century G. The year is divided into 12 parts called months. Their order and the number of days in each is shown in the TABLE. 1°. January 31 days. 7°. July 31 days. 60 minutes a 24 hours a 7 days tt 365 days -^ v. changes at every point ; as GH. /^ N^ A straight line is one whose direc- cubved line. tion does not change at any point ; as AB. A B Straight lines are parallel when they straight ldtb. have the same direction ; as CD and EF. ^ The length of a line is the number of ^ P times it contains a given straight line parallel lines. taken as a unit. 1°. LONG MEASURE. 11'7. This is used for measuring distances and dimen- sions of objects. TABLB. 12 inches {in.) make 1 foot ft. 3 feet " 1 yard yd. 5 J yards ** 1 rod rd. 40 rods *' 1 furlong . . .fur. 8 furlongs, or 320 rods " 1 mile mi. 3 miles " 1 league lea. It is found convenient in practice to reduce yards, feet, and inches to Jialf yards and inches ; we thus avoid the inconvenience of a scale in which one of the numbers is fractional. In this case, the first part of the preceding table may be replaced by the following : 18 inches make 1 half yard hf. yd. 11 half yards " 1 rod rd. TABLES. 147 The yard is the primary unit of English and American measures of length. By the act of Parliament already referred to, it was declared that the brass standard yard, then in the custody of the Clerk of the House of Commons, should be the imperial standard yard. From it we derive all other measures of length. Note. — In measuring doth, ribbons, and the like, the yard is sub- divided into halves, quarters, eighths, and sixteenths. 2°. SURVEYORS' MEASURE. 118. This is used in measuring land. The unit is a Gunter's Chain ; this chain is 4 rods, or 66 feet, in length, and is divided into 100 equal parts, called links. TABLE. 7.92 inches make 1 link U, 100 links " 1 chain ch, 80 chains " 1 mile . . ..mi. MEASUKES OF SUKFAOE. DEFINITIONS. 119. A Surface is a magnitude that has length and breadth, without thickness. A Plane is a surface such that if a straight line is applied to it in any direction it will coincide with the surface throughout. 130. An Angle is the opening between two lines that meet at a point ; as, BAG. The lines AB and AC are called sides, and the point A is called the vertex of the angle. If one straight line meets another so as to make the two adjacent angles equal, the first line is said to be perpendicular to the second, and the angles are called right angles ; thus, if the angles Bx\D and BAC are equal, they are both right angles and BA is perpendicular D A C *^ ^^- BI6HT ANaiiES. 148 COMPOtJN'D NUMBERS. 121. A Square is a plane figure bounded by four equal lines, called Sides, and having all its angles right angles ; as, ABCD. A Rectangle is bounded by four lines, paral- lel two and two, and having all its angles right angles. square. 132. A Unit of Surface is a square whose sides are equal to the unit of length. If each side is a yard, the square is called a square yard. The Area of a Surface is an expression for that sur- face in terms of a square unit. Note. — It is shown in geometry that the area of a rectangle is equal to the product of its length by its breadth ; that is, the number of square units in the surface is equal to the number of units in its length multiplied by the number of units in its breadth. 1°. SQUARE MEASURE. 123. This is used in measuring surfaces. TABLE. 144 square inches {sq. in.) make 1 square foot sq. ft. 9 square feet " 1 square yard . . ..sq. yd. 30J- square yards " 1 square rod sq. rd. 160 square rods " 1 acre A. It is found convenient in practice to reduce square yards, square feet, and square inches, to quarter square yards and square inches, to avoid using a scale containing a fractional number. In this case 324 square inches make 1 quarter square yard . . qr. sq. yd, 121 square quarters '' 1 square rod sq. rd. 160 square rods " 1 acre A, TABLES. 149 2°. LAND MEASURE. 124. This is used in measuring land. TABLE. 10000 square links (sq. U.) make 1 square chain sq. ch. 10 square chains *^ 1 acre A. 640 acres " 1 square mile sq.mL The area of land is also reckoned by the following TABLE. 40 square rods (sg. rds.) make 1 rood B. 4 roods *^ 1 acre A. In governmeut surveys, a square mile is called a section ; 36 sec- tions make one township. MEASURES OF VOLUME AND CAPACITY. DEFINITIONS. 125. A Volume is .4 magnitude that has length, breadth, and thickness or height. 126. -6- Cube is a volume or solid, bounded by six equal squares, called Faces ; the sides of the squares are called Edges of the cube. Thus, ABCD-E is a cube, ABCD is one of its faces, AB, BC, and BE are edges, A rectangular volume, or solid, whose paraixelopipedon. edges are not equal is a parallelopipedon. 127. A Unit of Volume is a cube whose edges are equal to the unit of length. If each edge is a lineal yard, the cube is called a cuUc yard. 150 COMPOUN^D NUMBERS. 128. The Content of a volume is an expression for that volume in terms of a cubic unit. Note. — It is shown in geometry that the content of a cube or of any rectangular solid is equal to the product of its three dimensions ; that is, the number of cubic units in the volume is equal to the num- ber of units in its length, multiplied by the number of units in its breadth, multiplied by the number of units in its height. 139. A Unit of Capacity is a measure having a determinate content, or capacity. I*. CUBIC MEASURE. 130. This is used in measuring volumes or solids. TABLE. 1728 cubic inches {cu. in.) make 1 cubic foot cu.ft.. 27 cubic feet " 1 cubic yard cu. yd, A Cord of Wood is a pile i^^^^^^^^^^^^^^^^^^^B&v Aft. \yide, 4/i^. " : . - mr^"^.-^^^- high, and 8 ft. long. A foot in length from such ^P^'vi^V eet/^^ a pile is called a ^^v4^i\%l Cord Foot. A .^ ^.^-^..NE^ cc^f ^ cord foot is 16 cubic feet. ^^^.^ TABLE. 16 cubic feet {cu.ft) make 1 cord foot C.ft. 8 cord feet or 128 cu.ft " 1 cord C\ 2°. DRY MEASURE. 131. This is used in measuring dry articles, as grain, fruit, salt, and the like. TABLE 2 pints {pt.) make 1 quart qt, 8 quarts '' 1 peck .ph 4 pecks ** 1 bushel hu. TABLES. 151 The primary unit is 1 bushel. The bushel (known as the Win- chester bushel) is a cylindrical measure 18|^ inches across and 8 inches deep ; it contains 3,150| cubic inches nearly. 133. 3°. LIQUID MEASURE. This is used in measuring liquids. TABLE. 4 gills (gi.) make 1 2 pints '' 1 4 quarts " 1 31^ gallons " 1 2 barrels " 1 2 hogsheads '^ 1 2 pipes " 1 The primary unit is 1 gaUon; it contains 231 (yubic inches. The pint pt quart qt. gallon ,gal. barrel bU, hogshead. . .hhd, pipe pi. tun tun. liquid quart is about | of a quart of dry measure. ANGULAK MEASURE AND LONGITUDE. DEFINITIONS. 133. A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the Centre. The bound- ing line is called the Circumference, and any part of this circumference is called an Arc of the circle. 134. A Diameter is a line passing through the centre and terminating in the circumference. 135. A Radius is a line drawn from the centre to any point of the circumference. Thus, AEDF is a circle, C its centre, AD a diameter, and CE a radius. 152 COMPOUND NUMBERS. \\ ANGULAR MEASURE. 136. In measuring angles, the Right Angle (Art. 120) is taken as the primary unit. The ninetieth part of a right angle is called a Degree. TABLE. 60 seconds (") make 1 minute 60 minutes " 1 degree ° 90 degrees " 1 right angle rt a, DEFINITION AND EXPLANATION. 137. The Longitude of a place is the angular distance of the meridian of that place from some stmidard meri- dian. It is measured on the equator, and is equal to the angle through which the earth turns on its axis whilst the sun is passing from the meridian of one place to that of the other. But the earth turns through an angle of 360° in 24 hours, that is, it turns through an angle of 15° in 1 hour ; hence, we have the following relations between the difference of longitude of two places and the difference of their local times. TABLE. 15° of arc make 1 hour of time. 15' of arc " 1 ininute of time. 15" of arc " 1 second of time. Note. — All longitudes referred to in this book are reckoned from the meridian of Greenwich, England. MISCELLANEOUS TABLES. 138. The following miscellaneous tables are often used in operating on compound numbers : TABLES. 153 l". COUNTING. 12 things make 1 dozen doz. 12 dozen " 1 gross gr. 12 gross " 1 great gross ff- ff^* 20 things " 1 score sc, 2\ PAPER. 24 sheets make 1 quire qr, 20 quires " 1 ream rea7n. 2 reams " 1 bundle bund. 2 bundles " 1 bale hale, Z\ BOOKS. A book in which each sheet is folded into 2 leaves is a folio. " " " " 4 « « quarto, or 4to. " « " " 8 « " octavo, or 8vo. " " " " 12 " " duodecimo, or 12ma " " " " 16 « " 16mo. " « u « 24 " " 24mo. « " « " 32 " " 32mo. THE METRIC SYSTEM. DEFINITIONS AND EXPLANATIONS. 139. The Metric System is a system of weights and measures based on a primary unit of length called a Meter. This system was first adopted by France and afterwards by vari- ous other countries. Since 1866 its use has been permitted in the United States, by act of Congress. The meter is approximately equal to i o o o^o o o o ^^ ^^^e distance from the equator to the north pole, measured on 154 COMPOUND NUMBERS. the meridian through Paris. It is nearly equal to 39.37 inches. The scales of all compound numbers in the metric sys- tem are decimal The names of units in the ascending scale are formed by prefixing* the following numerals to the names of the primary units : deca, ten ; hecto, one hundred ; kilo, one thousand ; and myria, ten thousand. The names of units in the descending scale are formed by pre- fixing the following numerals to the names of the primary units : deei, one tenth ; centif one hundredth ; and milli^ one thousandth. MEASURES OF LENGTH. 140. The primary unit is the meter. TABLE. 10 millimeters {mm.) make 1 centimeter (cm.) = .3937 »w. 10 centimeters " 1 decimeter (dm.) — 3.937 " 10 decimeters " 1 meter (m.) = 39.37 " 10 meters ** 1 decameter (decam.) = 393.7 " 10 decameters " 1 hectometer {hectom.) = S28 ft. 1 in. 10 hectometers '* 1 kilometer (kilom.) = S2S0 ft. 10 in. Lengths are usually expressed decimally in terms of some one of the units of the preceding scale. Small distances are expressed in millimeters, ordinary distances in meters, and long distances in kilometers. MEASURES OF SURFACE. 141. The primary unit of ordinary surfaces is a Square Meter. TABLE. 100 sq. centimeters make 1 sq. decimeter = 16.6 sq.in. 100 sq. decimeters " 1 sq. meter = 1550 " " For land measure the primary unit is the are (pron. ar) ; it is a square decameter, or 100 square meters. TABLES. 155 100 sq. meters make 1 are = 119.6 sq. yds. 100 ares " 1 hectare = 2.471 acres, MEASURES OF VOLUME AND CAPACITY. 143. The primary unit of ordinary volumes is the Stere (pron. stair)', it is a Cubic Meter. TABLE. 1000 cu. centimeters make 1 cu. decimeter = 61.026 cu. in, 1000 cu. decimeters " 1 cu. meter = 35.316 cu.ft. For wood measure the primary unit is the stere. 10 decisteres make 1 stere {st,) = .2759 cords. 10 steres " 1 decastere = 2.759 " For measures of capacity the primary unit is the liter (pron. leeter). It is a cubic decimeter. TABLE. Dry Measure. Liquid Measure. 10 centiliters W make 1 deciliter {dl.) = .0908 g?.= .1057 g^. 10 deciliters " 1 liter (I.) = .908 g^. = 1.0567 g^. 10 liters " 1 decaliter (s. = 9 27. 1| cwi^. + 17.25 Ihs. + 49 Z^>5. = F 28. £32.5 + 17.55. + 37^8. = f 29. 47.5 da. + 34.2 ^a. + "^f da. — f PRACTICAL PROBLEMS. 1. A merchant sent off the following quantities of but- ter : 47 cwt. 2 qrs. 7 Ihs. ; 38 mvt. 3 qrs. 8 Ihs. ; and 16 cwt. 2 qrs. 20 Ihs. ; how much did he send off in all ? 2. A silversmith has 3 parcels of silver : the first con- tains 7 Ihs. Soz. 16 dwts. ; the second contains 9 Ihs. 7 oz. 3 dwts. ; and the third contains 4 ZJs. 1 dwt. ; how much has he in all ? 3. A merchant sells cloth as follows: to A., 16^ yds.; to B., 90^ yds. ; and to C, 19033g- yds. ; how much does he sell to all ? 4. A man has three farms : the first contains 120 A. 74 sq. rds. ; the second contains 75 ^. 46 sq. rds.; and the third contains 97 A. 46 sq. rds. ; how much do they all contain ? 5. B. aged 14 yrs. 6 mos. goes out to service ; he lives at one place 1 yr. 9 mos., at another place 2 yr.^. 5 7nos., and at a third place 3 yrs. 9 mos. ; how old is he then ? ADDITION. 171 6. A man spent 111.26 francs for a vest, 62.17 /r. for a coat, and 38.29 /r. for a pair of boots; how many dollars did he spend in all ? 7. A man sold 4 cheeses: the first weighed 9.25 Icilog., the second 10.14 hilog., the third 11.16 kilog., and the fourth 10.77 hilog. ; how many pounds did they weigh ? 8. In a farm there are 5 fields : the first contains 18 A. 8 sq. ch., the second 12 J. 3 sq. cli., the third 9 ^. 4 sq. ch., the fourth 11^., and the fifth 16 J[. 2sq.ch.j what is the content of the farm ? 9. How many yards in 4 pieces of cloth measuring as follows: SOi tjds., 21!^ yds., 39-^ yds., and S'^iyds.? 10. A man bought 3 loads of wood : the first contained 1 C. 17 cu.ft.f the second 1 C. 115 cu.ft., and the third 1 0.2 C.ft. ; how much wood did he buy ? REVIE^A^ QUESTIONS. (149.) What is addition of compound numbers? (150.) What is the rule for addition of compound numbers ? IV. SUBTRACTION OF COMPOUND NUMBERS. DEFINITION. 151. Subtraction of Compound Numbers is the operation of finding the Difference of two numbers of the same kind. MENTAL EXERCISES. 1. What is the difference of l^qts. and 9qts.9 of 7phs. and 4:phs. ? of 18 yds. and 11 yds. ? of 30 ds. and 17 cts. 9 of 25 rds. and 14 rds. f of 16 mi. and 11 mi. f 172 COMPOUND NUMBERS. 2. What is the difference of Sid. and IBcl 9 How many shillings in 31^..? in 16d.f In the difference between did. and 16d.f What is the difference between 2^. 7d. and Is. 4:d. 9 2s. 7^. — Is. 4:d. = 9 3. How many yards in 26 ft. 9 in 16//. P What then is the difference between 8 yds. 2 ft. and h yds. 1ft. 9 between %yds. 1ft. and Q yds. 2 ft. 9 between 1 gals. Iqt. and 5 gals. 3 qts. 9 Note.— The principles used in subtraction of compound numbers are the same as those used in mbtraction of simple numbers. OPERATION OF SUBTRACTION OF COMPOUND NUMBERS. 152, Let it be required to find the difference between £9 4s. M. and £2 ISs. 6d. : Explanation.— We write the subtra- operation. hend under the minuend so that units £ S. d. of the same denomination shall stand 9 4 3 in the same column. Beginning at the lowest denomination, we see that 6d. 1" " . cannot be taken from M. ; we therefore £q k <, q^ add 12d. to Sd., which gives 15^., and then subtract 6d. from the sum ; the remainder, 9d., we set down, and to compensate for the 12d. added to the minuend, we add its equal. Is., to the next column of the minuend. The sum, 19s., being greater than 4«., we add 20^. to the latter and subtract 19s. from the sum ; the remainder, 5s., we set down and as before carry forward 20s., or its equal £1, and add it to the minuend, giving £3 ; this taken from £9 leaves £6 : hence, the required remainder is £6 5s. 9d. In like manner we may treat all similar cases ; hence, the fol- lowing RULE. /. Write the subtrahend under the minuend so that units of the same denomination shall stand in the same colunnn. SUBTRACTION. 173 //. Subtract each number in the lower line from the one above it and write the remainder in the line below. III. If any number in the lower line is greater than the one above it, increase the latter by as many units as make one of the next higher de- nomination, perform the subtraction and then add 1 unit to the next number in the lower line. EXAMPLES. Perform the following indicated subtractions : (1.) (2.) (3.) £ s. d. lbs. oz, dxvts. bu. pks. qts. From 14 14 3 6 11 14 65 1 7 Take 9 17 1 2 3 16 14 3 4 Eem. £4 175. 