UC-NRLF IN MEMORIAM FLOR1AN CAJORI AV^C, HIGHER AEITHMETIC BY WOOSTER WOODRUFF BEMAN H PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN AND DAVID EUGENE SMITH PROFESSOR OF MATHEMATICS IN THE MICHIGAN STATE NORMAL COLLEGE BOSTON, U.S.A., AND LONDON GINN & COMPANY, PUBLISHERS 1897 COPYRIGHT, 1897, BY WOOSTEB WOODRUFF BEMAN AND DAVID EUGENE SMITH ALL RIGHTS RESERVED CAJOW PREFACE. 1. THE present work has been prepared with the belief that it will be of service to progressive teachers in American high schools, academies, and normal schools. As indicated by its title, it is intended for those who are taking up the subject a second time with the desire to review and extend the knowledge previously acquired. The purpose of the authors is more fully set forth in the following statement of some of the distinctive features of the book. 2. The applied problems refer to the ordinary commer- cial life of to-day, or they deal with elementary questions arising in the laboratory, or they are inserted for general information. The fact that tradition has furnished the schools with a mass of inherited puzzles which give a false notion of business, that in an age of science and invention these subjects have found no place in the arithmetics, and that the common graphic methods of representing statistics are not seen in the schools, has not deterred the authors from attempting to modernize the subject. At the same time they believe that the exercises will be found much more straightforward and simple than those with which the average text-book has so long been encumbered. 3. Problems in pure arithmetic in the high school are intended to furnish training in mathematical analysis. This is almost their only justification. Hence, the attempt has been made to lead the pupil to a clear understanding of 911349 IV PREFACE. such subjects as the greatest common divisor, the multi- plication and division of fractions, the square and cube roots, etc. To this end it has been necessary to resort to the literal notation. Most students will know enough algebra for this purpose, but for those who do not the necessary foundation can be laid in two or three lessons. 4. The work being intended as a review, it has not been thought necessary to attempt a definition of every arith- metical term which is employed. A table of the most common terms is given on p. vii, with definitions and etymologies. 5. The work being intended only for those teachers who recognize in arithmetic an instrument for mental training, no rules are given. 6. .Teachers are urged to follow the suggestions laid down in the work, with respect to the omission of such chapters or portions of chapters as are not adapted to their pupils, and to change the sequence to suit their own tastes. It is hardly necessary to suggest that only a portion of the exercises should be attempted by any one class, the advantage of changing from term to term being evident. 7. While the authors have not failed to consult the leading French, German, Italian, and English arithmetics, they have taken but few problems from these sources. To Day's " Electric Light Arithmetic " (London, 1887) they are, however, indebted for several exercises. 8. In the work in mensuration several figures have been taken from the authors' "Plane and Solid Geometry" (Boston, Ginn & Co.). For the scientific treatment of that subject and for additional exercises, teachers are referred to the text-book mentioned. W. W. BEMAN. D. E. SMITH. JULY 1, 1897. TABLE OF CONTENTS. PAGE PREFACE iii TABLE OF CONTENTS ........ v DEFINITIONS AND ETYMOLOGIES ...... vii SYMBOLS AND ABBREVIATIONS ...... xvii CHAPTER I. NOTATION AND THE FUNDAMENTAL OPERATIONS 1 I. WRITING AND READING NUMBERS .... 1 II. CHECKS ......... 3 III. ADDITION ......... 4 IV. SUBTRACTION ........ 5 V. MULTIPLICATION 6 VI. DIVISION 10 AXIOMS ......... 13 FUNDAMENTAL LAWS ...... 14 CHAPTER II. FACTORS AND MULTIPLES .... 15 I. TESTS OF DIVISIBILITY . . . . . . 15 II. CASTING OUT NINES ....... 19 III. GREATEST COMMON DIVISOR ..... 21 IV. LEAST COMMON MULTIPLE ...... 24 CHAPTER III. COMMON FRACTIONS . . . 26 CHAPTER IV. POWERS AND ROOTS 32 I. INVOLUTION ........ 32 II. SQUARE ROOT 34 III. CUBE ROOT 39 CHAPTER V. THE FORMAL SOLUTION OF PROBLEMS. . 41 I. SYMBOLS 42 II. LANGUAGE ......... 44 III. METHODS 44 IV. CHECKS 49 CHAPTER VI. MEASURES 51 I. COMPOUND NUMBERS ....... 54 II. THE METRIC SYSTEM 59 III. MEASURES OF TEMPERATURE 64 VI TABLE OF CONTENTS. CHAPTER VII. MENSURATION 66 CHAPTER VIII. LONGITUDE AND TIME . . . . 80 CHAPTER IX. RATIO AND PROPORTION .... 87 I. RATIO 87 II. PROPORTION 96 CHAPTER X. SERIES 104 I. ARITHMETIC SERIES 105 II. GEOMETRIC SERIES 107 CHAPTER XI. LOGARITHMS ...... 110 CHAPTER XII. GRAPHIC ARITHMETIC . . . .122 CHAPTER XIII. INTRODUCTION TO PERCENTAGE . . 126 CHAPTER XIV. COMMERCIAL DISCOUNTS AND PROFITS . 132 CHAPTER XV. INTEREST, PROMISSORY NOTES, PARTIAL PAYMENTS 136 I. SIMPLE INTEREST ....... 136 II. PROMISSORY NOTES ....... 141 III. PARTIAL PAYMENTS 143 IV. COMPOUND INTEREST . . . . . 145 V. ANNUAL INTEREST 147 CHAPTER XVI. BANKING BUSINESS .... 148 I. DEPOSITS AND CHECKS ...... 148 II. LENDING MONEY 150 III. DISCOUNTING NOTES ....... 152 CHAPTER XVII. EXCHANGE 154 CHAPTER XVIII. GOVERNMENT REVENUES . . .163 I. THE UNITED STATES GOVERNMENT .... 163 II. STATE AND LOCAL TAXES 165 CHAPTER XIX. COMMISSION AND BROKERAGE . . 168 CHAPTER XX. STOCKS AND BONDS 171 CHAPTER XXI. INSURANCE 177 CHAPTER XXII. MISCELLANEOUS EXERCISES. . . .179 APPENDIX NOTE I . 189 II 190 III 192 DEFINITIONS AND ETYMOLOGIES. THE following list includes such definitions as are apt to be needed for reference, together with pronunciations and etymologies. The latter are those given by the Century Dictionary. KEY. L. Latin, G. Greek, F. French, ML. Mediaeval Latin, LL. Low Latin, AS. Anglo-Saxon, ME. Middle English, dim. diminutive, fern, feminine. a, fat, & fate, a'/ar, a, fall, a ask, a /are, e met, e mete, e her, i pin, I pine, o not, 6 note, o move, 6 nor, u tub, u mute, u. pull. s as in leisure. A single dot under a vowel indicates its abbreviation. A double dot under a vowel indicates that the vowel approaches the short sound of u, as in put. The numerals refer to pages in the arithmetic. Abscissa (ab-sis'a). L. cut off. A certain line used in determining the position of a point in a plane. 68. Abstract (ab'strakt). L. abstractus, drawn away. An abstract num- ber is a number not designated as referring to any particular class of objects. E.g., 7, as distinguished from 7 ft. which is a concrete number. Addition (a-dish'on). L. addere, to increase. The uniting of two or more numbers in one sum. Ampere (am-par'). Term adopted by the Electric Congress at Paris, 1881. Name of French electrician, Andre' Marie Ampere, d. 1836. The unit employed in measuring the strength of an electric cur- rent. 101. Analysis (a-nal'i-sis). G. a loosing, a resolution of a whole into its parts. A form of reasoning from a whole to its parts. Antecedent (an-te-se'-dent). L. antecedere, to go before. The first of two terms of a ratio. Antilogarithm (an-ti-logVrithm). G. anti-, opposite to, + logarithm. The number corresponding to a logarithm. 117. Vlll HIGHER ARITHMETIC. Approximation (a-prok-si-ma'shpn). L. ad", to, + proximare, to come near, (a) A continual approach to a true result, (b) A result so near the truth as to be sufficient for a given purpose. Are (ar or ar). L. area, a piece of level ground. A square deka- ineter, or 119.6 sq. yds. Area (a're-a). See Are. The superficial contents of any figure or surface. Arithmetic (a-rith'me-tik ; as adj., ar-ith-inet'ik). G. arithmos, num- ber. The theory of numbers, the art of computation, and the applications of numbers to science and business. Arithmetic series. See Series. 101. Associative law (a-so'shi-a-tiv). L. ad 1 , to, + sociare, to join. The law which states that certain operations give the same result whether they first unite two quantities A and B, and then unite the result to a third quantity C ; or first unite B and C, and then unite the result to J., the order of the quantities being preserved. Average (av'e-raj). Etymology obscure. The result of adding several quantities and dividing the sum by the number of quantities. Avoirdupois (av"or-du-poiz'). F. aver, goods, + de, of, + jpois, weight. A system of weight in which 1 Ib. = 16 oz. = approximately 7000 troy grains. Axiom (ak'si-om). G. axioma, a requisite, a self-evident principle. A simple statement, of a general nature, so obvious that its truth may be taken for granted. Bank discount (bangk). ML. bancus, bench. See Discount. Base (bas). LL. bassus, low. (a) The line or surface forming that part of a figure on which it is supposed to stand, (b) The base of a system of logarithms is the number which, raised to the power indicated by the logarithm, gives the number to which the loga- rithm belongs, (c) In percentage, the number which is multiplied by the rate to produce the percentage. Bond (bond). AS. bindan, to bind. An obligation, under seal, to pay money. It may be issued by a government, a railway corpora- tion, a private individual, etc. Broker (bro'ker). Originally, one who manages. An agent. Brokerage (bro'ker-aj). The fee or commission given to a broker. Cancel (kan'sel). L. cancelli, a lattice. Originally, to draw lines across a calculation. To strike out or eliminate as a common factor in the terms of a fraction, a common term in the two mem- bers of an equation, etc. DEFINITIONS AND ETYMOLOGIES. IX Cast out nines. 14. Characteristic (kar^ak-te-ris'tik). G. characterizein, to designate. 112. Check. ME. cheker, a chess board. To verify ; that is, to mark off as having been examined. Circle (sir'kl). L. circulus, dim. of circus (G. kirkos), a ring. A plane figure whose periphery is everywhere equally distant from a point within it, the center. Circulating decimal (ser'ku-lat-ing). 109. Circumference (ser-kum'fe-rens). L. circum, around (see Circle), + /erre, to bear. The line which bounds a circle. Cologarithm (ko-log'a-rithm). 118. Commission (kp-mish'pn). L. com-, together, + mittere, to send. The act of intrusting ; hence, a fee paid to one who is intrusted. Common denominator (kom'pn). L. communis, general, universal. A denominator common to two or more fractions. Common fraction. See Fraction. A fraction in which both terms are written out in full, as distinguished from a decimal fraction. Common multiple. See Multiple. A multiple of two or more expres- sions is a common multiple of those expressions. Commutative law (kp-mu'ta-tiv). L. com-, intensive, + mutare, to change. The law which states that the order in which elements are combined is indifferent. Complement (kom'ple-ment). L. com-, intensive, + plere, to fill. A number added to a. second number to complete a third. Complex fraction (kom'pleks). L. com-, together, + plectere, to weave. (See Fraction.) A common fraction whose numerator or denominator contains a common fraction. Composite number (kom-poz'it). L. com-, together, + ponere, to put. A. number which can be exactly divided by a number exceeding unity. Compound interest (kom'pound). See Composite. 145. Compound number. 54. Concrete number (kon'kret or kpn-kret'). L. concretus, grown together. A number which specifies the unit, as 3 ft. In the case of 3 ft. , 3 is strictly the number and 1 ft. is the unit ; it is, however, conven- ient to designate 3 as an abstract number and 3 ft. as a concrete number. Cone (kon). G. fconos, a cone. The elementary form considered in arithmetic is a solid generated by the revolution of a right-angled triangle about one of its sides as an axis. X HIGHER ARITHMETIC. Consequent (kon'se-kwent). L. com-, together, + sequi, to follow. The latter of the two terms of a ratio. Consignee (kon-sl-ne'). L. com-, together, + signare, to seal. One who has the care or disposal of goods upon consignment. Consignor (kon-si'nor or kon-si-n6r'). A person who makes a con- signment. Corporation (kor-po-ra'shon). L. corporare, to form into a body, from corpus, a body. An artificial person created by law from a group of natural persons and having a continuous existence irrespective of that of its members. Coupon bond (ko'pon). F. couper, to cut. A bond, usually of a state or corporation, for the payment of money at a future day, with severable tickets or coupons annexed, each representing an instal- ment of interest, which may be conveniently cut off for collection as they fall due. Cube (kub). G. kubos, a die, a cube, (a) A regular solid with six square faces, (b) To raise to the third power, (c) The third power of a number. Cube root. The cube root of a perfect third power is one of the three equal factors of that power. A number which has not a perfect third power has not three equal factors. It is, however, said to have a cube root to any required degree of approximation. Thus, the cube root of n to 0. 1 is that number of tenths whose cube differs from n by less than the cube of any other number of tenths. Cylinder (sirin-der). G. kylindros, from kyliein, to roll. The ele- mentary form considered in arithmetic is a solid generated by the revolution of a rectangle about one of its sides as an axis. Date line (daf lin). The fixed boundary line between neighboring regions where the calendar day is different. Decimal (des'i-mal). L. decem, ten. Pertaining to ten. Denominator (de-nom'i-na-tor). L. denominare, to name. 27. Diagonal (dl-ag'o-nal). G. dia, through, + gonia, corner, angle. A line through the angles of a figure, but not lying in its sides or faces. Difference (dif'e-rens). L. differ ens, different. The difference between two numbers is the number which added to either will produce the other. In arithmetic it is usually taken as the number which added to the smaller will produce the larger. Digit (dij'it). L. digitus, finger. The number represented by any one of the ten symbols 0, 1, 2, 9. The term is more often used to designate one of the ten symbols mentioned. DEFINITIONS AND ETYMOLOGIES. xi Directly proportional. 97. Discount (dis'kount). L. dis, away, + computare, to count. An allowance or deduction made from the customary or normal price, or from a sum due or to be due at a future time. Bank discount ' is simple interest paid in advance, reckoned on the amount of a note. Distributive law (dis-trib'u-tiv). 14. Dividend (div'i-dend). L. dividere, to divide, (a) A number or quantity to be divided by another called the divisor, (b) A divi- sion of profits to be distributed proportionately among stockholders. Divisibility (di-viz-i-bil'i-ty). The quality of being divisible without a remainder. Usually applied in speaking of abstract integers. Division (di-vizh'pn). 29. Divisor (di-vi'zor). See Dividend. Draft (draft). AS. dragan, to draw. A writing directing the pay- ment of money on account of the drawer. 155, 156. Drawee of a draft. One on whom the order is drawn. Drawer of a draft. One who draws the order for the payment of money. Duty (du'ti). Due + ty. Sum of money levied by a government upon goods imported from abroad. Equal (e'kwal). L. aequalis, equal. Having the same value. Equation (e-kwa'shpn). A proposition asserting the equality of two quantities and expressed by the sign = between them. In algebra, an equality which exists only for particular values of certain letters called the unknown quantities. Equilateral (e-kwi-lat'e-ral). L. aequus, equal, + latus, side. Hav- ing all the sides equal. Evolution (ev'o-lu'shon). L. evolvere, to unroll. The extraction of roots from powers. Exchange (eks-chanj'). ML. ex, out, + cambiare, to change. The transmission of the equivalent of money from one place to another, such equivalent being redeemable in the money of the place to which it is sent. Exponent (eks-po'nent). L. exponere, to set forth, indicate. A symbol placed above and at the right of another symbol (the base) to denote that the latter is to be raised to a power. Eor general meaning, see pages 32, 33. Extremes (eks-tremz'). L. extremus, outermost. The first and last terms of a proportion or of any other related series of terms. Xll HIGHER ARITHMETIC. Face (fas). L. fades, face. The principal sum due on a note, bond, policy, etc. Factor (fak'tor). L. facer e, to do. One of two or more numbers which when multiplied together produce a given number. Fraction (frak'shon). L. frangere, to break. 26. Fractional unit. One of the equal parts of unity. Geometric series (je-o-met'rik). G. geometria, geometry. 104. Grace (gras). L. grains, dear. 141. Gram (gram). 59. Greatest common divisor of two or more integers is the greatest integer which will divide each of them without a remainder. Hypotenuse (hi-pot'e-nus). G. hypo, under, + teinein, to stretch. The side of a right-angled triangle opposite the right angle. Improper fraction. A common fraction whose terms are positive, and whose numerator is not less than its denominator. Index notation. 1. Insurance (in-shor'ans). OF. enseurer, to insure. A contract by which one party for an agreed consideration undertakes to com- pensate another for loss on a specific thing. Integer (in'te-jer). L., a whole number. Interest (in'ter-est). L. interest, it concerns. A sum paid for the use of money. Inversely proportional (in-versli). 97. Involution (in-vo-lu'shon). L. involvere, to roll up. Multiplication of a quantity into itself any number of times. Isosceles (i-sos'e-lez). G. isos, equal, + skelos, leg. Having two sides equal. Least common denominator. The least denominator which can be common to two or more common fractions. Least common multiple. The least integer which is a multiple of two or more given integers. Lever (lev'er or leaver). L. levare, to raise. A bar acted upon at different points by two forces which severally tend to rotate it in opposite directions about a fixed point called the fulcrum. Liter (le'ter). G. litra, a pound. The unit of capacity in the metric system ; a cubic decimeter. 59. Logarithm (log'a-rithm). G. logos, word, 4- arithmos, number. The exponent of the power to which a number called the base (in the common system, 10) must be raised to produce a number. DEFINITIONS AND ETYMOLOGIES. Xlll Longitude (lon'ji-tud). L. longus, long. The angle at the pole be- tween two meridians, one of which, the Prime Meridian, passes through some conventional point and from which the angle is measured. Mantissa (man-tis'a). L., something left over. 112. Maturity (ma-tu'ri-ti), L. maturus, mature. The time when a note or bond becomes due. Means (menz). The second and third terms of a proportion. Measure (mezh'ur). L. mensura, measure, (a) A unit or standard adopted to determine the length, volume, or other quantity of some other object, (b) The determination of quantity by the use of a unit. Mensuration (men-su-ra'shon). The science of measuring. Meridian (me-rid'i-an). L. meridianus, belonging to mid-day. A semi-circumference passing through the poles. Meter (me'ter). G. metron, measure. Unit of length in the metric system. 59. Metric (met'rik). 59. Mikron (ml'kron). G. mikros, small. Millionth part of a meter. Minuend (min'u-end). L. minuere, to lessen. The number from which another number is subtracted. Mixed number. The sum of an integer and a fraction. Multiple (murti-pl). L. multus, many, + plus, akin to E. fold. A number produced by multiplying an integer by an integer. Multiplicand (murti-pli-kand). A number multiplied by another number called the multiplier. Multiplication (mul-ti-pli-ka'shon). 29. Multiplier. See Multiplicand. Net proceeds (net pro'seds). The sum left from the sale of a note or other piece of property after every charge has been paid. Notation (no-ta/shon)/ L. notare, to mark. A system of written signs of things and relations used in place of common language. Note (not). L. nota, mark. A written or printed paper acknowledg- ing a debt and promising payment. Number (num'ber). L. numerus, a number. An abstract number is the ratio of one quantity to another of the same kind. See Abstract, Concrete. Numeration (nu-me-ra'shon). L. numerare, to count. The art of reading numbers. XIV HIGHER ARITHMETIC . Numerator (nu'me-ra-tor). The number, in a common fraction, which shows how many parts of a unit are taken. Ohm (om). Named after G. S. Ohm, a German electrician. The unit of electrical resistance. 101. Ordinate (or'di-nat). L. ordinare, to order. 68. Parallelepiped (par-a-lel-e-pip'ed or pl'ped). G. parallelos, parallel, + epipedon, plane. A prism whose bases are parallelograms. Parallelogram (par-a-lel'o-gram). G. parallelos, parallel, + gramma, line. A quadrilateral whose opposite sides are parallel. Par value. L. par, equal. Face value. Per capita (per cap'i-ta). L., by the head. Per cent. L., by the hundred. 126. Percentage (per-sent'aj). (a) That portion of arithmetic which involves the taking of per cents, (b) The result from multiplying a number (called the base) by a certain rate per cent. Policy (pol'i-si). ML. politicum, a register. A contract of insurance. Poll tax (pol). ME. poll, head. A tax sometimes levied at so much per head of the adult male population. Premium (pre'mi-um). L. praemium, profit. (a) Amount paid to insurers as consideration for insurance, (b) Amount above par at which stocks, drafts, etc., are selling. Present worth of a sum. An amount which placed at interest at a given rate will amount to that sum in a given time. Prime number (prime). L. primus, first. An integer not divisible without a remainder by any integer except itself and unity. Two integers are prime to one another when they have no common divisor except unity. Prime meridian. See Longitude. Principal (prin'si-pal). L. princeps, first. A capital sum lent on interest. Prism (prizm). G. priein, to saw. A solid whose bases are parallel congruent polygons and whose sides are parallelograms. Problem (problem). G. problema, a question proposed for solution. Product (prod'ukt). L. pro-, forward, + ducere, to lead. The result from multiplying one number by another. Proper fraction. A common fraction whose terms are positive, and whose numerator is less than its denominator. Proportion (pro-por'shon). L. pro, before, + portio, share. 96. Pyramid (pirVmid). G. pyramis, a pyramid. A solid contained by a plane polygon as base, and other planes meeting in a point. DEFINITIONS AND ETYMOLOGIES. XV Pythagorean theorem (pi-thag'o-re-an). A theorem first proved by Pythagoras. 68. Quadrilateral (kwod-rl-lat'e-ral). L. quatuor, four, + latus, a side. A four-sided plane figure. Quantity (kwon'ti-ti). L. quantus, how much ? The being so much in measure or extent. Quotient (kwo'shent). L. quotiens, how many times ? The number which taken with the divisor as a factor produces the dividend. Radius (ra'di-us). L., rod. A line from the center of a circle to the circumference. Rate per cent. 127. Ratio (ra'shio). L., a reckoning. 87. Reciprocal numbers (re-sip'ro-kal). L. reciprocus, alternating. Two numbers which multiplied together make unity. Rectangle (rek'tang-gl). L. rectus, right, + angulus, angle. A quad- rilateral all of whose angles are right angles. Rectangular parallelepiped. A parallelepiped all of whose faces are rectangles. Reduction (re-duk'shon). L. re-, back, + ducere, to bring. Chang- ing the denomination of numbers. Reduction ascending, changing to a higher denomination, as from 144 inches to 12 feet. Reduc- tion descending, changing to a lower denomination. Registered bond (rej'is-terd). A bond bearing the name of the owner, the name and residence being registered on the books of the cor- poration issuing the bond. Remainder (re-man'der). L. re-, back, + racmere, to stay. The same as Difference. Root (rot or rut). ME. roote, root. The root of a number is such a number as, when multiplied into itself a certain number of times, will produce that number. For more extended definition, see Cube Root, Square Root. Series (se'rez or se'ri-ez). L. series, a row, from serere, to join together. 104. Share (shar). AS. scercm, to cut. One of the whole number of equal parts into which the capital stock of a corporation is divided. Significant figure (sig-nif i-kant). L. signum, a sign, + /acere, to make. The succession of figures in the ordinary notation of a number, neglecting all the ciphers between the decimal point and the figure not a cipher nearest to the decimal point. XVI HIGHER ARITHMETIC. Solid (sol'id). L. solidus, firm. Any limited portion of space. Specific gravity (spe-sif'ik grav'i-ti). L. species, kind, +/acere, to make. 92. Sphere (sfer). G. sphaera, a ball. A solid bounded by a surface whose every point is equidistant from a point within the solid, called the center of the sphere. Square (skwar). L. quatuor, four. (a) An equilateral rectangle, (b) The second power of a number, (c) To raise a number to the second power. Square root. 34. Standard time (stan'dard). ML. standardum, standard. A system of uniform time for a given section of country. 85. Stere (star). G. stereos, solid. A cubic meter; 35.31 cu. ft. Stock (stok). AS. stoc, post, trunk. The share capital of a corpora- tion. Subtraction (sub-trak'shon). L. sub, under, + trahere, draw. The operation of finding the difference between two numbers. See Difference. Subtrahend (sub'tra-hend). The number subtracted from the minuend. Sum (sum). L. summa, the highest part. See Addition. Surd (serd). L. surdus, deaf. A number not expressible as the ratio of two integers. Surface (ser'fas). L. superficies, the upper face. The bounding or limiting parts of a solid. Terms of a fraction (terms). L. terminus, limit. The numerator and denominator together. Theorem (the'o-rem). G. theorema, a sight. A statement of a truth to be demonstrated. Thermometer (ther-mom'e-ter). G. therme, heat, + metron, measure. An instrument by which temperature is measured. Trapezoid (tra-pe'zoid). G. trapeza, table, + eidos, form. A quad- rilateral having two parallel sides. Triangle (trrang-gl). L. tres, three, + angulus, angle. A three-sided plane figure. Unit (u'nit). L. unus, one. Any standard quantity by the represen- tation and subdivision of which any other quantity of the same kind is measured. Volt (volt). From Volta, an Italian physicist. The unit of electro- motive force. 101. Volume (vol'um). L. volvere, to roll round. Solid contents, SYMBOLS AND ABBREVIATIONS. THE following are used in this work, and are inserted here for reference. Other symbols are explained as needed. For historical note, see p. 43. E.g., I.e., ex., ax., th., Latin exempli gratia, for example. Latin id est, that is. exercise, axiom, theorem, g.c.d., greatest common divisor, l.c.m., least common multiple. Tt, Greek letter it, symbol for 3.14159 v, since. .