2d. 4 lbs. 7 oz. 18 dwts. 50 bu. 2pks. 3 qts. Proof. — The method of proof is the simple numbers. ! same as for subtraction of (4.) cwt. qrs. lbs. 7 3 13 (5.) hhds. gals. 112 23 qts. 1 (6.) yds. ft. in. 4 2 11 5 1 15 75 37 1 2 2 9 (7.) acres, sq. rds. 29 50 (8.) 23° 45' 54" (9.) fi) ! 3 3 grs. 35 7 3 1 14 24 65 r 49' 57" in. SI 17 10 6 1 18 10. From 4 rds. 2 yds. 1 ft. 9 ^ ibtract 2 rds. 3 yds. 1ft. 11 171. 174 COMPOUND LUMBERS. OPBBATION. rds. yds, ft. in, rds, lif. yds. in, 4219 = 4 5 3 2 3 1 11 = 2 7 5 1 rd. 8 lif.rds. 16 in, z=: 1 rd. 4 yds. 1 ft. 4 in. Ans. 11. From 12 rds. 2 yds, 2 ft. 1 in. subtract 3 rds. 3 yds. 2 ft. 10 in. 12. From Srds. l^yds. subtract 3 rds. 3 yds. 2 ft. 6 vi. Note. — To write a date as a compound number, we first write the number of the current year, then the number of the current month, counted from the beginning of the year (Art. 114), and then the number of the day. Thus, July 7th, 1839, is written 1839 yrs. 7 mos. 7 da. In computing the difference of two dates, a month is to be counted equal to 30 days. 13. What is the difference of time between October 16th, 1869, and Aug. 2d, 1873 ? OPERATION. August 2d, 1873 . . . 1873 yrs, 8 mos. 2 da. October 16th, 1869 . . 1869 yrs. 10 mos. 16 da. 3 yrs. 9 mos. 16 da. Ans. 14. How long from Sept. 25, 1871, to July 4, 1876 ? 15. How long from July 7, 1815, to Nov. 1, 1873 ? .16. How long from May 13, 1816, to June 25, 1859? 17. What is the difference between 22 hrs. 17 min. 4 sec, and 14 hrs. 9 min. 51 sec. 9 18. What is the difference between £1.5 and 7s. 6d,f 19. From -^ of 1 hhd. subtract f of 1 qt. 20. From 3.107 kilog. subtract 331.2 grams. 21. From 16 da. 21 hrs. 42 min. 13 sec, subtract 12 da. 22 hrs, 58 min. 39 sec. SUBTRACTION. 176 32. From 7 T, 14 cwt 3 qrs. 19 lbs, subtract 3 T. 18 cwL 1 gr. 4 lbs, 23. From 14 ?5s. 1 oz. 3 ^. 1 pk. 2 g^s. subtract 94 bu. 3 ^^5. 7 qts, PRACTICAL PROBLEMS. 1. A merchant bought a piece of cloth for £22 105. and sold one half of it for £14 18^. ; for what must he sell the rest to make £7 14s. M. 9 Ans. £15 6s. Sd. 2. A farm contains 273 ^. 1 i2. 5 sq. rds., hut only 111 ^. 2 i?. 38 sq. rds. was capable of tillage; how much of it was incapable of tillage ? 3. From a piece of cloth containing 39^ yds., there was cut off at one time 3f yds. and at another time 4J- yds. ; how much remained in the piece ? 4. A merchant has 183 cwt. 24 lbs. of butter, of which he ships 78 cwt. 3 qrs. 14 lbs. ; how much remains ? 5. How long from Jan. 20, 1873, to Nov. 14, 1875 ? 6. A man was born Jan. 10, 1803, and died Sept. 21, 1875 ; what was his age at the time of his death ? 7. The revolutionary war began April 19, 1775, and ended Jan. 20, 1783 ; how long did it last ? 8. How long from the discovery of America, Oct. 11, 1492, to the declaration of independence, July 4, 1776 ? 9. From a pile of wood containing 11 C. 4 C. ft., there was sold 4 (7. 5 C.ft. 12 cu. ft. ; how much remained? 10. The latitude of Albany is 42° 39' 3" N., and that of St. Petersburg is 59° 56' N. ; what is the difference ? Ans. 17° 16' 57". 176 COMPOUND NUMBEES. Note. — The Latitude of a place is its angular distance from the equator. If the place is north of the equator its latitude is marked N., if south, its latitude is marked S. If the latitude of two places are both nortJi or both south, their difference of latitude is found bj subtracting the less from the greater ; if the latitude of one place is north and the other south, their difference of latitude is found by adding the latitudes of both. 11. The latitude of New York is 40° 42' 45" N., that of the Cape of Good Hope is 34° 22' S ; what is the differ- ence? Ans, 75° 4' 45". 12. The latitude of St. Augustin is 29° 48' 30" N., and that of Gibraltar is 36° 7' N. ; what is the difference ? 13. The longitude of New York is 74^ 3' W., and that of San Francisco is 122° 26' 45" W. ; what is their differ- ence of longitude ? Ans, 48° 23' 45". Note. — Longitudes are reckoned both east and west from some assumed meridian, usually that of Greenwich, England. The method of finding difference of longitude is the same as for finding difference of latitude. 14. The longitude of Berlin is 13° 24' E., and that of Washington is 77° 0' 15" W. ; what is the difference ? 15. The longitude of Charleston is 79° 55' 38" W., and that of Boston is 71° 3' 30" W.; what is the difference ? 16. A farmer has 147 hu. 1 pK of oats ; he puts 49 lu. 3 pks. in one bin, 27 lu. 1 pJc. in a second bin, 32 hu. 3pks. in a third bin, and the rest in a fourth bin ; how many does he put in the fourth bin ? REVIE^A^ QUESTIONS. (151.) What is subtraction of compound numbers? (152.) Give the rule. How proved ? How are dates written ? What is the latitude of a place ? How reckoned ? What is the method of finding difference of latitude? Difference of longitude? MULTIPLICATION. 177 V. MULTIPLICATION OF COM- POUND NUMBERS. DEFINITION. 153. Multiplication of Compound Numbers is the operation of taking a compound number as many times as there are units in an abstract number. MENTAL EXERCISES. 1. How many inches are 7 times 8 inches? What is the product of 11 in. by 9 ? of Id. by 11 ? of 6 oz. by 14 ? oilQyds. by 8? 2. What is the product of M. by 11 ? How many shil- lings in the product and how many pence remain ? What is the product of %ft. by 16 ? How many yards and feet in the product ? What is the product of 7 qts. by 13 in pecks and quarts 9 3. What is the product of 15 in. by 9 ? How m^\ij feet and inches in 15 in.9 in 9 times 15 in.9 What then is the product of 1ft. 3 ill. by 9 ? What is the product of 4 iu, d2)ks. by 11? of U 7^. by 8 ? Note. — The principles used in multiplication of compound num- bers are the same as those used in multiplication of simple numbers. OPERATION OF MULTIPLICATION OF COMPOUND NUMBERS. 154. Let it be required to multiply £4 ^s. 6d. by 16 : Explanation. — Having written the operation. multiplier under the multiplicand, we £ S. d. multiply 5d. by 16, which gives SOd., 4 2 5 or 6s. 8d. ; setting down 8d, we carry .. ^ 6s. to the next column. We then multiply 2s. by 16 and add 6s. to the £65 18s. Sd. product, which gives 385., or £1 18s. ; setting down 18s., we carry £1 to the next column. Finallv, we 178 COMPOUND NUMBERS. multiply £4 by 16 and add £1 to the product, which gives £65. Hence, the required product is £65 18s. Sd. In like manner we may treat all similar cases ; hence, the fol- lowing RULE. /. Multiply the units of the lowest denomina- tion of the multiplicand by the multiplier, and divide the product by the number of the scale that connects this denomination with the next higher one; set down the remainder and carry the quotient to the next column. II. Multiply the units of the next higher de- nomination by the multiplier, add the units brought forward, and proceed as before, continu- ing the operation till all the parts of the given number have been multiplied, EXAMPLES. (1-) (3.) £ 8, d, cwt qrs, lbs. oz. Multiplicand. 17 15 9 8 3 19 Multiplier. . . 6 7 Product .... £106 145. ed, 61 cwt, 1 qr, 10 lbs. 15 oz, (3.) mi. rds. yds. ft. in. mi. rds. Tif. yds. in, 9 110 4 2 6 = 9 110 9 12 9 9 84 mi. 37 rds. 10 hf. yds. in. = 84 mi. 37 rds. 5 yds. Ans, Multiply Multiply 4. 5 mot. 2 qrs, by 7. 5. $8.75 by 24.5. MULTIPLICATION. 179 6. 65.35//-. by 46. 14. £| by 17.5. 7. 5Mkilog. by 12. 15. 10^. 1 M. by 11. 8. 15 yds. 1ft, by 21. 16. 3 lirs, 15\min. by 24. 9. 123.25 m. by 15. 17. 5 ciot. 2| qrs. by 24. 10. 6 whs, 3 da. by 13. 18. £3 14fyrs. at 1% is 1712 X .01 X 3 = $21.36. $21.86 ; but the given interest is $128.16, that is, it is 6 times $128.16 -j- $21.36 = 6^. as great ; hence, the required rate is six times 1 % , or 6 % . In like manner all similar cases may be treated ; hence, the fol- lowing RULE. Find the interest at 1% on the principal for the given time; then divide the given interest by the result. EXAMPLES. What is the rate per annum when 1. The interest on $950 for IQmos. is $88.66|? 2. The interest on $380 for 1 yr. 4 mos. is $22.80 ? 3. The interest on $8,726 for l^yrs. is $916.23 ? 4. The interest on $712 for 3 yrs. is $128.16 ? 5. The interest on $329.5 for 2 yrs. is $46.13 ? 6. The interest on 794 for S^yrs. is $194.53 ? 7. The interest on $450 for 3 yrs. 6 mos. 18 da. is $79.87i-? 220 PERCENTAGE AND ITS APPLICATIONS. 191. To find the Time when the Principal, the Rate, and the Interest are given. Let it be required to find the time iii which the interest on 11,200 at 6% will he equal to $120. Explanation.— The in- operation. terest on $1,200 for 1 year il200 X 06 = $72 is $72. But $72 is con- tained in the given interest !^ ^ ^ 2 y^^^^ = 1 yr. 8 mOS, $120, If times ; conse- $72 ^ ^ ^ quently, the required time is If times 1 year ; that is, it is 1 yr. 8 mo. In like manner we may treat all similar cases ; hence, the fol- lowing RULE. Find the interest on the pidiicipal for 1 year at the given rate ; then divide the given interest by the result. EXAM PLES. Find the time in which the interest 1. On $712 at 6% will be $128.16. 2. On $8,942 at Si% will be $2,470.22J. 3. On $329.50 at 7^ will be $46.13. 4. On $980 at 6% will be $44.10. 6. On $1,175 at 6% will be $82.25. 6. On $846 at 6% will be $63.45. 7. On $872 at 6% will be $915.60. 8. On $1,500 at 1% will be $210. 9. On $3,000 at 6% will be $600. 192. To find the Principal, when the Interest, the Rate, and the Time are given. Let it be required to find the principal that will give $65 interest in 20 mos. at 6% per annum. SIMPLE INTEREST. 221 OPBBATION. Explanation. — The interest on $1 for 20 mos. is 10 ds. ; now if $1 draws IQcts. in the given time, |1 X .06 X If = 10.10 it will require $650 to draw $65 in |g5 $65 . xTry = $650. the same time : hence, zttt-t- is »U.i 10 c^s., ^ equal to the number of dollars in •*• 1650. Ans. the required principal. In like manner we may reason on all similar cases ; hence the following RULE. Find the interest on $1 for the given time at the given rate ; then divide the given interest by the result. What is the principal on which the interest 1. At b% for l^mos. is $157.50 ? 2. At 6^ for %yrs. 6 mos. is 1450 ? 3. At 44^ for dyrs. 4: mos. is $412.50? 4. At S% for 27 mos. is $324 ? 5. At 7% for 20.4: mos. is $297.50 ? 6. At 6% for 15^^05. is $7.66|? MISCELLANEOUS EXAMPLES. 1. The interest on a certain sum for 4 years, at 7 per cent., is $266 ; what is the principal ? 2. The interest on $3,675, for 3 years, is $771.75 ; what is the rate ? 3. The principal is $459, the interest $183.60, and the rate 8 per cent. ; what is the time ? 4. The interest on a certain sum for 3 years, at 6 per cent., is $40.50 ; what is the principal ? 5. The principal is $918, the interest $269.28, and the rate 4 per cent. ; what is the time ? 222 PERCENTAGE AND ITS APPLICATIONS. 6. What sum of money must be placed at interest at 7^, for 3yrs. 9 mos., that the interest may be 1393.75 ? 7. In what time, at 7 per cent., will a mortgage of $8,000, whose interest is unpaid, amount to 19,120 ? 8. If I buy a house for 15,620 and receive $1,803 for rent in 2yrs. Smos. 15 da,, what rate of interest do I get for my money ? 9. What sum of money, at 6%, will produce, in 2 yrs, 9 mos. 10 da., the same interest that 1350 produces, at S%, in Syrs. lO-mos. 6 da J 10. In what time will 15,000 at 11%, produce the same interest as $9,625 at 6^^, in 4: yrs, 5 mos. 18 daJ ANNUAL I NTEREST. 193. Annual Interest is simple interest on the prin- cipal and also on each year's interest from the time it falls due to the time of settlement. This mode of computation is legal in some of the States when notes are made payable " with interest annually." EXAM PLES. 1. What is the interest on a note for $600 at 6%, payable in 3 years tvith interest annually? Solution.— The interest on $600 for 3 yrs. is $108 ; the interest on $36 (the first year's interest) for 2 yrs. is $4.32 ; and the interest on $36 {the second year's interest) for 1 yr. is $2.16 : hence, the entire interest is equal to $108 + $4.32 + $2. 16 = $114.48. A7is. 2. What is the interest on a note for $1,200 at 7^, pay- able in 4 yrs. with annual interest ? 3. What is the interest on a note for $980 at %%, payable in 4: yrs. with ammal interest? SIMPLE INTEREST. 223 NOTES. • 194. A Promissory Note is a written promise to pay a sum of money, either on demand, or at some specified time. The person who signs the note is called the Maker, and the party that has legal possession of it is called the Holder. 195. A Negotiable Note is one that is payable either to order, or to bearer. FORM OF A NEGOTIABLE NOTE. In this case John Doe, the person named in the note, is called the Payee ; he can transfer it by writing his name across the back ; he is then called an Indorser, and is obliged to pay the money when it falls due, if the maker of the note, Richard Roe, fails to do so. 196. The Face of a note, or bill, is the sum named in it. Thus, in the note just above described, the face is $375. PARTIAL PAYM ENTS. 197. A Partial Payment is a payment of a part of the amount due on a note, or other written obligation to pay money. The date and the amount of each partial payment is indorsed, that is, written on the back of the note, or obligation, and is to be taken into account in making the settlement. 224 PERCENTAGE AND ITS APPLICATIONS. METHODS OF SETTLEMENT. 198. The following method of settling a note, or other interest-bearing obligation on which partial payments have been made, has been sanctioned by the Supreme Court of the United States, and is now adopted in New York, Massachusetts, and many other States : SUPREME COURT RULE. /. Find the amount of the given principal up to the time when the sum of the partial payments is equal to, or exceeds, the interest then due ; from this result subtract the sum of the partial pay- ments to the time considered. II. Take the remainder for a new principal and proceed as before, continuing the operation to the time of final settlement. EXAMPLES. 1. On a note dated May 1, 1866, for f 1,200 at 6^, were the following indorsements: Nov. 1, 1866, $100; Mar. 1, 1867, $20; Sept. 1, 1868, 1180: what was due on the note Nov. 1, 1869 ? OPERATION, Given principal $1200 Int. to Nov. 1, 1866 (6 mo8.) $86 Amount $1236 1st payment $100 1st new principal $1136 Int. to Sept. 1, 1868 (22 moa) .... $1249 6 Amount $1260.96 Sum of 2d and 3d payments $200 Second new principal $1060.96 Int. to Nov. 1st, 1869 (14 mos) .... $74.267 Amount due Nov. 1st, 1869 $1135.227 SIMPLE INTEREST. 225 Explanation. — We compute the interest on $1,200 from the date of the note to the time of the first payment, and because the first payment is greater than the interest then due, we add the interest to the principal and subtract the first payment, which gives $1,136 for a new principal. We then see, by inspection, that the interest on the new principal from Nov. 1, 1866, to Mar, 1, 1867 {4:mos.), is greater than the second payment; we therefore compute the interest on the new principal to the time of the third payment, add it to the principal, and from the result subtract the sum of the 2d and 3d payments, which gives another new prin- cipal. We then find the amount of this principal up to the time of settlement.. 2. On a note dated May 1, 1866, for $35(> at 6%, were the following indorsements : Dec. 25, 1866, received 150. Sept. 1, 1868, " 120. June 13, 1869, " $100. What was due April 13, 1870 ? 3. On a note dated July 1, 1869, for $700 at 7^, were the following indorsements : July 1, 1870, received 1200. July 1, 1871, " $400. What is due July 1, 1872 ? 4. On a note dated May 1, 1860, for $2,000 at 1!%, were the following indorsements : May 1, 1861, received $500. May 1, 1862, " $450. May 1, 1863, « $750. May 1, 1864, « $400. What was due May 1, 1865 ? g Ans, S311.761. 226 PERCENTAGE AND ITS APPLICATIONS. 5. On a note dated Aug. 1, 1870, for $1,000 at 6%, were the following indorsements : Dec. 1, 1870, received 1310.25. April 1, 1871, " $225.50. Aug. 1, 1872, " $400.00. What was due Jan. 1, 1873 ? 6. On a note dated Sept. 18, 1873, for 17,000 at 6%, were the indorsements: July 6, 1874, $500; Sept. 24, 1875, $1,500; and Dec. 6, 1875, $1,000: what was due July 12, 1876? 7. On a note dated Jan. 6, 1875, for $1,280 at 7^, were the indorsements: July 18, 1875, $175; Dec. 12, 1875, $375 ; and July 24, 1876, $400 : what was due July 9, 1877 ? 8. On a note dated June 15, 1874, for $1,500 at 7^, were the indorsements: Dec. 15, 1874, $300; May 30, 1875, $300; and Dec. 18, 1875, $400: what was due July 15, 1877 ? When partial payments are made on interest-bearing obligations due witliin a year, the balance is usually adjusted amongst business men by the following rule, called MERCANTILE RULE. Find the amount of the principal from the date of the note to the time of settlement ; find the amount of each paym^ent from the time it ivas made to the time of settlement, and subtract their sum from the first result. Note. — In applying this rule, the times are reduced to days, and the interest is then computed by the rule for days. SIMPLE INTEREST. 227 9. On a note dated Jan. 1, 1873, for $1,000 at 1%, were the following indorsements : Feb. 15, 1873, received $200. . . May 16, 1873, " $400. What was due Aug. 14, 1873 ? SOLUTION. Amt. of $1000 for 225rf« $1043.75 Amt. of $200 for imda $207 Amt. of $400 for ^0 da $407 Sum of amts. of payments $614 Balance due Aug. 14, 1873 . . . $429.75 10. On a note dated Jan. 1, 1873, for $800 at 6^, are the following indorsements : Feb. 6, 1873, received $200. April 30, 1873, " $210. What is due on the note June 5, 1873 ? REVIEW QUESTIONS. (183.) What is interest? (184.) What is the principal? The rate? The amount? What is the legal rate? What is usury? How is time reckoned? (185.) What is simple inter- est? Compound interest? (186.) Rule for interest when the principal, rate, and time in years are given? (187.) When the time is given in months ? When in years, months, and days ? (188.) Rule for interest when the time is given in days? (189.) Rule for accurate interest? (190.) Rule for finding the rate? (191.) Rule for finding the time? (192.) Rule for finding the principal? (193.) What is annual interest? (194.) What is a promissory note? The maker? The holder? (195.) What is a negotiable note ? Payee? Indorser? (196.) What is the face of a note or other obligation? (197.) What is a partial payment? (198.) What is the Supreme Court rule ? The Mercantile rule ? 238 PERCENTAGE AND ITS APPLICATIONS. VII. COMPOUND INTEREST. DEFINITIONS. 199. Compound Interest is interest computed on the principal and also on the accrued interest as it falls due. Interest may be added to principal at the end of each year, half year, or other fixed period. Unless otherwise stated, the period is supposed to be a year. 200. From principles already explained we have the following RULE FOR COMPOUND INTEREST. I. Find the amount of the given principal for the first period ; then find the amount of this result for the second period ; and so on to the end of the given tiine ; the final result will he the total amount. II. From the total amount subtract the given principal and the remainder will be the compound interest. EXAMPLES. 1. What is the compound interest on $642 for 2 years at 6^ per annum, interest being compounded annually ? Explanation. — Here the period is 1 year, and the amount of $1 for that period is $1.06 ; multiplying $642 by 1.06, we find the amount for the first period, $680.52 ; multiplying this by 1.06, we find its amount for the second period, $721,351. Subtracting th,e original $642. 1.06 $680.52. 1.06 $721.351 . $642 OPERATION. . . . Principal, . . . 1st amount, . . Total amount, $79.351 . . . Compound int principal, we find $79,351 for the required interest. COMPOUKD IKTEREST. 2^9 2. What is the compound interest on $918 for 3 years at Q% per annum, interest being compounded annually ? 3. What is the amount of $650 for 4 years at 6% per annum, interest being compounded semi-annually ? Explanation. — Here the period is one half of a year, and the amount of $1 for each period is $1.03. Proceeding as before, we find the amount. Note. — The operation of computing the amount of any sum for a given time may be shortened by the use of the following TABLE, Sliowiny the amount of $1 at compound interest, for any number of periods from 1 to 20. PERIODS. 2%. ^fc. 3%. Hfc. 4%. 5%. 6%. 11%. ' 1 1.0200 1.0250 1.0300 1.0350 1.0400 1.0500 1.0600 1.0700 2 1.0404 1.0506 1.0609 1.0712 1.081G 1.1025 1.1236 1.1449 3 1.0612 1.0769 1.0927 1.1087 1.1249 1.1576 1.1910 1.2250* 4 1.0824 1.1038 1.1255 1.1475 1.1699 1.2155 1.2625 1.3108 5 1.1041 1.1314 1.1593 1.1877 1.2167 1.2763 1.3382 1.4026 6 1.1262 1.1597 1.1941 1.2293 1.2653 1.3401 1.4185 1.5007 7 1.1487 1.1887 1.2299 1.2723 1.3159 1.4071 1.5036 1.6058' 8 1.1717 1.2184 1.2668 1.3168 1.3686 1.4775 1.5938 1.7182 9 1.1951 1.2489 1.3048 1.3629 1.4233 1.5513 1 6895 1.8385 10 1.2190 1.2801 1.3439 1.4106 1.4802 1.6289 1.7908 1.9672, 11 1.2434 1.3121 1.3842 1.4600 1.5395 1.7103 1.8983 2.1049! 12 1.2682 1.3449 1.4258 1.5111 1.6010 1.7959 2.0122 2.2522 13 1.2936 1.3785 1.4685 1.5640 1.6651 1.8856 2.132912.4098 14 1.3195 1.4130 1.5126 1.6187 1.73171.9799 2.26092.5785 15 1.3459 1.4483 1.5580 1.6753 1.8009|2.0789 2.3966 2.7590 16 1.3728 1.4845 1.6047 1.7340 1.8730j2.1829 2.5404 2.9522 17 1.40021.5216 1.6528 1.7947 1.9479 2.2920 2.6928 3.1588 18 1.42821.55971.7024 1.8575 2 . 0258 2 . 4066 2 . 8543 3 . 3799 19 1.4568 1.5987 1.7535 1.9225 2.1068:2.5270 2.0256,3.6165! 20 1.4859 1.6386 1.8061 1.9898 2.19112.6533 2. 2071 j 3.8697! J 230 PEHCEiTTAGE AND ITS APPLICATIONS. 4. What is the amount of $820 for Qyrs. at 4^ per annum, interest being compounded semi-annually? Explanation. — In this case there are 12 periods and the rate for each period is 2 % . From the table, we find that |1 at the given rate amounts to $1.3682 in that time, and because $820 amounts to 820 times as much as $1, we have, $1 .2682 x 820 = $1,039,924. Ans. 5. What is the amount of $900 for 9 years at 7^ per annum, interest being compounded semi-annually? 6. What is the compound interest on $1,850 for 3 yrs, at 8^, interest being compounded quarterly ? 7. Find the amount of $800 for 14 years at 7^ per annum, interest being compounded annually. Ans. $2,062.80. Note. — If the last period is fractional, compute the amount to the end of the next preceding period, and then find the amount of that result for the fractional period. 8. What is the amount of $500 for 3 years 2 months, at Q% per annum, interest being compounded annually ? Explanation.— The amount for 3 years is $1,191 x 500 or $595.50, and this in two months amounts to $601,455. Ans. 9. What is the amount of $1,200 for 4ryrs, Smos. at 7%, interest being compounded annually ? Ans. $1,646,365. 10. What is the amount of $1,350 for 6 yrs. 4:mos. at 6%, interest being compounded semi-annually ? A71S. $1,850.55. Note. — In using the table we retain the 4 decimal places given. REVIE^A^ QUESTIONS. (199.) What is compound interest ? How often may interest be added to j^rincipal ? (200.) Wliat is the rule for compound interest V Explain the use of the table. DISCOUNT. 231 VIII. DISCOUNT. 301. Discount is a percentage deducted from the face of a bill, debt, or note. COMMERCIAL DISCOUNT. 203. Commercial Discount is a percentage de- ducted from the face of a bill of merchandise. The face of the bill is the Base and the difference be- tween this and the discount is called the Net Proceeds. From definitions and preceding principles we have the following RULE. /. Multiply the face of the hill by the rate per c&nt. and the product will he the discount. II. Subtract the discount from the face of the bill and the difference will he the net proceeds. Note.— The net proceeds may be found by multiplying the face of the bill by 1 minus the rate per cent. EXAM PLES. 1. What is the discount on a bill of $350 at h%, and what is the net proceeds ? Ans. Dis. = 117.50 ; net proceeds = 1332.50. 2. Sold a bill of merchandise amounting to 11,173, deducting 10^ for cash ; what was the net proceeds ? 3. Flour is sold on credit at $12.50 per barrel ; what is the cash price, the discount being 15^ ? 4. Find the discount on a bill of goods whose face is $1,200, at the rate of ^%, 232 PERCENTAGE AND ITS APPLICATIONS. 5. Sold a lot of goods amounting to $918, deducting 124^^ for cash ; what was the net proceeds ? 6. What is the net proceeds of 56 tubs of butter, each weighing 42J lbs. at 22 cts. a pound, b% off for cash ? 7. Sold 50 hUs. flour at 17.50 per barrel, deducting 7^^ for cash ; what was the net proceeds ? 8. Sold coal at 15.50 per ton, 10% off for cash ; what was the cash price ? 9. Sold 500 5w. of oats at Q,^cts. a bushel, b% off for cash ; what was the cash price ? PRESENT VALUE AND TRUE DISCOUNT. 203. The Present Value of a debt payable at a future time is a sum which, being placed at interest, will give an amount equal to the debt when it falls due. Thus, $100 is the present value of $107 due 1 year hence, interest being reckoned at 1%. If the debt does not bear interest, its amount is the same as its face ; if it bears interest, its amount is equal to its /crce together with the interest up to the time it is due. The True Discount is the difference between the amount of the debt and its present value. The method of finding the present value of a debt due at a future time is the same as finding the principal, when the amount, the rate, and the time are given ; hence, the following RULE. I. Divide the amount of the debt by 1 plus the product of the rate by the time in years; the quotient will be the present value. DiscouiTT. 233 II. Subtract the present value from the amount of the debt ; the remainder will be the true discount. EX A M PLES. 1. What is the present value of $1,500, due 1 yr. 4 mos. hence, money being worth 6^ per annum ? Solution.— The rate, .06, multiplied by the time in years, 1^, equals .08 ; hence, $1,500 h- 1.08 = $1,388,889. Arts, 2. What is the true discount on $1,200, due 2 years hence, money being worth 11% ? Ans. $1,200 — $1,200 -^ 1.14 = $147.37, 3. What is the present value of a debt of $1,760, due 3 yrs. 6 mos, hence, at the rate of 6^ per annum ? 4. Find the true discount on a debt of $1,141.25, due 7 mos. 15 days hence, at 6^ per annum. 5. What is the true discount on $730, due in 2 years, at b% per annum ? 6. Find the present value of a debt of $986, due 2 yrs. 8 mos. hence, at Q% per annum. 7. What is the true discount in the last example? 8. What is the present value of $1,200, due in lyr, 4 mos., at 7^^ per annum ? 9. A debt of $1,400 is due in 9 mos. ; what is the true discount, interest being computed at the rate of Q% ? 10. Find the present value of a note for $750, due in 1 yr. 8 mos. 12 da., at Q%. 11. What is the present value of a note for $1,300, due in 2 yrs. 8 mos. at 7^ ? 12. What is the present value of $10,000, due in 4 mos. 18 da., at ^1% ? 234 PERCENTAGE AND ITS APPLICATIONS. 13. What is the present yalue of $1,828.75, due in 1 year, and bearing 4^% interest ? 14. What is the present value of a note for $4,800, due 4 yrs. hence, at 5% interest ? 15. A. owes B. $3,456, payable Oct. 27, 1877; what ought A. to pay Aug. 24, 1877, interest being Q% per annum ? BANKS AND BANK DISCOUNT. 304. A Bank is an incorporated institution author- ized by law to deal in money. Some of tlie operations of banking are, receiving money for safe keeping, discounting notes and other evidences of indebtedness, issuing bills to circulate as money, buying and selling bills of exchange, gold and silver bullion, coin, and the like. Most banks are engaged in only a part of these operations. 205, Bank Discount is a percentage charged for advancing money on a note or other obligation payable at a future time ; it is simply interest in advance on what the note or obligation will yieeld when it becomes legally due. This operation of advancing money on notes or obligations not yet due is called Discounting. 206. A note is said to Mature when it becomes legally due, which is three days after it is nominally due. The three days that elapse after a note is nomi7ially due before it is legally due are called Days of Grace. The time at which a note falls due may be denoted by a double date. Thus, the expression, " Due July /(a/' written at the foot of a note shows that it is nominally due on the 7th of July, and that it is legally due on the 10th of July. Note. — If the last day of grace falls on Sunday, or on a legal holiday, the note is legally due in some of the States on the preced- ing day, and in others on the foUomng day. DISCOUNT. 235 METHOD OF DISCOUNTING A NOTE. 207. The ordinary method of discounting a note is illustrated in the following example : Form of Note. $600. New Yoek, July 7, 1877. Sixty days after date I promise to pay to the order of Lucius Slocum six hundred dollars at the Fourth National Bank. Value received. KoBEET Bull. Due Sept. Vg, 1877. Explanation. — This note is supposed to be discounted on the day of its date. The holder, Lucius Slocum, endorses it by writing his name across its back, and delivers it to the proper bank officer. Interest is then computed on the face of the note, $600, for 63 days, the days of grace being included, and at the legal rate, which in New York is 7%. This sum, S7.35, is the discount ; subtracting the dis- count from $600, we have $592.65, which is called the Proceeds, and this amount is paid over to Lucius Slocum. If the note is not paid before the close of the last day of grace, Sept, 8, a written notice, called a Protest, is sent to Lucius Slocum, and he then becomes liable for its payment. EXAMPLE. 1. A note for $1,400, payable 60 days after date, is dis- counted at the rate of 7^; what is the proceeds ? Explanation.— The interest of $1,400 for 63 days at 7% is $17.15 ; this is the discount : subtracting it from $1,400, we have $1,382.85, which is the proceeds. Note. — If a note is not discounted on the day of its date, the dis- count is reckoned from the time of discount to the time of maturity. EXAMPLE. 2. A note for $1,200, dated Sept. 5, 1877, and payable 90 days after date at 7^, is discounted Oct. 2, 1877; what is the proceeds ? Ans. $1,184.60. 23 G PERCEl^TAGE AlTD ITS APPLICATIONS. Explanation.— Here the note is due Dec. ^/^, and from Oct. 2 to Dec. 7 is 66 days The interest on $1,200 for 66 days at 7% is $15.40 ; hence, the proceeds equals $1,200 — $15.40, or, $1,184.60. From what precedes we have the following RULE. /. Compute interest on the face of the note, from the time of discount to the time of jnaturity ; this will be the discount. II. Subtract the discount from the face of the note ; the result will be the proceeds. Note. — If an interest-bearing note is discounted, the amount of the note at maturity is made the base on which discount is reck- oned. E X AM PLE S. 3. What is the bank discount on a note for $670, pay- able 60 days after date at Q% ? 4. What is the bank discount on a note for $350, pay- able 00 days after date at 7% ? 5. What is the proceeds of a note of $1,000, payable at bank, 60 days after date, at Q% ? 6. A. has a note against B. for $1,728, payable 90 days after date, which he gets discounted at the rate of 7% ; what does he receive ? 