-., therefore. %, per cent, hundredth, hun- dredths. See p. 126. +, plus, symbol of addition and of positive numbers. , minus, symbol of subtrac- tion and of negative numbers. plus or (and) minus, and absence of sign be- tween letters, denote multiplication, and fractional form de- note division, ratio, a special form of division, equal or equals, approaches as a limit. See p. 108. , x, ', : : , sometimes used in propor- tion as a sign of equality. >, is (or are) greater than. , is (or are) not greater than. <, is (or are) not less than. , and so on. a- 2 , a- 1 , a, a 1 , a 2 , a n , indi- cate powers. See pp. 2, 32. V~ and the exponent indicate square root. See p. 33. /- 1 V and the exponent - indicate n nth root. ( ), , symbols of aggregation, log, colog, antilog, see pp. 111- 118. For abbreviations of common measures, see pp. 52, 53. For abbreviations of metric meas- ures, see pp. 60, 61. n' is read n-prime. fi " / sub one, or /-one. t n u t sub n, or (if there can be no misunderstand- ing) t-n. H1GHEE AEITHMETIO. CHAPTER I. Notation and the Fundamental Operations. I. WRITING AND READING NUMBERS. THE universal notation among civilized nations at the present time is the Hindu or Arabic, the symbols of which, except the zero, originated in India before the beginning of the Christian era, and seem to have been the initial letters of the early numerals. The system derives its in- trinsic importance, however, from the zero, which renders possible the distinctive feature known as place value. Thus, in the number 302, the 3 stands for hundreds because it is in hundreds' place, a fact which could not be conveniently indicated without the symbol or its equivalent. The zero ' appeared about the fifth century A.D., and somewhat later the Arabs, coming from the East after the conquest of Spain, brought the new system with them. About the year 1200 these Hindu numerals began to be known in Christian Europe, but it was not until the fifteenth century that they were generally taught and used. The decimal point appeared about the opening of the seventeenth century, and through its influence the sub- ject of arithmetic, both pure and applied, has materially changed. The extent of this change may be seen in the number of cases in which the decimal fraction is used to-day. 2 HIGHER ARITHMETIC. The Roman numerals were in common use in Europe prior to the introduction of the Hindu system. As now written they have changed considerably from the form used at the beginning of the Christian era. In America they are at present employed chiefly in numbering the chapters of books, and hence are rarely used beyond one or two hundred. The old custom of printing the number of the year in Roman notation on the titlepages of books has practically ceased. In modern science numbers are often used which con- tain several zeros, for the reason that absolute accuracy of measurement is generally impossible. Thus, it is said that the distance from the earth to a certain star is 21,000,000,000,000 miles, but the distance even to within a billion miles is quite unknown. Similarly, the length of a wave of sodium light is said to be 0.0005896 of a millimeter, but the seventh decimal place is doubtful and the subsequent ones are unknown. The naming of these numbers is a matter of little importance, and the writing of them in full is usually unnecessary. Scientists often resort to an index notation, in which an integer, sometimes followed by a decimal, is multiplied by a power of 10. Thus, 21,000,000,000,000 may be written 2.1 X 10 13 , or 21 X 10 12 . And since 10" 1 means 0.1, and 10~ 2 means 0.01, etc., therefore 0.00000274 may be written 2.74 X 10" 6 . Since the index notation is now so extensively used in science, and since the limit of necessary counting in finan- cial affairs is met in the billion or trillion, no elaborate system of naming numbers is practically used. Attention should be paid, however, to the proper reading of the num- bers in common use, types of which are given in the exam- ples on p. 3. Thus, 123.4567 should be read, one hundred twenty-three and four thousand five hundred sixty-seven ten-thousandths. NOTATION AND FUNDAMENTAL OPERATIONS. d Exercises. 1. What are the various names given to the symbol ? 2. Read the numbers 0.0002, 0.00004, 0.400. 3. Also, 0.123, 100.023. 4. Also, 0.1246, 1200.0046. 5. In the numbers XV and 15, why are the Hindu characters said to have a place value and the Roman not ? 6. Express in the index notation the numbers in the following statements : (a) In a cubic centimeter of air there is 0.00001114 of a grain of water. (6) A centimeter is 0.0000062138 of a mile. (c) The distance to the sun is 93,000,000 miles. (d) The distance from the sun to Neptune is 2,788,800,000 miles. 7. Express in the common notation the numbers in the following statements : (a) The distance from the equator to the pole is 39.377786 X 10 7 inches. (6) The earth's polar radius is 6.35411 X 10 8 centimeters. 8. A syndicate is to bid on some government bonds ; to how many decimal places should they express their bid per $100 if they bid for $10,000 worth? $1,000,000 worth? $25,000,000 worth ? $100,000,000 worth ? II. CHECKS. A check on an operation is another operation whose result tends to verify the result of the first. E.g., if 11 7 = 4, then 7 + 4 should equal 11 ; this second result, 11, verifies the first result, 4. The verification is usually incomplete. If, as is said, "the result does not check," there must be an error in (1) the original operation, (2) the check, or (3) both. If, on the other hand, the result does check, there may have been an error in one operation which just balanced the other. Hence a check makes it more or less improbable that an error remains undiscovered. The secret of accu- rate computation largely lies in the knowledge and the continued use of proper checks. HIGHER ARITHMETIC. III. ADDITION. In adding the annexed column, the computer should say to himself, " Five, fourteen, eighteen ; one, seven, fourteen; nine, twelve" thus omitting all super- 374 fluous words. Bookkeepers, whose business leads 69 to rapid addition, omit much that would seem 805 necessary to the student, and not infrequently 1248 add two columns at once, a power gained only by practice in their profession. The most practical way to check addition is to perform the operation a second time. Experience shows, however, that if the operation is performed in the same way the mind tends to fall into the same error. Hence it is better to add the numbers in the reverse order. It is unnecessary to give many exercises in addition, since the student who finds himself in need of practice can easily prepare them. Exercises. 1. As an exercise, it is convenient to prepare a table of multiples of some number in this manner : Write any number, as 4197, on paper, and the same number on a small card, 4197 | ; place the card above the number and add, thus giving 2 X 4197 ; slide the card down and add again, thus giving 3 X 4197, and so onto 10 X 4197, when the work checks if the result is 41,970. 2. In the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, , each number after the first two is obtained by adding the two preceding numbers ; calculate the 30th number of the series. 3. Add 15, 214, 3962, 9984, 9785, 6037. Check. 4. Add 201, 76, 435, 7726, 8687, 8812, 8453, 1162. Check. 5. What is the sum of the first two odd numbers, i.e., 1 + 3 ? of the first 3 ? of the first 4 ? of the first 10 ? of the first 20 ? From these results, what would be inferred as to the sum of the first 100 odd numbers ? 6. Explain why 234 + 859 is 1000 more than the difference between 234 and 141. Also why 4396 + 8501 is 10,000 more than the difference between 4396 and 1499. NOTATION AND FUNDAMENTAL OPERATIONS. 5 IV. SUBTRACTION. There are three common methods of subtraction. In the annexed example, we may say, (1) 5 from 14, 9 ; 2 from 2, ; 3 from 12, 9 ; 1234 (2) 5 from 14, 9 ; 3 from 3, ; 3 from 12, 9 ; 325 (3) 5 and 9, 14 ; 2 and 1 and 0, 3 j 3 and 9, 12. 909 Each of these three methods is easily understood. The first is the simplest of explanation, and hence it is generally taught to children. The second is slightly more rapid than the first. But the third, familiar to all as the common method of " making change," is so much more rapid than either of the others that it is recommended to all computers. Since subtraction is the inverse of addition, the simplest check is the addition of the subtrahend and difference j the sum should equal the minuend. Exercises. 1. If not entirely familiar with the third method above given, use it with enough problems to become so, and state the reason of its advantage in rapidity over the others. 2. In subtracting 34,256 from 100,000, show that the subtraction can easily be made from the left by taking each digit from 9, except the 6, which must be taken from 10. 3. As in Ex. 2, show how to subtract 27,830 from 100,000 ; also 948,900 from 1,000,000. In Exs. 2 and 3, the results are called the complements of the given numbers, because they complete the next higher power of 10. 4. Show that the difference between 1234 and 5612 may be found by adding the complement of 1234 to 5612 and then subtracting 10,000. 5. Does a b equal a + (10 6) 10 ? Does a b equal a + (10" 6) 10" ? Show that this proves that the method of subtrac- tion given in Ex. 4 is general. 6. Is there any advantage in subtracting by means of adding the complement of the subtrahend in a case like 521 173 ? 7. Solve the problem 6872 - 4396 + 342 - 896 - 243 + 750 by adding the proper complements and finally subtracting the proper powers of 10. Is there any advantage in using the method of com- plements in this case ? b HIGHER ARITHMETIC. V. MULTIPLICATION. The multiplication table is usually learned to 10 X 10. This is all that is necessary for practical purposes. It is often convenient, however, to perform quickly multiplica- tions with certain larger numbers. There are many rules for such operations, most of them of little practical value. Some are, however, quite useful, and these are given. To multiply by 5, 25, 33-J-, 125 is the same as to multiply by i^ ioo.^ 10 Q_ } iOg--- It is easier to multiply by J^ Q , that is, to add a cipher and divide by 2, than to multiply by 5, and similarly for the other cases. Also, to divide by 5 is the same as to divide by *- or to multiply by T 2 Q, and it is easier to multiply by 2 and divide by 10 than to divide by 5. These short processes have especial value because 5%, 12.5%, 25%, 33-J-%, 50% are common rates in business and scientific problems. To multiply by 9 is to multiply by (10 1). Hence, it suffices to annex a cipher and subtract the original number from the result, a method not always more convenient than ordinary multiplication. To multiply by 11 is to multiply by (10 + 1). Hence, it suffices to annex a cipher and add the original number ; or, what is more convenient, to add the digits as in the follow- ing example : 11 X 248, 8 + = 8, 4+8 = 12, 2 + 4 + 1 = 7, 2 + = 2; therefore, the result is 2728. To multiply by numbers differing but little from 10". For example, to multiply by 997 is to multiply by (10 3 3) ; that is, to annex three ciphers and subtract 3 times the multiplicand. E.g., 995 X 1474 = 1,474,000 - i of 14,740 = 1,474,000 - 7370 = 1,466,630. NOTATION AND FUNDAMENTAL OPERATIONS. 7 When two factors lie between 10 and 20 the product is readily found as follows : 14 Xl6 = 10(14 + 6) + 4 X 6 = 224. To prove that this process is general : (1) Any number from 10 to 19 may be represented by 10 + a, 10 + &, (2) (10 + a) (10 + b) = 100 + 10 -a + 10 1 + ab, = 10 (10 + a + b) + ab. To square numbers ending in 5. While not as practical as the problems already given, this has some value. The method is illustrated by 65 2 ; it is merely necessary to say, 6 X 7 = 42, 5 X 5 = 25, .-. 65* = 4225. To prove that this process is general : (1) Any number ending in 5 may be represented by 10 a + 5, where a may equal 0, 1, 2, 9, 10, (2) (10 a + 5) 2 = 100 a 2 + 100 a + 25, = 100 a ( + !) + 25. (3) That is, the result ends in 25, and the number of hundreds is a (a + 1). Applications of the formulas for (a + 6) 2 , ( + &)( &). In a few problems there is an advantage in recalling the identities (a + fl) = a a + 2 aft + 2 , (a + l>)(a 1) = a 2 l\ For example, 62 2 = 3600 + 240 + 4 =3844. 23 X 17 = (20 + 3) (20 3) = 391. Oral Exercises. 1. Multiply 12,564 by 5; by 25. 2. Multiply 4239 by 0.33*. 3. Multiply 5148 by 5; by 2.5 ; by 33*; by 12.5. 4. Square 25, 35, 45, 95, 105, 115. 5. Multiply 14 by 19; 17 by 15 ; 16 by 18; 12 by 19 ; 13 by 17. 6. Multiply 22 by 99 ; by 98 ; by 97. 7. Multiply 12,345 by 11. 8. Explain the short process of dividing by 33* ; by 125. 8 HIGHER ARITHMETIC. Arrangement of work. In multiplication, there is no practical advantage in beginning with, the lowest order of units of the multiplier ; 437.189 in fact, there is a decided advantage in 26.93 beginning with the highest order, as is 8743.78 clearly apparent in approximations in 2623.134 multiplication. The arrangement of work 393.4701 would then be as shown in the annexed 13.11567 example. Since 20 X 0.009 = 0.18, the 11773.49977 position of the decimal point is at once known, and the rest of the process is apparent. Approximations in multiplication are frequently desired. This arises from the fact that perfect measurements are rarely possible in science, and that results beyond two or three decimal places are seldom desired in business. Thus, if the radius of a wheel is known only to 0.001 inch, it is not possible to compute the circumference with any greater degree of accuracy ; hence, 10.48 labor would be wasted in seeking a product 3.1416 to more than three decimal places. In 31.44 such cases all unnecessary work should be 1.048 omitted, as in the annexed example. This 0.419 . represents the multiplication in the solu- 0.010 . . tion of the following problem : To find the 0.006 . . . circumference of a steel shaft of which the 32.92 diameter is found by measurement to be 10.48 centimeters. Since it was practical to carry the original measurement only to 0.01 centimeter, the result need be sought only to 0.01. In order to be sure that the result is correct to 0.01, the partial products are carried to 0.001, and in multiplying by 0.04, for example, the effect of the fourth decimal place on the third is kept in mind. Checks. The best check on multiplication is the " cast- ing out of nines," explained in Chap. II. NOTATION AND FUNDAMENTAL OPERATIONS. 9 Exercises. 1. Multiply 42.3531 by 3.1416, carrying the result to 0.001, that is, " correct to 0.001." 2. Multiply 126 by 0.3183, correct to 0.1. 3. If V2 = 1.414 , find the values of 4.324 V, 0.057 V2, and 8.346 V2, correct to 0.01. 4. If V3 = 1.732 , find the values of 2.35 V, 42.89 V, and 0.869V3, correct to 0.1. 5. Find V2 VI, correct to 0.01. 6. Find the interest on $1525.75 for one year at 6|%, correct to $0.01. 7. If the circumference of a circle is 3.1416 times the diameter, and if the radius of the earth is found by measurement to be 6,378,249.2 meters, find the circumference to the necessary number of decimal places, the earth being supposed spherical. Express the result in the index notation. 8. Similarly, find the circumference of a wheel the diameter of which is found by measurement to be 6.3 ft. 9. Similarly, find the circumference of a shaft the radius of which is found by measurement to be 4.32 in. 10. If 1 yard is found by measurement to be 0.914 of a meter, find, to the necessary number of decimal places, the number of meters in 23.463 yds. 11. Similarly, if 1 meter is found by measurement to be 3.28 feet, find the number of feet in 3.476 meters. 12. If the number of cubic units of volume in a sphere is f of 3. 1416 times the third power of the number of units of radius, find the volume of a sphere whose radius is found by measurement to be 3.27. 13. Find the value of 9 X 12,345,678 + 9 ; of 9 X 1,234,567 + 8 ; of 9 X 123,456 + 7. 14. Find the value of 9 X 98,765,432 + ; of 9 X 9,876,543 + 1 ; of 9 X 987,654 + 2. 15. Find the value of 8 X 123,456,789 + 9 ; of 8 X 12,345,678 + 8 ; of 8 X 1,234,567 + 7. 16. Show that 8,212,890,625 2 terminates in 8,212,890,625. 17. If a railway uses a freight car belonging to another company it pays the owner 0.6 of a cent a mile for the distance run ; during one year the freight cars of the United States were used in this way over a total distance of about 12,000,000,000 miles. How much rental did they earn their owners ? 18. Prove that to multiply by 625 one may move the decimal point four places to the right and then divide by 2 four times. 10 HIGHER ARITHMETIC . VI. DIVISION. In division there is an advantage in placing the quotient above the dividend. The decimal point is then easily fixed, although it is unneces- sary to carry it through the work. In case a decimal 15040.0 point appears in both I25fifi*4. dividend and divisor, it 247360 is better to multiply 219912 each by such a power of 10 as shall make the divisor integral. Thus, . ^ oo no 23.152 remainder. in the case of 32.92 -f- 3.1416 it is better to multiply both numbers by 10 4 , and divide 329200 by 31416, as above. The work may be further abridged 10.478 by omitting partial products and 31416)329200 decimal points, keeping only the 15040 partial dividends, as here shown. 24736 It is advisable in extended cases 27448 of division to prepare a table of 23152 multiples of the divisor, as in the following division of 4,769,835 by 291 : 1 291 2 582 16391 3 873 4769835 4 1164 1859 5 1455 1138 6 1746 2653 7 2037 345 8 2328 54 remainder. 9 2619 NOTATION AND FUNDAMENTAL OPERATIONS. 11 Approximations in division. In division as in multipli- cation, approximations are often necessary. For example, if the circumference of a shaft is found by measurement to be 32.92 centimeters and it is required to know the diameter, it would be a waste of time to attempt to find the diameter beyond 0.01. Since 10's divided by 10,000's < 0.01's, the last two figures of the dividend will not affect the quotient within two decimal places and hence may be neglected. Hence, also, the divisor may be con- sidered as 3142 and may be continually contracted. The process is apparent by first examining the complete form in the example below. The student should note how much better for practical purposes the last form is than the others, and he is recommended to become so familiar with it as to use it in all cases where only approximate results are required. 10.48 31416)329200 3142 = approximately 10 X 3141(6) 126 = 0.4 of 314(16) 24 24 = 0.08 of 31(416) 10.48 31416)329200 150 24 Checks. The best check on division is the "casting out -of nines " explained in Chap. II. Since the dividend equals the product of the quotient and divisor, plus the remainder if any, the work may also be checked by one multiplication and one addition, 12 HIGHER ARITHMETIC. Exercises. 1. Divide 42,856,731,275,834 by 574,238, correct to the tens' place. 2. Divide 100 by 3.1416, correct to 0.01. 3. Divide 5,080,240 by 40,467, correct to 0.1. 4. Divide 1 by 3.14159, correct to 0.001. 5. The population of British India is about 225,000,000, and the area is about 965,000 square miles ; what is the average population per square mile, to the nearest unit ? 6. In a certain year it cost $357,231,799 to pay the expenses of the United States government, which was $5.346 to each person ; what was the population in that year, to the nearest 1000 ? 7. In a certain year the revenue of the United States government was $403,080,983, which was $6.577 to each person; what was the population in that year, to the nearest 1000 ? 8. A hectare is 2.471 acres, and a liter is 61.027 cubic inches ; express the rainfall of 1 liter per hectare in cubic inches per acre, correct to 0.1. 9. Knowing that the circumference of a circle is 2 X 3.14159 X the radius, find, correct to 0.1, how many revolutions a mill-wheel 12 feet in diameter makes per minute when the speed of the periphery is 6 feet per second. 10. How many revolutions per mile are made by a locomotive drive-wheel 4.5 feet in diameter ? (Correct to units.) 11. 84.25 liters of water are drawn through a pipe every 4.5 minutes from a tank containing 23,711 liters ; how long will it take to empty the tank ? (Correct to 1 minute.) 12. If 41 liters of water weigh as much as 51 liters of alcohol, and 1 liter of water weighs 1 kilogram, how much will 1 liter of alcohol weigh ? (Correct to 0.01 kilogram.) 13. The horse-power of an engine is usually calculated by the formula , where p, Z, a, n are abstract numbers representing oo,000 the pressure in pounds per square inch on the piston, the length of the stroke in feet, the area of the piston in square inches, and the number of strokes per minute. Calculate the horse-power, to the nearest unit, of each of these engines : (a) p = 20, 1= 6, a= 400, n = 60. (6) p = 8, J = 11, a = 3600, n = 40. (c) p = 25, 1= 3, a= 100, n = 90. (d) p = 18, 1= 5, a= 200, n = 50. NOTATION AND FUNDAMENTAL OPERATIONS. 13 Axioms. There are a number of general statements of mathematics the truth of which may be taken for granted. Such statements are called axioms. The following are the axioms most frequently used in elementary arithmetic and algebra, the first, second, third, sixth, and seventh being especially important : 1. Numbers which are equal to the same number or to equal numbers are equal to each other. If 5 x = 3, and 1 + x = 3, then 5 x = 1 + x. 2. If equals are added to equals, the sums are equal. If x 2 = 7, then x 2 + 2 = 7+2, or x = 9. 3. If equals are subtracted from equals, the remainders are equal. If x + 2 - 9, then x - 7. 4. If equals are added to unequals, the sums are unequal in the same sense. If x + 2 > 8, then x + 2+5>8 + 5. 5. If equals are subtracted from unequals, the remainders are unequal in the same sense. If x + 5<16, then x< 11. 6. If equals are multiplied by equal numbers, the products are equal. If ~ = 6, then x = 18. o 7. If equals are divided by equals, the quotients are equal. If 2 x = 6, then x = 6 + 2 = 3. 8. Like powers of equal numbers are equal. If x = 5, then x 2 = 25. 9. Like roots of equal numbers are arithmetically equal. If x 2 = 25, then x = 5. (From algebra it should be remembered thatx= 5.) 14 HIGHER ARITHMETIC. Fundamental laws of elementary operations. There are also certain fundamental laws of number, the strict proof of which for cases involving fractions, surds, nega- tive numbers, etc., is properly a part of algebraic analysis. Since their treatment is too advanced for an elementary work, their validity is here assumed. They are not, how- ever, axioms, because they are not generally taken for granted in mathematics. These laws, as well as the axioms on p. 13, are so fre- quently used in subsequent discussions that their formal statement is necessary. They are as follows : 1. The associative law for addition and subtraction, that the terms of an expression may be grouped in any way desired. E.g., a + b-c + d = a + (b-c) + d = (a + b)-(c-d) = 2. The commutative law for addition and subtraction, that these operations may be performed in any order desired. E.g., a + b c + d = a c + d + b = d c + a + b= 3. The associative law for multiplication and division, that these operations may be grouped in any way desired. E.g., a-b-c + d-e = a-(b'c) + (d-7- e) = a-b-(c-r d)-e = 4. The commutative law for multiplication and division, that these operations may be performed in any order desired. E.g., a-b-r-c = a + c-b = b + c-a= Of course it is absurd to say that 72 times $3 is the same as $3 times 72, since the latter has no meaning by the common idea of multiplication. All that the commutative law asserts is that 72 $3 = 72 3 $1 = 3 72 $1 = 3 $72. 5. The distributive law for multiplication and division, that m(a b) = ma mb, and that - = m mm CHAPTER II. Factors and Multiples. I. TESTS OF DIVISIBILITY. IN speaking of factors and multiples, only integers are considered. Thus, 2 and 3.5 are not considered factors of 7, nor is 9 considered a multiple of 2, although 4J X 2 = 9. In practical computations, cancellation enters exten- sively, requiring a knowledge of the factors of numbers. There is no general process for determining large factors, and hence elaborate factor tables have been prepared for those who have extensive computations to make. But there are simple methods for determining small factors, methods not only valuable in a practical way, but also for the logic involved in their consideration. Fundamental theorems. The theory of factors and multiples depends largely on two theorems : I. A factor of a number is a factor of any of its multiples. 1. Let n be any number of which/ is a factor, q being the quotient 2. Then, since = 7, 3. .-. - + - + ..... m times = q + q + ..... m times, Ax. 2 4. Or -r mq. That is, / is a factor of mn, a multiple of n. 16 HIGHER ARITHMETIC. II. A factor of each of two numbers is a factor of the sum or the difference of any two of their multiples. 2. Then = aq, and = bq'. Th. 1 3. Suppose an > bn', 4. Then an ^ W = aq bq'. Axs. 2, 3 That is, / is a factor of an 4- &M' and of an bn', the sum and the difference of two multiples of n and ri. Tests of divisibility. I. 2 is a factor of a number if it is a factor of the number represented by its last digit, and not otherwise. 1. Any number has the form 10 a + 6, where 6 has any value from to 9 inclusive, and a has any value from and including 0. (E.g., 7036 = 10 X 703 + 6, and 3 = 10 X + 3.) 2. 2 is a factor of 10, and hence of 10 a. Th. 1 3. .-. 2 is a factor of 10 a + b if it is a factor of 6. Th. 2 4. Otherwise 2 is not a factor of 10 a + 6, for Wa + b - 5 a + |, 7 and - is not an integer. II. 4 i s a factor of a number if it is a factor of the num- ber represented by its last two digits, and not otherwise. Any number has the form 100 a + 10 6 + c, where, etc. The proof, which is similar to that of I, is left for the student. III. 8 is a factor of a number if it is a factor of the number represented by its last three digits, and not other- wise. The proof, which is similar to the proofs of I and II, is left for the student. IV. 5 is a factor of a number if it is a factor of the num- ber represented by its last digit, and not otherwise. The proof, which is similar to that of I, is left for the student. TESTS OF DIVISIBILITY. 17 V. 9 is a factor of a number if it is a factor of the sum of the numbers represented by its digits, and not otherwise. 1. Any number may be represented by a + 106+ 100 c + 1000 d + 10,000 e + where a represents the units' digit, 6 the tens', (E.g., in 7024, a = 4, b = 2, c = 0, d 7, e = 0, ) 2. Or by a +96 + 6 + 99c + c + 9992 X II 2 X 31. 3. 121,968 = 3 2 X II 2 X 2 4 X 7. 4. And since the greatest common divisor is the greatest factor common to the three numbers, it is 3 2 X II 2 , or 1089. Exercises. 1. Find, by factoring, the g.c.d. of 153, 891, and 1008. 2. Also of 32,760, 1170, and 1573. 3. Also of 720, 336, and 1736. 4. Also of 837, 1134, and 1377. 5. Also of 187, 253, and 341. 6. Also of 1331 and 4,723,598. 7. Also of 231, 165, 451, 4004, and 2827. 8. Also of 117, 143, 221, 338, and 650. 9. Determine mentally that 49, 81, 121, and 4936 are prime to one another. 10. Similarly for 429, 490, and 12,347. 22 HIGHER ARITHMETIC. 4356)9801 8712 4 1089)4356 4356 The Euclidean or long division method may also be illus- trated by a single example, the one already considered. 1. v the g.c.d. is contained in each number it ^j> 9801. 3 2. The g.c.d. ^ 9801, v that 9801)33759 is not a factor of 33,759. 29403 2 3. v the g.c.d. is a factor of 9801 and 33,759, it is a factor of 4356. P. 16, th. 2 4. .'. the g.c.d. y> 4356 and it is 4356 if that is a factor of 9801, 33,759, and 121,968. 5. But the g.c.d. ^ 4356 112 V that is not a factor of 9801. 1089)121968 6. V the g.c.d. is a factor of 1089 4356 and 9801, it is a factor 1306 of 1089. P. 16, th. 2 1089 7. .-. the g.c.d. y> 1089 2178 and it is 1089 if that is a 2178 factor of 4356, 9801, 33,759, and 121,968. 8. v 1089 is a factor of 4356, it remains to find whether it is a factor of 9801, 33,759, and 121,968. 9. v 1089 is a factor of itself and of 4356, it is a factor of 9801. P. 16, th. 2 10. .-. it is a factor of 33,759. P. 16, th. 2 11. And 1089 is a factor of 121,968, by trial. 12. .'. 1089 is the greatest common divisor of the three numbers. Exercises. 1. Find, by the Euclidean method, the g.c.d. of 6961 and 9976. 2. Also of 8673 and 23,989. 3. Also of 2827, 3341, and 11,565. 4. Also of 5187, 14,421, and 3249. GREATEST COMMON DIVISOR. 23 Abbreviations of the Euclidean form will readily suggest themselves. Only two will, however, be considered. 1. If a factor is common to two numbers it must be a factor of their greatest common divisor. Hence, if seen, it may be suppressed in order to shorten the work, and introduced in the result. For example, the factor 2 in the problem below. 2. If a factor of either number is not common to the other it cannot be a factor of their greatest common divisor. Hence, if seen, it may be suppressed in order to shorten the work. For example, the factor 3 2 in the problem below. Required the g.c.d. of 33,282 and 73,874. 2133282 19 1849 > 36937 1844T l 1806)1849 42 Since a composite factor, like 9, may not be common to two numbers and yet may contain a factor, as 3, which is common, only prime factors should be rejected. Exercises. 