7. A note for $1,620, dated July 7, 1877, and payable 90 days after date, is discounted July 25, 1877, at S%; what is the proceeds ? Note. — In what precedeo, interest has been computed on the sup- position that 360 days make a >oar. If accurate interest is required, as it is in many of the States, it may be found by diminishing the interest, computed as above, by its 7V }>a''t, or, better still, by means of tables constructed for the purpose. DISCOUNT. 237 208. To find the Face of a Note payable at a Future Time, whose Proceeds shall equal a Given Sum. To find the face of a note, payable 90 days after date, that will yield $500, the rate being 6%. Explanation.— The proceeds of $1 for 93 days at 6% is $.9845. But the entire proceeds must be $500 ; hence, the face of the note must be as many dollars as .9845 is contained in 500, that is, it must be 1500 -5- .9845, or $507,872. In like manner all similar cases may be treated ; hence, the fol» lowing RULE. Divide the given sum by the proceeds of $1 for the given time and rate; the quotient will he the numher of dollars required. EXAMPLES. 1. Find the face of a note payable in 90 days at 1%, so that the proceeds shall be $2,050. Ans. 12,087.747. 2. Find the face of a note payable 60 days after date at Q% that will yield $500. 3. Find the face of a note payable 90 days after date at 8^ that will yield $750. 4. Find the face of a note payable 30 days after date at 10% that will yield $1,000. REVIEW QUESTIONS. (201.) What is discount? (202.) What is commercial dis- count ? Rulg ? (203.) What is the present value of a debt ? What is true discount ? Rule for present value and true discount ? (204.) What is a bank ? What operations may be carried on by a bank ? (205.) What is bank discount ? What is discounting ? (206.) When does a note mature ? What are days of grace ? (207.) Explain the process of discounting a note. When and how is a note protested ? Give rule for finding the bank discount and proceeds of a note. (208.) What is the rule for finding the face of a note that will yield a given sum ? 238 PERCEl^TAGE AND ITS APPLICATIONS. IX. STOCKS AND BONDS. DEFINITIONS. 209. A Corporation is an association of persons authorized, under certain restrictions, to transact business as an individual. 210. The Capital Stock is the money used in car- rying on the business of a corporation. It is divided into equal parts called Shares. The owners of the shares are called Stockholders. 211. A Certificate of Stock is a certificate signed by proper authority, showing that the party therein named owns a certain number of shares of the capital stock. 212. The Par Value of a stock is the value named on the face of the certificate. This is usually $100 per share, but it may be either more or less. 213. The Market Value of a stock is the price it will bring in open market. If the market value is greater than the par value, the stock is said to be at a Premium, or Above Par ; if the market value is less than the par value, it is said to be at a Discount, or Below Par. 214. Dividends are percentages on the capital stock paid to stochholders as profits on the business. Assess- ments are percentages that stockliolders are called on to pay to meet losses and extraordinary expenses. 215. A Bond is a properly authenticated obligation to pay a sum of money at or before a certain time, with interest at fixed periods. STOCKS AND BOI^DS. 239 There are two classes of bonds : 1\ Bonds for whose payment the public faith is pledged, as National and State bonds ; and 2°. Bonds for whose payment the property of some incorporated com- pany is pledged, as Railroad bonds. 216. The face of a bond is usually $1,000, but it may be any sura, either greater or less. Stocks and bonds are bought and sold in open market. The reported price, or Quotation, is the number of per cent, that the selling price is of the par value. Thus, if a stock or bond is quoted at 87|, its selling price is 87|% of its par value. UNITED STATES BONDS. 217. The interest-bearing debt of the United States is represented by bonds issued at separate times, payable at different dates, and bearing various rates of interest. Some bonds carry certificates of interest, called Coupons, and pass from hand to hand like bank bills ; others are Registered on the books of the treasury, and can only be transfered by a change of record. United States Bonds are designated by giving their date of pay- ment, and, if necessary, their interest. Thus, the term ** TJ. S. 6's, 81 R." stands for " Registered 6% Bonds of the United States, pay- able in 1881;" the term "U. S. 5-20's, 65 C." stands for *' United States Coupon Bonds, payable at the option of the government at any time between 5 and 20 years after their date," that is, at any time between 1870 and 1885 ; the tenn " U. S. 10-40's " stands for " United States Bonds, payable at any time between 10 and 40 years after their date," which is 1864. All the 5-20's bear 6% interest, and all the 10-40's bear 5 % interest, payable in gold. 218. All problems relating to the buying and selling of stock and bonds are solved by the rules for percentage and interest. Note. — In what follows the par value of each share of stock is supposed to be $100, and the par value of each bond $1000 ; the brokerage, which is paid by the party for whom the purchase or sale 240 PERCENTAGE AKD ITS APPLICATIONS. is made, is supposed tohe^fo, and is always computed on the par value of the stock or bond. EXAM PLES. 1°. Applications of the Rules for Percentage (Arts. 165-168). 1. What is the market value of 120 shares of bank stock @ 53}^ ? Ans. $12,000 x .5375 = 16,450. 2. What is the cost, including brokerage, of 44 shares of bank stock @ 129^^ ? Ans. 14,400 x 1.2975 = 15,709. 3. Sold 15,000 in gold @ lVi\%] what was the net pro- ceeds after deducting brokerage. A7is. $5,600. 4. A. sold 160 shares railroad stock @ 92|^, and pur- chased with the proceeds bank stock @ 73|-%', paying brokerage on both sa,le and purchase ; how many shares did he receive ? Ans, 160 x 92^ -^ 74 = 200. 5. What is the premium on 88 shares of bank stock @ 114^? Ans. 18,800 x .14 = $1,232. 6. What is the discount on 190 shares of railroad stock which is sold @ 89^ ? 7. A. sold 93 bonds, U. S. 5-20's, R. @ 113^^, paying brokerage on the sale ; what did he receive ? 8. A speculator buys 225 shares of Erie stock @30J^ and sells it again @ 31^;^, paying brokerage on both trans- actions ; what does he gain ? 9. If 170 railroad bonds cost (including brokerage) $192,100, what is their market rate ? 10. What is the cost, including brokerage, of $9,000 U. S. 6V81 C. @112i? 11. Sold $42,000 U. S. 5-20's C. for $46,830 net, after paying brokerage ; what was the market price ? STOCKS AND BONDS. 241 12. How much gold @ lllf can be bought for $8,930 in currency, brokerage not considered ? 13. A broker sells 30 shares of bank stock @ 96^%, and 120 shares of railroad stock at 105^, retaining the broker- age ; how much does he pay to his principal ? Ans. $15,457.50. 2°. AppUcatmis of the rules for interest. 14. If I buy a 6% stock @ 90^, what rate of interest do I receive on the investment ? Explanation. — Here the principal, $90, yields an interest of $6 per annum ; hence, we have, $6-f-$90 = 6|%. Ans. 15. The market rate of a 6% stock is 85^% ; if the pur- chaser pays brokerage, what rate of interest does he receive on his investment ? 16. An 8% stock sells @ 112^ ; what rate of interest does it yield to the purchaser ? 17. At what rate must an 8% stock be purchased to yield the purchaser 7^ interest ? In this case the price of one share is a principal which, at the rate of 7%, yields $8.00 in one year; hence, (Prin. 5°, Art. 164), we have, $8.00 -^ $.07 = 114f %. Ans. 18. I wish to purchase a 6% stock on such terms that it will give me 7% on my investment ; how much can I pay for the stock, including brokerage ? 19. If I buy U. S. 6's, '81, at 112|, including brokerage, and sell the gold interest at 107|-^ for currency, what rate of interest do I get on my investment ? Ans. $6 X 1.071 -r- 112^ = 5.75^, nearlt/. 24:2 PERCENTAGE A^D ITS APPLICATIONS. 20. At what rate must I buy a 6% stock to get the same rate of interest as from a 6% stock at 116% ? Ans. 90%. REVIEW QUESTIONS. (209.) What is a corporation? (210.) What is the capital stock? Shares? Stockholders? (211.) What is a certificate of stock? (212.) What is par value of a stock? (213.) What is the market value ? When is a stock at a premium ? At a discount ? (214.) What are dividends? Assessments? (215.) What is a bond ? Explain the two classes. (2 16.) How are bonds and stocks quoted? (217.) Give an account of U. S. bonds. What are cou- pons and coupon bonds? Registered bonds? (218.) By what rules do we solve problems in stock operations ? X. EXCHANGE. DEFINITIONS. 219. Exchange is a method of making payments at distant places by means of drafts, or bills of exchange. The theory of settling accounts by exchange may be illustrated by the following simple case : A flour merchant of Chicago forwards $5,000 worth of flour to a shipper in New York, and at the same time a dry goods merchant of Chicago buys $5,000 worth of mer- chandise from a New York importer. The flour merchant draws his draft for $5,000 on the shipper and sells it to the dry goods mer- chant ; the latter forwards it to the importer, who presents it to the shipper and receives the money called for. In this way the debts in both cities are liquidated without the necessity of sending any money from either. Dealers in exchange usually buy drafts on distant places and send them forward as a basis of credit ; they then sell their own drafts drawn against this credit in sums to suit their customers. 220. A Draft or Bill of Exchange is a written order from one party to another to pay to a third party a certain sum of money at a specified time. EXCHANGE. 243 A Sight Draft is one that is payable on presentation ; a Time Draft is one that is payable at a specified time, or at a certain time after presentation. Note. — In the latter case three days grace are allowed, but not in the former. 231. The Drawer or Maker is the party that draws the bill ; the Drawee is the one on whom it is drawn ; and the Payee is the party to whom the money is to be paid. A party that buys a bill of exchange is called a Buyer, or Re- mitter ; if the payee, or any other party, writes his name across the back of the bill, he is called an Indorser ; and the party that has legal possession of the bill at the time of payment is called the Holder. 223. An Acceptance is an agreement on the part of the draicee to pay the draft at maturity. If he agrees to pay it, he writes the word Accepted across its face and signs his name ; he is then called an Acceptor. 223. An Inland or Domestic Bill is one in which both drawer and drawee reside in the same country. A Foreign Bill is one in which the drawer and drawee reside in different countries. 224. The Par of Exchange between two places is the relative value of the principal units of currency of the two places. Thus, £1 Sterling is equal in value to $4.8665, and this is the par of exchange between London and New York ; the Franc is equal to $0,193, and this is the par of exchange between Paris and New York ; the Mark is equal to $0,238, and this is the par of exchange between Berlin and New York. Note.— If $4.8665 in New York will just purchase a bill of £1 on London, exchange on London is at par ; if it will buy a larger bill, exchange is against London and in favor of New York ; if it requires 244 PERCENTAGE AND ITS APPLICATIONS. more than $48665 to buy a bill of £1, exchange is in favor of Lon- don and against New York. 225. The Course of Exchange is the variation in price of bills of exchange. These variations are shown in the daily quotations published in the papers. Thus, on the 10th of August sterling exchange was quoted at $4.86, sight; that is, a sight bill on London was worth $4.86 for each pound sterling of its face. INLAND OR DOMESTIC EXCHANGE. 226. Inland or Domestic Exchange is the method of making payments at distant places in the same country by means of drafts or inland hills of exchange. Form of a Draft. 1500. Troy, N. T., Aug. 10, 1877. At sight, pay to the order of W^illiam Whitman Five Hundred Dollars, value received, and charge the same to the account of John S. Wesley. To the Fourth National Bank, New York City. Note.*— To change the above to a time draft, say for 10 days, the words " at sight" are replaced by " at ten days sight." 227. All problems relating to inland exchange can be solved by the rules for percentage and bank discount. In applying the rules for percentage, the Face of the Bill is the base, the Premium or Discount is the raie per cent., and the Cost of the Bill is the amount. EXAMPLES. 1. What is the cost of a sight draft when exchange is 1^% premium ? Solution.— $500 + 1^% of $500 = $507.50. Am. EXCHAI^GE. 245 2. What is the cost of a sight draft on New York for 11,250, exchange being 1^% discount ? Ans. $1,234.37|-. 3. What is the market price of a sight draft on New York for $890, exchange being worth 101 J^ ? Explanation. — Here the rate of premium is l|^%, and 1 plus the rate, that is, the amount per cent, is 1.0125 ; hence, the draft is worth $890 x 1.0125 = $901. 12|. Ans. 4. Find the market value of a sight draft on New York for 11,800, exchange being 99^. A71S. 11,782. Note. — Both exchange and bank discount are computed on the face of a time draft. They may be computed separately, or together, as is more convenient. 5. Find the cost of a draft on New York for 11,400, payable 60 days after sight, exchange being worth 102-J-^, and interest being reckoned at '7%. OPERATION. Amount of $1400 at 2^% premium . . . $1435.00 Interest on $1400 for 63 days at 7% . . . 17.15 Difference $1417.85 Ans. 6. What is the cost of a draft on New York for 12,400, payable 90 days after sight, interest being 10^ and ex- change 103^ ? Ans. 12,410. 7. Find the value of a draft on New York for $1,650, payable 60 days after sight, exchange being 98|^ and interest 6%. Ans. $1,650 x .9745 = $1,607,925. 8. If exchange is 101^^, how large a sight draft can be bought for $7,900 ? A71S. $7,900 -^ I.OIJ = $7,783,251. 9. What is the face of a sight bill that can be bought for $5,000, when exchange is 98^^ ? Solution.— $5,000 -^ .985 --= $5,076.14^. Ans. 246 PERCENTAGE AND ITS APPLICATIONS. 10. What is the face of a draft at 60 days sight which costs 11,000, exchange being 103^ and interest 6% ? Explanation. — The interest on $1 for 63 days is $0.0105, and this taken from $1.03 gives $1.0195 as the cost of $1 of exchange. Hence, the face of the bill is $1,000 -^ 1.0195 - $980.87^^. Arts. 11. What is the face of a draft at 30 days, which costs $2,000, exchange being 102% and interest 6% ? FOREIGN EXCHANGE. 228. Foreign Exchange is the method of making payments in foreign places by means of bills of exchange. Form of Set of Exchange. ^^^^' New York, Aug. 11, 1877. At sight of this first of exchange {second and third of same date and tenor unpaid), pay to the order of Salmon Stoddard Five Hundred Pounds Sterling, value received, and charge the same to Johnson Lummis. To, Copely & Brothers, London. Note. — Three copies constitute a set, each of which is forwarded by a different mail. The other copies differ from the one above given, in no respect, except that in one the words first and second change places, and in the other the words first and third change 229. Exchange betweeen the United States and for- eign countries is computed by the method of equivalents. The equivalents are made known, both for sight and time bills, by the current or daily quotations. But as these equivalents are in gold, they must be converted into cur- rency by the ordinary rules for percentage. EXCHAN^GE. 247 EXAM PLES. 1. What is the cost of a sight bill for £87, when £1 is worth $4.82 in gold, gold being worth 106^ currency ? Explanation. — Because £1 is worth |4.82 in gold, £87 is worth $4.83 X 87 = $419.34, and this is converted into currency by multi- plying it by 1.06 ; hence, $419.34 x 1.06 = $44450. Ans. 2. What is the cost of a sight bill on London for £750, when £1 is worth $4.85, gold being quoted at 108^ ? 3. What is the cost of a sight bill on London for £300, £1 being worth $4.86, gold 107^ currency ? 4. Find the cost of a 60 day bill on London for £315, when £1 at 60 days is worth $4.80, gold being quoted at 105 ? 5. How large a bill on London can be bought for $2,000 in currency, when sterling exchange is quoted at $4.85 and gold at 106 ? Explanation. — Here we convert $4.85 in gold into currency, which gives $4.85 x 1.06 = $5.14. Then, because it takes $5.14 in currency to buy £1 of exchange, we have $2,000 -4- 5.14 = £389 2s. Id. Ana. 6. How large a 60 day bill on London can be bought for $3,000, when 60 day bills are quoted at $4.84, and gold at 110 ? 7. What is the cost of a bill on Paris for 3,000 francs, when exchange is at the rate of $1 to ^.16 francs, and gold is worth 110 ? Ans. ($5,000 ^ 5.15) x 1.10 = $1,067,96^. 8. What is the cost of a draft on Paris for 30,000 francs, when exchange is at the rate of $1 for 5.25 francs, and gold is worth 106 ? 9. How large a bill on Paris can be bought for $1,000 in gold, when exchange is at the rate of $1 to 5.20 francs ? 248 PERCENTAGE AKD ITS APPLICATIONS. REVIEW QUESTIONS. (219.) What is exchange? (220.) What is a draft or bill of exchange? A sight draft? A time draft? (221.) What is the drawer ? The drawee ? The payee ? A remitter or buyer ? An indorser? A holder? (222.) What is an acceptance? How made? An acceptor? (223.) What is an inland bill? (224.) Define par of exchange. Illustrate. (225.) What is the course of ex- change? Illustrate. (226.) What is inland or domestic exchange? How are problems in inland exchange solved? (228.) What is foreign exchange ? (229.) How is foreign exchange computed ? XI. EQUATION OF PAYMENTS. DEFINITIONS. 230. Equation of Payments is the operation of finding the time at which several debts due at different times may be paid without loss of interest to either party. The time at which the payment may be made is called the Equated Time, and the corresponding date is called the Equated Date. 231. The time that a debt has to run is called its Time of Credit, and the date from which this time is reckoned is called the Initial Date. A credit of II for a unit of time is called a Unit of Credit ; the product of a debt by its time of credit is its Amount of Credit. The unit of time may be 1 day, or 1 month, but it must always be the same throughout the same problem. 232. The solution of every problem in Equation of Payments depends on the following PRINCIPLE. Tlie amount of credit of the sum of several debts is equal to the sum of the amounts of credit of the debts taken separately. OPERATION. 400 X 9 = 3600 800 X 6 = 4800 600 X 4 = 2400 EQUATIOI^ OF PA-XMENTS. 249 There may be three cases : V, the debts may be on one side, and have the same initial date ; 2'', the debts may be on one side, and have different initial dates ; 3°, some of the debts may be on one side, and some on the other, the dates of payment being different. 233. When the Debts are on one side and have the same Initial Date. On the 1st of January, 1877, A. owes B. 1400 payable in 9 months, $800 payable in G months, and $600 payable in 4 months ; at what date may the whole debt be paid without loss to either party ? Explanation.— In this case the unit of credit is $1 for 1 month. Multiplying the different sums by their terms of credit 1800 ) 10800 ( 6 and adding the pro- A..^ \ Equated time = 6 months, ducts, we find the ' ] Equated date, July 1, 1874. sum of the amounts of credit equal to 10,800 units ; but this is equal to the amount of credit of $1,800 for the required time ; hence, 10,800 divided by 1,800 will give the number of months in the equated time. Adding this to the initial date we have the equated date. Since all similar cases may be treated in the same manner, we have the following RULE. /. Multiply each debt by its time of credit and find the sum of the products; divide this by the sum of the debts and the quotient will be the equated time. II. Add the equated time to the initial date and the result will be the equated date* 250 PERCENTAGE AKD ITS APPLICATIONS. EXAMPLES. 1. A merchant owes $2,400, of which $400 is payable in 6 mos., 1800 in 10 mos., and $1,200 in 16 7nos.; what is the equated time? Ans. 12^ months. 2. A. owes B. $2,400, of which $800 is payable in 6 months, $600 in 8 months, and $1,000 in 12 months; what is the equated time ? Ans. 9 months. 3. A merchant owes $300, payable as follows : $80 in 22 days, $100 in 60 days, and $120 in 75 days; what is the equated time ? Ans. 56 days. Note. — The rule gives 55f| days. In such cases we shall neglect the fraction when it is less than ^, and add 1 to the whole number when it is equal to, or greater than ^. If the unit of time is 1 month, the fractional part will be reduced to days at the rate of 30 days per month, and fractional parts of the latter will be treated as just explained. 4. A. owes B. $1,000, payable as follows: $200 at present, $400 at 6 months, and $400 at 15 months ; what is the equated time ? Ans. 8 months 12 days. 5. A. owes a sum of money, of which \ is payable at 30 days, ^ at 60 days, and the rest at 90 days ; what is the equated time for the payment of the whole ? 234. When the Debts are on one side, and have different Initial Dates. A. sells goods to B. on a credit of 60 days, as follows : Jan. 14, 1874, a bill of $2,000 Feb. 10, '' '' '' " $1,500; March 25, " '' " '' $3,1)00 Kequired the equated time and date of payment. EQUATION OP PAYMENTS. ^1 Explanation. — The operation. first payment is due March 15, 2000 X 0= 00000 March 15, the second April 11,1500x27= 40500 "^ April 11, and the third ^ 24,3000x70=210000 May 24. Assuming the "^ date of the earliest pay- 6500) 250500(38.538 ment as the initial date, j Equated time, 39 days. theotherpaymexitswill ^^- "j Equated date, April 23, 1874. have 27 and 70 days to > r ^ run. Proceeding as before, we find the equated time greater than 38.5 days ; calling it 39 days, according to the rule, and adding it to the assumed initial date, we find the equated date to be March 15 + 39(^a.rr April 23. Since all similar cases may be treated in the same manner, we have the foUowmg RULE. Find the date of each payment, and assume the earliest as an initial date ; then find the time of credit of each item, and proceed as in the last case. EXAMPLES. 1. A. bought goods as follows : May 5, 1500 on 4 months' credit ; May 25, $750 on 4 months' credit ; June 27, $800 on 6 months' credit ; what is the equated date of payment of the whole debt? Ans. Oct. 26. Note. — When credit is given by months, calendar months are understood, without reference to the number of days they contain ; in applying the preceding rule, the times of credit, if not exact months, are reduced to days, counting the actual number of days in each month. Thus, in the example just given, the items mature Sept. 5, Sept. 25, and Dec. 27, and the times of credit, counting from Sept. 5 as the initial date, are 0, 20, and 113 days. 2. There are three notes payable as follows: the Ji7'st for 1500, Feb. 12, 1877 ; the second for $400, March 12, 1877; and the third for $300, April 1, 1877; what is the equated date for the payment of all ? A ns. March 5,1877. 252 PERCEKTAGE AND ITS APPLICATIONS. 235. When each party owes the other, the times of payment being different. The difference of the sums of the deUts and credits of an account is called the Balance of Account. The balance must be added to the smaller sum. By preceding rules, all the items on either side may be reduced to a single item, payable at a specified time. Hence, every account may be reduced to two items, one on eacli side. The following suppositions illustrate every case that can arise : 1°. Suppose that A. owes B. 1500 payable July 16, 1877, and that B. owes A. 1200 payable July 1, 1877. 2°. Suppose that A. owes B. 1200 payable July 16, 1877, and that B. owes A. $500 payable July 1, 1877. In both cases it is required to find the date at which the 'balance of account may be paid without loss to either party. In each example let the date of the earlier payment, July 1, 1877, be taken as the initial date. 1°. Explanation. — On the 1st operation. of July, A. was entitled to 500 x 15, 5qq >^ j^g __ 750O or 7,500, units of credit ; but at that 7500 time the balance due B. was but = 25. $300 ; hence, the time before the ^^^ balance became due was 3^^^, or Ans. July 26, 1877. 25 days ; adding this to the initial date, we find that the balance was due July 36, 1877. 2°. Explanation.— On the 1st of July, A. was entitled to 200 x 15, or 3,000, units of credit ; but B. was ^^0 X 15 = 3000 also entitled to the same amount ; 3000 hence, the balance due him, $300, oqq = 1^^* was entitled to credit for ^V(r» or a j 01 10*7 »v 10 days before the 1st of July, that ^^^- ^^^"6 4,1, 18 77. is, from the 21st of June, 1877. EQUATION OF PAYMENTS. 353 Since all similar cases can be treated in the same manner, we have the following RULE. /. Reduce each side to a single term, and take the date of the earlier payment as an initial date ; then multiply the side of the account that falls due last by the time between the dates of payment and divide the product by the balance of the account; the quotient will be the equated time. II. If the greater side of the account falls due last, add the equated time to the initial date; if the smaller side falls due last, subtract the equated time from the initial date; the result will be the equated date at which the balance is payable. EXAM PLES. 1. A. owes B. $8,750 payable July 21, 1877, and B. owes A. $6,500 payable June 9, 1877; when, and to whom, is the balance payable ? Ans. To B. Nov. 19, 1877. 2. A. owes B. $6,500 payable July 21, 1877, and B. owes A. 18,750, payable June 9, 1877; to whom is the balance payable, and what is its equated date ? Ans. To A., and the equated date is Feb. 8, 1877. EQUATION OF ACCOUNTS. 236. Equation of Accounts is the operation of finding the time at which the balance of an account should be made payable in order that there may be no loss of in- terest to either party. This time is called the Equated Time. :e54 PERCENTAGE AND ITS 4PPLICATIONS. 237. Let it be required to find the equated time of payment of the balance of the following account : Dr. Hekry Ahl in acct. with Benj. Barrol. Or. 1878. 1873. — April 1. Merchandise. $875 00 April 20. Cash. . . $500 00 " 18. " 250 00 May 20. 5. cost? Note. — If all the numbers have the same unit, the nature of the question will show which is to be the third term. 6. If a piece of property worth $3,250 is taxed $35.75, what should be the tax on a house worth $17,350 ? Explanation. — Here the answer is to be the tax on $17,350 ; hence, the third term must be the tax on $3,250. Statement.— $3,250 : $17,350 : : $35.75 : x. 7. Solve the proportion 3 : 4 : : 21 : ic. 8. If 3 pairs of socks cost $1.41, what will 7 pairs cost ? 9. If ^ tons of hay will keep 2 cows for the winter, how many cows can be kept on 24f tons ? 10. If 18| bags of coffee contain 758 lbs. 8 oz., how many bags are there in 12,136 lbs.? 266 RATIO AND PROPORTIOIS". 11. How long will it take to travel 1,290 miles, at the rate of 306f miles in 20|- days ? 12. If 2^ yds. 3 qrs. of carpeting, 1 yard wide, will cover a room, how many yds. of carpeting. If yds. wide will it take to cover the same room ? Explanation.— It will take fewer yards of the latter width than of the former ; hence, the fourth term is less than the third. Statement.— If ^ to the remainder bring down the second period for a dividend. III. Double the root found and place it on the left for a d^ivisor. See how many times this divisor is contained in the dividend, exclusive of the right hand figure, and place the quotient in the root and also at the right of the divisor. 278 ROOTS. IV. Multiply the divisor, thus augmented, by the last figure of the root already found, subtract the product from the dividend and to the remainder bring down the next period for a new dividend. V. Double the root already found for a new divisor, and continue as before, until all the periods have been brought down and operated on. Notes. — 1. If any quotient figure proves too large, let it be diminished until it gives a product less than the partial dividend. 2. If the last remainder is 0, the given a number is a perfect square and the root is exact ; if not, the root is true to within less than 1. 3. The square root of a simple fraction is equal to the square root of its numerator divided by the square root of its denominator, EXAMPLES. 1. Find the square root of 8836. Explanation. — The two periods are 88 operation. and 36 ; the greatest perfect square in 88 ftSSfiCQi is 81, (table. Art. 260), and its square root — is 9 ; this we write as the first figure of the "^ root and place its square 81 under the first 18/4)73/6 period ; subtracting, we have 7 for a remain- no q der, to which we bring down the period 36 for a dividend ; doubling 9 we have 18, which we place on the left for a divisor, and this is contained 4 times in 73 ; we therefore place 4 on the right of 9 and also on the right of 18 ; multiplying 184 by 4 we find 736, which taken from 736 gives for a remainder ; hence, the squaie root of 8836 is 94. Perform the following indicated operations : 2. V9604. 6. V14641. 10. A/«i. 3. \/T3225. 7. V37636. 11. V^TT- 4. V342225. 8. V41616. 12. VfU. 5. V944784. 9. V52441. 13. V^T- SQUARE ROOT. 279 Note. — If there is a remainder the operation may be continued by annexing periods of decimal ciphers ; for each period thus annexed there will be one decimal figure in the root. Thus, /v/i87 = ^^187.0000 = 13.67. Here the approximate root is true to within less than .01. Find the square roots of the following numbers to two decimal places : 14. 229. Ans. 15.13. 16. 450. Ans, 21.21. 15. 354. Ans. 18.81. 17. 592. Ans. 24.33. Note. — The square root of a decimal may be found by the pre_ ceding rule. In this case we begin to point off periods at the deci. mal point and proceed toward the right. Any simple fraction may be changed to a decimal and then operated upon by the rule. Find the square roots of the following numbers to three places of decimals : 18. .0249. Ans. .157. 21. .152881. Ans. .391. 19. .69. Ans. .830. 22. .326041. Ans. .571. 20. .1051. Ans. .324. 23. .010404. Ans. .102. PRACTICAL PROBLEMS. 1. A general forms an army of 117,649 men in a square; how many men are there in each rank and how many ranks in the square ? Ans. 343. 2. In a square pavement there are 48,841 stones, each 1ft. square ; what is the length of the pavement and what is its breadth? Ans. 221fL 3. A square farm contains 640 acres ; how long is each gide? Ans. S20 rods. 4. A square field contains 160 acres; what will it cost to build a wall around if each rod of wall cost $2 ? Ans. $1,280. 