1. Find, by the Euclidean method, suppressing factors whenever it is advantageous, the g.c.d. of 845,315 and 265,200. 2. Also of 4,010,401 and 4,011,203. 3. Also of 16,897 and 58,264. 4. Also of 40,033 and 129,645. 5. Also of 29,766 and 208,362. 6. Also of 376, 940, 1034, and 1081. 7. Reduce to lowest terms the fractions TT and 8. Also the fractions \%\\ and ff \\. 9. Also the fractions fff f and y^V 10. Also the fractions T 9 because if the unit is divided into -th as many c ft c a parts, each part (and hence the fraction) is a times as large. V. The value of a fraction is not altered by multiplying or dividing both terms by the same number. Since n x r = \- > by III, and n X ^ = ^ by IV, bo rib o a na a na , n x T = n x > or -r = by ax. 7. b nb b nb J Addition and subtraction of fractions. I. When they have a common denominator, as -;-;> 7 _ ad where As $a $6 = | (a 6), the unit $1 being the same, so - - = - the unit - being the same. add d II. When they have not a common denominator, they may be reduced to fractions having a common denominator (by fundamental property V), and preferably to fractions having the least common denominator. COMMON FRACTIONS. 29 Multiplication of fractions. To multiply one number by another is to perform that operation upon the first which being performed on unity produces the second. Since the primitive notion of multiplication is the taking of a num- ber a certain number of times, as 5 times $2, and since it is meaning- less to take a number 3 inches times, multiplication is considered as an operation in which the multiplicand may be either abstract or concrete, but in which the multiplier is always abstract. To illustrate the definition, 2 X $3 = $6 ; since 1 is added to itself to produce the multiplier, $3 is added to itself to produce $6. Similarly in the case of =- X - To produce the multiplier = from 1, 1 must be divided into 6 parts, and a of those parts taken. So to produce the product from the multiplicand - the fraction J must be divided into 6 parts (each being by IV), and a of those parts ac taken, giving The symbols of multiplication, X and , are usually read "of" after a pure fraction, but " times " after an integer or a mixed number. Thus, I- x $5 is read "f of $5," but 1-j- X $5 is read " 1| times $5," this being the abridged form for " once $5 and of $5." (See p. 42.) Division of fractions. Division may be defined as the operation of finding one of two factors, the quotient, when the product and the other factor are given. For example, 2 X $5 = $10, .-. $10 -=-$5 = 2, and $10 -f 2 = $5. To divide r by is, therefore, to find a quotient q, such that .-. = q, by multiplying equals by d, and dividing by c, by axs. 6 oc and 7, and fundamental properties III and IV. It therefore appears that the quotient equals the product of the dividend and the reciprocal of the divisor, and can, therefore, be obtained by multiplication. Hence the familiar rule, "Invert the divisor and multiply," a rule that is always valid for abstract divisors. 30 HIGHER ARITHMETIC. a Complex fractions of the form - c may be considered as the equivalent of - -r- -, and treated accordingly. And since nXX- nXX^- = nXX~-^- d c_ d b d b d d d nX l and n X - X = n X - , by def . of division ; nx a a a c c - n x - d d That is, fundamental property V applies to complex fractions as well as to simple fractions. Practical suggestions as to the treatment of fractions. I. Make free use of fundamental property V, multiplying or dividing the terms by the same number. For example, the fraction T \ 9 g 3 f should have the factors 9 and 11 suppressed at once, the fraction reducing to T 7 ^. In the case of the complex fraction -^ the factor 8 should be introduced, the fraction reducing to f , or T 5 , an operation much simpler than division. II. In reducing to lower terms, it is best to reject simple factors at once, without attempting to find the greatest com- mon divisor by the long process. For example, in the case of the fraction f/-/ 7 above considered. III. Feel free to use the common fraction or the decimal as may be the more convenient in the computation in hand. For example, it is better to multiply or divide by \ than by 0.25, and by j- than by 0.125. But 0.2 is an easier operator than , and 0.04 is easier than . COMMON FRACTIONS. 31 Exercises. 1. Explain the reduction of 5f to - 3 T 7 . 2. Also the reduction of Q to 7|. 3. Perform at sight the following multiplications: 0.125 of 640, 0.33i of 903, 0.25 of 500, and 0.5 of 720. 4. Also the following divisions: 840^0.125, 69 -r 0.33, 200 -r 0.25, and 68 -r 0.5. 5. Reduce to lowest terms the fractions f f , /J^, $|j[$, and f f . 6. Simplify^, |, y, and|- 7. Add 2/31, ^f y ! 3\4> If ft- 3 ( Th e Actors of 4199 are 13, 17, and one other.) 8. Also the fractions ifoi ? 259^ an( j ^59^ 9. A + A of T 4 T + T 5 2 of T 4 T of A + & of A of A of f + T 5 of A of T 3 o of f of 8 = ? 10. | of ^of^of ^=? 11. 1|- x 2^ x 3 x 4^ x 5 = ? 12. Divide f of | of 3^ by f of | of 4. 13. Divide ^ of & by ^5 of ^. 14. Of the three fractions |, ^f , and f f , which is greatest ? which least? 15. Which is greater, f or f f ? | of | or f of f ? 16. What is the effect of adding the same number to both terms of a proper fraction on the value of the fraction ? Prove it for the general case of T' where a < 6. 17. Investigate the same for the fraction -rt when a > 6. 18. Show that the fraction lies between the greatest and o ~r o "r / the least of the fractions f , f , f . 19. A " magic square " is a square array of numbers such 438 that the sums of the numbers in the rows, columns, and two 951 diagonals are equal, as in the annexed illustration. Insert 270 the fractions to complete the following magic square : If * H CHAPTER IV. Powers and Roots. THE cases are few in practical business where either involution or evolution is used. In scientific work, num- bers often have to be raised to powers, and roots have to be extracted, but the operations are usually performed with the help of tables of powers, roots, or logarithms. The value of the subject may, therefore, be said to lie largely in the exercise of the reasoning powers. Hence, in the present chapter more attention is directed to the reasons for the various steps than to short methods of securing results. I. INVOLUTION. Symbolism, a 2 is read " a square," or " a to the second power," and means a a ; a 3 is read " a cube," or " a to the third power," and means a a a ; a 4 is read " a to the fourth power," and means a a f a* a; and, in general, a n is read " a to the nih power," and means a a (n times). This symbolism and the notion of power have been extended, thus : v a* is obtained by dividing a 3 by a, so a 2 a 1 is defined as or a, and is read " a to the first power," a a 1 and a is defined as or 1, and is read " a to the zero a INVOLUTION. 33 a 1 power," and a~ l is denned as or > and is read " a to the a a a~ l 1 minus first power," and a~ 2 is defined as or > and is a as read " a to the minus second power/' and, in general, a~ n is defined as > and is read " a to the minus nth power." A further extension has also been made to include frac- tional powers, thus : V a 2 = V& 4 , and a i = Va 2 , so a* is defined to mean Va, and is read " a to the -J- power " or " the square root of a " ; 1 n, 2 and,' in general, a n is defined as v, and a as the mth power of the ^th root of a. Thus, a 1 - 25 means the 125th power of the 100th root of a. Raising numbers to high 2 4 = 16 powers. It occasionally 2 4 = 16 becomes necessary to raise 2 8 = 256 a number like 2 to some ?!_ 256 high power, as in the case 2 16 = 65536 of 2 30 . Here the computer 2^ = 65536 should recall the fact that 2J = 4294967296 a m -a n = a m+n , and proceed 2 30 = 2 32 -=- 2 2 = 1073741824 as indicated in the annexed multiplication. Powers of binomials. In the extraction of roots by the method to be considered it is necessary to know the cor- responding powers of binomials. The student may expand the following : =/ 3 + 3/% + 3/n 2 + n. (/+ n) 4 = (?). Obtain it by squaring (/ + n) 2 . Obtain it by squaring (/ + n). 34 HIGHER ARITHMETIC. Exercises. 1. Square 41 by using the formula for (/+ n) 2 . 2. Cube 22 by using the formula for (/ + n) 3 . 3. Similarly, find the values of II 2 , II 3 , II 4 , and from the results find the values of II 5 and II 7 by single multiplications. 4. Express 2~ 4 , 4" 1 , 5~ 2 , and 10~ as decimal fractions. 5. Express 0.04 and 0.03125 as negative powers of integers. 6. Prove that no number ending in 2, 3, 7, or 8 can be a perfect square. 7. Prove that the square of a number ending in 5 ends in 025, 225, or 625. 8. Prove that a square must end in 0, 1, 4, 5, 6, or 9. 9. Prove that a cube may end in any of the digits. 10. Prove that the cube of a number ending in 5 must end in 125, 375, 625, or 875. 11. Prove that the 5th power of a number ends in the same digit as the number itself. 12. If the student has taken the subject of imaginary numbers in algebra, but not otherwise, he may solve the following : (a) (- i + i V^3) 2 = ? (6) (- i - i V^3) 2 = ? (c) (-i + W-3) 3 = ? (d) (-i-iV-3) = ? II. SQUARE ROOT. The square root of a perfect second power is one of the two equal factors of that power. In case a number is easily factored, the square root may be found by this means, as explained on p. 37. A number which is not a perfect second power has not two equal factors. It is, however, said to have a square root to any required degree of approximation. Thus, the square root of n to 0.1 is that number of tenths whose square differs from n by less than the square of any other number of tenths. E.g., the square root of 2 to 0.1 is 1.4, to 0.01 is 1.41, etc. It should be observed that under these definitions only abstract numbers can_ have square roots. Thus, 4 is the product of 2 and 2, hence 2 = \/4 ; but no number multiplied by itself equals $4, or 4 feet, or 4 square meters. SQUARE ROOT. 35 The general theory of this subject is best understood by following the solution of a problem. Suppose the square root of 547.56 be required. Let / = the /ound part of the root at any stage of the operation, and n the next digit to be found. Then ' (/+n)*=/* + 2fii + n\ The greatest square in 500 is v 20 has been found, and 20 2 , or / 2 , subtracted, this 147.56 must contain 2 fn+ ri 2 , or 2 20 n + n 2 . .-. by divid- ing by 2 20, or 40, n can be found approximately. .-. n = 3. .-. 2fn + w 2 , or 2-20-3 + 3 2 = 2 3. 4 5'47.56 400. 1 47.56 contains 2fn + n 2 . that is, the square of 20, or/ 2 . 1 29 / is now 20 because that is all that has been found. n is now 3, the next digit. = 2fn + n 2 . v 23 has been found, and 23 2 , or/ 2 (= 400 + 129), sub- tracted, this 18.56 must con- tain 2/n + w 2 , or 2 23 n + n 2 . .-. by dividing by 2 23, or 46, n can be found approxi- mately. .-.n = 0.4. .-.2/n+n 2 , or2-23-0.4 + 0.4 2 18.56 contains 2/n + n 2 . / is now 23 because that has been found. n is now 0.4, the next digit. = 18.56 = 2/n + n 2 . The actual computation may be conveniently arranged in either of the following ways, the first being preferable for the majority of students. 40 43 2 3. 4 5'47.56 400 1 47 1 29 46 46.4 18.56 18.56 43 46.4 2 3. 4 5'47.56 1 47 18.56 For those who desire a complete explanation of the process a more extended discussion appears on p. 36. 36 HIGHER ARITHMETIC. 2 3. 4 5<47.56 /! 2 = 4 00.00 2/i =40 147.56 contains 2j ?!! + V / = 20 2/ 1 + w 1 = 43 1 29.00 = (( 2/ 2 =46 18.56 contains 2j ^2 + ^2 2 /2 = 23 2/ 2 -f ?i 2 = 46.4 18.56 = " ^ 2 = 0.4 1. v the highest order of the power is 100's, the highest order of the root is 10's, and it is unnecessary to look below 100's for the square of 10's. 2. Similarly, it is unnecessary to look below 1's for the square of 1's, below lOOths for the square of lOths, etc. [These places may be indicated by points ('), as in the above example.] 3. The greatest square in the 100's is 400, which is the square of 20, which may be called f\ (read u /-one"), the first found part of the root. 4. Subtracting, 147.56 contains 2/n + n 2 because/ 2 has been sub- tracted from/ 2 + 2fn + n 2 , where /stands always for the found part and n for the next order of the root. 5. 2/n, + n 2 is approximately the product of 2 /and n, and hence, if divided by 2/, the quotient is approximately n. .. n = 3. 6. .-. 2/ + n 2 X 20 + 3 = 43, and this, multiplied by n, equals 2/n + n 2 . 7. v / 2 has already been subtracted, after subtracting 2/n + n 2 there has been subtracted/ 2 + 2/n + n 2 , or (/+ n) 2 , or 23 2 . 8. Calling 23 the second found part, / 2 , and noticing that /a =/i + i, it appears that 23 2 , or/ 2 2 , has been subtracted. 9. .-. the remainder 18.56 contains 2/ 2 n 2 + n. 10. Dividing by 2/ 2 for the reason already given, n 2 = 0.4. 11. .-. 2/ 2 + n 2 = 46.4, and 18.56 = 2/ 2 n 2 + n 2 2 , as before. 12. Similarly, the explanation repeats itself after each subtraction. 13. Students will remember from algebra that every number has two square roots, one + and the other . .-. V547.56 = 23.4, but the positive root is the only one likely to be needed in practice. The subject of square root is still further discussed in the Appendix, Note 1. SQUARE ROOT. 37 Common fractions. There are three general methods for extracting the square root of a common fraction. 1. The square root of both terms may be extracted, as is advisable when each is a square number. E.g., 2. The fraction may be reduced to the decimal form, as is advisable when this can easily be done. E.g., V^ = Vo! = 0.447 ..... 3. The fraction may be reduced to an equal fraction whose denominator is a square number. E.g.,^ = VJf = \ Vl4 = \ of 3.741657 = 0.534522, an easier method in most cases than either of the two just mentioned. Factoring method. The cube root of a perfect third power is one of the three equal factors of that power, and similarly for the fourth, fifth, ..... nth roots. As mentioned on p. 34, such roots can often be found by factoring. E.g., 85,766,121 = 3-3-3-3-3-3-7-7-7-7-7-7; .-. V85,766,121 = 3-3-3-7-7-7 = 9261, %/85,766,121 = '3 3 7 7 = 441, and V85,766,121 =3-7 = 21. Even in case a number is not a perfect power the factoring method can often be advantageously used. E.g., V882 = V2 3 2 7 2 = 3 7 V2 = 21 V2 = 21-1.4142 = 29.698 Exercises. 1. The student may test his knowledge of the general theory by answering the following questions : (a) If you separate into periods of two figures each, where do you begin ? Consider, for example, the square root of 14.4. (b) Why does the remainder contain 2 fn + n 2 the first time ? the second ? (c) Why is 2 / always taken as the trial divisor ? (d) Why is n added to 2 / to make the complete divisor ? (e) In the example on p. 35, why does 129 equal 2 fn + n 2 ? (/) In that example, how can 129 and 18.56 each equal 2fn + n 2 ? 38 HIGHER ARITHMETIC. 2. Extract the square roots of the following numbers, writing out the solutions in the full form given on p. 35 : (a) 80.4609. (&) 8226.49. -(c) 1280.9241. (d) 0.21224449. (e) 12.8881. (/) 0.49112064. (g) 592330.3369. (h) 32.26694416. 3. Extract the square roots of the following numbers, abridging the solution as in the first at the foot of p. 35 : (a) 40509.6129. (6) 0.501361708761. (c) 234.579856. (d) 96.27534400. (e) 1.47403881. (/) 416.05800625. (g) 28597039.6644. (h) 8260.628544. (i) 85747600. 4. Extract the square roots, to 0.001, of the following numbers : (a) 0.0068. (6) 20. (c) 2. (d) 951. (e) 680. (/) 809. (g} 13. (h) 1000. 5. Extract the square roots, to 0.00001, of the following numbers : (a) 976. (6) 887. (c) 0.565. (d) 3. 6. Decide which of the three methods for extracting the square root of a common fraction is the best for each of the following num- bers, giving the reason, and extract the root accordingly, to 0.001. (a) TV ( & ) A- ( c ) I - (d) mi- (e) &. (/) iff*. (g) f. (h) l\. (0 *flk- 0') T h- (*) 50 ~ 2 - (0 M-. 7. By separating into factors, find the square roots of 2304, 9216, 396,900, 194,481, 11,025, 117,649. 8. Similarly, the cube roots of 46,656, 91,125, 1,953,125, 11,390,625, 250,047,000, 85,766,121, 1,771,561. 9. Similarly, the fourth roots of 15,752,961, 43,046,721, 59,969,536, 96,059,601. 10. Similarly, the fifth roots of 59,049, 4,084,101, 9,765,625, 3,486,784,401. 11. Similarly, the sixth roots of 34,012,224, 113,379,904, 177,978,- 515,625. 12. The following sums are the squares of what numbers ? (a) II 2 + 60 2 . (6) 8088 2 + 1,022,105 2 . (c) 13,552 2 + 936,975 2 . (d) 18 2 + 19 2 + 20 2 + 21 2 + 22 2 + 23 2 + 24 2 + 25 2 + 26 2 + 27 2 + 28 2 . 13. Draw a square whose side is/ + n (f may be taken as of an inch and n as J of an inch). From this figure, show that the square on / + n is made up of the square on /, plus the square on n, plus two rectangles which are / long and n wide, thus illustrating the fact that 14. From the figure of Ex. 13, show that after / 2 is taken away there remains 2/n + w 2 . CUBE ROOT. 39 III. CUBE BOOT. The complete definition of cube root may be inferred from that of square root. Suppose the cube root of 139,798,359 be required. Since the theory so closely resembles that of square root, the explanation is given in analogous form. 5 1 9 139,798,359 / 3 = 125,000,000 3/n 3/ 2 + 3/n 14,798,359 contains 3/ 2 n + 3/n 2 + n 3 . 3/ 2 + n 2 + n 2 /i = 500. 7,651,000 = 3/ 2 n + 3/n 2 + n 3 . 750,000 15,100 765,100 m = 10. 7,147,359 contains 3/ 2 n + 3/n 2 + n 3 . / 2 = 510. 780,300 13,851 794,151 7,147,359 = 3/ 2 n + 3/n 2 + n 3 . n 2 = 9. 1. v the highest order of the power is hundred-millions, the highest order of the root is 100's (why ?), and it is unnecessary to look below millions for the cube of 100's. (Why ?) 2. Similarly, it is unnecessary to look below 1000's for the cube of 10's, below 1's for the cube of 1's, etc. (These periods may be indi- cated by points as in square root, if desired.) 3. The greatest cube in the hundred-millions is 125,000,000, the cube of 500. .-. 500 may be called/. 4. Subtracting, 14,798,359 contains 3/ 2 n + 3/n 2 + n 3 . (Why ?) 5. This is approximately the product of 3/ 2 and n, and hence if divided by 3/ 2 the quotient is approximately n. .-. n = 10. 6. .*. 3/n + ?i 2 = 15,100, and 3/ 2 + 3/n + n 2 = 765,100, and this, multiplied by n, equals 3/ 2 n + 3/n 2 + n 3 . 7. v / 3 has already been subtracted, after subtracting 3/ 2 n + 3/n 2 + n 3 there has been subtracted (/ + n) 3 , or 510 3 . 8. Calling 510 the second found part, / 2 , it appears that / 2 3 has been subtracted. .-. the remainder contains 3/ 2 n + 3/n, 2 + n 3 . 9. The explanation now repeats itself as in square root. 40 HIGHER ARITHMETIC. In practice, the work is usually arranged somewhat as follows : 5 1 9 139,798,359 125 7500 7651 14798 7651 780300 794151 7 147 359 7 147 359 The subject of cube root is still further discussed in the Appendix, Note II. Exercises. 1. (a) If you separate into periods of three figures each, why do you do so ? Where do you begin ? Why ? Consider, for example, the cube root of 13.31. (6) Why does the second remainder contain 3/ 2 w + 3/n 2 + n 3 ? (c) Why is 3/ 2 always taken as the trial divisor ? (d) Why is 3/n + n 2 added to 3/ 2 to make the complete divisor ? (e) In the example on p. 39, why does 7,651,000 equal 3/ 2 n + 3/n 2 + n 3 ? (/) In that example, how can 7,651,000 and 7,147,359 each equal 3/% + 3/w 2 + n 3 ? 2. Extract the cube roots of the following numbers, writing out the solutions in, the full form given on p. 39 : (a) 139,798,359. (6) 248,858.189. (c) 0.004657463. (d) 19.902511. (e) 0.000091733851. (/) 731.432701. 3. Extract the cube roots of the following numbers, abridging the solution as suggested in square root and at the top of this page : (a) 553,387,661. (6) 381.078125. (c) 997.002999. (d) 0.051064811. (e) 0.0001851930. (/) 0.876467493. 4. Extract the cube roots, to 0.001, of the following numbers : (a) 251. (6) 455,000. (c) 0.57. (d) 0.27. (e) 998. (/) 0.007. (g) 0.194104601. (h) 0.47637955. 5. Explain three methods of extracting the cube root of a common fraction, analogous to those given for square root. 6. Decide which of these three methods is the best for each of the following numbers, giving the reason, and extract the root accord- ingly, to 0.001 : CHAPTER V. The Formal Solution of Problems. THE most important portion of arithmetic, considered from the business standpoint, has already been completed, especially essential being that part which treats of addi- tion, subtraction, multiplication, and division of integers and fractions. The subsequent portions of the subject are taught for the business and scientific principles involved, but largely as an exercise in logic. And since the opera- tions mentioned have been the subject of extensive drill in the lower grades, it is neither necessary nor advisable to preserve them, after checking the result of each computa- tion, in the treatment of applied problems. The solution should now be logically arranged in steps numbered for reference, the complete operations being preserved when- ever the teacher advises. In this way, the logic of the solution stands out prominently, while on the other hand there is no loss in the way of arithmetical computations. At every stage of the solution time and energy should be economized by resort to factoring and cancellation. It is a good rule, never multiply till you have to, always factor if you can. The advantages of this rule are seen in the problem solved on p. 47. The student should also be advised as to the proper use of symbols and language, and to this end a few suggestions may be of value. 42 HIGHER ARITHMETIC. I. SYMBOLS. The common symbols for multiplication are X and , the latter being preferable for students sufficiently mature not to confuse it with the decimal point. It is advisable to write the multiplier first because (a) it is usually read first, (&) the tendency among leading writers is to place it first, and (c) in an algebraic expression like 4 x the first factor is usually looked upon as the multiplier. Thus, if 1 book costs $2, 123 books, at the same rate, will cost 123 $2. This would be indicated by the step 123 $2 = $246, but the actual multiplication would of course be 2 123, on the principle that 123 2 $1 = 2 123 $1. The symbols may therefore be read as follows : 2 $3, or 2 X $3, "2 times $3," or " 2 into $3 " ; $3 2, or $3 X 2, " $3 multiplied by 2." The word " times " in this connection has a much broader meaning than that assigned when arithmetic was in its infancy. Thus, we say "2 times $4," meaning thereby "2 times $4 and $ of $4." It is not customary, however, to use the word after a proper fraction ; thus, &-$4 is read " of $4," and ft. is read "f of a foot." Hence, 2 times $4 has acquired a meaning; but to look out of the window 2 times is nonsense. The general agreements of mathematicians as to the relative weight of symbols should also be understood. The usage varies in different countries, however, and occasion- ally is not entirely settled in any one. The following may be taken as indicating the rules followed by the leading writers of the day. The absence of a sign between two letters, and the frac- tional notation, indicate operations to be performed before any others. Thus, in the expression a -r- cd -f- 'the multiplication cd is first y performed ; then the division of e by g ; then the other divisions in order, beginning at the left. THE FORMAL SOLUTION OF PROBLEMS. 43 The word " of " following a fraction stands next as to weight. Thus, in a -f- ^ of cd, the multiplication cd is first performed ; then the multiplication by % ; then the division of a by the result. The symbols , X, -f-, / stand next, one having the same weight as another. Thus, inl-3x4 + 2X 6/3, the operations are performed in order from left to right, the result being 12. The symbols +, stand next, one having the same weight as the other. Thus, 2 + 3-6-5 + 4-8-|- = 35. The symbol : , when used as a symbol of ratio, stands next. Thus, 2 + 3:4 + 1 = 5:5 = 1. But since the symbol is one of division, and in some countries is the leading symbol of that opera- tion, it is frequently given the same weight as the -r. In that case, 2 + 3 : 4 + 1 = 3f , while (2 + 3) : (4 + 1) = 1. The other common symbols are sufficiently understood already, or are explained elsewhere in this work. HISTORICAL NOTE. The symbols + and were used by Wid- inann in an arithmetic published at Leipzig in 1489, = by Recorde in an algebra published in 1557, X by Oughtred in 1631, the dot () as a symbol of multiplication by Harriot in 1631, the absence of a sign between two letters to indicate multiplication by Stifel in 1544, : as a symbol of division by Leibnitz, + as a symbol of division by Rahn in an algebra published at Zurich in 1659, > and < by Harriot in 1631. The symbols =56, ^>, <, indicating "not equal," etc., are recent. Parentheses were first used as symbols of aggregation by Girard in 1629. The decimal point came into use in the seventeenth century ; it seems to have appeared first in a work published by Pitiscus in 1612, but it was not extensively employed until more than a century later. Positive integral exponents in the present form were first used by Chuquet in 1484. The symbol V~ was first used in this form by Rudolf! in 1525. 44 HIGHER ARITHMETIC. II. LANGUAGE. The student should also guard against statements like the following : " 2 times greater than 3 " for " 2 times as great as 3"; "3 + 1 equals to " "$4 + 3" for $4 + $3" ; 2 X 3 = $6 " for " 2 X $3 = $6 " ; "2 is contained in $6 $3 times " for $6 divided by 2 equals $3," or " of $6 equals $3." III. METHODS. There is no general method of solution covering all prob- lems ; if there were, the subject would lose substantially all of its value as an exercise in logic. The student should feel encouraged to put into the work all the individuality possible, only being sure (1) that each statement is true, (2) that each result is checked, (3) that his solution involves no undue labor. 1. Analysis in general. The solution of any problem of applied arithmetic requires analysis of some kind ; in other words, the application of a student's common sense. Two types are here given, and it will be seen that if the steps are properly arranged the oral analysis is a simple matter, beginning at each stage with a " since " and reasoning to a " therefore,' 7 Problem. If the average velocity of a bullet in going from a gun to a target is 1342 ft. per sec., and that of sound is 1122 ft. per sec., how much time will elapse, on a range of 1000 yds., between the time the bullet strikes the target and the time that the sound of the dis- charge reaches the target ? Solution. 1. 1000 3 ft. = 3000 ft. 2. The bullet goes 1 ft. in y^ sec. 3. .-. it goes 3000 ft. in ffff sec., or 2.24 sees. 4. Similarly, sound requires ff f sec., or 2.67 sees. 5. 2.67 sees. 2.24 sees. = 0.43 sec. THE FORMAL SOLUTION OF PROBLEMS. 45 Analysis, v 1 yd. = 3 ft., .-. 1000 yds. = 1000 3 ft. v the bullet goes 1342 ft. in 1 sec., .-. it goes 1 ft. in yJ^ of 1 sec., and 3000 ft. in 3000 T ^ 7 of 1 sec. Similarly, sound goes 3000 ft. in 3000 T1 ^j of 1 sec. The difference in time is evidently the result required. Problem. At what time between 1 and 2 o'clock are the hands of a clock at right angles to each other ? Analysis. (The student should first draw the figure.) v they are together at 12 it is readily seen that the minute hand must gain 60 minute-spaces on the hour hand to bring them together again. .-. it must gain (60 + 15) minute-spaces or else (60 + 45) minute- spaces to bring them at right angles between 1 and 2. v at 1 o'clock the minute hand is at 12 and the hour hand at 1. .-. the former gains 55 minute-spaces in 1 hr., or 1 minute-space in & hr - .-. to gain 75 or 105 minute-spaces it requires 75 -fa hr. = 1 hr. 21 r 9 T mins., or 105 -fa hr. = 1 hr. 54 T 6 T mins. Exercises. 1. What is the speed in feet per sec. of a train moving uniformly at the rate of 20 mi. per hr. ? 60 mi. per hr. ? 