280 ROOTS. CUBE ROOT. 262. The Cube Root of a Number is one of its three equal factors. Thus, 5 is the cube root of 125. If a number is not a perfect cube its cube root is only approx- imate. All the perfect cubes less than 1,000, with their cube roots, are written in the following TABLE. Perfect cubes, 1 8 27 64 125 216 343 512 729; Cube roots, 12345 6 7 8 9. Note. — The sign, ^ , shows that the cube root of the number under it is to be taken. Thus, /y/125 denotes that the cube root of 125 is to be taken. The number 3 written over the sign is called an Index. METHOD OF EXTRACTING A CUBE ROOT. 263, The method of finding the cube root of a num- ber depends on the principles of algebra, (see Manual of Algebra, Art. 111). In accordance with these principles we have the following RULE. I. Separate the numher into periods of three fig- ures each, heginning at the right; the left-hand pe7%od will often contain less than three figures. II. Find the greatest perfect cube in the first period on the left, and set its root on the right after the manner of a quotient in division; subtract the cube of this root from the first peHod and to the remainder bring down the first figure of the next period for a dividend. CUBE ROOT. 281 III. Take three times the square of the root thus found for a divisor, find how many times it is con- tained in the dividend, and place the quotient for a second figure of the root. Cube the number thus found, and, if its cube is greater than the first two periods, diminish it successively by 1 until its cube is less than the first two periods ; then subtract the result from the first two periods and to the remain- der bring down the first figure of the next period for a new dividend. IT. Take three times the square of the root found for a new divisor and proceed as before, continuing the operation tilt the periods have been operated on. Notes. — 1. If the last remainder is the number is a perfect cube and the root is exact ; if not, the root is true to within less than 1. 3. The cube root of a simple fraction is equal to the cube root of its numerator divided by the cube root of its denominator. 3. The cube root of a decimal or the approximate cube root of an Imperfect cube, may be found by a process entirely similar to that employed in finding the square root in similar cases. EXAM PLES. 1. Find the cube root of 804357. Explanation.— The num- opebation. ber having been separated 804 357 ( 93 into periods, we find the ^g ^J^ greatest cube in 804 to be ^ — ^"^^ 739 and its cube root, 9, is 3 ^ 92 _ 243 ) 753 the first figure of the root ; —"oTUq^ taking 739 from 804 and ^^ — ^^^^^ ^ bringing down 3, we have Q 753; dividing this by 3 times the square of 9, or 243, we get 3 for the second figure of the root ; cubing 93 we find the result equal to the given number ; hence 93 is the required root. 282 PEOGRESSIONS. Perform the following indicated operations : 2. ^531441. Ans. 81. 6. ^/W. Ans, 4.06. 3. ^^970299. Ans. 99. 7. 'V^lot ^^5. 4.7. 4. v^SSM?. ^ws. 33. 8. '^206. Ans. 6.9. 5. \^224755712. AnsMS. 9. ^1^585. ^^s. 8.36. REVIE^A/■ QUESTIONS. (259.) What is a root of a number? What is a perfect square? A perfect cube ? (260.) What is the square root of a number ? What is the radical sign ? What does it show ? (261 .) What is the rule for extracting the square root of a number ? Of a simple fraction ? How do you find an approximate value of a square root ? How do you find the square root of a decimal to any number of places ? Of a simple fraction reduced to a decimal ? (262.) What is a cube root? Its sign? What is an index? (263.) Give the rule for extracting the cube root of a number. III. PROGRESSIONS. DEFINITIONS. 264. A Progression is a series of numbers that in- crease, or decrease, according to a common law. The numbers forming a progression are called Terms j the first and last terms are called extremes and all the rest are called means. Note. — Progressions are of two kinds, arithmetical and geo- metrical. 1°. ARITHMETICAL PEOGRESSION. DEFINITIONS. 265. An Arithmetical Progression is one in which each term, after the first, is equal to the preceding term increased, or diminished, by a given number. This num- ber is called the Common Difference. ARITHMETICAL PROGRESSION. 283 If a progression is formed by the continued addition of a com- mon difference it is increasing ; if it is formed by the continued subtraction of a common difference it is decreasing. The first of the following progressions is increasing and the second is decreasing : 2 4 6 8 10 Increasing progression. 10 8 6 4 2 Decreasing progression. If the increasing progression is inverted, that is, if it is taken in a reverse order, it becomes a decreasing progression. TO FIND ANY T^RM. 366. From the preceding definitions it is obvious that we may find any term by the following RULE. Multiply the commofb difference by the number of terms that precede the required term ; if the progression is increasing, add the product to the -first term; if the progression is decreasing, sub- tract the product from the first term. EXAM PLES. 1. The first term of an increasing arithmetical pro- gression is 3, and the common diiference is 3 ; what is the 9th term ? Ans. 3 f 3 x 8 = 27. 2. The first term of a ^ecre«5m^ arithmetical progres- sion is 36, and the common difference is 6 ; what is the 5th term ? Ans. 36 — 6 x 4 = 12. 3. In an increasing progression the first term is 4, and the common difference is 2 ; what is the 20th term ? Ans. 42. 4. In a decreasing progression the first term is 45, and the common difference 4 ; what is the 8th term ? Ans. 17. OPERATION. 3, 6, 9, 12, 15, 18 18, 15, 12, 9, 6, 3 284 PKOGRESSIOKS. TO FIND THE SUM OF THE TERMS. 267. A rule for finding, the sum of the terms may be deduced by inverting the progression and proceeding as in the following OPERATION. 18 . . , Given progression. Same inverted. 21 +21 + 21 +21+21 + 21 . . . Sum of both. Explanation. — The sum of the terms in both progressions is obviously equal to twice the sum of the terms of the given pro- gression ; hence, the sum of the terms of the given progression is -V- X 6, or 63. Since all similar cases may be treated in the same manner, we have the following RULE. Multiply half the sum of the extremes by the number of terms. EXAM PLES. 1. The first term of a progression is 3, the last term is 27, and the number of terms is 9 ; what is the sum of the terms ? ' Ans. 135. Note.— If the last term is not given, it may be found by the rule of Article 206. 2. The first term of a decreasing progression is 36, the common difference is 6, and the number of terms is 5; what is the sum of the terms ? Ans, 120. 3. In a decreasing progression, the first term is 45, the common difference is 4, and the number of terms is 8; what is the sum of the terms ? Ans. 248. GEOMETRICAL PROGRESSION^. 285 4. What is the sum of the natural numbers, 1, 2, 3, &c., up to 99, inclusive ? Ans. 4,950. 5. The first term of a decreasing progression is 15, the last term is 5, and the number of terms is 6 ; what is the sum of the terms? Ans. 60. 6. The first term of an increasing progression is 15, the common dificrence is 3, and the number of terms is 6; what is the sum of the terms ? Ans. 135. 7. What is the sum of the terms of the progression 1, 2, 3, 4, etc., up to 12 inclusive ? Ans, 78. 8. The first term of an increasing progression is 7, the common difference is 4, and the number of terms is 7; what is the sum of the terms ? Ans. 133. 2°. GEOMETEICAL PROGRESSION. DEFINITIONS. 268. A Geometrical Progression is one in which each term, after the first, is equal to the preceding term multiplied by a given number. This number is called the Ratio of the progression. If the ratio is greater than 1, the progression is increadng ; if less than 1, the progression is decreasing. Thus, 3, 4, 8, 16, is an increasing progression, and 16, 8, 4, 3. is a decreasing progression. The ratio in the first case is 3 and in the second case it is \ ; in all cases, the ratio is equal to the quotient obtained by dividing the second term by the first. TO FIND ANY TERM. 269. In accordance with the preceding definitions, we may find any term by the following 286 PROGRESSIONS. RULE. Raise the ratio to a power whose exponent is the number of terms that precede the required term and multiply the first term by the result. EXAM PLES. 1. In a progression the first term is 3, and the ratio is 3; what is the 6th term ? Ans, 3 x 3^ = 729. 2. The first term of a progression is 64, and the ratio is i; what is the 5fch term ? Ans. 64 x (i)^ = 4. 3. Find the 10th term of the progression 2, 4, 8, &c. A?is. 2 X 29 — 1,024. 4. What is the 5th term of the progression 243, 81, 27, &c..? Ans. 3. TO FIND THE SUM OF THE TERMS. 270. Let it be required to find the sum of 4 terms of the series 2, 8, &c. 2 -f 8 + 32 + 128 Indicated Bum of the terms ; 8 + 32 4- 128 + 512 4 times tlie sum; 512 — 2 3 times the sum; 512-2 128x4-2 .^_ ^ r^: = i7U. . . Required sum. o o Explanation. — Having indicated the sum of the terms, we mul- tiply each by 4 and set the products one place toward the right ; the sum of these results is 4 times the sum of the given series ; subtracting the latter from the former, we find 512 — 2, which is 3 times the required sum ; dividing by 3, we have 170, which is the required sum. Since all similar cases may be treated in like manner, we have the following GEOMETRICAL PROGRESSIOIS". 287 RULE, Multiply the last term by the ratio; take the difference between the product and the first term; multiply this by the difference between 1 and the ratio. EXAMPLES. 1. The first term of a progression is 3, the last term is 729, and the ratio is 3 ; what is the sum of the terms ? . 729 x 3-3 ^ „„„ Ans, T = 1,092. Z Note. — If the first term and ratio are given, the last term may be found by the preceding rule. 2. The first term of a progression is 2, the ratio is 4, and the number of terms is 5 ; what is the sum of the terms? Ans. 682. 3. The first term of a geometrical progression is 3 and the ratio is 2; what is the sum of 6 terms ? Ans. 189. 4. The first term of a geometrical progression is 64 and the ratio is J ; what is the sum of 6 terms ? Ans. 126. 5. What is the sum of 7 terms of the progression 2, 6, 18, etc. ? Ans. 2,186. REVIE^A^ QUESTIONS. (264.) What is a progression? What are terms ? Extremes? Means ? (265.) What is an arithmetical progression ? An increas- ing progression ? A decreasing progression ? (266.) What is the rule for finding any term? (267.) For finding the sum of the terms ? (268. ) What is a geometrical progression ? What is the ratio ? When is the progression increasing and when decreas- ing? (269.) How do you find any term? (270.) The sum of any number of terms ? 271. Mensuration is the operation of finding how many times any given magni- tude contains its unit of measure. The unit of measure of a magnitude is always a magnitude of the same kind. The unit of a line, or the linear unit, is a straight line of given length ; as one foot : the unit of a surface, or the superficial unit, is a square whose sides are equal to the linear unit ; as one square foot : the unit of a volume, or the cubic unit, is a cube whose edges are equal to the linear unit ; as one cubic foot. Note. — The rules for mensuration depend on the definitions and principles of Geometry, some of which have already been given. In what follows, the references are to the Manual of Geometry. In these references B. stands for Book and P. for Proposition. 273. A Polygon is a plane figure bounded on all sides by straight lines. Each of the bounding lines is called a Side of the polygon, and the point at which any two sides meet is called a Vertex of the polygon. ^^ 273. The Area of a Polygon is the number of superficial units that it contains. 274. A Triangle is a polygon of three sides ; as ABC ; the side AB on which it MENSUBATIOl^. 289 EU A K B PARALLELOGRAM. E F RECTANGLB. is supposed to stand is called its Base, and the shortest distance, CB, from the opposite vertex to the base is called its Altitude. A Right-angled Triangle is a triangle that has one right angle. Thus, ABC is a right-angled triangle. The side AC, opposite the right angle, is called the Hypothe- nuse. *^75. A Parallelogram is a polygon of four sides, parallel two and two. A „ Rectangle is a right- angled parallelogram. The figure ABCD is a par- allelogram whose ham is AB and whose altitude, or breadth, is KD ; the figure EFGH is a rect- angle whose base is EF and whose altitude, or breadth, is FG. 276. A Trapezoid is a polygon of four sides, only two of which are parallel. The figure ABCD is a trapezoid; the longer one of its parallel sides, is its lower base, the shorter one its upper base, and the perpendicular distance between them is its altitude. 277. A Prism is a solid bounded by two parallel poly- gons called Bases, and by parallelograms called Lat- eral Faces. Prisms are named from their bases. The figures in the mar- gin show a quadrangular prism and a hexagonal prism. C E F TRAPEZOID. QUADRANGULAR FBISH. The Altitude of a Prism is the shortest distance between its bases. 10 HEXAQONAIi PRISM. 290 MENSUKATIOJS". FBUSTUM. 278. A Pyramid is a solid bounded by a polygon called the Base, and by three or more tri- angles called Lateral Faces. The lateral faces meet at a point which is called the Vertex of the pyr- amid. If a pyramid is cut by a plane parallel to the base, the part included between this plane and the base is called a Frustum of a Pyramid. The Altitude of a Pyramid is the shortest distance from the vertex to the base. The Altitude of a Frustum is the shortest distance between its bases. The figure shows a pyramid and a frustum of a pyramid. 279. A Cylinder is a solid bounded by two equal and parallel circles called Bases, and by a curved surface called the Convex Surface. The Altitude of a Cylinder is the shortest distance between its bases. The figure shows a cylinder. 280. A Cone is a solid bounded by a circle called the Base, and by a curved surface called the Convex Surface. The convex surface ta- £%v., pers uniformly from the ^ conb. rRusTUM. MENSUKATION. 291 base to a point which is called the Vertex of the Cone. If a cone is cut by a plane parallel to the base, the part included between this plane and the base is called a Frustum of a Cone. The Altitude of a Cone is the shortest distance from the vertex to the base. The Altitude of a Frustum is the shortest distance between its bases. The figure shows a cone and the frustum of a cone. 281. A Sphere is a solid every point of whose surface is equally dis- tant from a point within called The Centre. A straight line from the centre of a sphere to any point of the surface is called a Radius. A straight line through the centre and terminating at both ends in the surface is called a- Diameter. A plane through the centre of a sphere cuts from the sphere a Great Circle. Any plane that inter- sects the sphere but does not pass through the centre, cuts from the sphere a Small Circle. PROPERTY OF THE RIGHT-ANGLED TRIANGLE. 282. It is shown in geometry (B. 4., P. 8), that the square of the hypothenuse of a ri^it-angled triangle is equal to the sum of the squares of the other two sides. Calling the sides about the right- angle, the Base and the Altitude, we have the following relations : 292 MENSURATION. 1°. Hypothenuse = ^{Basef + {AltUude)\ 2°. Base — "s/ (Hypoiheniisef — {Altitudef. 3°. Altitude = V {Hypothenusef — {Base)\ EXAMPLES. 1. Find the hypothenuse of a right-angled triangle whose base is l^ft. and whose altitude is 24//. Solution.— Hypothenuse = ^'(18)^ + (24)2 ^ 39^)5. ^^5. 2. The hypothenuse of a right-angled triangle is VZ^yds. and its altitude is 10 yds. ; what is its base ? Solution.— Base = ^y{l^f - (lO)^ = 11^ yds. Ana. 3. The hypothenuse of a right-angled triangle is ^^ft, and its base is 4J//. ; what is its altitude ? Solution.— Altitude = ^/{^f — mf = 6ft. Ans, 4. A room is 30//. long and 22 j^ ft. wide ; what is the distance between two opposite corners? Ans. 37^//. 5. A flag-staff is perpendicular to a level plain and a rope 71i//. long reaches from the top of the staff to a point of the plain 42f //. from the foot of the staff; what is the height of the staff? Ans. 61! ft. 6. A pair of rafters are each 22^ ft. long, and the build- ing on which they are placed is 36//. wide ; how high is the ridge above the plane of the eaves ? Ans. IS^ft. 7. A. and B. set out from the same point at the same time ; A. travels due north at the rate of 6 miles an hour, and B. travels due east at the rate of 4-J miles an hour; how far apart are they at the end of 3 hours ? Ans, 22 J miles. MENSURATION. 293 LENGTH OF A CIRCUMFERENCE. 283. It is shown in Geometry (B. 5, P. 11), that the circumference of a circle is equal its diameter multiplied by 3.1416; that is, Circumference = Diameter x 3.1416. EXAMPLES. 1. What is the length of a circumference whose diameter is 12 feet ? Ans. 12 ft. x 3.1410 = 37.6992/?f. 2. What is the length of a circumference whose diameter is 6.75 /if. ? Ans. 21.8058/A 3. Find the length of a circumference whose radius is 8.5 in. Ans. 53.4072 in. 4. What is the circumference of a circle whose diameter is 20 yards ? Ans. 62.832 yds. 5. What is the diameter of a circle whose circumference is 78.54/if. ? Ans. 78.54/^.-^3.1416 = 26 ft. AREA OF A TRIANGLE. 284. It is shown in Geometry (B. 4, P. 4), that the area of a triangle is equal to half the product of its base and altitude ; that is, Area of triangle = Base x Altitude -7-2. EXAMPLES. 1. The base of a triangle is 8 feet and its altitude is 6 feet ; what is its area ? Ans. 24 sq. ft. Note. — By tlie term product of tico lines we always mean a rect- angle whose lengtli is one of the lines and whose breadth is the other. Hence we say that the product of a line by a line is a surface. 294 MENSURATION. 2. What is the area of a triangle whose base is 16 yards and whose altitude is 3 J yards ? Ans. 28 sq, yds, 3. What is the area of a triangle whose base is S^ yds. and whose altitude is 14 yds. ? Ans, 59^ sq. yds. 4. The area of a triangle is 74 sq,ft, and its base is 9^/1. ; what is its altitude ? -4?i5. IQft. AREA OF A PARALLELOGRAM. 285. It is shown in Geometry (B. 4, P. 3), that the area of a parallelogram is equal to the product of its base and altitude; that is, Area of parallelogram = Base x Altitude. EXAM PLE s. 1. The base of a parallelogram is 14 yards and. its alti- tude is 5 yards ; what is its area ? Ans. 14 yds. x 5 yds. = 70 sq. yds. 2. Find the area of a parallelogram whose base is 13/^. and whose altitude is 7J/^. Ans. 97^ sq.ft. 3. A rectangle is 7 J- rds. long and 5 J rds. wide ; what is its area Ans. 41 J- sq. rds. 4. A rectangular field contains 4 acres and its length is 32 rods ; what is its breadth ? A71S. 20 rds. AREA OF A TRAPEZOID. 286. It is shown in Geometry (B. 4, P. 5), that the area of a trapezoid is equal to the half sum of its bases multiplied by its altitude ; that is, Area of trapezoid = J x ( Upper dase-\- Lower base) x Altitude. MENSURATIOl^. 295 EXAMPLES. 1. The parallel sides of a trapezoid are 14 yds. and 20 yds. and the altitude is 7 yds. ; what is its area ? Ans. ^ of (14 ^6^5.4-20 yds,) x 7 yds. = 119 sq. yds. 2. Find the area of a trapezoid whose parallel sides are ISft. and 22/^. and whose altitude is 111 ft. Ans. 3^0 sq.ft. 3. A board 14/i^. long is 18 in. wide at one end and 12 in. at the other end ; how many square feet in its area ? Ans. 171 ^Q' fi' 4. The parallel sides of a trapezoidal field containing 2J acres are respectively 48 rods and 32 rods ; what is the altitude, or breadth, of the field ? Ans. 10 rds. AREA OF A CIRCLE. 287. It is shown in Geometry (B. 5, P. 11), that the area of a circle is equal to the square of its radius multi- plied by 3.1416 ; that is. Area of circle = {RadiusY x 3.1416. EXAMPLES. 1. What is the area of a circle whose radius is 13 inches ? Ans. 13 in. x 13 in. x 3.1416 =530.9304 sq. in. 2. What is the area of a circle whose radius is 2.5 rods 9 Ans. 19.635 sq. rds. 3. The radius of a circular fish-pond is 75 feet ; what is its area ? Ans. 17, 671 J sq. ft 4. The area of a circle is 176.715 sq.ft.; what is its radius. Ans. V176.715-t-3.1416 = 7.5 //J. 296 MENSURATION. SURFACE OF A SPHERE. 288. It is shown in Geometry (B. 8, P. 7), that the surface of a sphere is equal to 4 times the square of its radius multiplied by 3.1416; that is, Surface of sphere = 4 x {Radius) ^ x 3. 1416. EXAMPLES. 1. What is the area of the surface of a sphere whose radius is 12 inches ? Atis. 4 X 12 in. x 12 in. x 3.1416 = 1809.5616 sq. in. 2. Find the area of the surface of a sphere whose radius is 4 feet. ' Ans. 201.0624 sq.ft. 3. A ten-pin hall has a surface of 78.54 sq. in. ; what is its radius ? Ans. '\/78.54 sq. m. -^(4 x 3.1410) = 2 J in. 4. The radius of a billiard ball is l-J in. j what is the area of its surface ? Ans. 15.904 sq. in. VOLUME, OR CONTENT, OF A PA R ALL ELOPI PE DO N, PRISM, OR CYLINDER. 289. It is shown in Geometry (B. 7, Propositions 13 and 14 ; B. 8, P. 1), that the content of a parallelopipedon, prism, or cylinder, is equal to the product of its base by its altitude ; that is, for either of these solids we have Content = Base x Altitude. Note.— The area of the base may often be found by one of the preceding principles. EXAMPLES. 1. The base of a parallelopipedon is 24 sq. ft. and its altitude Sft.; what is its content? Ans. 192 cu.ff. MEKSURATIOK. 297 Note.— By the term product of a surface and line we always mean a parallelopipedon whose base is equal to the surface, and whose altitude is equal to the line. The number of cubic units in the volume is equal to the number of superficial units in the sur- face multiplied by the number of linear units in the height. Hence, we say, the product of a surface and line, or the continued product of three lines, is a volume. 2. The base of a parallelopipedon is 81 sq.ft. and its altitude 4/^, ; what is its volume ? Ans. 324 cu.ft 3. Find the contents of a prism whose base is 86 sq.ft., and whose altitude is 7/if. Ans. 602 cu.ft. 4. The base of a cylinder is equal to 80 sq.ft. and its altitude is equal to ^ft.; what is its content ? Ans. 4:00 cu.ft. 5. The radius of the base of a cylinder is 2.5 ft. and its altitude is 14/^.; what is its volume or contents ? Ans. (2.6 ft.)^ X 3.1416 x 14//.= 274.89 cu.ft. 6. A stick of hewn timber is 27 ft. long and its cross section is 1.5 sq.ft. ; what is its content ? Ans. 4:0^ cu.ft, CONTENT OF A PYRAMID, OR OF A CONE. 390. It is shown in Geometry (B. 7, P. 17, and B. 8, P. 2), that the volume of a pyramid or of a cone is equal to the product of its base by ^ of its altitude ; that is, for either of these solids we have . Content = Base X Altitude X i. EXAM PLES. 1. A base of a pyramid is 49 sq.ft. and its altitude is 4: ft. ; what is its volume ? Ans. 49 sq.ft. x 4//.-^3 = 65.3333 cu.ft. 298 MENSURATION. 2. The base of a cone is 15.9 sq. ft. and its altitude is 6//. ; what is its content ? Ans. 31.8 cu.ft. 3. The altitude of a cone is 18/^. and the radius of its base is ^ft; what is its content? Ans. 301.5936 cu.ft. CONTENT OF A SPHERE. 391. It is shown in Geometry (B. 8, P. 8), that the yolume or content of a sphere is equal to f times the cube of the radius multiplied by 3.1416 ; that is, Content of spliere = ^x {RadiusY X 3.1416. EXAMPLES. 1. The radius of a sphere is bft. ; what is its volume ? . 4x5//^.x5/if. x5/2f. X3.1416. _^ _ ., Ans. ^- — - — ^ = 523.6 cu.ft, 2. Find the volume of a sphere whose radius is 11.5 /jf. Ans. 6370.6412 cw. /if. 3. What is the volume of a sphere whose radius is 1\ in. 9 Ans. 1767.15 cu. in. 4. The content of a sphere is 696.9116 cu. in. ; what is its radius ? Ans. 5^ in. BOAR D M EASU RE. 292. A board foot is a solid one foot long, one foot wide, and one inch thick. It is equal to one-twelfth of a cubic foot. This unit is used in measuring boards, planks, and some kinds of timber. Note. — Boards and planks are of uniform thickness throughout, but they are often of different widths at the two ends ; in this case the half sum of the widths at the ends is taken as the width of the board, or plank. The width and thickness are usually expressed in inches, but the length is given in feet. MElTStJRATlON". 299 The number of board feet in a board, plank, or stick of timber may be found by the following RULE. Multiply the length in feet by the product of the breadth and thickness, both in inches, and divide the result by 12. EXAMPLES. 1. How many board feet in a board 13 feet long, 16 inches wide, and 1} inches thick ? Ans. 13 X 16 X li -T- 12 = 21f. 2. A board is 1 7 feet long, 13 inches wide at one end, 17 inches wide at the other end, and 1 inch thick ; how many board feet does it contain ? A7is. 21 J. 3. A plank 16 feet long and 2J inches thick is 16 inches wide at one end and 18 inches wide at the other; how many board feet does it contain ? Ans. 56|. 4. How many board feet in a piece of scantling 18//. long, 4 in. thick, and 9 i7i. wide ? Ans. 54. TIMBER MEASURE. 293. Timber, when not measured in board measure, is usually measured in cubic feet. Timber may be Round, that is, it may have a circular cross section ; or it may be Hewn, that is, it may have a rectangular cross section. Tlie cross section may be the same throughout, or it may be greater at one end than at the other. The Mean Cross Section is the cross section midway between the two ends. \ 300 MENSURATION-. The cross section of a round stick of timber at any point can be found when we know its girt at that point. The Girt is the circumference after the bark is removed. The cross section in square inches may be found by the RULE. Multiply the square of the girt in inches by .0796. EXAMPLES. 1. The girt of a round stick of timber is 42 inches; what is its cross section ? A71S. 422 X .0796 = 140.41445^. in, 2. Find the cross section of a round stick at a point where the girt is 52 inches. Ans. 215.23845^. in. 3. If the girt is 60 inches, what is the cross section ? Ans. 286.56 sq. in. The cross section of a hewn stick of timber at any point may be found by the following RULE. Multiply the breadth of the stick by its thickness at that point, both in inches ; the product is the cross section in square inches. EXAMPLES. 4. The breadth of a square stick of timber at its larger end is 14 inches and its thickness is 13 inches ; what is its greatest cross section ? Ans. 182 sq. in. 5. The breadth of the same stick at its smaller end is 12 inches and its thickness is 10 inches; what is its smallest cross section? Ans. 120 sq. in. MUNSU RATION". 301 6. The breadth of the same stick at the middle of its length is 13 inches and its thickness is 11| inches ; what is its mean cross section ? Ans. 152| sq. in. Knowing the two end sections and the mean section in square inches, and the length in feet, we can find the number of cubic feet in a stick of timber by the following RULE. To the sum of the end sections add four times the mean section, all in square inches, and multiply the result by the length in feet ; then divide by 86 4> and the quotient will be the number of cubic feet required. EXAM PLES. 7. The end sections of a stick of timber are 182 and 120 55'. in., the middle section is 152^ sq. in., and its length is soft. ; what is its content ? 864 -^ 8. The end sections of a stick of timber 4:0ft. long are 46O and AOO sq. in., and the mean section is 440^5'. in.; what is its content ? A71S. 121.2963 cw. /if. METHOD OF DUODECIMALS. 294. If the linear dimensions are expressed in feet and inches, areas and volumes may be found by the Method of Duodecimals. In this system of numbers the primary unit is 1 foot; it may be a linear foot, a square foot, or a cubic foot. One twelfth of a foot is called a Prime, one twelfth of a prime is called a Second, and one twelfth of a second is called a Third, as shown in the following 302 MENSUKATIOK. TABLE. 12 thirds '" make 1 second ". 12 seconds " 1 prime , '. {ft- 12 primes " 1 foot X sq.ft. { cu.ft. The scale of the system is uniform, that is, it is 12, 12, 12. In accordance with the principles laid down for multi- plying lines by lines, and surfaces by lines, w^e see that feet multiphed by feet give feet ; feet " " 'primes " primes; feet " " seconds " seconds; primes " " primes " seconds; primes " " seconds " thirds. OPERATION OF MULTIPLICATION. 295, Let it be required to find the continued product of 3 ft. 5 in., 2 ft. 6 in., and 4: ft. 7 in. : Explanation. — Having written the operation. first two numbers so that units of j^i / n m the same name stand in the same ^ ' column, we begin at the left hand and multiply all the parts of the 2 6. multiplicand by 2, writing the pro- g Tq ducts, without reduction, in their proper columns according to the prin- -*-" ^^ ciples explained in the last article. 8 6 6 We then multiply all the parts of the a « multiplicand by 6, and place (the pro- ducts in their proper columns, as de- 32 24 24 termined by the rules in the last arti- 56 42 42 cle. We next add the partial products '^ ~ ^ ^ by the rule for addition of compound numbers, which gives 8 sq. ft. 6' 6". = 39-^ cu.ft. MENSURATIOI^. 303 We now multiply 8 sq. ft. 6' 6" by 4: ft. 7 in., in the same manner as before and find for the required product 39 cu. ft. 1' 9" 6", that is (39 + ^ + Yf^ + XTaF)> cu.ft., which is equal to 39//^ cu.ft. In like manner we may multiply in all similar cases ; hence, the following RULE. /. Wi'ite the numbers so that units of the same nam^ shall stand in the same column. II. Multiply all the parts of the multiplicand by each part of the multiplier and write the cor- responding partial products, ivithout reduction, in their proper columns. III. Add the partial products by the rule for addition of cornpound numbers. EXAMPLES. Multiplicanc ft- I.... 3 2 n ft' 5 (2.) 7 Multiplier . . .... 5 7 7 10 15 10 35 49 21 14 50 70 Product.,.. ...17 8 2 43 8 10 = ] Vmsq. ft' = 434f.^./if. 3. Multiply Zft 7 in. by dft. 4: in. Ans. 33 sq.ft. 5' 4" = 33^ sq. ft 4. Multiply 6 ft. ll*?i. by 16 ft. 2 in. Ans. 95 sq.ft. 7' 10" = 95^ sq.ft. 6. Find the continued product of 3 ft. 4: in., 2 ft. 11 in., and 6 ft. 11 in. Ans. 59 cu.ft. 6' 8" 8'" = m^\ cu.ft. 304 MEKSURATION. 6. Find the continued product of ^ft 3 in,, hft. 2 in., and 6fL 5 in, Ans. 140 cu,ft 10' 9" 6'" = 140|f| ciufL PRACTICAL PROBLEMS. 1. How many square feet in a ceiling 17/^. 3 m. long and 11/^. 5 in, wide ? ^7^5. 196|| s^. /^. 2. How many square feet in a pavement 12ft. 6 in. long and lO/jJ. 2 in. wide ? ^^5. 127^ sq. ft. 3. Find the capacity of a box 3/^. 3 in. long, 2/^. 9 in. wide, and 1/^. 11 in. deep. ^W5. Vl-^cu.ft. 4. Find the contents of a stick of timber 42/^. 6 m long, 1 ft. 7 *w. wide, and 1 ft. 4 in. thick. 5. What is the capacity of a bin 1ft. 3 m long, 4/"/. 2 i'^. wide, and Zft. 5 m. deep ? Ans. 103^ cu. ft 6. How many cords of wood in a pile 13/^. 3 in, long, 4 /if. 2 m. wide, and Zft. 6 in. high ? Ans. \C. 65iicu.ft, 7. How many cords in a pile 20 ft. 4 in. long, 4//f. 3 in. wide, and 5/^. 2 in. high ? ^;^5. 3 C. 62f| cz^/if. 8. How many cubic feet of stone in a wall 27 ft. 6 in, long, 3 ft. 3 m. thick, and 4: ft. 2 in. high ? ^/^5. 372i|c«^./^. 9. What is the area of a rectangle whose length is 9 ft. 7 in., and whose breadth is 7 ft. 4 in. 9 Ans. 70-^ sq.ft. 10. The base of a cylinder is 24: sq. in,, and its altitude is 2 ft. 9 in. ; what is its content ? Ans. Q% cii. ft. 11. What is the content of a room whose length is l^ft. MENSURATION. 305 6 in., whose breadth is 12 ft 4: in., and whose height is 10ft. 2 in.? Ans. 2,319ff cu. ft. 12. What is the area of a floor whose length is 25 ft. 3 in., and whose breadth is 20 ft. 6 in. f Ans. 611!