2. The earth's center moves about the sun at the average rate of 101,090 ft. per sec.; how many miles per hr.? 3. An elastic ball rebounds to a height which is f of that through which it fell ; on the third rebound it rises to a height of T 4 r ft. ; from what height did it first fall ? 4. Each of the two arctic zones covers 0.02 of the earth's surface, and each of the temperate zones 0.26 ; what part is covered by the two torrid zones together ? 5. A locomotive consumes ^ of its tankful of water every mile ; it starts with only f of a tankful ; how many miles has it gone when it has T 2 5 of a tankful left ? 6. A steam engine using 28.5 tons of coal in 30 working days has an improvement effected rendering it necessary to use only 4.8 tons a week of 6 working days ; how much is saved in a year of 300 working days, coal costing $5.60 a ton, not considering the cost of the improve- ment ? 7. If the pressure of air on the surface of a lake is 15 Ibs. per sq. in., and if 1 cu. ft. of water weighs 1000 oz., find the pressure per sq. ft. at the depth of 100 ft. 46 HIGHER ARITHMETIC, 8. The average daily motion of the earth about the sun is 59' 8.3" a day ; that of Mars is 31' 26. 5" ; if they moved in the same plane and kept these rates, how many days would elapse from the time they were in the same straight line on the same side of the sun to the time when the earth was again directly between Mars and the sun ? Draw a diagram illustrating the problem. 9. Similarly for earth and Mercury, the latter's daily rate being 4 5' 32.5". 10. Similarly for earth and Venus, the latter's daily rate being 1 36' 1.1". 11. Similarly for earth and Neptune, the latter's daily rate being 21.5". 12. Similarly for Mercury and Venus (see Exs. 9, 10). 13. Similarly for Venus and Neptune (see Exs. 10, 11). 14. What are the relative positions of the earth, the moon, and the sun at the time of a new moon ? The average daily motion of the moon about the earth is 13. 1764 ; the apparent daily motion of the sun in the same direction is 0.98565 ; required the time from one new moon to another. 15. It is estimated that a cannon-ball leaving the earth at the rate of 500 mi. per hr., and continuing that rate to the nearest fixed star, would require about 4,500,000 yrs. for the journey, (a) Express in index notation, giving only the first two significant figures, the dis- tance to the star. (6) Knowing that light travels about 186,000 mi. per sec., find, to 0.1, the number of years required for the light of the star to reach the earth. (Take 365J da. = 1 yr.) 16. Sound travels 65,400 ft. per min.; how far away is a gun whose report is heard 15 sees, after firing ? 17. The average cost per day for tuition in the common schools of the United States is 8.2 cts. Estimating the number of pupils at 14,000,000, what is the total cost per day ? What is the cost per year of 150 school days ? 18. The velocity of light being 186,330 mi. per sec., how long does it take the light from the sun to reach the earth, the distance being 93, 165,000 mi.? 19. A factory is insured for $2500 in one company, $3500 in another, and $2000 in another ; it is damaged by fire to the extent of $4875 ; what portion of the loss should each company bear ? 20. A man left by will to four persons the sums of $1000, $950, $800, $750, respectively ; his estate produced only $2900 ; how much should each legatee receive ? THE FORMAL SOLUTION OF PROBLEMS. 47 2. Unitary analysis is so called because the student analyzes the problem by passing to one or more units. The method is very advantageous in the solution of many problems which, although of no especial value in business or in science, are still found in most text-books. While such problems are foreign .to the spirit of the present work, a few are given by way of illustration. In all these cases the words " at the same rate " are to be understood. Problem. How many pumps working 12 hrs. per da. will be required to raise 7560 tons of water in 14 da., if 15 pumps working 8 hrs. per da. can raise 1260 tons in half that number of days ? Solution. 1. 1260 t. raised in 7 da. of 8 hrs. each require 15 times the work of 1 pump. 2. 1 t. in 7 da. of 8 hrs. requires T ^Q times the work of 1 pump. 3. 1 t. in 1 da. of 8 hrs. " 7 T jfo " 4. 1 t. in 1 da. of 1 hr. " 8 7 T ^ " 5. 7560 t. in 1 da. of 1 hr. " 7560 8 7 T |f^ " 6. 7560 1. in 14 da. of 1 hr. " T ^ 7560 8 7 T || - " 7. 7560 t. raised in 14 da. of 12 hrs. each require Ta ' i? ' 756 ' 8 7 T if o times the work of 1 pump = the work of 30 pumps. In actual practice it would be better to pass from step 1 to step 4, and then directly to step 7. It would be a waste of energy to per- form the operations at each step; by waiting until the last step numerous cancellations simplify the computation. Exercises. 1. If 10 yds. of cloth yd. wide cost $6.25, how much will 15 yds. of that cloth 1 yd. wide cost at the same rate per sq. yd. ? 2. How long will it take 12 men to do a piece of work which 8 men can do in 54 da. ? 3. If a 5-ct. loaf of bread weighs 1.5 Ibs. when wheat is 75 cts. per bu., what should it weigh when wheat is $1.00 per bu.? 4. If 5 compositors in 16 da. of 10 hrs. each can set up 20 sheets of 24 pages each, 40 lines to a page and 40 letters to a line, in how many days of 8 hrs. each can 10 compositors set up a volume com- posed of 40 sheets of 16 pages to the sheet, 60 lines to a page and 50 letters to a line ? 48 HIGHER ARITHMETIC. 3. The simple equation. Few mathematicians now assert that the distinction between arithmetic and algebra lies in the use of letters as symbols of quantity. It is impossible to study exhaustively the science of number without using literal notation. The simple equation is now used in all of the grades of many grammar schools, and it is no innova- tion to suggest its use in a work of this nature. No more difficult equation is necessary than one of the following type : 1. Given ax + b = c, to find x. 2. ax = c 6, by subtracting 6 from these equals. Ax. 3 3. x = by dividing these equals by a. Ax. 7 4. Check : Putting - - f or x in step 1, a h b = c. a a/ A single illustration may be given : What sum gaining 0.06J of itself in a year amounts to $157.50 in 2 yrs. ? 1. Let x = the sum. 2. 2 O.OGi = 0.12|. 3. .-. x + 0.12z = $157.50. 4. .-. 1.12ix = $157.50. 5. .-. x = $140. Exercises. 1. What number is that which divided by 17 equals 2.1? 2. Divide 10 into two parts such that twice one part equals 3 times the other. 3. A fulcrum is to be placed under a 3-ft. lever so as to divide it into two parts such that 1.2 times the first shall equal 4.8 times the second ; how far is it from either end ? 4. Alcohol as received in the laboratory is 0.95 pure ; how much water must be added to a gallon of this alcohol so that the mixture shall be half pure ? 5. Air is composed of 21 volumes of oxygen and 79 volumes of nitrogen ; if the oxygen is 1.1026 times as heavy as air, the nitrogen is what part as heavy as air ? CHECKS. 49 IV. CHECKS. A good computer checks his work at every step, and the student who does this has no need of the printed answers to problems involving only numerical calculations. Checks have already been given for the fundamental operations, the most valuable one being that of casting out nines. One other deserves especial mention in tins con- nection : Always form a rough estimate of the answer before beginning a solution. Beach as close an approximation as possible in a short time. This will be found to check most large errors and to do away with the absurd results often given by careless students. E.g., if the interest on $475 for 1 yr. at 4|% is required, the student should at once think that it is a little less than half the inter- est on $1000, that is, a little less than half of $45 ; he might therefore make the estimate $20. The interest is really $21.38. In the following exercises form a rough estimate of the answers and write down the approximations. Then solve. Exercises. 1. At 12| cts. a pound, how much will 2f Ibs. of cheese cost ? (In solving note that 12 = J-f^.) 2. At 37| cts. a yard, how much will 13 yds. of cloth cost ? (In solving note that 37| f of 100.) 3. At $3.50 a barrel, how much will 68 bbls. of flour cost ? 4. At $1.70 a barrel, how much will 126 bbls. of apples cost ? 5. At 45 cts. a yard, how many yards of cloth can be bought for $6.75 ? 6. If a person's taxes are 5.8 mills on $1, how much will they be on $8500 ? 7. If 41 qts. of water weigh as much as 51 qts. of alcohol, and 1 qt. of water weighs 2.2 Ibs., how much will 1 qt. of alcohol weigh ? 8. Bronze contains by weight 91 parts of copper, 6 of zinc, and 3 of tin ; how many pounds of each in 700 Ibs. of bronze ? 9. In drawing a picture of a tower which is 160 ft. high and 35 ft. in diameter, the diameter is to be represented by 5 in. ; by how many inches should the height be represented ? 50 HIGHER ARITHMETIC. Exercises. 1. How much water must be added to a 5% solution of a certain medicine to reduce it to a 1% solution ? 2. If sound travels 5450 ft. in 5 sees, when the temperature is 32, and if the velocity increases 1 ft. per sec. for every degree that the temperature increases above 32, how far does sound travel in 8 sees, when the temperature is 70? 3. 84J qts. of water are drawn through a pipe every 4 mins. from a tank containing 237 qts. ; how many minutes will it take to empty the tank, supposing the water to continue to run at the same rate ? 4. The total debt of the United States government Jan. 1, 1897, was about $1,785,412,641, and the estimated population on that day was 74,036,761 ; what was the debt per capita ? 5. A clock is set on Monday at 7 A.M.; on Tuesday at 1 P.M., cor- rect time, it is 3 mins. slow ; how many minutes will it be behind at 7 A.M., correct time, on Saturday ? 6. If a railroad charges $18 for transporting 12,000 Ibs. of goods 360 mi., how much ought to be charged for transporting 15,000 Ibs. of goods 280 mi. at the same rate ? 7. If in. on a map corresponds to 7 mi. of a country, what dis- tance on the map represents 20 mi. ? 8. How much pure alcohol must be added to a mixture of alcohol and $ water, so that ^ of the mixture shall be pure alcohol ? 9. If 40 pupils use 6 boxes of crayons, 200 in a box, hi 3 mo., how many boxes, 150 in a box, will be required, at the same rate, to supply 75 pupils for 2 mo. ? 10. If the velocity of electricity is 288,000 mi. per sec., how long will it take electricity to travel around the earth, 24,900 mi. ? 11. The respective rates per sec. at which sound travels through air, water, and earth are approximately 1130 ft., 4700 ft., and 7000 ft. ; at these rates, in what time could sound be transmitted a distance of 6 mi. through each of these media ? (1 mi. = 5280 ft.) 12. A certain sum of money gains of itself, the total amount then being $728 ; what is the sum gained ? 13. One of the trains on the Caledonian railway from Carlisle to Stirling, 117f mi., makes the run in 124 mins.; the "Empire State Express" makes the run from Syracuse to Rochester, 80 mi., in 84 mins. ; what is the average rate of each per hr. ? 14. The total debt of the various states and territories at the time of the eleventh census was $1,135,210,442, which was $18.12 per capita; compute the total population at that time, correct to 1000. CHAPTER VI. Measures. THE earlier business arithmetics contained a large num- ber of tables of measures, a necessity when the world was divided into relatively small states, each with its own system of coinage, weights, etc. As an example of the number of tables in use in a single country, there were nearly four hundred ways of measuring land in France at the close of the eighteenth century. Moreover, certain trades adopted special measures, thus adding to the con- fusion. The result is seen in the tables of Troy, avoirdu- pois and apothecary weights, the wine, beer, apothecary and common measures of capacity, besides numerous special units practically obsolete in general business in America, as the stone, long hundredweight, etc. For common use to-day, only a few tables are needed. If one is to enter some trade which continues to use special measures, as that of druggist, the tables should be learned at that time as part of his technical education. Similarly, in an exchange office one must learn a consider- able number of money systems, but for general information three or four suffice. The problems here set for review require only those tables in general use in business or in the sciences. The tables on pp. 52 and 53 are inserted chiefly for reference. They include those which the student will most need to review. The metric tables are given on pp. 61 and 62. 52 HIGHER ARITHMETIC. TABLES OF COMMON MEASURE NEEDING REVIEW. COUNTING BY 12. 1 dozen (doz.) = 12. 1 gross (gro.) = 12 2 . 1 great gross (gt. gro.) = 12 3 . COUNTING SHEETS OF PAPER. 24 sheets = 1 quire. 20 quires, or 480 sheets = 1 ream. COMMON MEASURES OF LENGTH. 12 inches (in.) = 1 foot (ft.). 3 feet = 1 yard (yd.). 6i yards, or 16|ft.= 1 rod (rd.). 320 rods, or 5280 ft. = 1 mile (mi. ). SURVEYORS' MEASURES OF LENGTH. 7.92 inches = 1 link (li.). 100 links = 1 chain (ch.). 80 chains = 1 mile. MISCELLANEOUS MEASURES OF LENGTH. 4 inches = 1 hand. 6 feet = 1 fathom. 1.15 miles, nearly, = 1 knot, or 1 nautical or geographical mile. LIQUID MEASURE. CAPACITY. 4 gills (gi.) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.) = 231 cubic inches. Barrels and hogsheads vary in size. DRY MEASURE. CAPACITY. 2 pints = 1 quart. 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.) = 21 50.42' cu. in. AVOIRDUPOIS WEIGHT. 16 ounces (oz.) = 1 pound (lb.). 100 pounds = 1 hundredweight (cwt.). 2000 pounds = 1 ton (t.). 2240 pounds = 1 long ton, little used in America except in whole- sale transactions in mining products, and not generally there. COMMON MEASURES OF SURFACE. 144 square inches (sq. in.) = 1 square foot (sq. ft.). 9 square feet = 30 J square yards = 160 square rods 640 acres = 1 mile square = square yard (sq. yd.). square rod (sq. rd.). acre (A.). square mile (sq. mi.). section. 36 square miles = 1 township. 100 square feet = 1 square (of roofs, etc.). MEASURES. 53 CUBIC MEASUKE. 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.). 128 cubic feet = 1 cord. The word "cord" is generally used, however, to mean a pile of wood 8 ft. long and 4 ft. high, the price depending (other things being the same) on the length of the stick. 1 cubic yard = 1 load (of earth, etc.). 24f cubic feet = 1 perch. ENGLISH MONEY. 12 pence (d.) = 1 shilling (s.) = $0.243 +. 20 shillings = 1 pound () - $4.8665. FRENCH MONEY. 100 centimes = 1 franc (fr.) =.$0.193. The French system is also used in several other countries, as in Belgium, Switzerland, Italy, etc., but the names are not uniform, in Italy, for example, the franc being called a lira. GERMAN MONEY. 100 pfennigs = 1 mark (M.) = $0.238. APOTHECARIES' WEIGHT. 20 grains (gr.) = 1 scruple (sc. or 9). 3 scruples = 1 dram (dr. or 5)- 8 drams = 1 ounce (oz. or ). 12 ounces = 1 pound (lb.). 6760 grains = 1 pound. The table of apothecaries' weight is used in selling drugs at retail. TROY WEIGHT. 24 grains (gr.) = 1 pennyweight (pwt. or dwt.). 20 pennyweights = 1 Troy ounce. 12 Troy ounces = 1 Troy pound. 437.5 grains = 1 Avoirdupois oz. 7000 grains = 1 Avoirdupois lb. 480 grains = 1 Troy oz. 5760 grains = 1 Troy lb. Troy weight is used for precious metals. 54 HIGHER ARITHMETIC. I. COMPOUND NUMBERS. When a concrete number is expressed in several denomi- nations it is called a compound number. E.g., 3 ft. 6 in. But 3.5 ft. and -$2.25 are not compound numbers. Reduction of compound numbers is a process so familiar to the student that two examples will satisfy for illustra- tion. (1) Reduction descending. Eeduce 365 da. 5 hrs. 48 mins. to minutes. Explanation and solution. 1. v 1 da. = 24 hrs. 2. .-. 365 da. = 365 X 24 hrs. = 8760 hrs. 3. 8760 hrs. + 5 hrs. = 8765 hrs. 4. v 1 hr. = 60 mins. 5. .-. 8765 hrs. = 8765 X 60 min. = 525,900 mins. 6. 525,900 mins. + 48 mins. = 525,948 mins. Practical calculation. (The notes in parentheses explain the opera- tions.) 365 (Multiply by 3)= 1095 (For 24 = 3 X 8.) (Multiply by 8 and add 5)= 8765 (i.e., 365 X 24 hrs. + 5 hrs.) (Multiply by 60 and add 48)= 525,948 (i.e., 8765 X 60 mins. +48 mins.) (2) Reduction ascending. Eeduce 525,948 mins. to days, etc. Explanation and solution. 1. v 1 min. = ^ hr. 2. .-. 525,948 mins. = 525,948 X ^ hr. = 8765 hrs. and ff hr. or 48 min. 3. v 1 hr. = J da. 4. .-. 8765 hrs. = 8765 X J da. = 365 da. and ^ da., or 5 hr. 5. .-. 365 da. 5 hrs. 48 mins. Practical calculation. (The notes in parentheses explain the opera- tions.) 60|525948 24 1 8765 48 (525,948 X ^ hr.) 365 5 (8765 X ^ da.) COMPOUND NUMBERS. 55 Exercises. 1. Eeduce 43 wks. 5 hrs. 49 mins. 57 sees, to seconds. 2. Keduce 4,568,657 sees, to weeks, days, etc. 3. Reduce 625 cu. yds. 19 cu. ft. 1609 cu. in. to cubic inches. 4. Reduce 1,847,638 ft. to miles, yards, and feet. 5. Reduce 25 mi. 459 yds. 31 in. to inches. 6. Reduce 12,563,257 sq. in. to acres, square yards, etc. 7. Reduce 5 gals. 3 pts. to pints. . 8. Reduce 341 qts. to gallons. 9. Reduce 150 Ibs. to ounces. (Avoirdupois.) 10. Reduce 274 oz. to pounds and ounces ; to pounds and decimals of a pound. (Avoirdupois.) 11. Reduce 36 gt. gro. to gross; to units. 12. Reduce 15 reams to quires ; to sheets. 13. Reduce 142,872 sheets to quires ; to reams. 14. Reduce 19,436 cu. ft. to cubic yards. 15. Reduce 2 wks. 2 da. 19.2 hrs. to the fraction of a month of 4 wks. 16. Reduce 14 hrs. 15 mins. to the fraction of 3 da. 17. Reduce T 8 T pt. to the fraction of a gallon. 18. Reduce 3 qts. 1 pt. to the decimal of a gallon. 19. Reduce 18 hrs. 30 mins. 30 sees, to the fraction of a week. Compound addition and subtraction differ so little in theory from the addition and subtraction of simple abstract numbers as to require 9 Ibs. 15 oz. no extended review. The cases arising 10 12 in actual practice rarely involve more _8 6 than two denominations, the tendency 29 1 being to reduce the lower units to deci- mals of the higher. Thus, while it was 29 Ibs. 1 oz. formerly not unusual to add numbers 10 12 like 27 rds. 5 yds. 2 ft. 11 in., it is now 18 5 more common to deal with numbers like 463.4 ft., 1.27 A., 4.345 mi., etc. In the annexed example in addition the computer should say, "6, 18, 33, 1 ; 2, 10, 20, 29." In the example in subtraction he should proceed as with simple numbers, remembering that 16 oz. = 1 lb., and should say, " 12 and 5 are 17 ; 11 and 18 are 29." 56 HIGHER ARITHMETIC. Exercises. 1. What check should be used in compound addi- tion ? in compound subtraction ? 2. Add 9 Ibs. 7 oz., 52 Ibs. 6 oz., 91 Ibs. 12 oz., 7 Ibs., 5 Ibs. 2 oz., 13 oz. (Avoirdupois.) 3. Add 13 t. 450 Ibs., 12 t. 700 Ibs., 342 t., 44 1. 1500 Ibs., 1200 Ibs. 4. Add 25 gals. 3 qts., 47 gals. 2 qts. 1 pt., 15 gals. 1 qt., 1 pt., 9 gals. 5. Add 5 yds. 2 ft., 6 yds. 1 ft. 7 in., 9 yds. 2 ft. 5 in., 1 ft. 6. Add 6 bu. 3 pks., 9 bu. 2 pks., 5 bu. 1 pk., 3 pks. 7. Add 10 da. 5 hrs. 42 mins. 7 sees., 23 hrs. 10 mins. 2 sees., 11 da. 4 mins., 5 hrs. 4 mins. 5 sees., 1 da. 15 sees. 8. Add 7 mo. 15 da., 5 mo. 14 hrs. 3 sees., 1 mo. 7 da. 5 mins. 57 sees., 2 mo. 54 mins., 9 hrs., 7 da., 2 mo. 9. Add 12 mi. 3 rds. 2 ft., 3 mi. 75 rds. 10 ft., 4 mi. 12 ft., 3 rds. 6ft. 10. From 13 Ibs. 9 oz. subtract 9 Ibs. 10 oz. (Avoirdupois.) 11. From 7 mo. 9 da. subtract 5 mo. 15 da. 12. From 25 gals. 2 qts. subtract 10 gals. 3 qts. 1 pt. 13. From 5 yds. 1 ft. 7 in. subtract 2 yds. 2 ft. 10 in. 14. From 6 bu. 2 pks. subtract 5 bu. 3 pks. 15. From 7 da. 5 hrs. 27 mins. 42 sees, subtract 5 da. 5 hrs. 27 mins. 43 sees. 16. From 87 cu. yds. 8 cu. ft. 924 cu. in. subtract 35 cu. yds. 23 cu. ft. 1688 cu. in. Compound multiplication. The remarks already made concerning practical problems in addition and subtraction apply with equal force to multiplication and division. The general theory is evident from the analysis of the following problem. Required the product of 10 X 3 Ibs. 4 oz. CALCULATION. Analysis. 1. 10 X 4 oz. = 40 oz. 2 Ibs. 8 oz. 3 Ibs. 4 oz. 2. 10 X 3 Ibs. = 30 Ibs. 10 3. 30 Ibs. + 2 Ibs. 8 oz. = 32 Ibs. 8 oz. 32 Ibs. 8 oz. Compound division. The definition of division, already given on p. 29, should now be recalled for the purpose of distinguishing between the two general cases. COMPOUND NUMBERS. 57 Division is the operation of finding one of two factors, the quotient, when the product and the other factor are given. Hence, there are two general cases illustrated by the following example. Since $10 is the product of the factors 2 and $5, 1. .-. $10 -T- $5 = 2, the idea of measuring, being contained in, con- tinued subtraction. That is, $5 is contained in $10 2 times. 2. $10 -r 2 = $5, the idea of separation, partition, multiplication by a fraction. That is, $10 divided by 2 equals $5, $10 has been separated into two parts. 1. When dividend and divisor are compound numbers of the same kind, they may be reduced to the same denomination and the division performed in the ordinary way. For example, how many times does 32 Ibs. 8 oz. contain 3 Ibs. 4 oz. ? In this case it is more simple to reduce to pounds, thus : 1. 32 Ibs. 8 oz. = 32.5 Ibs. 2. 3 Ibs. 4 oz. = 3.25 Ibs. 3. 32.5 Ibs. -r 3.25 Ibs. = 10. But it is usually easier to reduce to one of the lower denominations, especially where more than two denominations are involved. For example, how many times does 29 t. 87 Ibs. 2 oz. contain 3 t. 4 cwt. 54 Ibs. 2 oz. ? 1. 29 t. 87 Ibs. 2 oz. = 58087.125 Ibs. 2. 3 t. 4 cwt. 54 Ibs. 2 oz. = 6454.125 Ibs. 3. 58087.125 Ibs. -f 6454.125 Ibs. = 9. II. When the divisor is an abstract number. For example, divide 29 t. 87 Ibs. 2 oz. by 9. CALCULATION. 9 ) 29 t. 87 Ibs. 2 oz. 3 t. 454 Ibs. 2 oz. Analysis. 1. 29 t. -J- 9 = 3 t., and 2 t., or 4000 Ibs., remainder. 2. 4000 Ibs. + 87 Ibs. = 4087 Ibs. 3. 4087 Ibs. -r 9 = 454 Ibs., and 1 lb., or 16 oz., remainder. 4. 16 oz. + 2 oz. = 18 oz. 5. 18 oz. -r 9 = 2 oz. 6. .-. 3 t. 454 Ibs. 2 oz. 58 HIGHER ARITHMETIC. Exercises. 1. Multiply 27 gals. 3 qte. 1 pt. 3 gi. by 36, checking by division. 2. Also by 236, checking by division. 3. Multiply 17 wks. 4 da. 13 hrs. 27 mins. 36 sees, by 9, checking by division. 4. Also by 79, checking by division. 5. Multiply 23 cu. yds. 6 cu. ft. 459 cu. in. by 8, checking by division. 6. Multiply the result in Ex. 5 by 9, checking by division. 7. Multiply 512 rds. 2 yds. 2 ft. 2 in. by 6, checking by division. 8. Multiply 2 sq. yds. 3 sq. ft. 9 sq. in. by 10, checking by division. 9. Divide 878 wks. 4 da. 15 hrs. 37 mins. 36 sees, by 56, checking by multiplication. 10. Divide 4285 cu. yds. 6 cu. ft. 1689 cu. in. by 23, checking by multiplication. 11. Also by 85, checking by multiplication. 12. Divide 5863 gals. 3 qts. 1 pt. 3 gi. by 8, checking by multiplica- tion. 13. Also by 75, checking by multiplication. 14. How many jars, each containing 2 gals. 3 qts. 1 pt. 3 gi., can be filled from a cask containing 285 gals. ? Check the result by multi- plication. 15. Divide 346 da. 18 hrs. 34 mins. 32 sees, by 1 da. 7 hrs. 45 mins. 56 sees., checking the result by multiplication. 16. A carriage wheel revolves 3 times in going 11 yds. ; how many times will it revolve in going of a mi. ? 17. If 277,280 cu. in. of water weigh 10,000 Ibs., how many cubic feet (approximately) will weigh 1000 oz.? 18. If a clock gains 12 mins. a day, what is the average gain per min. ? 19. Supposing the distance traveled by the earth about the sun to be 596,440,000 mi., what is the average hourly distance traveled, taking the year to equal 365J da. ? 20. Supposing the distance from the earth to the sun to be 91,713,000 mi. and that the sun's light reaches the earth in 8 mins. 18 sees., what is the velocity of light per sec. ? 21. From the data of Ex. 20 and that on p. 3, find how long it would take the sun's light to reach Neptune. Express the result in hours, minutes, and seconds. THE METRIC SYSTEM. 59 II. THE METRIC SYSTEM. Soon after the opening of the nineteenth century, France legalized a uniform system of measures generally known as the Metric System. This is now used in practical business by most of the highly civi- H E lized nations, except the United States and J E England and her dependencies. In scientific work it is generally used by all countries, and there is every reason to believe that it will also become universal in business. a Units. The system is based on the unit of * length, called the meter (meaning measure), f which is 0.0000001 (or 10~ 7 ) of the distance a | from the equator to the pole. 5: | The unit of capacity is the liter, a cube 0.1 of S * a meter on an edge. 2. The unit of weight is the gram, the weight of cube of water 0.01 of a meter on an edge. Through an error in fixing the original units, they are not exactly as stated, but the system loses none of g its practical advantages on this account. The original | units are preserved at Paris. The prefixes set forth on p. 60 must be thoroughly g memorized, after which the metric system offers few difficulties. Some of these prefixes are never used with g. certain units in practice, just as the only units generally g. used in speaking of United States money are dollars and ^ cents. We never say, "4 eagles 2 dollars 5 dimes and - 3 cents " for $42.53. So in the metric system the myria- liter and kiloliter are never used, and the dekameter and hektometer rarely. In the following tables, the units most commonly used are, therefore, printed in bold-faced type. The abbreviations of the metric system are not uniform even in France. Those here given have been adopted by the International Committee of Weights and Measures and by other international asso- ciations, and are therefore given as the most approved now in use. 60 HIGHER ARITHMETIC. THE PREFIX MEANS myria- 10000 1000 100 TABLES. AS IN WHICH MEAXS kilo- g hekto- deka- .2 deci- g >3 centi- | milli- Greek. mikrO- 10 1 0.1 0.01 0.001 myriameter 10000 kilogram hektoliter dekameter decimeter centigram millimeter 0.000001 mikrometer 0000 meters. 1000 grams. 100 liters. 10 meters. 1 0.1 of a meter. 0.01 of a gram. 0.001 of a meter. 0.000001 of a meter. TABLE OF LENGTH. A myriameter = 10,000 meters. A kilometer (km) = 1000 " A hektometer = 100 " A dekameter = 10 " Meter (m) A decimeter (dm) = 0.1 of a meter. A centimeter (cm) = 0.01 u A millimeter (mm) = 0.001 " A mikron 0") 0.000001 " TABLE OF SQUARE MEASURE. A square myriameter = 100,000,000 square meters. " kilometer (km 2 ) = 1,000,000 " " hektometer " dekameter = Square meter (m 2 ) A square decimeter (dm 2 ) = " centimeter (cm 2 ) = " millimeter (mm 2 ) = 10,000 100 0.01 of a square meter. 0.0001 0.000001 The square dekameter is also called an are; and since there are 100 dm 2 in 1 hm 2 , a square hektometer is called a hektare. These are used in measuring land. TABLE OF CUBIC MEASURE. A cubic myriameter = 10 12 cubic meters. " kilometer 10 9 " " hektometer =1,000,000 " " dekameter 1000 Cubic meter (m 3 ) THE METRIC SYSTEM. 61 A cubic decimeter (dm 3 ) = 0.001 of a cubic meter. " centimeter (cm 3 ) = 0.000001 " " millimeter (mm 3 ) = 0.000000001 " The cubic meter is also called a stere, a unit used in measuring wood. TABLE OF WEIGHT. A metric ton (t) = 1,000,000 grams. A quintal (q) = 100,000 " A myriagram = 10,000 " A kilogram (kg) = 1000 " A hektogram 100 " A dekagram = 10 " Gram (g) A decigram 0. 1 of a gram. A centigram (eg) = 0.01 " A milligram (mg) = 0.001 " A mikrogram (7) = 0.000001 " The metric ton is the weight of 1 m 3 of water ; the kilogram of 1 dm 3 or 1 liter of water ; and the gram of 1 cm 3 of water. TABLE OF CAPACITY. A hektoliter (hi) = 100 liters. A dekaliter =10 " Liter (1) A deciliter = 0. 