^ sq.ft. 13. What is the content of a box Ift. 6 in. long, 1ft. 3 in. wide, and 1ft. 1 in. deep ? Ans. 2-^cu.ft. 14. What is the content of a cube, each edge of which is 3 ft. 4 in. 9 Ans. 37^ cu. ft. REVIE^A^ aUESTIONS. (271.) What is mensuration? (272.) What is a polygon? Sides? Vertices? (273.) What is the area of a polygon? (274.) Define a triangle. Its base. Its altitude. (275.) What is a parallelogram? A rectangle? (276.) What is a trapezoid? Its lower and its upper bases? Its altitude? (277.) What is a prism ? Its bases ? Its lateral faces ? Its altitude? (278.) What is a pyramid? Its base? Its lateral faces? Its altitude? What is a frustum of a pyramid? (279.) What is a cylinder? Its bases? Its convex surface? Its altitude? (280.) What is a cone ? Its' base ? Its convex surface ? Its vertex ? Its altitude ? What is a frustum of a cone? (281.) What is a sphere? Its centre? A diameter? A radius? (282.) What is the relation between the sides of a right-angled triangle? (283.) What is the length of a circumference? (284.) The area of a triangle? (285.) Of a parallelogram ? (286.) Of a trapezoid? (287.) Of a circle ? (288.) Of the surface of a sphere ? (289.) What is the content of a parallelopipedon, prism, or cylinder? (290.) Of a pyramid or cone ? (291.) Of a sphere? (292.) What is a board foot? How do you find the number of board feet in a board or plank ? (293.) How is timber measured ? (294.) What is the method of duodecimals ? Define primes, seconds, and thirds. (295.) Give the rule for multiplication. 1. Find the product of the sum and difference of 25 and 16. 2. Divide the difference between 1296 and 441 by the sum of 36 and 21. 3. What are the prime factors of 9,800 ? 4. Eesolve 3,990 into prime factors ? 5. Find the g, c. d, of 2,290 and 458. 6. What is the g, c. d. of 1,435, 1,085, and 2,135 ? 7. Find the I c. m. of 15, 18, 24, 40, and 50. 8. What is the I c. m. of 508 and 889 ? 9. Add I, J off, and M. 10. Subtract ^ of 4^ from f of 9^. 11. Multiply \ of 2-J by i of 3f 12. Divide 2f by If 13. Multiply 3.31 + 4.06 by 8.13—3.43. 14. Divide 3.8 + 2.05 by 8.6-3f. 15. A man bought a horse and carriage ; the horse cost I as much as the carriage, and both together cost $640 ; what was the cost of the horse ? 16. At a certain election the successful candidate had a majority of 120, which was -}■ of all the votes cast ; how many votes did the opposing candidate receive ? MISCELLANEOUS EXAMPLES. 307 17. Divide 1357 among A., B., and C, so that B. shall receive 2^ times as much as A., and C. as much as A. and B. together. 18. A. can do a piece of work in 3 days, B. can do it in 4 days, and C. can do it in 5 days ; how long will it take them to do it together ? 19. How many bushels of oats can be raised on 4J acres, if each acre produces 47 iti. 3 pks. f 20. How many bushels of wheat at 11.75 per bushel will it take to pay for 3 civt, of pork at 17 per civt. f 21. A grocer mixes 120 lis. of sugar at 10 cts. a pound, 140 lbs, at 12 cts., and 60 lbs. at 14 cts. ; at what rate must he sell it to clear 20^ on its cost ? 22. Divide 11,000 among 3 persons in the proportion of 5, 7, and 8. 23. Bought a horse for $312 and sold him at a loss of 121%; ^^^^ . I. I. 2. J" M- 4. ii?« 5. y'k* 6. 7. i. 8. 3 ■ y 9- ?• 10. i^. II. U- 12. If. 13. 14. M* 15. ?¥• 16. 17. !|: 18. i. 19. |. 20. 21. :|: 22. T' 23- i« 24. 25- ■Jg. 26. tf* 27. It* 28. "A"* 29' |. ^- 1 ANSWEBS, 32. f 33. 34. '• 55. ff. 36. 37. 1*. 38. A. 39. H- ^r«. 71. Page 90. 2. 3. ft; 4. nnr* 5. 6. W' 7. AV 8. *• 9'M Art. 72. Page 91. 15 30 18 3Ty» 30» 3^* 72 81 48 TTF8r> TTJ¥' TTI¥) 168 64 140 ^2Tj ^^¥» ¥^¥» 495 780 1001 "JTT^J ^Tl?» ^T¥¥' 80482 7315 12005 1564 5681 516>8 TT^7' 7¥¥7» 7¥^9- 19720 17340 3451 3 3¥^¥>33 5^¥>¥¥5^¥' 1716 8993 3861 TYT^^J 57T7» ¥TT¥- 416 234 588 11T» ¥^¥' ¥^T- 2116 2898 4554 ?5^^» ^^¥1?> 7^¥^- Page 92. 24 30 72 18 ?!¥? ¥T» FT» ¥¥> 164 90 176 54 !¥■?» T¥¥> T7¥» TITF- 234 198 390 66 ¥55> FF5» ¥55> 5BF* 108 90 140 T¥TJ> T'SIS' T??TF- 840 252 264 TJ¥» ¥S^«^» arj^« 270 63 88 680 330 1848 H. ii ^. If > /?. If. 3 19 85 ¥¥' ¥Tr> ¥IT' MH, A*^, ^V^. |Vt, ifl, A\- TITOT' 3 0011 TTTffT* ^°- T^¥T» 1%¥4. T-gi?- IQ f4-l ^2 886 42 3 ^9* f¥3^> ¥¥7» TFT* f tl 2n JjaO 7 8 2 8 5 9 2T ^5- i-O 4 5 4 8 5 6/1 ¥Tr> ^17» ¥11' try- 22 _6 6 6 8 6 6 4 9 22- ^T» ^3T> fST* 131^ Mf. 2-5 4-8 5 1 328 78 1 ^> f 6> 6^1 SW-y -¥¥-• OA J6 7.8 5 7 3 5 6 4 ■''J' T6¥> Te?' 16#» T6 8" 26 128 1 195 9 8 124 27. Z^^>^^^> 3-00., 28. ^v-,lil!iif,AV 29. Wt\JA¥-, m. 30. -VW->^Vt¥-, W/, Mi. -4^f. 75. Pa^e 5<^. 1 23 ItVo- 1 1 1 117 15|f. II. 2ft^. ~36t 12. 2i|. 13. 2fH. 14. 2A. 19. 22^^. 20. 672/^. 21. 624|. 22. 106/y. 23. 1.357x«^. 24. 134/^. 25. llii§. 26. 20AV 27. 15, 28. 1.^ 30. 14f^. 31. 23f|. ^3: ISr ANSWERS. 321 34. 34H. 27 ^%mi. 35. 36||. 28 $215^. 36. 9^^. 29 48| T. 37. lO^m. 30 79f ^. 38. mi 31 82f 1 in. 39. 19i^. 32 milrds. 4o. lO/A. 33 $124|-;. 41. mn- 34 201^ rds. 42. 07,%V Ws. 35 454i hu. 43. 486J-I. Problems. 44- 3253^,=,, yds. Pages 08-99. Problems. 4- %^r^,yds. Page 96. 5- 94i| A. 3. m^mi. 6. 42ff^6«^. 4. 57,%^Ar«. 7. 94f .V tons. 5- .$394,641. 8. 241 yds. 6. $43f. 9- $39/0. 7. 123|f^^&*. 10. $3^-. 8. 5-||fA ^o;^s. II. '^I'^lyds. 9- IBS/ffVy^*. 12. 33.1 yni. 10. im\A. 13. 16 1 gcd. Art. 77. Page 97. 14. 15- 16. 54:lbs. $11H. gain^li^. I- A. 17. 9,r, m^-. 2. iV- 18. mi^ yds. 3. 4^1,. ^' Ik 5. 4,3^. 6. 3A. 7. IIA. 19. lost .$131 x'^. Art. 79. Pages 101, 102. 6. __7_ 8. 65/j. 7. 7|f. 9. 31il. 8. 9|f. IO. ^. 9. 65^. ir. 52f. 10. 45H4. 12. 84i§. II. 34. 13. 8A. 12. 66. 14. 2H. 13. 70^. 15. 5.V 16. 749^?A. 14. 15- 121-1. 17. $5^V- 16. l,648i. 18. m^ft. 17. 10/7. 19- ^m- 18. M 20. 11- ^d«. 19. 30^^. 21. 20|^ pds. 20. §. 22. $94 fV. 21. 2fV. 23. |12fVo. 22. 21^. 24. 51H|. 23- 13^ 25. 582|fl. 24. 6. 26. llOpw. 25- 4,V.. 11 26. 9,333ft^. >7>7 355 27. T¥T6. 28/ ll. 29- m- 30. lf||. 31. 24,450f. 32. tV 33. 561f^. 34. 1.841 ,V 35. 31,79011. 36. 8,3471^. 37. 6,199t|. 38. 56,455|t. 39. 99,151*. 40. 19,166^1. Problems. 102. 10 J. 5. 49^% ^*' 7. 13ff hu. 8. $43f. 9. |17,llli|i. 10. gain $13. 13. 333f ^. 14. 28| mi. 15. 66^ yrs. 16. $§: 17. 297| 2^^ 188 20 21. ^M 22. 224 ll — r5¥* 23. 1 24. 5t=V 25. M. 26. 8||. 27. sHi)- 28. M- 29. 3^1^. 30. 1//^. 31. IN,' 32. 34|ff i 33. ^eV 35. Hi 36. 14 37. 31 17. 5^Ty. 18. 16. 19. 39Mf. 38. 6^¥^. 39. 9tV. 40 ^vv. Problems. Pages 107, 108. 7- 10t\. 8. 5M. 9. 10. |!04. II. 419^1 ;&5.; 12. 131A lbs. 13- li e?«. 14. d&l 39 2m lbs. 15. 16. 45 da. 17. 7^ mi.; 42 mi. 18. $49,200. 19. 11 da. 20. 6^V ^' 21. $2i 22. 12 mi 23. $23,^A- 24. 973 men. 25- $30,000. Art. 82, Page 109. 3. 9,850. 4. 93,100. 322 AKSWERS. 5. 203,075. 6. 255,025. 7. 109,650. 8. 394. 9. 3,724. 10. 3,501. 11. 1,854. 12. 6,961. 13. 1,012.3. 14. 11,425. 15. $60,425. 16. $12,500. 17. 948,400 /if. 18. 2S4,rmyds. 19. $545,625. 20. 426,437iyJi0.0027. 20. 20.4905 lbs. 21. 84.5688. 22. 0.3737. 23. 412.5508. 24. O.nm yds. 25. 0.5103 rds. 26. 87.4894. 27. $1,713,782. 964.5215. 72.5641. 29,170.4499. 3.3538. $1,485. 81.648 ^&s. 34. 1,594.974^<7«. 38. $117.33. 39. 9,189.866 ?&s. 40. $4,812,975. 41. 5dd.su yds. 42. 3,512.7728. Problems. Pages 12G-128. 1. $108.37i. 2. $117.56^. $200.59i. $129.68|. $2,675.75. $12.75. $27.18f. $14,114.25. $41,781. $5,699. $23,751. $192,231. $81.56i. 32.045 mi. 264.1875 mi. 65.515 mi. $13,737. $111.58. $14.396^am. 297.2538 mi. 3. 4. 5- 6. 7- 8. 9- 10. II. 12. 13- 14. 15- 16. 17- 18. 19 20. ANSWERS. Art. 98. Pages 130, 131. 4. 0.0025. 5. $1-39. 6. 31.5434. 7. 0.25. 8. 87.5. Q. 4.75. 10. 112.8767. 11. 356.1111. 12. 12.5 /i(. 13. 12.24. 14. 1,485.6016. 15. $0,167. 16. 153.8462. 17. $0,061. 18. S2.27 yds. 19. 0.0268. 20. 0.916. 21. $0.67. 22. 2.36. 23. 79.52. 24. 123.107. 25. 7.54. 26. $70.55. 30. $16,196. 31. $1,439. 32. 7.1364/15. 33. 84.775 ytZs. 34. 3.1034 lbs. Problems. Pages 131-133. 5. 48.5 A. 6. $8.50; $53.55. 7. 19 0.; 3.7 C. 8. 11.5yds.\ 35.6365 yds. 9. 149 11)8. 10. S7. 5 bbls. 11. 13.5 m*.; 15.25 m 12. $3. 13. 58.8666 w^.; 559.2333 wj. 14. 306.675 m. 15. lOAhrs. 16. $128.50. 17. $1.12|; $7.31i. 18. $16.08tV 19. $67.68. 20. 29^ct8. 21. 11 hrs. 22. 72 bu. 23. 18 cts.; $5,624. 24. 51 bu. 25. 2Slbu. 26. 215.18 m. 27. 10.5. 28. 7 hrs. 29. $27.50. 30. 10.5(7. 31. $5.89. 32. 41 T. 33. 7 -yes^^. ^*'*. J 00. Pages 134, 135. 3. $62.50. 4. $18. 5. $3.75. 6. $23. 7. $3.12^. 8. $9.12A. 9. $4.25. 10. $23.40. 11. $15.66f. 12. $9,061. 13. $29.10. 14. $39.37|. 15. $23.33^. 16. $7. Art. 101. Pages 135, 136. 2. $5.17/g. 3. $356. 66|. 4. $54,461 5. $5,025. 6. $34.33i. 7. $35.77: 8. $118.14. 9. $262.23. 10. $82.87i. 11. $219.60. 12. $37.80. 13. $69. 14. $70.61. Art. 102. Page 136. 2. $23 59t^. 3. $99,533^. 4. $79.67^. 5. $407.10. 6. $1,029.60. 7. $12.96. 8. $63,602. 9. $57,833. 10. $66.18. 11. $70.18. 12. $8.32|-. 13. $22.06^. 14. $301,883. Art. 103. Pages 138, 139. 1. $102.58A. 2. $111.83. 3. $51. 4. $6.62. 5. $89.87. 6. $49.81. Art. 104. Page 140. 1, $86.28 Gr. 2. $247.68 Cr. Art, 146. Pages 159, 160. 1. 4,382 /ar. 2. 21,268 5rrs. 3. 37,740 min. 4. 5,iS6 yds. 5. 10,890 sq.fi. 6. 6615 pts. 7. 2,739,600 sec. 8. 30,183 /ar. 9. 1,365 in. 10. 175,000 5^. E 11. 200 cu. ft. ; 3,200 cu. ft. 12. 664: qts. 15. 17s. 6d. 16. 11 OS. 5 dwts. 17. 2 cwt. 1 qr. 12 lbs 8oz. 18. 4 sq.ft. 7.2 sq. in. 19. £2 7s. 6d. 20. 3 da. 12 7ir5. 15 min, 21. 2 ^?'*. 37 w. 5 da. 6 hrs. 324 ANSWEES. 22. 23. 24. 25. 26. 27, 28. 29. 30. 31- 32. 33- 34. 35- 36. 37. 38. 39. 40. 41. 4 r. 11 cwt d T. 16 cwt. 121 lbs. 2 mi. 6 fur. dOrds. 7 fur. 31 rds. Sj^ in. 3 &w. 2 ^/fcs. 3 qts. 43 2/^«. 6| m. 11 yds. i ft. Uin. 4 M kilom.', 18 30 12.0701 kUom. 43.3247 g^s. 105.668. 119.2391 1. ; 280.1173 I. 25.077^.: 1.7988^. 163.1404 ;&s. 1,124.346 lbs. 4.1385(7. 4.8283 C, • 618,0224 C2«./^. £0.8906. 2.S 277 7526 /6s.; 21.0812 oz. 0.5144^A;s. 3s. 8d 3.52 /or. 4 fur. 2^ rds. 11 ft. 2.64*71. Pa^cs 168-170. 83 &w. 6 g^s. 2 &?^. 2 i>A;s. 2i qts. 27 ^ff?s. 2 g^s. 1 pt. 16 yds. 2 ft. 5 in. IdA.S sq. ch. 1926 sq. li. l^da. 2^ hrs. 23 min. £35 15s. 107Z&S. 4o2. 10c?«jfe. 12 mi. 306 r^. £92 Id 157 bu. 4 g^s. £1 8 5.0987 da. £49 2s. 2|d 2 ^&s. 6 <>2. 19 dwts. 17 ^rs. lib 31 33 23 18 grs. 1 T. 10 cwt. 2 qrs 4 S>s. 39 wks. 15 hrs. 25 mi. 253 ?*c?s. 1 yd. 1ft. 6 in. ANSWERS. 325 24. 591 yds. 25. 321 bu. 2 pks. 5 qts. 26. 84.023 T. 27. 180.5S57 lbs. 28. £35.2375. 29. 89.1 da. Problems. Pages 170, 111. 1. 103 ciot. 10 ?6«. 2. 21 Z6s. 4 02. 3. 297| 2^c?«. 4. 293 A. 6 ,. 6. $23,718. 7. 91.094 ?&«. 8. 67 A. 7 «(7. cA. 9. \^^_%yds. 10. 4 a36cw./iJ. Art, 152. Pages 173-175. 4. 3 cwt. 1 g-r. 33 Tbs. 5. 36 Mc?«. 49 gals. 6. 2 2^c?5. 2 in. 7. 4 ^4. 145 sq. rds. 8. 15^^ 55' 57". 9. 17tt) 81 43 33 1 6 f/rs. 11. 8 rds. 4 yds. 9 w. 12. 4 r(?s. 3 yds. 1ft. 14. 4 yrs. ?«o«. 9 (Za. 15. 58 yrs. 3 ?Aios. 34 da. 16. 4:3 yrs. Imo. 12 da. 17. 8 7irA\ 7 mm. 13 sec. 18. £1 2s. 6d. 19. 16 gra/*. 3 qts. 3t« 20. 2,775.S grams. 21. 3 ^64.11. $554.40. $45.10. $432.64. 11. $210. 12. $158.60. 14. $4,416,494. 15. £179 15s. M. nfar. 17. $517.50. 18. $4,795.56. 19. £3,166 136-. M. 20. $8,718.75. 21. $1,030.40. 22. $2,115.80. 23. $3,392.50. 24. $11,732.58. 25. $18,597.96. 26. $1,080. 27. $1,200. Art. 187. Pages 21Jr-217. $58,892. $28.35. $66 36. $1,387.80. $4,639.70. $2,955,767. $18.23. $2.65. $729,646. $44.50. $82.25. $393.42|. 17. $63.45. 18. $54,371. 19. $918.60. 20. $1,012.65. 23. $510.86. 24. $191,205. 25. $266.07. 26. $328.32. 27. $190,151. 28. $44,289. 29. $7.66-^. $55,417. $193.61. £52 10s. $79,875. $190,517. £12,170. 38. $372.60. 39. $4,606.87^. 40. $6,974.57*. 41. .If 843. $297.50. $3,875.37*. $964,219.'' $328. $74,025. $71,867. $532.50. $4,205,331. $5,385. Art. 188. Pages 217,218. 2. $0,526. 3- 4- 5- 6. 7. 10. $1,302. $1,449. $11,828. $7.92 ; . $13.20. 11. $4.37; $7.29. 12. $1,008; $2,176. 13. $1,408; $4,224. 14. $1,087; $1,521. Art. 189. Page 219. 2. $7.84. 3. $20.16. 4. $66.21. 5. $1,167. 6. $4,792. 7. $23,123. Art. 190. Page 219. I. 7%. 4i%. 7%. 6%. 7%. 7%. 5%. 328 ANSWEES. Art. 191. 3 yrs. '3^ yrs. 2 yrs. 9 mos. 1 yr. 2 mos. 1 yr. 3 mos. 171 yrs. 2 yrs. 4 yrs. Art. 192. Page 221. 1. $2,100. 2. $3,000. 3- '13,750. 4. $1,800. 5. $2,500. 6. $100. Miscellaneous. 1. $950. 2. 7%. 3. 5 yrs. 4. $225. 5. lyr-H. A mo. 6. $1,500. 7. 2r'«- 8. 14^.. 9. $646,333. 10. 1 yrs. 11 mo. 24: da. Art. 19:i. Page 222, 2. $371.28. 3. $351,232. Art. 198. Pages 225, 220. 2. $252,074. 3. $200.55. 5. $128,152, 6. $5,081,628. 7. $474,887. 8. $700,452. 10. $405.44. Art. 200. Pages 229,231 2. $175,253. 3. $823,401. 5. $1,671.75. 6. $496.17. Art. 202. 231, 23i $1,055.70. $10.62t. $30. $803.25. $494,491. $346. 87i. $4.95. " $.59#. Art. 20s. Pages 233,234. 3. $1,454,546. $41.25. $66,364. $136. $1,090,909. $60,287. $680,581. $1,095,506. $9,830,425. $1,750. $4,000. $3,420,089. Art. 207- re 230. 3. $7,035. 4. $6,329. 5. $989.50. 6. $1,696,752. 7. $1,593. Art. 20s. Page 237. 2. $505,306. $765,853. $1,009,251. Art. 218. Pages 240, 241. 6. $2,090. 7. $105,322.50, 8. $168.75. 9- $112!%. 10. $10,125. 11. lllf%. 12. $8,000. 15. 5fft^. 16. 71%. 18. 71|%. Art. 227. Page 24G. II. $1,971.4U. Art. 229. Page 247. 2. $3,928.50. 3. $1,560.06. 4. $1,587.60. 6. £563 9.s'.8M 8. $6,057,143. 9- 5,200 /r. Art. 23S. Page 250. 5. 70 da. Art. 246. Pages 200, 261. 2. o. 3. 5. 4. 3. 5- \. 6. 10. 7. 30. 9- i. 10. I 11. 28. 12. ,-v 13. 2 lbs. 10 oz. 14. 3 Ihs. 1 oz. 15. $3. 16. 20 yds. Art. 250. Page 203. 3. 20 yds. 4. 27. 5- 96. 6. 9. 7. i 8. 0.26. 9. $4.20. 10. £3i. II $270. 83i. 12. $7.50. 13. £4 8."?. 6fZ. 14. 3i C. 15. $165.79. 16. 49. 17. $5. 18. 641. 19. $1.50. 20. $524. 21. $20,909. 22. 6| 2/^s. Art. Pages 3. a'= 4. x= 5. a? 6. a?= 7. a'= 8. 0?= 9. X— 10. a;= 11. a;= 12. X— 13. «-- 14. x= 15. 0!= 16. «!= 17. X— 18. a;= 19. X— 20. 0?=: 205-208. 643 J W2I. 58.75. $125,797. $190.85. 28. $3.29. 11. 300. 86/0% da. l^ yds. 48 men. 11 yds. $3. 20 hi(. $8.22.i. 1(\% da. $9:95. ANSWERS. 329 21. a;=%ld2. 22. a'=112| hii. 23. x=3d.l5fr. 24. ir=35 7«. 25. ir=$820. 26. X =162 da. 27. .T=$3,597.2'2|. 28. »•= $9,972.9:^%, 29. a;=S7,200. 32. a;=10^ws€s, ^»f. 253. Page 269. 5. A., $180; B., $85 ; C, $240. 6. A., $37-1 ; B., $621. 7. A., $130; B, $195 ; C, $325. 8. A., $150; B., $225 ; C, $270 ; D., $375. Art, 258. Page 276, 1. 64. 2. 625. 3. 2,744. 4. 625. 5. 9,604. 6. .0081. 7. .0225. 8. 15,625. 9. .0359. 10. 11.56. II. (J4- 12. tf- 13. 15|. 14. 34fi. 15. 410,V Art, 261, Page ooo- 2. 98. 3. llo. 4. 585. 5. 972. 6. 131. 7. 194. 8. 204. 9. 229. 10. M. 11. ^^.. Miscellaneous Examples. 1. 369. 2. 15. 3. 2. 2. 2. 5. 5 7.7. 4. 2. 3. 5. 7. 19, 5. 458. 6. 35. 7. 1,800. 8. 3,556. 9- '^m- 10. 11. TT 11 12. l-V- 13. 34.639. 14. 1.2002. 15. $240. 16. 480. 17. A., $51 ; B., $127-J- ; C, $1781 18. lUda. 19. ^Wihu.^pks. 4:qts. 20. 12 bu. 21. IShfj Ct8. 22. $250 ; $350 ; $400. 23. $273. 24. 10 mi. 25. 47,250 lbs. 26. 60 da. 27. $4,070. 28. $160. 29. $4,350. 30. $898.80. 18%. 25%. 40%. 53 mi. S15. 120 rds. 195 bu. 30 men. $830. 214 sheep. 25,920 men. 250 melons. 216 ft. S., $660 ; J., $577.50 ; B., $412.50. $2,604i. $4,102fi 18 ba^es. 2| mo. $36.75. $131.35. 1st, $15 ; 2d, $22.50 ; 3d, $7.50. n,66^id.ft. 21 a l3i cu. ft- A., $128; B., $168; C, $104. $1 per yd. 50 ft. $32,000. 25%. 2,400 9 iceeks. 3,468 «g./^. Son, $8,000;! eldest dmi., $6,000; younger dav. $4,000. 286,688. 120 men. 14. 5 hrs. 27-i\ min. 750 men. 68. 69. 70. 71. 72. 73. 74- 75- 76. 77. 78. 79. 80. 66 yrs. 28f da. 12 hrs. 10 m. P.M. 125 reams. 12 da. 75 cu. yds. 11 da. 23^. 82. 83. 85. 86. 87. 94. 95. 96. 97. 98. 99. 100. 15.76 m. 167.4. 39.06 Mlog. Man 6. 15 fr.; worn. 2.05 fr. A., $380 ; B., $430. W., $4,420. B., $3,080. A., $4,080 ; B., $3,100 ; C, $1,930. A., $7^19.84; B., $5,086. 70-f\; C, |5,(43,05-,\. 5 better, 4 poorer, 16i da. Elder,$5.A00; younger, $4,600. $1 ,500. $4,800. 16 men. 72 in. B., $3,750 ; C, $3,125. Capt., $243; man, $162; , boy, $54. $l,666f. 190. A., $445 ; B.,$230; C, $335. lO^Wiy. 5 hrs. 20 m. 45 yrs. 65. YB 35831 i iO. 54)475 UNIVERSITY OF CAUFORNIA LIBRARY