1 of a liter. A centiliter =0.01 " A milliliter (ml) = 0.001 " A mikroliter (X) = 0.000001 " TABLE OF EQUIVALENTS. In general, the metric system is used by itself, as in scientific work, and the common English-American system by itself. Hence, there is little demand for reducing from one to the other. Such reductions are, however, occasionally necessary, and hence a few of the common equivalents are here given. These equivalents are only approximate. A meter = 39.37 inches = 3 feet nearly. A liter = 1 quart nearly. A kilogram = 2.2 pounds nearly. A kilometer = 0.62 of a mile = 0.6 of a mile nearly. A gram = 15.43 grains = 15| grains nearly. A hectare = 2.47 acres = 2| acres nearly. 62 HIGHER ARITHMETIC. Oral Exercises. 1. What is the meaning of hekto- ? myria- ? centi-? kilo-? deci- ? deka- ? milli- ? mikro- ? 2. What is the prefix which means 10,000? 0.1? 10? 100? 0.01? 1000? 0.001? 3. About when was the metric system established ? Where ? How extensively is it used at present, (a) in business, (6) in science ? What are its advantages over the older systems ? 4. How was the length of the meter fixed ? How was the liter fixed ? the gram ? 5. What is the weight of a liter of water ? (This is only approxi- mate and it refers to distilled water at its maximum density.) 6. What is the weight of a cubic centimeter of water ? of a cubic decimeter ? of a cubic meter ? 7. How many mm in a km ? in a hektometer ? in a myriameter ? 8. How many cm 2 in a m 2 ? in a km 2 ? 9. How many mm 3 in a cm 3 ? in a liter ? in a m 3 ? 10. How many g in 125 kg ? in a metric ton ? 11. How many dm 3 in 5 steres ? in a cubic dekameter ? 12. Reduce 17 km to m ; to mm ; to dm ; to /*. 13. Reduce 5 dekaliters to 1 ; to cm 3 ; to X. 14. Reduce 300 ha to a ; to m 2 ; to km 2 . 15. Reduce 45,000 m 2 to ha ; to a ; to a fraction of a km 2 . 16. Reduce 0.573 m 2 to cm 2 ; to mm 2 . 17. Reduce 15 km 2 to m 2 ; to cm 2 ; to ha. 18. Reduce 27 m 3 to dm 3 ; to mm 3 ; to 1. 19. 25 kg are how many Ibs. , to the nearest unit ? 20. 300 km are how many mi. , to the nearest unit ? 21. 65 1 are how many qts., to the nearest unit ? 22. 30 ha are how many acres, to the nearest unit ? 23. 50 acres are how many ha, to the nearest unit ? 24. 20 qts. are how many 1, to the nearest unit ? 25. 50 mi. are how many km, to the nearest unit ? 26. 220 Ibs. are how many kg, to the nearest unit ? 27. 325 ft. are how many m, to the nearest unit ? 28. The Eiffel tower at Paris is 300 m high ; this is about how many feet ? 29. The papers report the rainfall at Berlin, for a given period, to be 11.0 cm ; this is how many inches, to the nearest tenth? 30. What is the pressure in grams per cm 2 of a column of water 1 m deep ? THE METRIC SYSTEM. 63 Written Exercises. 1. The length of a wave of sodium light is 5893 X 10" 8 cm j how many such wave-lengths in 1 m ? in 1 /* ? 2. Cast copper being 8.8 times as heavy as an equal volume of water, what is the weight of 5 dm 3 ? 3. A stream flowing uniformly 1 km per hr. flows how many cm per sec. ? 4. A liter of mercury weighs 13.596 kg ; how many mm 3 of mer- cury weigh 1 g ? 5. A man takes 120 steps in walking 100 m ; what is the average length of each step ? 6. At the rate given in Ex. 5, how many steps will be taken in walking 6.2 km ? 7. How many mm 2 in dm 2 ? in of a decimeter square ? 8. Air being 0.001276 as heavy as an equal volume of water, what is the weight of air in a room containing 600 m 3 ? 9. A train traveling 1 km per min. travels how many m per sec. ? 10. Sound travels 332 m per sec. ; how long will it take it to travel 1 km ? Answer to 0.01 sec. 11. Granite being 2.7 times as heavy as water, what is the weight of a block containing 2.50 m 3 ? 12. Show that an inch is nearly 2.54 cm, and use this equivalent in Exs. 13, 14. 13. Express the following readings from a barometer in centi- meters : 29.9 in., 30.0 in., 30.1 in., 30.2 in. 14. Also the following in inches, to the nearest 0.1: 71.119cm, 73.659 cm, 74.929 cm. 15. Olive oil being 0.92 as heavy as an equal volume of water, and petroleum 0.7, and alcohol 0.83, what is the weight of a liter of each ? 16. The distance from Paris to Rouen is 136 km ; the prices of tickets are, 1st class 15.25 francs, 2d class 11.40 francs, 3d class 8.40 francs ; what is the price for each class per kilometer ? 17. The United States government lays down these regulations concerning foreign mail : letters weighing 15 g or less require 5 cts.; packets of samples of merchandise may be sent not exceeding 350 g in weight, 30 cm in length, 20 cm in width, and 10 cm in height ; express these measures in the common units. 18. The Eiffel tower at Paris is 300 m high and cost about 5,000,000 francs; if enough 20-franc gold pieces each 1J mm thick could be piled one above another to equal this sum, would the pile equal the height of the tower ? 64 HIGHER ARITHMETIC. BOrUNG POINT OF WATER 212 FREEZING POINT 32 III. MEASURES OF TEMPERATURE. Temperature is ordinarily measured by a thermometer FAHRENHEIT using one of two scales, (a) the Fahrenheit, used in this country in ordi- nary business, or (&)the Centigrade, quite gene- rally used in science. The Fahrenheit (Fah.) scale was suggested by Fahrenheit in the early part of the eighteenth cen- tury. He took for a tem- perature which he obtained by mixing ice and salt, and for 212 the boiling point FIG. i. of water, thus bringing the freezing point at 32, a very unscientific arrangement. The Centigrade (C., from Latin centum, hundred, and gradus, degree) scale was adopted by Celsius in 1742, and is often called by his name. It places at the freezing, and 100 at the boiling point of water. Degrees above on each scale are usually indicated by the sign + ; below, by the sign . To reduce from Fahrenheit to Centigrade. V 212 Fah. 32 Fah., or 180 Fah. = 100 C., 100 C. 5 C. 9 r It must also be remem- bered that the freezing point of water is at -f- 32 Fah. E.g., + 80 Fah. corresponds to what temperature C. ? 1. + 80 Fah. = + 80 Fah. - 32 Fah. above freezing, or 48 Fah. above freezing. KO r* 2. lFah. =^, '1 C 3. .-. 48 Fah. = 48 X - = 26.67 C. MEASURES OF TEMPERATURE. 65 Also, 10 Fah. corresponds to what temperature C. ? 1. - 10 Fah. - 32 Fah. = - 42 Fah. KO p 2. lFah. = !L ward (positive pressure) ; A ' | A in the second it is upward l v (negative pressure). There is a law in physics that, if p', w' represent the number of units of distance AF, FB, respectively, and p, w the number of units of pressure and weight, respectively, then *-*-, = 1. In the first figure p, w, p', w' are all considered as positive ; in the second figure p is considered as negative because the pressure is upward, and w' is considered as negative because it extends the other way from F. Hence, the ratio pp' : ww' = 1 in both cases. Example. Suppose AF = 25 in., FB = 14 in., in the first figure ; what pressure must be applied at A to raise a weight of 30 Ibs. at B ? 1. By the law of levers ^ = 1. 2. .-. p = 16.8, and .-. the pressure must be 16.8 Ibs. Exercises. 1. Two bodies weighing 20 Ibs. and 4 Ibs. balance at the ends of a lever 2 ft. ong ; find the position of the fulcrum. 2. The radii of a wheel and axle are respectively 4 ft. and 6 in. ; what force will just raise a mass of 56 Ibs., friction not considered ? 3. What pressure must be exerted at the edge of a door to counter- act an opposite pressure of 100 Ibs. halfway from the hinge to the edge ? one-third of the way from the hinge to the edge ? 4. The length of the spoke of a capstan is 6 ft. measured from the axis, and the radius of the drum is 1 ft. ; find the weight of an anchor that can just be raised by 6 men, each exerting a force equal to 100 Ibs. at the end of a spoke, friction not considered. 5. In each figure, what must be the distance AF in order that a pressure of 1 kg may raise a weight of 100 kg 3 dm from F ? RATIO. 95 (c) Boyle's law. It is proved in physics that if p is the number of units of pressure of a given quantity of gas, and v is the number of units of volume, then p varies inversely as v when the temperature remains constant. This law was discovered in the seventeenth century by Robert Boyle. E.g., if the volume of a gas is 10 dm 3 under the ordinary pressure of the atmosphere (" under a pressure of one atmosphere "), it is as much when the pressure is n times as great, n times " " " " " " - " n the temperature always being considered constant. Example. A toy balloon contains 3 1 of gas when exposed to a pressure of 1 atmosphere ; what is its volume when the pressure is increased to 4 atmospheres ? decreased to of an atmosphere ? 1. v the volume varies inversely as the pressure, it is i as much when the pressure is 4 times as great. 2. Similarly, it is 8 times as much when the pressure is i as great. 3. /. the volumes are 0.75 1 and 24 1. Exercises. 1 . Equal quantities of air are on opposite sides of a piston in a cylinder that is 12 in. long ; if the piston moves 3 in. from the center, find the ratios of the pressures. Draw the figure. 2. If a cylinder of gas under a certain pressure has its volume increased from 20 1 to 25 1, what is the ratio of the pressures ? 3. A cubic foot of air weighs 570 gr. at a pressure of 15 Ibs. to the square inch ; what will a cubic foot weigh at a pressure of 10 Ibs. ? 4. A mass of air occupies 18 cu. in. under a pressure of 1\ Ibs. to the square inch; what space will it occupy under a pressure of 25 Ibs. ? 5. If the volume of a gas varies inversely as the height of the mer- cury in a barometer, and if a certain mass occupies 23 cu. in. when the barometer indicates 29.3 in., what will it occupy when the barometer indicates 30.7 in. ? 6. A liter of air under ordinary pressure weighs 1.293 g when the barometer stands at 76 cm ; find the weight when the barometer stands at 82 cm, the weight varying inversely as the height of the barometer. 7. A certain gas has a volume of 1200 cm 3 under a pressure of 1033 g to 1 cm 2 ; find the volume when the pressure is 1250 g. HIGHER ARITHMETIC. II. PROPORTION. A proportion is an expression of equality of ratios. Thus, | = |, H = ^= ^, $3.50 : $7 = 4 books : 8 books, 3/4 = $6/$8, are examples of proportion. There is another symbol formerly much used to express the equality of ratios, the double colon (: :). There may be an equality of several ratios, as 1:2 = 4:8 = 9 : 18, the term continued proportion being applied to such an expression. There may also be an equality between the products of ratios, as f f = \ - *-, such an expression being called a compound proportion. Arithmetic uses continued proportion but little, and problems in compound proportion are more easily solved by the unitary analysis. In the proportion a:b = c:d, a, b, c, d are called the terms, a and d the extremes, and b and c the means. If three of the four terms of a proportion are known, the other can be readily found by multiplying or dividing equals by equals. Thus, x 4 7 *4 if - = - then x = -r- = 3.5. If this should seem to require multi- 7 o o plying by a concrete number, the difficulty may be avoided by making $4 4 each term of either ratio abstract; this is permissible because S = ' Exercises. 1. Given r = -> the terms being abstract numbers, to prove that the product of the means equals the product of the extremes. 2. In the proportion - = - prove that x = bc/d ; that is, that one extreme equals the product of the means divided by the other extreme. 3. In which of these proportions is the value of x the more easily . x 17 23 5 found, and why? - = -,_ = -. x 7 4. Find the value of x, correct to 0.1, in the proportions - = z:3 = 4:5, 0.3:7 =x : 1.52, ^ = y' 5. Also in the proportions x : 1.273 = 0.4 : 2.3, 1.7 : 3 = x : 7. PROPORTION. 97 If one quantity varies directly as another, the two are said to be directly proportional, or simply proportional. E.g., at retail the cost of a given quality of sugar varies directly as the weight ; the cost is then proportional to the weight. Thus, at 4 cts. a pound 12 Ibs. cost 48 cts., and 4 cts. : 48 cts. = 1 Ib. : 12 Ibs. If one quantity varies inversely as another, the two are said to be inversely proportional. E.g., in general, the temperature being constant, the volume of a gas varies inversely as the pressure, and the volume is therefore said to be inversely proportional to the pressure. Exercises. 1. State which of the following, other things being equal, are directly proportional and which are inversely proportional : (a) Volume of gas, pressure. (6) Volume of gas, temperature. (c) Distance from fulcrum, weight. (d) Cost of carrying goods, distance. (e) Weight of goods carried for a given sum, distance. (/) Amount of work done, number of workers. (g) Price of bread, price of wheat. 2. Given 1.43 : x = 4.01 : 2, find, correct to 0.01, the value of x. . x 63 3. Also in - = 7 x 4. Also in 27 : x = x : 48. The applications of proportion are found chiefly in geometry and physics. While several types of business problems were formerly solved by this means, other methods are now generally employed. In the two illustrative examples on p. 98, the first three steps are explanatory of the statement of the proportion and may be omitted in practice. In the first problem the ratios are written in the fractional form in order that the reasons involved may appear more readily. The symbol for the unknown quantity may be placed in any term and the proportion arranged accordingly ; but if the solution is to be explained, it will be found more con- venient to place the x in the first term. (See p. 96, Ex. 3.) 98 HIGHER ARITHMETIC. Examples, (a) The time of oscillation of a pendulum is propor- tional to the square root of the number representing its length ; the length of a 1-sec. pendulum being 39.2 in., what is the length of a 2-sec. pendulum ? 1. Let x = the number of inches of length. 2. Then ^r = the ratio of the lengths. o9.2 3. And f = the ratio of the corresponding times of oscillations. . 4. v the time is proportional to the square root of the number representing the length, . V39.2 5. ... JL- = i, whence x = 39.2 4 =' 156.8. o9.2 6. v x = the number of inches, .. the pendulum is 156.8 in. long. (b) A mass of air fills 10 dm 3 under a pressure of 3 kg to 1 cm 2 ; what is the space occupied under a pressure of 5 kg to 1 cm 2 , the temperature remaining constant ? (See Boyle's law, p. 95.) 1. Let x the number of dm 3 under a pressure of 5 kg to 1 cm 2 . 2. Then x : 10 = the ratio of the volumes. 3. And 5:3 = the ratio of the corresponding pressures. 4. v the volume is inversely proportional to the pressure, .-. x : 10 = 3 : 5. 5. .-. = 10-3 : 5 = 6. 6. v x = the number of dm 3 , .-. the space is 6 dm 3 . Exercises. 1. The distance through which a body falls from a state of rest is proportional to the square of the number representing the time of fall ; if a body falls 176.5 m in 6 sees., how far does it fall in 3.25 sees. ? 2. Also in 1 sec. ? in 2 sees. ? in 4 sees. ? 3. It is proved in mechanics that, neglecting friction, the power acting parallel to an inclined plane and neces- sary to support a weight is to that weight as the height to which the weight is raised is to the length of the incline. (In the figure, p : w = h : I = h': I'.) If the height is of the length, what power will support a 20-lb. weight (neg- lecting friction in this and similar problems) ? FIG. 18. PROPORTION. 99 4. A train weighing 126 tons rests on an incline and is kept from moving down by a force of 1500 Ibs. ; the road rises 1 ft. in how many feet of its length ? 5. On a plane rising 3 ft. in every 5 ft. of its length, how many pounds of force exerted parallel to the plane will keep a mass of 10 Ibs. from sliding ? 6. What must be the length of an inclined plane in order that a man may roll a 500-lb. cask into a wagon 3.5 ft. high by the exertion of a force of 350 Ibs. ? 7. How long is a pendulum which oscillates 56 times a minute ? 8. If a pipe 1.5 cm in diameter fills a reservoir in 3.25 mins., how long will it take a pipe 3 cm in diameter to fill it ? 9. If a projectile 8.1 in. in length weighs 108 Ibs., what is the weight of a similar projectile 9.37 in. long ? (Answer to 0.1 Ib.) 10. The masses of two solids remaining unchanged, their attraction for each other is inversely proportional to the square of the number representing the distance between their centers of gravity ; if at a distance of 2 m they are attracted by a force of 1 mg, what will be their attraction at a distance of 1 km ? 11. A body weighs 15 Ibs. 5000 mi. from the earth's center (i.e., about 1000 mi. above the surface); how much will it weigh 4000 mi. from the center ? 12. If a body weighs 1 kg at the level of the sea, how much will it weigh at an elevation of 1 km ? (Take the radius as 6370 km ; give the result correct to 0.001 kg.) 13. What is the height of a tower which casts a shadow 143 ft. long at the same time that a post 4.5 ft. high casts a shadow 6.1 ft. long? 14. Of two bottles of similar shape one is twice as high as the other ; the smaller holds 0.5 pt., how much does the larger hold ? 15. The amount of light received on a given surface being inversely proportional to the square of the number representing its distance from the source of light, if the amount per sq. in. at a distance of 1 ft. is represented by 1, what will represent the amount at a distance of 0.1 in. ? (Answer to 0.01.) 16. If the interest received on a certain sum for 1.5 yrs. is $27.50, how much is the interest on the same sum at the same rate for 2 mo.? 17. If the interest received on a certain sum for a certain time is $53 at 6 cts. for every dollar, how much is the interest on that sum for the same time at 4 cts. for every dollar ? 100 HIGHER ARITHMETIC. 18. When the barometer stands at 30 in., the pressure of the atmos- phere is 14.7 Ibs. per sq. in. ; what is the pressure when the barometer stands at 28 in. ? 19. What is the width of a stream if a pole 64 ft. high 9 ft. from its bank casts a shadow which just reaches across the stream and the shadow of a nail in the pole 8 ft. from the ground just touches the bank? 20. A water tower 160 ft. high and 35 ft. in diameter is to be represented in a drawing as 10 in. high ; how many inches in the representation of the diameter ? 21. The Washington monument casts a shadow 223 ft. 6.5 in. when a post 3 ft. high casts a shadow 14.5 in. ; find the height of the monu- ment. 22. If a triangle whose base is 2 in. long has an area of 3 sq. in., what is the area of a similar triangle whose corresponding base is 8 in. long? 23. If a metal sphere 10 in. in diameter weighs 327.5 Ibs., what is the weight of a sphere of the same substance 14 in. in diameter ? 24. A cube of water 1.8 dm on an edge weighs how many kg ? 25. If a sphere whose surface is 16 it cm 2 weighs 5 kg, what is the weight of a sphere of the same substance whose surface is 32 it cm 2 ? 26. If the length of a 1-sec. pendulum be considered as 1 m, what is the time of oscillation of a pendulum 6.4 m long? 62.5 m long? 27. On a map constructed on a scale of T ^oWo tne distance from Detroit to Chicago is 112.86 in. ; how many miles between these cities ? 28. The ratio of immigrants from the United Kingdom to those from the rest of Europe during the decade from 1881 was 5 : 16 ; the total number from these two sources was 6,192,000 ; how many from each ? (Answer correct to 1000.) 29. Kepler showed that the squares of the numbers representing the times of revolution of the planets about the sun are proportional to the cubes of the numbers representing their distances from the sun. Mars being 1.52369 as far as the earth from the sun, and the time of revolution of the earth being 365.256 da., find the time of revolution of Mars. 30. Similarly, find the time of revolution of each of the following planets, the numbers representing relative distances from the sun, the distance to the earth being taken as the unit: Mercury 0.39, Venus 0.72, Jupiter 5.20, Saturn 9.54, Uranus 19.18, Neptune 30.07. PROPORTION. 101 Problems in electricity. The great advance in elec- tricity in recent years renders necessary a knowledge of such technical terms as are in everyday use. Problems involving these terms belong to proportion, but may be omitted without interfering with the subsequent work. When water flows through a When electricity flows through pipe some resistance is offered a wire some resistance is offered, due to friction or other impedi- This resistance is measured in ment to the flow of the water. ohms. An ohm is the resistance offered by a column of mercury 1 mm 2 in cross section, 106 cm long, at C. A certain quantity of water flows through the pipe in a second, and this may be stated in gallons or cubic inches, etc. A certain pressure is necessary to force the water through the pipe. This pressure may be meas- ured in pounds per sq. in., kilo- grams per cm 2 , etc. Hence, in considering the water necessary to do a certain amount of work (as to turn a water-wheel) it is necessary to consider not merely the pressure, for a little water may come from a great height, nor merely the volume, nor merely the resistance of the pipe; all three must be consid- ered. A certain quantity of electric- ity flows through the wire. This quantity is measured in amperes. An ampere is the current neces- sary to deposit 0.001118 g of silver a second in passing through a cer- tain solution of nitrate of silver. A certain pressure is necessary to force the electricity through the wire. This pressure is measured in volts. A volt is the pressure necessary to force 1 ampere through 1 ohm of resistance. Hence, in considering the elec- tricity necessary to do certain work it is necessary to consider not merely the voltage, for a little electricity may come with a high pressure, nor merely the amper- age, nor merely the number of ohms of resistance ; all three must be considered. The names of the electrical units mentioned come from the names of three eminent electricians, Ohm, Ampere, and Volta. If) 2 JSGHEK ARITHMETIC. It is proved in physics that the resistance of a wire varies directly as its length and inversely as the area of a cross section. That is, if a mile of a certain wire has a resistance of 3.58 ohms, 2 mi. of that wire will have a resistance of 2-3.58 ohms, or 7.16 ohms. Also, 1 mi. of wire of the same material but of twice the sectional area will have a resistance of $ of 3.58 ohms, or 1.79 ohms. From these laws and definitions, the most common prob- lems and statements concerning electrical measurements will be understood. The student should not feel obliged, however, to use the proportion form in the solutions. Ordinary analysis, the unitary analysis, or the equation may be employed. Exercises. 1. If the resistance of 1 mi. of a certain electric-light wire is 3.58 ohms, what is the resistance of 5 mi. of wire of the same material but of twice the sectional area ? Also of 1 mi. of wire of the same material but of twice the diameter ? 2. If the resistance of 700 yds. of a certain- cable is 0.91 ohm, what is the resistance of 1 mi. of that cable ? 3. If the resistance of 100 yds. of a certain wire is 5 ohms, what length of the same wire would have a resistance of 13.2 ohms ? 4. The resistance of a certain wire is 9.1 ohms, and the resistance of 1 mi. of this wire is known to be 1.3 ohms ; required the length of the wire. 5. If the resistance of 130 yds. of copper wire y 1 ^ in. in diameter is 1 ohm, what is the resistance of 100 yds. of copper wire -fa in. in diameter ? 6. What is the resistance of 1 mi. of copper wire 1.14 mm in diam- eter, if the resistance of 1 mi. of copper wire 1.4 mm in diameter is 8.29 ohms ? 7. What is the length of copper wire 1 mm in diameter which has the same resistance as 6 m of copper wire 0.74 mm in diameter ? 8. What must be the length of an iron wire of sectional area 4 mm 2 to have the same resistance as a wire of pure copper 1000 m long whose sectional area is 1 mm 2 , taking the conductivity of iron to be \ of that of copper ? PROPORTION. 103 9. How thick must an iron wire be in order that for the same length it shall offer the same resistance as a copper wire 2.5 mm in diameter ? (Answer to 0.01 mm.) 10. If the resistance per cm 8 of a certain metal is 1.356 10~ 5 ohm, what is the resistance of a wire of this metal 1 m long and 2 mm thick ? (Answer to 10~ 5 ohm.) 11. What is the ratio of the resistances of two wires of the same metal, one of which is 30.48 cm long and weighs 35 g, while the other is 18.29 cm long and weighs 10.5 g ? 12. The resistance of 1 m of pure copper wire 1 mm in diameter is 0.02 ohm, and the resistance of a certain specimen of copper wire 3 mm in diameter and 10 m long is 0.025 ; what is the ratio of their resistances ? 13. The resistance of 1 mi. of a certain grade of copper wire whose diameter is 0.065 in. is 15.73 ohms, and the resistance of a wire of pure copper 1 ft. long and 0.001 in. in diameter is 9.94 ohms ; what is the ratio of their conductivities ? 14. The resistance of a certain dynamo is 10.9 ohms and the resist- ance of the rest of the circuit is 73 ohms ; the electro-motive force of the machine being 839 volts, find how many amperes flow through the circuit. v 1 volt forces 1 ampere through 1 ohm of resistance, 839 volts will 839 amperes force 839 amperes through 1 ohm of resistance, or through (73 + 10.9) ohms of resistance. 15. The resistance of a dynamo being 1.6 ohms and the resistance of the rest of the circuit being 25.4 ohms, and the electro-motive force being 206 volts ; find how many amperes flow through the circuit. 16. Three arc lamps on a circuit have a resistance of 3.12 ohms each; the resistance of the wires is 1.1 ohms, and that of the dynamo is 2.8 ohms ; find the voltage necessary to produce a current of 14.8 amperes. 17. Three arc lamps on a circuit have a resistance of 2.5 ohms each ; the resistance of the wires is 0.5 ohm, and that of the dynamo is 0.5 ohm ; find the voltage necessary to produce a current of 25 amperes through the circuit. 18. The resistance of a certain electric lamp is 3.8 ohms when a current of 10 amperes is flowing through it ; what is the voltage ? 19. If 31.1 volts force 35.8 amperes through a lamp, what is the resistance ? CHAPTER X. Series. A series is a succession of terms formed according to some common law. E.g., in the following, each term is formed from the preceding as indicated : 1,3,5,7, , by adding 2; 7, 3, 1, 5, , by subtracting 4, or by adding 4; 3, 9, 27, 81, , by multiplying by 3, or by dividing by ; 64, 16, 4, 1, , , by dividing by 4, or by multiplying by i; 2, 2, 2, 2, , by adding 0, or by multiplying by 1. In the series 0, 1, 1, 2, 3, 5, 8, 13, , each term after the first two is found by adding the two preceding terms. It is evident that the number of kinds of series is unlimited. An arithmetic series (also called an arithmetic progres- sion) is a series in which each term after the first is found by adding a constant to the preceding term. E.g., 7, 1, 5, 11, , the constant being 6, 2, 2, 2, 2, , " " " 0, 98, 66, 34, 2, , " " " -32. A geometric series (also called a geometric progression) is a series in which each term after the first is formed by multiplying the preceding term by a constant. E.g., 2, 5, 12i, 31 J, , the constant being 2-, 3, -6, +12, -24, , " -2, 10, 5, 2i, 1, , " " " i, 2, 2, 2, 2, , " " " 1. SERIES. 105 By custom, and because of their simplicity, arithmetic considers only arithmetic and geometric series. It should be stated, however, that the presence of this subject in applied arithmetic is merely a matter of tradition. It properly belongs to algebra, and hence may be omitted from the present course if Chap. XI is not taken. Since its application to business and to elementary science is slight, only a few of the more simple cases are considered. I. ARITHMETIC SERIES. Symbols. The following are in common use : n, the number of terms of the series. s, " sum " " " " ti, 2 , 3 , ..... t n , the terms of the series. In particular, a, or t\, the 1st term, and Z, or n , the nth or last term. d, the constant which added to any term gives the next ; d is usually called the difference. Formulae. There are two formulae in arithmetic series of such importance as to be designated as fundamental. 1. t n , or I = a + (n 1) d. Proof. 1. t 2 = a + d by definition. t s = t 2 + d = a + 2 d. ti t s + d = a + 3 d. 2. .-. 4 = t n -i + d = a + (n - l)d. 3. Or I = a + (n - 1) d. Proof. 1. s = a + (a + d) + (a + 2 d) + ..... (I d) + 1. 2. Hence, s = I + (I d) + (I 2 d)+ ..... (a + d) + a, by revers- ing the order. 3. .-. 2 s (a + l) + (a + I) + (a + J), by adding equations 1 and 2. 4. .'.2s = n(a + l), v there is an (a+ 1) in step 3 for each of the n terms in step 1. 5. ... ias - 106 HIGHER ARITHMETIC. It is evident that from formulae 1 and 2, various others can easily be obtained. E.g., from I = a + (n 1) d it follows that I - (n - 1) d = a, ^y = d, etc. From s = ^-r - it follows that 2s - - = n, etc. From I = a + (n 1) d and s *-r - it follows that _ n [a + a + (n - 1) d] S ~~ ~~2~ Exercises. 1. From s = *-r - i find a in terms of s, n, I. 2. From I = a + (n 1) d, find n in terms of I, a, d. 3. Find t w in the series 540, 480, 420, 4. Find 100 in the series 1, 3, 5, 5. Find s, given a = 1, I = 100, n = 100. 6. Find s, given a = 10, n = 6, d = 4. Write out the series. 7. Find s, given a = 40, n = 113, d = 5. 8. Find n, given s = 36,160, a = 40, J = 600. 9. What is the sum of the first 50 odd numbers ? the first 100 ? the first n ? 10. What is the sum of the first 50 even numbers ? the first 100 ? the first n ? 11. What is the sum of the first 100 numbers divisible by 5 ? by 7 ? 12. $100 is placed at interest annually on the first of each January for 10 yrs. , at 6% ; find the total amount of principals and interest at the end of 10 yrs. 13. A body falling in vacuum would fall 4.9 m in the first second, and in each succeeding second it would fall 9.8 m farther than in the preceding second ; how far would it fall in the llth second ? the 17th ? 14. From the data of Ex. 13, how far would it fall in 11 sees. ? 17 sees. ? 57 sees. ? 15. How long has a body been falling when it passes through 53.9 m during the last second ? SERIES. 107 II. GEOMETRIC SERIES. Symbols. The following are in common use : ?i, s, a, Z, and t\, t 2j ..... t n , as in arithmetic series ; r, the constant by which any term may be multiplied to produce the next ; r is usually called the rate or ratio. Formulae. There are two formulae in geometric series of such importance as to be designated as fundamental. or l = ar n ~\ Proof. 1. t 2 = ar by definition. ts = t^r = ar 2 . 4 = t s r = ar 3 . 2. .-. 4 = 4-ir = ar"- 1 . 3. Or I = ar"- 1 . _ s n a Ir a Proof. 1. s = a + ar + ar 2 + ..... + ar"~ 2 + ar"- 1 . 2. .-. rs = ar + ar 2 + ..... + ar"- 2 + ar"- 1 + ar n , by multiplying by r. 3. .-. rs s = ar" a, by subtracting, (2) (1). 4. .-. (r l)s = ar" a, and s = :- , by dividing by (r 1). 5. And v ar n = ar"~ J r = Zr, .-. s = --- Exercises. 1. From I = ar"- 1 , find a in terms of J, n, r. 2. Also r in terms of w, Z, a. 3. From s = r- find a in terms of r, n, s. 4. Find n in the series 1, 2, 4, 8, ..... ; also in the series 1, , i, ..... 5. Find s, given a = 2, r = 4, w = 3; also, given t s = 50, r = 5, n =6. 6. To what sum will $1 amount, at 4% compound interest, in 5 yrs. ? (Here o = $l, r = 1.04, n = 6.) 7. To what sum will $1 amount in 5 yrs. , at 4% a year, compounded semi-annually ? 108 HIGHER ARITHMETIC. If the number of terms is infinite and r < 1, then s approaches as its limit This is indicated by the symbols s = _ n being infinite. The symbol == is read "approaches as its limit." Proof. 1. v r T is reduced to the decimal form, the result is 0.272727 , and similarly the fraction 1^ = 0.152777 The former constantly repeats 27, and the latter constantly repeats 7 after 0.152. When, beginning with a certain order of a decimal frac- tion, the figures constantly repeat in the same order, the number is called a circulating, repeating, or recurring deci- mal, and the part which repeats is called a circulate, repetend, or recurring period. These various names are used, the subject being of too little prac- tical importance to establish a uniform custom. A circulate is usually represented by placing a dot over its first and last figures, thus : 0.272727 is represented by 0.27 ; 0.152777 " " " 0.1527. A circulating decimal may be reduced to a common fraction by means of the formula s = ^ , as in the following examples : 1. To what common fraction is 0.27 equal ? 1. 0.27 = 0.27 + 0.0027 + 0.000027 + 2. This is a geometric series with a = 0.27, r = 0.01, n infinite. 0.27 27 3 1 - 0.01 99 11 2. To what common fraction is 0.1527 equal ? 1. 0.1527 = 0.152 + 0.0007 + 0.00007 + 2. Of this, 0.0007 + 0.00007 + is a geometric series with a = 0.0007, r = 0.1, n infinite. . 0.0007 _ 7 "1-0.1 ~9000' 4. To this must be added 0.152, giving 0.152, or $j$, or \\. Exercise. Express as common fractions : (a) 0.9. (d) 2.476. (6) O.Oi. (e) 0.003714. (c) 0.247. (/) 0.123450. CHAPTER XL Logarithms. ABOUT the year 1614 a Scotchman, John Napier, invented a system by which multiplication can be per- formed by addition, division by subtraction, involution by a single multiplication, and evolution by a single division. This chapter might properly have followed the work on the four fundamental operations. By reserving it until this time, however, the practical application to scientific problems and the relation to series are more evident. It is not necessary for the understanding of the subsequent chapters and may, therefore, be omitted if desired. For the student who proposes to take even an elementary course in physics, however, the subject will be found of much value. In considering the annexed series 90 1 of numbers it is apparent that, 2 1 2 2 7 = 128 1. v 2 3 2 6 = 2 8 , 2 2 = 4 2 8 = 256 ... 8 32 = 2 8 = 256 ; 2 3 = 8 2 9 = 512 .-. the product can be found by adding the 2* = 16 2 10 = 1024 exponents (3 + 6 = 8) and then finding what 2 5 = 32 2 11 = 2048 2 8 equals. 2. v 2 9 : 2 3 = 2 6 , .-. 512 : 8 = 64 ; .-. this quotient can be found from the table by a single subtraction of exponents. 3. v (25)2 = 2 5 2 5 = 2 10 , .-. 32 2 - 1024. 4. v V2 = \/2 5 2 5 = 2 s , .-. V1024 = 32. 5. The exponents of 2 form an arithmetic series, while the powers form a geometric series. LOGARITHMS. Ill In like manner a table of the powers of any number may be made and the four operations, multiplication, division, involution, evolution, reduced to the operations of addition, subtraction, multiplication, and division of exponents. For practical purposes, the exponents of the powers to which 10, the base of our system of counting, must be raised to produce various numbers are put in a table, and these exponents are called the logarithms of those numbers. In this connection the word "power" is used in its broadest sense, 10 W being considered as a power whether n is positive, negative, inte- gral, or fractional. The logarithm of 100 is written "log 100." E.g., 10 3 = 1000, .-. log 1000 = 3. 10 = 1, .-. log 1 =0. 102 = 10 Q, ... log 100 = 2. 10-i = _!_, ... i g o.l = - 1. 101 = 10, .-. log 10 = 1. 10-2 = , .-. log 0.01 = - 2. , that is, the thousandth root of 10 801 , is nearly 2, .-. log 2 = 0.301 nearly. Although log 2 cannot be expressed exactly as a decimal fraction, it can be found to any required degree of accuracy. In the present work logarithms are given to 4 decimal places ; logarithms to 5 or 6 decimal places are sufficient for ordinary computations of considerable length. Exercises. 1. What are the logarithms of these numbers ? (a) 100,000; (6) T oV the number of centimeters of radius. \4i7t ' 7.8 / log 3 = 0.4771 log 14,300 = 4.1553 colog 4 = 9.3979 10 colog it 9.5029 10 colog 7.8 = 9.1079-10 3 2.6411 log 7.593 = 0.8804 .-. radius = 7.593 cm. Exercises. 1. Find the value of (12.8 -r 0.07235)*. 2. Find the value of (42 9.37)* -f 0.127*. 3. Find the value of (4.376/Tzr)*. 4. Given x : 4.127 = 0.125 : 2736 ; find x. 5. Given x 3 = x 5 : 5 ; find x. 6. Find the value of 27^. 7. Given 117,600 = 7"- 1 ; find n, correct to 0.1. 8. Find the value of Vjr-4.927. LOGARITHMS. 121 9. Given 0.47 : x = x : 1.249 ; find z. 10. Find the value of 0.00234 72.28 5.126 -10-7. 11. What power will just raise a weight of 17.5 Ibs., the fulcrum of the lever being 1.73 ft. from the weight and 4.19 ft. from the power ? 12. At what distance from the fulcrum must a power of 91 Ibs. be exerted to raise a weight of 7493 Ibs. 2 ft. 3 in. from the fulcrum ? 13. It is shown in studying the strength of materials that a cylin- drical iron shaft 5f in. in diameter and 5 ft. 7.5 in. long between its supports will support a load at the center of 0. 726 -d- 8700 Ibs. /, where d = the number of inches of diameter and I = the number of feet of length. Perform the computation. 14. 2240 Ibs. of chalk occupy 15.5 cu. ft., and a cubic foot of water weighs 62 Ibs. ; what is the specific gravity of chalk ? 15. The surface of a sphere is 4 sq. in. ; what is its volume ? 16. The volume of a sphere is 1 cu. ft. ; what is its radius ? 17. What is the specific gravity of a substance of which a sphere of radius 9 cm weighs 15 kg ? 18. What is the weight of a silver cone of radius 2 cm and height 3.6 cm, the specific gravity of silver being 10.5 ? 19. If the intensity of light varies inversely as the square of the distance and directly as the illuminating power of its source, what is the ratio of the intensities of a candle 3.75 ft. distant and a 41.5 candle lamp 13.2 ft. distant ? (Answer to 0.01.) 20. If the intensity of the light of the full moon is found to be equal to that of a candle at a distance of 4 ft. , what is the equivalent in candle power of the moon ? (See Ex. 19 ; take the distance of the moon as 2.4 10 5 mi.) [Omit the following if the problems in electricity were not taken.] 21. The resistance of 1 cm 3 of copper is 1.6-10- 8 ohm at 0. and the resistance increases by 3.9- 10 ~ 3 of this for each degree rise in temperature ; find the resistance of a wire 10 m long and 1 mm in diameter at 25 C. 22. An incandescent lamp takes a current of 0.71 ampere and the electro-motive force is 98.5 volts; what is the resistance of three such lamps ? 23. If an incandescent lamp of 81 ohms resistance takes a current of 0.756 ampere, what is the voltage ? 24. A mile of telegraph wire 2 mm in diameter offers a resistance of 12.85 ohms ; what is the resistance of 439 yds. of wire of the same material 0.8 mm in diameter ? CHAPTER XII. Graphic Arithmetic. WHEN the mind seeks to clearly appreciate the relations between several measurements, it is of great value to resort to a graphic representation, accurately drawn to a scale. Thus, to say that the distance in millions of miles from the Sun to Mercury is 36, to Earth 92, to Jupiter 481, and to Neptune 2778 is not nearly as expressive as when accom- panied by the following graphic representation of these measurements : 111 20 15 10 NO. OF REPORT! JUT OF 100 230 T The annexed curve has been plotted to represent graphi- cally the statistics compiled by a state board of health with reference to typhoid fever and the condi- tion of wells for the various months in the year. The dotted line shows the average number of inches above the surface of water in the wells, II , \ 220 200 INCHES ABOVE WATER 21. GRAPHIC ARITHMETIC. 123 and hence is highest when the water is lowest ; the black line shows the number of reports, out of every 100, in which typhoid fever was mentioned as prevalent. The effect of low water upon typhoid fever is thus much more clearly seen than it would be from the mere numbers. -$150,000,000 -$100,000,000 $50,000,000 1861-2-3-4-5 1890-1-2-3-4-5-6 FIG. 22. The above curve represents the sums paid by the United States government for pensions in various years as follows (by millions of dollars) : 1862, 0.8 ; 1863, 1 ; 1864, 5 ; 1865, 8 ; 1870, 28 ; 1880, 57 ; 1890, 106 ; 1891, 119; 1892, 141; 1893, 158; 1894, 141; 1895, 141; 1896, 139. Exercises. 1. In cases of contagious diseases the premises should be isolated and disinfected. In a certain part of the country, where this was neglected, the number of cases and deaths to each outbreak of diphtheria averaged respectively 13.78 and 3.81 ; in the same state, but where these precautions were taken, the corresponding results were 2.45 and 0.69. Represent graphically by four straight lines. 2. Represent graphically the following per capita indebtedness as given in recent government reports : Austria-Hungary $71, France $116, Prussia $37, Great Britain and Ireland $88, Italy $76, Russia $31, Spain $74, United States $15. 124 HIGHER ARITHMETIC. 3. A foot-ton is the measure of force required to raise 1 ton 1 ft. Plot the five curves corresponding to the following statistics and show where the growth has been the most marked. MILLIONS OF FOOT -TONS DAILY. FOOT-TONS YEAR. HAND. HORSE. STEAM. TOTAL. INHABITANT. 1820 753 3,300 240 4,293 446 1840 1,406 12,900 3,040 17,346 1,020 1860 2,805 22,200 14,000 39,005 1,240 1880 4,450 36,600 36,340 77,390 1,545 1885 6,406 55,200 67,700 129,306 1,940 4. Represent the following statistics graphically : MILLIONS OF FOOT-TONS DAILY. FOOT-TONS PER. IN- HABIT ANT. HAND. HORSE. STEAM. TOTAL. United States 6,406 55,200 67,700 129,306 1,940 Great Britain 3,210 6,100 46,800 56,110 1,470 Germany 4,280 11,500 29,800 45,580 902 France 3,380 9,600 21,600 34,580 910 Austria 3,410 9,900 9,200 22,510 560 Italy 2,570 4,020 4,800 11,390 380 Spain 1,540 5,500 3,600 10,640 590 5. Of those persons in England and Wales marrying in 1843, 327 out of every 1000 of the men could not write and 490 out of every 1000 of the women ; the numbers for each succeeding tenth year to 1893 were as follows : 304 and 439, 238 and 331, 188 and 254, 126 and 155, 50 and 57 ; represent these statistics by two curves. 6. The average annual mortality from smallpox in Sweden has been as follows : 1774-1801, before vaccination, 2045 1802-1816, vaccination allowed, 480 1817 to the present, vaccination compulsory, 155 Represent graphically. GRAPHIC ARITHMETIC . 125 7. Represent graphically the annexed statistics concerning the population of the United States. 8. The population of the world is estimated at 1480 millions dis- tributed as follows (in millions) : Europe 357.4, Africa 164, Asia 826, Australia 3.2, the Americas 121.9, Oceanica and the Polar regions 7.5. Represent graphi- cally, first arranging in order of magnitude. 9. The distance from the sun to the earth is about 93 -10 6 mi., to Neptune 30 times as far, to the nearest fixed star 256 10 11 ; rep- resent these graphically by meas- urements on a single straight line. 10. In a certain city, before the strict enforcement of the law re- quiring milk and cream to be of a certain grade, the following are the per cents of samples found below grade for the various weeks from May 1 to Aug. 1 inclusive : 50, 64, 42, 45, 42, 35, 35, 38, 37, 39, 45, 37, 28, 50 ; for the corre- sponding weeks of the next year, when the law was strictly enforced, the per cents are 4, 3, 2, 2, 6, 4, 7, 7, 7, 5, 10, 5, 7, 5. Represent the two by broken lines on the same diagram. 11. The indebtedness of the United States government at the various periods named was as follows (in tens of millions of dollars) : 1791, 7.5; 1800, 8.3; 1810, 5.3; 1816, 12.7; 1820, 9.1; 1830, 4.9; 1835, 0.0004 ; 1840, 0.5 ; 1850, 6.3 ; 1860, 6.4 ; 1862, 52.4 ; 1863, 112 ; 1865, 268 ; 1866, 277 ; 1867, 268 ; 1868, 261 ; 1870, 248 ; 1880, 213 ; 1890, 155; 1896, 179. Represent these statistics graphically, noting that the differences in dates are not uniform. 12. The tonnage of the merchant ships of America and England for the various years is here stated in millions ; represent the statistics by two curves on the same diagram. America : 1850, 3 ; 1860, 5 ; 1870, 4 ; 1880, 4 ; 1890, 4.4 ; 1892, 4.8 ; 1894, 4.7 ; 1896, 4.7. Eng- land : 1850, 4 ; 1860, 6 ; 1870, 7 ; 1880, 8 ; 1890, 11.6 ; 1892, 12.5 ; 1894, 13.2 ; 1896, 13.6. POPULATION OF THE UNITED STATES. DECADES. TOTAL WHITES. FOREIGN WHITES. 1750 1,040,000 1760 1,385,000 1770 1,850,000 1780 2,383,000 1790 3,177,257 1800 4,306,446 44,282 1810 5,862,073 96,725 1820 7,862,166 176,825 1830 10,537,378 315,830 1840 14,195,805 859,202 1850 19,553,068 2,244,602 1860 26,922,537 4,138,697 1870 33,589,377 5,507,229 1880 43,402,970 6,679,943 1890 54,983,890 9,249,547 CHAPTER XIII. Introduction to Percentage. WITH the introduction of the metric system throughout a large part of the world and the almost universal use of the decimal system of money save in Great Britain and some of her dependencies, the subject of decimal fractions has in modern times become one of great importance. It has come to be common to reckon by tenths, hundredths, and thousandths, and the subject of computation by hun- dredths has received the special name percentage. The subject requires no principles differing from those used in operating with common and decimal fractions, and the problems require no methods for solution other than those already discussed. There is, therefore, no reason for not treating percentage with decimal fractions, as was done to some extent, except that it is especially needed in the business arithmetic which is now to be considered. Common terms. Per cent means hundredths (hundredth , of a hundredth), the words, as used in America, always being interchangeable within grammatical limits. The symbol % means, therefore, either per cent or hundredths (hundredth, of a hundredth). if TF> 6%, 0.06 are each read 6 per cent, or 6 hundredths. !$*% 0.06%, 0.0006 " " 6 hundredths per cent, 6 hundredths of a hundredth, 6 hundredths of 1 per cent, or 6 ten-thousandths. PERCENTAGE. 127 0.01, 1%, Y^Q- are each read 1 per cent or 1 hundredth. 0.00|, *% " " " * of a hundredth, * per cent, or * of 1 per cent, and each equals ^. The words "per cent" are sometimes taken to mean * out of 100," 6% then meaning "6 out of 100." 200% would not, however, be so clearly understood by this explanation. If a certain per cent (meaning a certain number of hun- dredths) of a number is to be taken, as 6%, the 6% is called a rate, the 6 being called the rate per cent. Thus, if the rate of interest in a certain bank is 4%, the rate per cent of interest is 4. Illustrative Exercises. I. 1.55 is what per cent of 15* ? 1. Let r% = the rate. 2. Then r% of 15* = 1.55. 1 fifi 3. .-. r% = y^ = .10, by dividing equals by 15*. Ax. 7 II. 69.35 is 9*% of what number ? 1. Let n = the number. 2. Then 9*% of n = 69.35. 3 - ' n = Sir = 730 > bv dividing equals by 9*%. Ax. 7 o.oy* III. What is 10% of $634 ? 10% of $634 = $63.40. The question is simply, " What is 0.1 of $634 ? " No more analysis should be required than in the problem, "Find what 2 X $3 equals." IV. After deducting 9*% of a number there remains 660.65 ; required the number. 1. Let n = the number. 2. Then n - 9*% n, or 90*% n, equals 660.65. fifift ft^ 3. .-. n = ^ f- = 730, by dividing equals by 90*%. Ax. 7 V. After adding 9*% of a number to that number the sum is 799.35 ; required the number. 1. Let n = the number. 2. Then n + 9*%n, or 109*% n, equals 799.35. 3 - ' n = T^T = 730 ' fe y dividing equals by 109*%. Ax. 7 '' 128 HIGHER ARITHMETIC. Exercises. 1. The United States silver dollar weighs 26.729 g and the Japanese silver dollar (or yen) weighs 26.9564 g ; each is 0.900 fine (i.e., 90% pure silver); how many grams of fine silver (i.e., pure silver) in each ? How do you check your result ? 2. An English sovereign weighs 7.9881 g and is 0.916 fine ; how many grams of fine gold does it contain ? 3. The British nautical unit of length is the knot, 6080 ft. ; the common mile is what per cent of the knot ? 4. A fathom being strictly 0.1% of a knot, this is what per cent of the 6-f t. fathom ? 5. Of 1486 graduates of women's colleges in England, recently questioned, 680 were teachers, 208 were married, 13 were physicians or nurses, and the rest were in various professions or trades. What per cent of the graduates were teachers ? married ? physicians or nurses ? in other work ? How do you check your result ? 6. The following table shows the values of the total exported mer- chandise of the United States for the several years, and of the manu- factured part. Find what per cent the manufactured part is of the total in each year. YEAR. TOTAL. MANUFACTURED. 1860 $316,242,423 $40,345,892 1870 455,208,341 68,279,764 1880 823,946,353 102,856,015 1890 845,293,828 151,102,376 1895 793,397,890 183,595,743 7. From the data of Ex. 6 find the rate of increase of each amount over that of the preceding period. 8. The ratio of the arid and semi-arid regions of the United States (excluding Alaska) to the rest of the country is about 24 : 25 ; at this ratio, what per cent of our territory is arid and what per cent is semi-arid ? 9. From the following table showing the wealth of the United States and the average wealth of each inhabitant, compute the rate of increase in each from period to period. CENSUS. MILLIONS OF DOLLARS. DOLLARS PER INHABITANT. 1820 1,960 205 1840 3,910 230 1860 16,160 514 1880 43,642 870 1890 65,037 1,039 PERCENTAGE. 129 10. Cinnabar consists of two substances, sulphur and mercury, in the ratio of 7 parts (by weight) of the former to 44 of the latter. The weight of the sulphur is what per cent of the weight of the mercury ? That of the mercury is what per cent of that of the sulphur ? The weight of each is what per cent of the weight of the cinnabar ? How many grams of each in 178.5 g of cinnabar ? 11. A dealer is obliged to sell sugar so that for 43.5 Ibs. he receives as much as 36 Ibs. cost ; did he gain or lose, and what rate per cent? 12. At the time of a recent census in Ireland 38,121 people, or 0.81% of the total population, could speak the Irish language only ; required the population at that time, correct to 1000. 13. In 1894 the population of London was 4,349,116, an increase of about 3.28% over the population in 1891 ; required the population in 1891, correct to 1000. 14. National banks are required to keep on hand 25% of their deposits ; find if these banks have complied with the law and give the per cent in each case : SPECIE ON OTHER LEGAL ^ HAND. TENDER. ^POSITS. (a) BankofN.Y. $2,050,000 $1,200,000 $12,170,000 (6) Manhattan Bank 2,608,000 3,219,000 16,154,000 (c) Nassau Bank 192,000 539,400 2,885,000 (d) German Exchange Bank 267,900 587,400 3,140,500 (e) Germania Bank 511,400 541,400 4,196,300 15. A certain bank has on hand $294,600 in specie and $325,400 in other legal tender, and this sum is 26.4% of the deposits ; find, correct to $1000, the amount of the deposits. 16. A report made by the banks of New York City shows $164,172,200 in cash on hand, this being 31.3% of the total deposits ; find, correct to $1000, the amount of the deposits. 17. In one year the imports of specie into New York amounted to $83,233,962, and the exports to $102,487,994 ; the difference was what per cent of the imports ? of the exports ? 18. An insurance company charges a premium of $22.50 for insur- ing a house for $1500 for 3 yrs. ; what is the rate for the 3 yrs.? 19. A book agent sells during the summer 300 books at $2.75 each ; he is allowed 40% of the receipts ; how much does he earn ? 20. A man invests $3000 in property which he rents for $228 a year. The taxes are $33, insurance is $18, water tax $5, repairs $47; what per cent does he receive on his investment ? 130 HIGHER ARITHMETIC. 21. What is the per cent of attendance in a schoolroom of 59 pupils when there are 29-J- da. of absence in 4 school weeks ? when there are 10 da. of absence in 1 school week ? 22. A man sold two horses for $125 each ; on the purchase price of one he gained 20% and on that of the other he lost 20% ; what was his total gain or loss ? 23. The distances between the following cities by the present routes of sea travel and also by the proposed Nicaragua Canal are given below ; required the gain per cent by the canal over the present route, in each case. MILES VIA BETWEEN MILES, PRESENT NICARAGUA ROUTE, VIA CANAL (a) New York and San Francisco Cape Horn 15,660 4,907 (6) New York and Puget Sound Magellan 13,935 5,665 (c) New York and Hong Kong Cape G. H. 13,750 10,695 (d) New York and Melbourne Cape Horn 13,760 9,882 (e) Liverpool and San Francisco Cape Horn 15,620 7,627 (/) New Orleans and San Francisco Cape Horn 16,000 4,147 24. A man has the following investments : $2000 which yields 4%, $450 which yields 6%, and $1200 which yields 6|%. He can invest the whole amount so that it will yield 5% ; would he gain or lose by so doing, and what per cent of the whole sum ? 25. What is the premium for insuring a house for $3000 for 3 yrs. at l-J-% for the whole time ? at 2% ? at 75 cts. per $100 ? at $7 per $1000 ? 26. It is estimated that about 600,000,000 passengers are carried on steamboats in the United States in one year and that about 70 are killed ; what per cent are not killed ? 27. About 590,000,000 passengers are carried on railways in the United States in one year and about 300 are killed ; what per cent are not killed ? 28. About 75f % of the number of patents granted by the United States in the 60 yrs. beginning with 1837 represents the number of patents refused ; the total number of applications was 993,953 ; how many patents were granted and how many refused ? (Answer correct to 1000.) 29. By the eleventh census the number of Indians on reservations under control of the Indian Office was 106% as large as the rest of the Indians ; the total number was 249,000 ; how many were in each of these two groups ? (Answer correct to 1000.) PERCENTAGE. 131 30. About 74% of the territory of the United States, excluding Alaska and the Indian reservations, is cleared land, and 495,000,000 acres are forest; what is the total area, excluding the portions specified? 31. What per cent of the 343,267 immigrants landing in the United States in a certain year did not rank among either the 46,807 skilled laborers or the 2324 professional men ? 32. The chief export of the United States is unmanufactured cotton, which is worth about 8 cts. a pound, and the value of which is 22% of the total exports of merchandise which amounted in a certain year to $863,200,487 ; how many pounds of cotton were exported ? (Answer correct to 1,000,000.) 33. If the total value of merchandise imported into the United States in a certain year was $731,969,965, 13% being coffee of which there were 652,000,000 Ibs. ; what was the average price per pound of coffee ? 34. The cost of collecting the customs revenue of $160,021,752 in a certain year being 4.52%, and of collecting the internal revenue of $146,762,865 being 2.62%, the former netted the government how much more than the latter ? 35. The world's total production of wool in a certain year being 2,582,103,000 Ibs., and the four largest producers being Russia with 290,000,000 Ibs., Argentina with 280,000,000 Ibs., the United States with 272,475,000 Ibs., and Great Britain with 135,000,000 Ibs., find what per cent of the total these countries produced, both collectively and separately. (Answer correct to 0.1%.) 36. The following table shows the receipts and certain disburse- ments of "old line" life assurance companies reporting to the N. Y. Insurance Department : 1,00MB. POUd. s 1871 $113,490,562 $28,773,041 $14,624,608 $20,242,707 1895 266,897,200 84,791,622 15,297,604 62,052,872 Find the rate of increase in each column. 37. Ascertain the population of the city or village in which you reside, according to the last three census reports ; represent the statis- tics graphically and compute the rate of increase or decrease of popu- lation for each period. 38. Similarly for the average annual attendance of your school for the past five years. 39. The radius of the sun being 10,856% of that of the earth, the latter being 6370 km, compute the volume of the sun in cubic kilo- meters. CHAPTER XIV. Commercial Discounts and Profits. IT is the custom of manufacturers, publishers, and whole- sale dealers to fix a price for their products and then to allow a discount under certain conditions. E.g., a book may be published at $2.00 with a discount of 25% to dealers, the book costing them $2.00 25% of $2.00, or $1.50. The $2.00 is known as the list price, the $1.50 as the net price. It frequently happens that wholesale houses issue expen- sive catalogues in which prices are specified. But as the cost of production varies, these prices change, and in order not to issue a new catalogue a house will print a new list of discounts for its customers. In some lines, indeed, the catalogue price has been so long fixed as to be several hun- dred per cent above the actual price, the latter being fixed by the discounts, of which there are often several. E.g., paper bags are quoted at a certain price, but the bill sent to the retailer may read "Less 70% 25% 10%, 30 da., and 2% off in 10 da." This means that they can be produced so much cheaper than formerly that the purchaser is allowed a discount of 10%, then 25% from that price, then 70% from that, and finally, if he pays within 10 days instead of waiting 30 days, he is allowed a further discount of 2%. Hence, if the list price was $100, the net price would be $90 after deducting 10% of $100.00, 67.50 " " 25% " 90.00, 20.25 " " 70% " 67.50, 19.84 " " 2% " 20.25. COMMERCIAL DISCOUNTS. 133 In this case, the catalogue price is over 500% of the actual price paid. In Ex. 6 it is shown that it is immaterial in what order these discounts are taken. On the bill heads of wholesale houses there is usually a note show- ing what discounts, if any, are allowed. For example, "Terms: 30 da. net, 1% 10 da." ; " Terms : 60 da., or 2% if paid within 10 da." ; " Terms : Net 60 da., or 2% disct. if paid in 10 da." Illustrative problems. 1. The list price of some goods is $62.70, a discount of 10% 6% 3% being allowed ; required the net price. Solution. 0.97 0.94 0.90 of $62.70 - $51.45. Analysis. 1. Let I = list price, $62.70. 2. Then I 0.10 1 = 0.90 J, the remainder after the first discount. 3. Then 0.90 1 0.06 0.90 1 = 0.94 0.90 1 = the remainder after the second discount. 4. Similarly, 0.97 -0.94 0.90 1 = the remainder after the third dis- count = net price. Application of logarithms. If the student has studied Chap. XI, this furnishes an application, the answer requiring no more than four figures and thus coming within the range of the table on pp. 114, 115. log 0.97 = 0.9868 - 1 log 0.94 = 0.9731 1 log 0.90 = 0.9542 - 1 log 62.70 = 1.7973 log 61.45 = 1.7114 Unless logarithms are used, which is not advisable in practice, it is, of course, better to take 10% of $62.70 and subtract, then 6% of this remainder and subtract, and then 3% of this remainder and sub- tract, than to perform the multiplication by 0.97-0.94-0.90. II. A merchant sells goods at a discount of 25% from the marked price and still makes a profit of 25% on the cost ; at what per cent above cost did he mark them ? 1. Let c = the cost, and m = the marked price. 2. Then m 0.25 m = c + 0.25 c. 3. .-. 0.75 m = 1.25 c. 5. .-. he must mark them 66% above cost. 134 HIGHER ARITHMETIC. Exercises. 1. The list prices and rates of discount being as fol- lows, find the cost : LIST PRICE. RATES OF DISCOUNT. (a) $1271.50 33%, often called " a third off." (6) 3.00 25%, " " "a quarter off." (c) 125.00 15%. (d) 37.50 20% 12i% 6%. (e) 2107.50 30% 8%. (/) 403.80 25% 10% 4%. (g) 3462.95 10% 3%. (h) 178.65 12|% 8% 2%. (i) 83.90 15% 7% 3%. (j) 623.30 8% 2% 1%. (k) 375.00 25% 10% 6%. (/) 150.00 a third off 40%. 2. In each case of Ex. 1, suppose the buyer had sold the articles at the list price, what would have been his rate of gain on the cost ? 3. In Ex. 1, what one rate of discount would have been equivalent to the several rates mentioned in (d), (e), (I) ? 4. Suppose a dealer buys goods at " a third off " and sells them at " a quarter off " the list price, what is his rate of gain on the cost ? 5. Show that the discounts 10% 8% 3% are equivalent to the dis- counts 3% 8% 10%, but not to the single discount 10% + 8% + 3%. 6. Generalizing Ex. 5, show that the discounts r\% r 2 % r s % are equivalent to the discounts r 8 % r 2 % ri%, or r 2 % r 3 % ri%, etc. ; that is, that it is immaterial in what order the discounts are taken. 7. The cost of certain goods and the rates of discount being as fol- lows, find the list prices : COST. RATES OF DISCOUNT. (a) $1827.40 12*%. (6) 436.90 25% 10%. (c) 49.63 30% 12% 6%. (d) 2341.50 30% 10% 2%. (e) 693.49 25% 10% 3%. (/) 127.90 33*% 6%. (g) 647.00 20% 8% 1%. 8. A merchant buys goods listed at $250, on which a discount of 15% 10% 3% is allowed ; he sells the goods for $225 ; what rate of profit does he make on the cost ? COMMERCIAL DISCOUNTS. 135 9. Prove that if the rates of discount are ri, r 2 , the equivalent single rate of discount is r\ + r 2 r\- r 2 , and hence that the single rate equivalent to two rates of discount equals their sum minus their product. 10. A bill of merchandise amounting to $327.50 was bought Oct. 1, "Terms: 3 mo. or 5% off 60 da., or 10% off 30 da." How much money would settle the bill Jan. 1 ? Nov. 20 ? Oct. 27 ? 11. What is the cost of a bill of hardware amounting to $1027.40, discounts 40% 10% 3%, freight being $10.60 ? 12. What is the net value of one case of prints containing 3000 yds. @ 6 cts. per yd., less 8% discount ; package $0.40, freight $0.95 ? 13. A merchant buys goods at a discount of 30% 20% and sells them at % off the list price ; what is his gain per cent on the cost ? 14. Which is the better for the buyer to take, a discount of (a) 30% 15% 10% or one of 47% ? (6) 10% 10% 5% 23%? (c) 15%12|% " 25%? (d) 12% 8% 1% 19%? (e) 10% 6% 2% 17%? (/) 30% 30% 30% " 66%? 15. At what per cent above cost must goods be marked in order to take off from the marked price (a) 10% and still make a profit of 8% on the cost ? (6) 25% " " 20% (c) 20% " " 30% " (d) 10% " " 50% (e) T!% " " r 2 % (/) 15% and lose 5% (9) 20% 15% (ft) n% " r 2 % (i) 30% and neither gain nor lose ? 16. A dealer purchases some goods listed at $281.50, off and 5% for cash ; if he pays on delivery, what is the net price, and what per cent discount can he give on the list price in order to make a profit of 15%? 17. Three rates of discount, 10% 10% r% are equivalent to the single rate 27.1% ; find r. 18. What are the three equal rates of discount equivalent to the single rate 48.8% ? CHAPTER XV. Interest, Promissory Notes, Partial Payments. I. SIMPLE INTEREST. THERE is practically a single type of problem in this subject, given the principal, rate, and time, to find the interest. Bankers and others who frequently meet this problem find the interest by the help of printed tables. These tables are usually based on 360 days to the year, and in using them it is customary to reckon exact days between dates. Some banks use tables based on 365 days to the year, this being the fairer method although yielding less interest. People generally, working without tables, reckon 360 days to the year, but they find the difference between dates by subtracting months and days, calling 30 days 1 month, and this is the method used in this work. Thus, to find the time from July 2 to Sept. 2, it is customary, if one has no interest table, to say 9 mo. 2 da. 7 " 2 " 2 mo. But if an interest table is at hand, it will readily appear that Sept. 2 is the 245th day of the year, and July 2 " 183d " " and that the difference in time is 62 da. SIMPLE INTEREST. 137 For exercises in analysis certain other problems are usually given in school, though rarely met in business. A few such have been inserted, types appearing in Problems II, III, IV, VI, on p. 138. Interest is reckoned as a certain per cent of the prin- cipal. When a rate is specified, the words " for one year " are to be understood unless the contrary is stated. 7.e., if it is said that the rate of interest is 6%, it means that the interest for 1 yr. is 6% of the principal. Occasionally, however, interest is quoted by the month, as 1% a month. Interest table. The following represents the first part of a page from an interest table such as bankers use. This particular page, if printed in full, would give the interest at 6% for 3 mo. and any number of additional days from to 29. This portion gives only 0, 1, 2, 3 da. in excess of 3 mo., this being sufficient for illustration. 3 MONTHS. TOTAL DAY*. 1000 2000 3000 4000 5000 6000 7000 8000 9000 DAYS OVER 3 Mo. 90 15.00 30.00 45.00 60.00 75.00 90.00 105.00 120.00 135.00 91 15.17 30.33 45.50 60.67 75.83 91.00 106.17 121.33 136.50 1 92 15.33 30.67 46.00 61.33 76.67 92.00 107.33 122.67 138.00 2 93 15.50 31.00 46.50 62.00 7750 93.00 108.50 124.00 139.50 3 The method of using the table will be seen from the following computation of the interest on $3975 for 93 da. at 6% : Int. on $3000 = $46.50 900 = 13.95, T V of int. on $9000. " " 70= 1.09 " 5= .08 $61.62 It will be noticed that the interest on ordinary sums can be told by merely glancing at the table. Thus, the interest on $250 for 3 mo. is $3.00 + $0.75 = $3. 75. 138 HIGHER ARITHMETIC. Illustrative Problems. I. What is the interest on $360 for 1 yr. 6 mo. 10 da. at 6% ? 1. 1 yr. 6 mo. 10 da. = lf yrs. 2. Int. for 1 yr. = 6% of $360. W 10 3. .-. int. for lf yrs. = 0.06 $200 = $33. pp II. The interest on $360 for 1 yr. 6 mo. 10 da. is $33 ; required the rate. 1. 1 yr. 6 mo. 10 da. = lf yrs. = ff yr. 2. v the int. for ff yr. = $33, ... i* u lyr . =$33 -5.^ = 1$. |33. 3. vr%of$360 = f-$33, 3 5 10 III. How long will it take the interest on $360 at 6% to equal $33 ? 1. The int. for 1 yr. = 6% of $360. 2. .-. " " tjrs.=t- 6% of $360 = $33. $33 0.06 -$360 3ir ' 4. .-. it will take lf yrs., or 1 yr. 6 mo. 10 da. IV. On what sum of money will the interest for !$ yrs. at 6% equal $33 ? 1. Let p = the principal. 2. v the int. for If! yrs. at 6% on p is $33, 3. .-. V. Find the amount (principal plus interest) of $360 for 1 yr. 6 mo. 10 da. at 6%. 1. By Prob. I, amt. = $360 + f f 0.06 $360 = (1 + f f . 0.06) $360 = $393. VI. What principal will amount to $393 in |f yr. at 6% ? 1. Rateforffyr. =ff-0.06. 2. Let p the principal ; then, p + ff -0.06 p = $393, or (1 + |f -0.06) p = $393. SIMPLE INTEREST. 139 Exercises. 1. Find the interest on $10,000 from July 2 to Sept. 2, at 6%, reckoning the time as follows : (a) Subtract the months and days, calling 30 da. = 1 mo., and 360 da. = 1 yr. (&) Take exact days between dates, but let 360 da. = 1 yr. (c) Subtract as in (a), but let 365 da. = 1 yr. (d) Take exact days between dates, but let 365 da. = 1 yr. 2. Of the four plans given in Ex. 1, (a) Which gives the most interest ? (6) Which is easiest without interest tables ? (c) Which is the fairest ? (d) Which result differs most from the fairest result ? (e) Why do people generally, without interest tables, use method (a) ? (/) Why do bankers generally use method (6) ? (g) Why is method (c) not used ? (h) Why do the government and some banks use method (d) ? 3. Find the interests on the following principals for the times and at the rates specified : (a) $250 for 2 yrs. 4 mo. 8 da. at 6%. (6) $40 " 1 " 3 " " 5%. (c) $125 " 8 " 15 " " 7%. (d) $350 " 3 " 6 " " 4%. (e) $820 " 6 " 4i%. (/) p " t " " r%. 4. Find the rates at which the following principals will yield the interests mentioned in the respective times : (a) $300 yields $45 interest in 2 yrs. 6 mo. (6) $175 " $10.50 " 1 " 6 " (c) p " i " t " 5. Find the times in which the following principals will yield the interests mentioned at the respective rates ; (a) $450 yields $24 at 6%. (6) $125 $6.25 " 4%. (c) p i r%. 6. Find the principals which will yield the following interests at the times and rates mentioned : (a) $62.50 interest in 1 yr. 3 mo. at 4%. (6) $5 " 1 " 1 10 da. * 6%. (c) i " t " " r%. 140 HIGHER ARITHMETIC. 7. Find the principals which will amount to the following sums at the times and rates specified : (a) $280 in 2 yrs. at 6%. (6) $45.25 " 1 " 10 mo. 15 da. " 7%. (c) a " t " " r%. 8. From the portion of the interest table given on p. 137, find the interest at 6% on : (a) $275 for 3 mo. (6) $750 " 3 " 3 da. (c) $9275 " 91 " (d) $5750' " 93 " Short methods. Before interest tables were common, short methods of computing interest were valuable. At present those who have much work of this kind use these tables. One method is, however, of enough value to be mentioned, especially as the most common rate of interest is 6%, and as most notes run for 90 days or less. Required the interest on $250 for 63 da. at 6%. 1. v the rate for 1 yr. is 6%, 2. .-. " $ " , or 2 mo., is 1%. 3. 1% of $250 = $2.50, interest for 60 da. 4. ^ "$2.50 = $0.12i, " 3 " 5. .-. $2.62 = " 63 " In practice it is merely necessary to put down these figures, the vertical line representing the decimal point : $2150 60 da. |l2j 3 " $2|62i 63 da. For 7%, add \ of $2.62^, and similarly for other rates. Exercises. Find the interests on the following sums for the times and at the rates specified : 1. $144, 30 da., 6%. 2. $750, 93 " 6%. 3. $125, 60 " 6%. 4. $250, 93 6%. 5. $400, 33 " 7%. 6. $50, 90 " 5%. 7. $150, 60 '" 8%. PROMISSORY NOTES. 141 II. PROMISSORY NOTES. Most promissory notes between individuals are of sub- stantially the following form : f 500. Chicago, 111., Dec. 3, 1900. Thirty days after date, I promise to pay to John Jones, or order, five hundred dollars, value received, with interest at 5%. John Smith. 1. In this case John Smith is the maker, John Jones the payee, $500 the face, and the face plus the interest is the amount, or future worth. 2. This note is negotiable, and may be sold by the payee, the trans- fer being indicated by his indorsing the note, that is, by writing his name across the back. Notes payable to the payee "or bearer" are also negotiable. 3. By indorsing the note the payee becomes responsible for its pay- ment in case the maker does not pay it. But if the buyer is willing to take the note without this guarantee, the indorser may be released by first writing the words " Without recourse" across the back, and then his name. 4. The indorsement may be made in blank, that is, the payee may merely write his name across the back, or in full, that is, the payee may specify the person to whose order it is to be paid, thus : "Pay to the order of John Brown. John Jones." 5. A note matures on the day when it is legally due. When the time is specified in days, exact days are counted in ascertaining maturity ; when in months, calendar months are counted. 6. Many states, following an old custom, allow three days of grace for the payment of notes. That is, a note dated Dec. 3, the time being " 30 days after date," is legally due Jan. 2 + 3 days, or Jan. 5, a fact indicated by writing "Due Jan. 2/5." A considerable number of states have abolished these days of grace, and the custom will in time become obsolete. Where the law still allows them it is quite common for notes to bear the words " Without grace." 7. The law as to the time of payment of notes due on legal holidays varies in different states. 142 HIGHER ARITHMETIC. 8. If a note reads " with interest," but does not specify the rate, it draws the rate specified by the law of the state. If it does not call for interest, it draws none until it is due and payment is demanded, after which it draws the legal rate. 9. In some states the law specifies what is called the " legal rate," and then specifies a maximum rate above which no contract for interest is legal. In other states the u legal rate " is also the maximum. Some states specify no maximum, allowing the parties to the contract to fix any rate they wish. Interest above the maximum rate allowed by law is called usury, and the taking of usury is punished according to the laws of the various states in which it is forbidden. Notes payable at a bank are discussed in Chap. XVI on Banking Business. The protest of notes is also discussed in that connection. Exercises. 1. In your state, are days of grace allowed on prom- issory notes ? When are notes which mature on legal holidays pay- able in your state ? 2. What is the " legal rate " of interest in your state ? Is there a maximum rate beyond this ? What is the punishment for usury in your state ? 3. What is the rate at which money is usually loaned to responsible persons in your vicinity ? 4. Write a 90-day note, signed by Peter Brown and payable to your order, bearing the rate which you found in Ex. 3 ; indorse it so that it shall be payable to the order of Kobert Jones. 5. Find the dates of maturity of, the interests on, and the amounts of promissory notes for the following sums, at the specified rates, supposing the notes paid when due ; add the days of grace if such is the law in your state, otherwise not : (a) $500, dated Feb. 7, due 6 mo. after date, at 6%. (6) $250, " Mar. 1, " 1 yr. 6 mo. " 5%. (c) $100, " July 15, " 90 da. " 7%. (d) $750, " Sept. 7, " 2 yrs. " 4|%. (e) $1275, " Aug. 10, " 60 da. " 6%. (/) $350, " June 3, " 4 mo. " at the rate found in Ex. 3. (g) $50, " Dec. 10, " 2 " " " (h) $200, " Oct. 5, " 4 " " PARTIAL PAYMENTS. 143 III. PARTIAL PAYMENTS. If a note or other obligation draws simple interest, and partial payments have been made at various times, the sum due at any specified date is usually computed as follows : 1. The interest on the principal is found to that time when the payment or payments which have been made equal or exceed this interest. 2. The payment or payments are then deducted and the remainder is treated as a new principal. These directions constitute what is known as the United States Rule of Partial Payments, the only legal method in most states. A few states, however, require other methods, and in these the teacher should explain the law and require the problems solved accordingly. The United States Rule, and the reason for the first sentence, will be understood from a single problem : A note for $1000 is dated Jan. 2, 1900, and draws 6% interest ; the following payments have been made, Jan. 2, 1901, $1 ; July 2, 1901, $89; Sept. 2, 1901, $500; required the amount due Jan. 2, 1902. 1. On Jan. 2, 1901, the $1000 amounts to $1060. 2. If the $1 were now deducted the new principal would be $1059. 3. But then the borrower would be paying interest on $59 more than he agreed. 4. .-. it would not be right to deduct the $1, or any other sum which might be paid, unless it should equal at least $60. 5. The practical solution usually appears in the following form : 1901 1900 7 mo. 1 " 2 da. 2 " Int. to July 2, Amt. Paymts. " New prin. " Int. to Sept. 2 Amt. " Paymt. " New prin. " Int. to Jan. 2, Amt. " 1901, ($1 + $89), , 1901, 1902, $1000. 90. 1 1902 1901 6 9 7 2 2 $1090. 90. $1000. 10. 2 1 9 2 2 $1010. 500. $510. 10.20 4 $520.20 144 HIGHER ARITHMETIC. Payments less than the accrued interest are seldom made. When they are made it is usually possible to detect the fact that they are less than the interest, before computing the amount due on that date. Partial payments are usually indorsed on the note, that is, written across the back, as, for example : "Jan. 2, 1902. Rec'd $10. July 5, 1902. Rec'd $50." Exercises. Find the amounts due at the dates of settlement specified : DATE OF NOTE. FACE. BATE. PARTIAL PAYMENTS. SETTLED. 1. Jan. 10 $603 6% June 1, $100; Aug. 15, $50; Sept. Oct. 15 20, $30 2. Apr. 4, 1900 $125 7% Aug. 1, $5; Oct. 16, $30; Jan. 10, Apr. 1, 1901 1901, $75 3. Nov. 1, 1899 $50 6% Dec. 12, $5; Jan. 10, 1900, $40; Apr. 10, 1901 Feb. 1, $1 4. Jan. 11, 1899 $500 5% Jan. 11, 1900, $20; July 11, $15; Mar. 5, 1901 Jan. 11, 1901, $50 5 June 14, 1897 $375 6% Sept. 10, $3.50; Nov. 15, $4.75; June 21, 1899 Jan. 7, 1898, $51.75; Jan. 11, 1899, $200 6. May 1, 1897 $1000 5% Sept. 1, $5; Nov. 1, $3; Mar. 1, Sept. 1,1898 1898, $100; July 15, $275 7. Apr. 1, 1901 $200 4i% July 1, $50; Jan. 16, 1902, $5; Sept. 1,1902 June 10, 1902, $2; July 1, 1902, $75 8. June 1, 1898 $800 5% Jan. 3, 1899, $35; Aug. 1, $15; Dec. 1,1902 Nov. 3, 1899, $70; Aug. 16, 1900, $100; Feb. 1, 1901,'$125; Sept. 1,1902, $180 9. Apr. 1, 1901 $500 6% Jan. 1, 1902, $100; Aug. 7, 1903, Jan. 1,1904 $25 10. Jan. 1, 1900 $2000 6% Jan. 1, 1901, $500; Apr. 1, 1902, Nov. 13, 1905 $250; Dec. 16, 1903, $100; Jan. 1, 1905, $600 11. Feb. 5, 1904 $675 5% Apr. 1, 1905, $25; Aug. 5, 1905, Jan. 20, 1907 $100; Sept. 5, 1905, $50; Jan. 20, 1906, $200 12. May 2, 1900 $575 5% July 1, 1901, $75; Sept. 3, 1901, Sept. 17, 1904 $100; Jan. 1, 1902, $50; Apr. 1, 1902, $100; July 1, 1902, $100; Sept. 17, 1903, $50 COMPOUND INTEREST. 145 IV. COMPOUND INTEREST. Savings banks usually add the interest to the principal at the end of the interest period, say every six months. The whole amount then draws interest, the depositor thus receiving interest on interest, or compound interest. Other- wise, the subject is not often met in a practical way, although banks, by loaning their interest as it is received, really have all of the benefits of compound interest. As in simple interest, there is a single case of practical value given the principal, rate, and time, to find the com- pound interest or the amount. E.g., what is the amount of $500 for 3 yrs. at 3% compound interest ? 1. The amt. of $500.00 and int. for 1 yr. = $515.00. 2. " $515.00 " " =$530.45. 3. " $530.45 " " =$546.36. 4. .-. " $500.00 " 3 yrs. = $546.36. Similarly, what is the amount of $150 for 3 yrs. at 4%, interest compounded semi-annually ? 1. The amt. of $150 and int. for 6 mo.= $150 + 0.02 $150 = 1.02 -$150. 2. .. " 1.02 -$150 " = 1.02 -1.02 -$150 = 1.02 2 -$150. 3. .-. " 1.02 2 -$150 " = 1.02* $150. 4. And finally, the amount for 6 six-month periods = 1.026 -$150 = $168.93. Interest tables. While compound interest is not in general use, it frequently happens that large investors wish to compute the amount resulting from reinvesting all interest as it becomes due; in other words, they wish to ascertain the amount of a certain sum at compound interest. For this purpose they resort to compound-interest tables, a specimen of which is given on p. 146. A table of loga- rithms evidently answers the same purpose. 146 HIGHER ARITHMETIC. AMOUNT OF $1000 AT COMPOUND INTEBEST. YEABS. 2% 2% 3% 4% 5% 6% 1 1020.00 1025.00 1030.00 1040.00 1050.00 1060.00 2 1040.40 1050.63 1060.90 1081.60 1102.50 1123.60 3 1061.21 1076.89 1092.73 1124.86 1157.63 1191.02 4 1082.43 1103.81 1125.51 1169.86 1215.51 1262.48 5 1104.08 1131.41 1159.27 1216.65 1276.28 1338.23 6 1126.16 1159.69 1194.05 1265.32 1340.10 1418.52 If the interest is at the rate of 4%, 5%, or 6% per year, but com- pounded semi-annually, the amount is evidently the same as if the rate were 2%, 2-J-%, or 3%, respectively, compounded annually for a period twice as long. jEJ.gr., what is the amount of $2750 for 3 yrs. at 5%, compounded semi-annually ? Amt. of $1000 for 6 yrs. at 2|% compounded annually = $1159.69. " $2750 = 2.75 X $1159.69 = $3189.14. The subject of compound interest is still further dis- cussed in the Appendix, Note III. Exercises. Find the amounts of the following sums for the times and rates of compound interest specified : 1. $50, 2 yrs. 6 mo., 3%, compounded semi-annually. " annually. u semi-annually. " quarterly. 4%, $168, 4 yrs. 3 mo., $1200, 3 yrs. 2 mo., $350, 1 yr. 8 mo., 4%, " $p, 1 yr., r%, " annually; also for 2 yrs., 3 yrs., t yrs. 6. From Ex. 5, find p, the principal, which at r%, compounded annually for t yrs. , amounts to a. 7. From Ex. 6, find p, given a = $123.73, t - 3, r% = 4%. 8. From the compound-interest table, find the amount of $500 at 4%, compounded annually for 5 yrs. 9. Also of $2500 at 5%, compounded semi-annually for 2 yrs. 10. Also of $350 at 3%, compounded annually for 6 yrs. 11. Also of $4000 at 2%, compounded annually for 4 yrs. ANNUAL INTEREST. 147 V. ANNUAL INTEREST. In some states, if a note or bond contains the words "with interest payable annually" this interest, if left unpaid, also draws interest to the day of settlement, or until cancelled by payment. The note or bond is then said to draw annual interest. E.g., to find the amount due on a $500 note dated Jan. 1, 1900, drawing annual interest at 6%, no payments made until the day of settlement, Jan. 1, 1904. 1. Face of note = $500. 2. Int. on $500 for 4 yrs. at 6% = 120. 3. Int. on $30 for 3 yrs. + 2 yrs. + 1 yr. at 6% = 10.80 4. Amt. due Jan. 1, 1904 = $630.80 Unless annual interest is allowed in the state in which this book is used, this subject may be omitted. Semi-annual or quarterly interest is treated hi a similar manner. Exercises. 1. What is the amount due at the end of 3 yrs. on a $1000 note bearing annual interest at 5%, no payments having been made? 2. In the western states coupon notes are often given, that is, notes bearing coupons which are themselves promissory notes for the interest due, and also drawing interest, often at a higher rate. Find the amount of a coupon note for $1000 at the end of 5 yrs., the principal drawing 6%, the coupons representing the interest due annually and drawing 8% remaining unpaid. 3. A coupon note draws 6%, the coupons being due semi-annually and drawing 10% if unpaid ; the face of the note being $800, and no payments having been made, find the amount due at the end of 4 yrs. 4. What is the amount due at the end of 4 yrs. on a $300 note bearing 5% interest, payable semi-annually, no payments having been made ? 5. A coupon note draws 6%, the coupons being due semi-annually and drawing 8%, if unpaid ; the face of the note being $500, and no payments having been made, find the amount due at the end of 3 yrs. 6. In Ex. 5, supposing the first three coupons had been paid when due, find the amount due at the end of 3 yrs. CHAPTER XVI. Banking Business. THE ordinary business of a bank is largely included under the following heads : 1. Receiving deposits and paying from the same on presentation of checks signed by the depositor. 2. Lending money upon promissory notes or (chiefly in the case of savings banks) upon bonds and mortgages. 3. Discounting notes which individuals may own and upon which they wish to realize money before the notes are due. 4. Selling drafts on other banks, and collecting drafts drawn by one person or corporation on another. (See Chap. XVII.) I. DEPOSITS AND CHECKS. If a person deposits money in a savings bank (or in the savings department of a bank having both savings and commercial departments), he usually receives a book in which are written the sums deposited and drawn out. If he wishes to draw out any money, he presents his book for the debiting of the amount and is usually required to sign a receipt. Savings banks usually pay from 2% to 4% interest compounded semi-annually. Ordinary deposits in other banks do not draw interest, the deposit being made for convenience and safety. When the depositor wishes to draw upon his deposit, he makes out a check, of which the following is a common form : BANKING BUSINESS. 149 (gfiicctyo, Jf,_ to tfiv oidei of f. A check is usually made payable to : 1. " Self," in which case the drawer alone can collect it. 2. The payee or bearer, or merely to "Bearer," in which cases any one can draw the money. 3. The order of the payee, in which case the payee must indorse it. Most banks also receive money and issue Certificates of Deposit, of which the following is a common form : Certificate of Deposit FIRST NATIONAL BANK OF DETROIT. Detroit, Mich,, G^**<*z^ , -/9C>C>. r^a^&n. &++ k as deposited in this Bank ^>*. c^****'/^, Teller. <-^*'c^=z^-^c^Ue. v Cashier. Exercises. 1. Write a check for $54.75. 2. Also one payable to the order of yourself, and properly indorse it so that it can be collected only by Richard Roe or his order. 3. Write a certificate of deposit payable to your order, for $75, dated Jan. 4, drawing 3% if left 3 mo., or 3i% if left 6 mo. Compute its value on Nov. 19 ; also on May 1 1 ; also on Mar. 23. 150 HIGHER ARITHMETIC. II. LENDING MONEY. If a person wishes to borrow money from a bank, and the bank is willing to lend it to him, he usually gives a promissory note. This note may be secured by depositing with the bank some evidences of value, as stocks, bonds, etc., usually known as " Collateral," or by having some responsible person indorse the note. At present banks frequently loan money without an indorser to persons of unquestionable financial standing, a custom formerly not common. Since the borrowing of money on an indorsed promissory note is the method most commonly followed, this is the only one here discussed. A bank note is usually in the following form : ftet date, to fray to tfo oidvi of ' J&o*^*s J%<^ f 73. at fy Jtnrt National Bank, Boston, Such notes are usually made payable in 1, 2, or 3 mo., or in 30, 60, or 90 da., so that the bank can get its interest often, the interest then being reloaned. It was formerly the general custom to add three " days of grace," as mentioned on p. 141 ; but as already stated, a considerable number of states have abolished this custom. In the above note the words " without grace " make the note mature July 5; otherwise it would mature July 8, drawing interest to that date. As a rule no interest is specified in such notes, but interest is paid in advance and is called discount. The bank thus gets interest on interest, but this is allowed, in such cases, by law. BANKING BUSINESS. 151 In the case of the above note, John Doe, the maker, wishes to borrow $75.00. He makes the note payable to the order of Richard Roe, with whose financial standing the bank is satisfied. Richard Roe indorses it, thereby promising to pay it if John Doe does not. The maker then takes it to the bank and receives $75.00 less the interest (or discount) on $75.00 for 2 mo. at the usual rate. Since Richard Roe indorses this note for the accommodation of the maker, he is called an accommodation indorser. On July 5, if John Doe does not pay this note, the bank places it in the hands of a Notary Public, who sends to Richard Roe, the indorser, a Notice of Protest. If this is not sent promptly, the indorser may assume that the note has been paid, and he is released by law. If this notice is placed in a properly addressed sealed envelope and deposited in the post office by the notary, the demands of the law have been fulfilled. The law of protest varies, however, in different states. In discounting notes, banks count the time by months or days according as the note specifies, and then compute the interest by the help of tables usually based on 360 da. to the year, calling 30 da. 1 mo. Exercises. 1. Are "days of grace" allowed by law in your state ? (If so, always add them hi solving the problems in this sec- tion ; otherwise not.) 2. What is the day of maturity and the discount on the following notes : DATE. TIME NAMED. FACE. KATE OF DISCOUNT. (a) Apr. 1, 60 da., $250, 6%. (6) Oct. 17, 3 mo., $5000, 5%. (c) May 10, 90 da., $125, 7%. (d) Dec. 12, 2 mo., $50, 6%. (e) July 7, 4 mo., $600, 5%. 3. What is the usual rate of discount on bank notes in your vicinity ? Using that rate, find the discounts on the following notes : DATE. TIME NAMED. FACE. (a) Apr. 15, 4 mo., $1000. (5) Jan. 3, 60 da., $500. (c) Aug. 5, 90 da., $750. (d) Dec. 9, 3 mo., $50. (e) Oct. 8, 2 mo., $75. 4. Write and properly indorse bank notes subject to the conditions stated in Ex. 3. 152 HIGHER ARITHMETIC. III. DISCOUNTING NOTES. Merchants frequently take notes from their customers, running 1, 2, or 3 mo. or even longer, and drawing inter- est. Such notes are often made payable at the bank in which the merchant keeps his account so that, in case he needs the money on a note before it is due, and sells it to the bank, the latter can the more easily collect it. In case of sale, the seller indorses the note and the bank discounts it ; that is, the bank pays the sum due at maturity, less the discount (interest) on that sum, this difference being called the proceeds. It will be seen that so far as the bank is concerned this process of discounting a note held by a customer is essentially that already described of lending money on an indorsed note. There are, how- ever, two differences : 1. The indorser is not now an accommodation indorser ; he is the owner of the note and he receives the money from the bank. If, however, the maker does not pay the note when it becomes due it is protested like any other note and the indorser is held responsible as explained on p. 151. 2. "The note usually draws interest and it frequently is not dis- counted on the day of its date. The discount is therefore reckoned on the face of the note plus the interest, or on the future worth, for the time between the day of discount and the day of maturity. This time is occasionally computed in exact days, but more often in months and days. E.g., a merchant takes from a customer a note for $755.50, dated Apr. 16, due in 90 da. without grace, at 6%. Needing the money on May 1 he indorses the note and deposits it in his bank. If the bank is discounting at 6%, it gives him credit for the proceeds. The computation is as follows : Face of note = $755.50 Int. for 90 da. = 11.33 Future worth = $766.83 Disc't for 75 da. = 9.59 Proceeds = $757.24 BANKING BUSINESS. 153 Exercises. In the following problems take the rate of discount usually charged by banks in your vicinity, except as otherwise speci- fied, allowing days of grace or not according to their custom. The first exercise includes the practical business problems ; the rest are of value merely for the analysis. 1. Find the discount and the proceeds on the following notes : FACE. DATE. TIME TO RUN. (a) $136.75, Feb. 7, 3 mo., (6) $75.50, May 10, 60 da., (c) $352.00, Oct. 5, 2 mo., (d) $50.75, July 8, 90 da., (e) $800.00, Jan. 10, 4 mo., (/) $62.25, Dec. 8, 3 mo., INTEREST. DISCOUNTED. 7%, Apr. 1. 6%, June 2. 6%, Oct. 5. None, Sept. 1. 5%, Feb. 10. None, Dec. 27. INT. DISCOUNTED. RATE OF DISCOUNT. 6%, Mar. 4, 7%. 5%, July 20, 6%. 7%, Dec. 18, 6%. None, Apr. 1, 6%. 2. Find the face of the following notes : PROCEEDS. DATE. TIME TO RUN. (a) $75.24, Feb. 4, 3 mo., g (6) $81.46, July 13, 2 mo., 6 (c) $101.56, Oct. 3, 4 mo., | (d) $39.85, Apr. 1, 2 mo., f 3. For how long is a note for $74.60 discounted at 6%, if the pro- ceeds are $73.85? 4. At what rate is a note for $125.50 discounted for 4 mo., if the discount is $2.09 ? 6. Do you know of any savings bank in your vicinity ? If so, what rate of interest does it pay and how often is this compounded ? Under these conditions, what would be the amount at the end of 3 yrs, of $100 invested July 1 ? 6. If you had a check on a bank in your vicinity, payable to your order, what would be the steps necessary to get the money ? What would be the steps necessary to transfer it to another person so that the money could be drawn only on his order ? 7. Find the difference in the amount of $100 invested in a savings bank at 4%, compounded semi-annually, and the amount of the same sum at simple interest at 4%, the time being 5 yrs. in each case ; also for 6 yrs. ; also for 7 yrs. 8. Which yields the better income on $100 in 10 yrs., 4%, com- pounded semi-annually, or 5% simple interest ? 9. What principal will amount to $29,588.62 in 3 yrs. 3 mo. 3 da. at 6% ? CHAPTER XVII. Exchange. IF a person in one place owes a debt in another, he can settle it in a variety of ways. 1. He may send the money (a) By an unregistered letter ; this is liable to be lost or stolen, although with our present postal service this liability is slight. (6) By a registered letter ; in case of loss this can be traced to the one at fault ; in case of loss by accident no recovery is possible, but otherwise the government, while not holding itself responsible, requires the one at fault to make good the loss. (c) By express or other messenger, in which case the company or messenger is liable for loss. 2. He may cancel the debt by sending (a) A check on his home bank where he has money deposited, in which case the creditor may have to pay a bank for collecting it. (&) A draft drawn by some bank on a bank in a large city like New York or Chicago ; such a draft, especially if presented by a customer, is usually cashed without discount at any bank. (c) A postal money order ; this is not as safe as (a) or (&) since the government cannot be sued in case of payment to the wrong person ; identification is required, however, unless the sender waives it. (d) An express money order, issued by various express companies and costing the same as the postal order ; in case of loss or of payment to the wrong person these companies can be sued. (e) A telegraphic order ; this method is the most expensive, but the most rapid. The subject of Exchange relates to the second of these plans and includes the five methods named. EXCHANGE. 155 (a) The check. This instrument has already been described on p. 149. If a check is drawn by John Doe of Albany on a bank in that city, payable to the order of Richard Roe of Cleveland, the latter on receiving it indorses it and deposits it in the bank where his account is kept. This bank will probably collect it for him without charge. This is the usual plan, and a large part of the indebtedness of the country is settled by checks. If Richard Roe has no bank account, the bank to which he takes the above check will require his identification, will charge him exchange, that is, a small sum for collecting it, and will probably not pay him the money until it has been received from Albany. (b) The draft. Drafts are usually in substantially this form : fast Iati0nal anh of Prang. fo ffo oidei of To Mercantile National Bank, \ New York aty. } It will be noticed that a draft is quite like a check, but it is signed by the cashier of some bank and is drawn on some bank in a large commercial center. In the case mentioned under (a), John Doe might have purchased a draft for the amount of his indebtedness, payable to his own order or to the order of Richard Roe ; if to his own order, he would have indorsed it payable to the order of Richard Roe. He might have to pay a slight premium to the bank, usually 10 cts. to 15 cts. for drafts under $100. On receipt of the draft, Richard Roe would indorse it and receive the money, usually without any discount, at a bank. 156 HIGHER ARITHMETIC. The drafts already described are sometimes known as bankers' drafts to distinguish them from commercial drafts. The latter are extensively used by merchants, though rather as a means of demanding and collecting payment for a debt through the agency of banks than as a system of domestic exchange. The great majority of such drafts are substantially in the following form : fit bigfit feay to tfiv otdel of 1. In the above case suppose John Doe has bought goods of Richard Roe to the amount of $53.75, say on 30 da. credit, and does not pay the bill when due. Roe may then make out a draft as above, payable to the order of his bank, and deposit it for collection. 2. The Cleveland bank would send it to some Albany bank, asking it to collect and remit. 3. The Albany bank would send a messenger to John Doe and demand payment. In some states 3 days of grace are allowed on a sight draft, though not in New York. In such case, or in case of a time draft (ie., a draft payable a certain number of days after sight), Doe writes "Accepted, July 8, 1898" (if that is the date) across the face and signs it ; at the proper time it is again presented by a mes- senger from the bank and payment is demanded. 4. In case Doe declines to accept it, or to pay it if due immediately, the draft is returned to the Cleveland bank and Roe is notified ; he must then take other means for payment. 5. If it is paid, the Albany bank remits by draft to the Cleveland bank, deducting a small amount for making the collection. EXCHANGE. 157 The fluctuation of exchange. For small sums, say for $500 or less, New York or Chicago exchange always sells at a premium of about 0.1%. This is to pay the bank for its trouble and for the expense of shipping the money when its balance at New York or Chicago gets low. Banks usually buy New York or Chicago drafts at par, that is, at their face value, thus making no charge for cashing them. But on large sums the rate of exchange varies. If the San Francisco banks owe the New York banks $1,000,000, they must send that amount by express, an expensive pro- ceeding. If a man in San Francisco at that time wished to buy a draft on New York for $10,000 they would charge him more than usual because they would have to express that much more to New York. But if a man in New York wished to buy a draft on San Francisco he might buy it for $9999 or less because they would get their money at once and the risk and expense of transmitting it would be saved. The premium or discount is usually quoted as a certain per cent of the face of the draft, but sometimes as the amount on $1000. Thus, the quotation of % premium is the same as that of $2.50 premium. Exercises. 1. New York banks are selling drafts on New Orleans banks at 0.1% discount; which city is owing the other the more money? How much would a draft on New Orleans for $5000 cost in New York ? 2. Denver is selling drafts on Boston at i% premium ; which city is owing the other the more money ? How much would a draft on Boston for $15,000 cost in Denver ? 3. Suppose the balance of trade between Cincinnati and Chicago is said to be largely against Cincinnati, what does this mean ? In which place would the drafts on the other certainly be at a premium ? 4. Suppose that in Denver drafts on New York are selling at \% premium, on Chicago at 0.1% premium, on San Francisco at 0.1% discount ; what is the probable balance of trade between Denver and each of the other cities ? 158 HIGHER ARITHMETIC. The clearing house. If the draft shown on p. 155 is indorsed by John Doe payable to the order of Richard Roe of Cleveland, Roe will take it to his Cleveland bank to be cashed or placed to his credit. The Cleveland bank will send it to the New York bank with which it does business, say the Chemical National, and will receive credit for it. The next morning the Chemical National will send it to the Clearing House, where the leading New York banks send representatives to transfer the drafts held by each on the others. There it goes to the representative of the Mercantile National Bank on which it is drawn, and is paid. By the Mercantile National it is finally returned to the First National Bank of Albany which drew it. Since both the Chemical arid Mercantile National Banks have drafts on one another, and so for all other banks in the Clearing House, only a comparatively small balance is necessary to settle all accounts. Every large city has its own clearing house, but the one at New York is the largest in the country. In fact, its exchanges aggregate more than those of all the others together, averaging about $100,000,000 daily. The banks do not pay the balances to one another ; but if a bank owes a balance of $100,000 to all the others, it pays this to the clearing house; and if another bank has a balance in its favor of $50,000, the clearing house pays it that sum. In this way, the amount coming into the clearing house must equal the amount to be paid out. Exercises. 1. The exchanges in the New York Clearing House for one week were $623,405,190, and the actual balances paid were $36,951,619 j what was the average of each per day, and the balances for the week were what per cent of the exchanges ? 2. Since 1880 the highest average daily clearings for any one year at the New York Clearing House were $159,232,191 for 1881 ; the average daily balance paid in money was 3.5%; how much was this? 3. In the same period the lowest average daily balance paid in money was $4,247,069, which was 5.1% of the average daily clearings ; to how much did the average daily clearings amount ? EXCHANGE. 159 (c) The postal money order is substantially the same as a draft, except that instead of being drawn by a bank cashier on a bank in some large city it is drawn by one postmaster on another. These orders are always sold at a premium and cashed at par. The premium (price) varies from 3 cts. to 30 cts. depending on the amount, which may be from 1 ct to $100. (d) The express money order is substantially like the postal money order. (e) The telegraphic money order. Telegraph companies receive money at one office and telegraph (usually through the office of the manager of this part of the business) to another office to pay out an equal sum to the person named. A higher fee is charged than for drafts, but this method is employed when great promptness is necessary. Exercises. 1. If you owed $25 in Syracuse, N. Y., which of the five methods named would you take to pay it ? Why ? 2. What would be the cost of a $500 draft at 0.1% premium ? at par ? at 0.1% discount ? 3. If the fee for a money order over $75 and not exceeding $100 is 30 cts., what is the total cost of a money order for $87.50 ? 4. To send money by telegraphic order costs the double rate for a 10- word message and 1% of the sum sent ; if the rate for a 10-word message is 40 cts. , what would be the total cost of a telegraphic order for $87.50 ? 5. When a New York draft for $40,000 can be bought in St. Louis for $39,950, is exchange at a premium or a discount ? What is the rate ? How is the balance of trade ? 6. Find the cost of each of the following drafts : FACE. EXCHANGE. FACE. EXCHANGE. (a) $4350, i%prem. (/) $1276.90, i% prem. (6) $9275, % disct. (g) $2493.60, par. (c) $2450, 1% (h) $4275.75, i% prem. (d) $7500, $1.25 prem. (i) $10,023.60, $1 disct. (e) $8556.75, 75 cts. disct. 0') $9870, $1.50 prem. 160 HIGHEB, ARITHMETIC. 7. The cost of a draft including the premium of 0.1% is $4254.25 ; what is the face ? (Business men usually merely subtract the premium from the cost, a process which, while not accurate, gives a sufficiently close result on sums below $10,000. In this and the following exer- cises find both the correct result and the business approximation.) 8. Find both accurately and by the business approximation the face of each of the following drafts (see Ex. 7) : COST. EXCHANGE. COST. EXCHANGE. (a) $5244.75, $1 disct. (d) $5012.50, i% prem. (6) $1757.80, i% " (e) $4268.07, 25 cts. " (c) $13,593.29, i%prem. (/) $14,518.13, $1.25 " Foreign exchange is subject to the same general laws as domestic exchange, differing chiefly as to the currency and the manner of quoting the rate of exchange. Thus, the par of exchange on London is 4.8665, that is, 1 in gold is worth $4.8665 in gold. If exchange is selling at 4.90 it is above par, a draft for 1 costing $4.90 ; while if it is selling at 4.84 it is below par. Exchange on London and other cities in Great Britain and Ireland is always quoted at so many dollars to the pound. Exchange on Paris, and other cities in France and in countries like Belgium and Switzerland which use the French monetary sys- tem, is usually quoted at so many francs to the dollar, the quotation 5.14 meaning that $1 will buy a draft for 5.14 francs. It is some- times, and more conveniently, quoted at so many cents to the franc, the quotation 19.8 meaning that 19.8 cts. will buy a draft for 1 franc. The par of exchange is about 5.18, or 19.3. Exchange on German cities is usually quoted at so many cents to 4 marks, the quotation 96 meaning that 96 cts. will buy a draft for 4 marks. It is sometimes, and more conveniently, quoted at so many cents to the mark, the quotation 23 meaning that 23-J cts. will buy a draft for 1 mark. The par of exchange is about 95.2, or 23.8. Foreign drafts are usually called bills of exchange, a bill at 30 days' sight being a draft due 30 days (or 30 days + 3 days of grace) after sight. It is the custom with foreign bills, and occasionally with domestic drafts (sometimes called inland bills) between distant cities, to make out duplicates, as follows : EXCHANGE. 161 50. New York [date] , No. 147638. At sight of this first of exchange (second of the same tenor and date unpaid) pay to the order of Brown Brothers & Co. fifty pounds sterling, value received, and charge the same to the account of To Brown, Shipley & Co., "1 ' f J j r John Doe. London, Jbmgland. J 50. New York [date] , No. 147638. At sight of this second of exchange (first of the same tenor and date unpaid) pay to the order of Brown Brothers & Co. fifty pounds sterling, value received, and charge the same to the account of To Brown, Shipley & Co., "1 T j -ci i j f Jonn London, England. J Such duplicate drafts form a set of exchange. Formerly three such drafts were made out ; at present the leading dealers in foreign exchange draw their bills in duplicate ; recently, in the case of express company drafts used by tourists, only a single bill is demanded, and the custom is extending. Exercises. 1. What is the cost of a draft on London for 100, exchange 4.90 ? Is the balance of trade, judged by this quotation, against this country or against England ? 2. What is the cost of a draft on Paris for 1000 francs, exchange 5.20 ? exchange 5.13 ? Which result is the greater ? In which case is the balance of trade against this country ? In which is it in favor of this country ? 3. What is the cost of a draft on Paris for 1000 francs, exchange 19.8 ? exchange 19 ? Which result is the greater ? In which case is the balance of trade against this country ? In which case is it in favor of this country ? 4. What is the cost of a draft on Leipzig for 500 marks, exchange 97 ? exchange 94 ? Consider the balance of trade in each case. 162 HIGHER ARITHMETIC. 5. What is the cost of a draft on Hamburg for 200 marks, exchange 23| ? exchange 94 ? 6. Find the cost of each of the following drafts : FACE. DRAWN ON. BATE OF EXCHANGE. (a) 700, London, 4.84. (&) 2750 francs, Paris, 5.19. (c) 6280 marks, Frankfort, 95. (d) 525 8 shillings, Liverpool, 4.88. (e) 1425 francs, Paris, 19. (/) 800 marks, Berlin, 23f. (g) 25 4 shillings, London, 4.90. (h) 750 francs, Brussels, 5.18. (i) 575 marks, Leipzig, 24. (j) 50, Glasgow, 4.87. (k) 8760 marks, Munich, 95f 7. A New York merchant owes the following sums to foreign dealers ; if he remits by draft, what is the face in each case ? AMOUNT OWED. DRAFT PAYABLE AT. BATE OF EXCHANGE. (a) $2435, London, 4.87. (6) $1920, Paris, 5.20. (c) $2400, Leipzig, 96. (d) $1958, Liverpool, 4.89|. (e) $81.62, Paris, 19. (/) $117.50, Berlin, 23f 8. An express company sells travelers' checks payable in various countries of Europe. It charges \% premium, and every $20 check allows the owner to draw 4 Is. 6d. in England, or 102.50 francs in France, or 82.50 marks in Germany. The company also has the use of the money until the checks are paid, an average of 2 mo., the use of the money being worth at the rate of 5% a year. On $500 of checks, paid in England, how much does the company make above the par of exchange ? 9. In Ex. 8, suppose the checks paid in Germany. 10. In Ex. 8, suppose the checks paid in France. 11. The rates for foreign money orders, payable in the currency of the country to which they are sent, are : for sums not exceeding $10, 10 cts.; $10-$20, 20 cts.; $20-$30, 30 cts.; $90-$100, $1. What would be the cost of the following money orders : (a) $75 payable in London ? (6) $62.50 " Paris? CHAPTEE XVIII. Government Revenues. CERTAIN revenues are necessary for the support of the governments of the United States, the various individual states, the counties, the cities, etc. The methods of obtaining these revenues are prescribed by law and vary for these different kinds of governments. I. THE UNITED STATES GOVERNMENT. The expenses of our general government are about a million dollars a day, and our income should be about the same or enough more to gradually reduce our indebtedness. Some of our sources of income and our principal expendi- tures are as follows, although all of the items vary from year to year : INCOME. Internal revenue Spirits $90,000,000 Tobacco $30,000,000 Fermented liquors $32,000,000 Oleomargarine and other penalties $2,000,000 Customs revenue, $160,000,000 to $200,000,000 Total, including the above and the income from public lands, etc., $325,000,000 to $400,000,000 EXPENDITURES. War dept. $50,000,000 $30,000,000 $10,000,000 $135,000,000 $30,000,000 Navy " Indians Pensions Interest on debt Diplomatic and consular service, and miscellaneous, $100,000,000 Total,$325,000,000 to $400,000,000 164 HIGHER ARITHMETIC. The customs revenue (tariff, duty) is collected at custom houses situated at ports of entry established by law. Merchandise brought into the country (1) is on the free list (i.e. } it is not subject to duty), or (2) is subject to ad valorem duty .(a certain per cent on the value at the place of purchase), or (3) is subject to specific duty (a certain amount by number, measure, etc.), or (4) is subject to both ad valorem and specific duty. E.g., by the tariff of 1883 apples were on the free list; by the tariff of 1890 they paid a specific duty of 25 cts. a bushel ; by the tariff of 1894 they paid an ad valorem duty of 20%. By the tariff of 1890 oriental rugs paid a specific duty of 60 cts. per sq. yd., and an ad valorem duty of 40%. Ad valorem duty is, if honestly collected, the more fair ; but on account of undervaluation by the importer there is much more chance for fraud in the collection. Exercises. 1. Taking the total revenue of our general govern- ment for one year as $326,926,200, and our internal revenue as $146,762,865, what per cent of our income was of this class ? 2. In one year the internal revenue was $143,421,672, including $79,862,627 from spirits, $29,707,908 from tobacco, and $31,640,618 from fermented liquors ; what per cent was derived from these three classes ? 3. The revenue of the post office department for a certain year was $76,983,128, and the expenditures were $86,790,172 ; the excess was what per cent of the revenue ? 4. Mathematical instruments pay a duty of 35%; what is the invoice price of an instrument which pays a duty of $6.30 ? 5. The duty on cheese is 4 cts. per Ib. ; how much does a city which consumes 40,000 Ibs. of French cheese a year pay to the government for this privilege ? 6. The duty on aniline dyes being 25%, what is the valuation at the custom house on a package of dyes which pays $59.38 ? 7. The duty on fine blankets being 35%, what is the invoice price of a shipment of blankets which cost the importer $786.73, including the duty and $12.50 freight ? 8. Rubber coats pay a duty of 40% ; how much is the duty on 100 doz. invoiced at 1 Is. a dozen, reckoning the pound at $4.86f-? GOVERNMENT REVENUES. 165 9. Ready-made woolen clothing pays a duty of 50% ; how much less would a $20 suit cost if it were on the free list, not considering the freight and profit ? 10. Cutlery valued from $1.50 to $3 a dozen pays a duty of 75 cts. a dozen and 25% ad valorem ; what is the duty on 100 doz. Sheffield knives invoiced at $2.25 a dozen ? 11. English books pay a duty of 25%; how much less would you have to pay for an English book which costs you $8, including the duty and 50 cts. postage, if it were not for this tariff ? II. STATE AND LOCAL TAXES. The method of collecting taxes varies in different states, but in general it may be said that a valuation is placed upon the property of corporations, of land owners, and of persons possessing any considerable amount of personal property. Upon this assessed valuation a certain rate of taxation is fixed. The expression rate of taxation is usually applied to the number of mills of tax on each dollar of valuation. Thus, if the rate is 5-J mills, the tax is 5^- mills on each dollar. The rate of taxation is, therefore, found by dividing the amount to be raised by the number of dollars of valuation. E.g., if a village has to raise $12,575, and if the valuation is $2,465,685, the rate of taxation is ^f^ = $0.0051 on $1. In addition to the tax already mentioned, male citizens over 21 years of age are frequently required to pay a poll (i.e., head) tax. If taxes are not paid when due a fine is usually imposed in the form of a certain per cent of increase of the tax. E.g., if a man's taxes are $12, and he does not pay them when due, the law may require him to pay 5% additional, thus making his tax $12.60. 166 HIGHER ARITHMETIC. Tax collectors usually prepare a table similar to that given below. For this table, the rate of 5-J- mills on $1 has been taken. TAX TABLE. KATE 5 MILLS ON $1. 1 2 3 4 5 6 7 8 9 0000 0055 0110 0165 0220 0275 0330 0385 0440 0495 0550 0605 0660 0715 0770 0825 0880 0935 0990 1045 1100 1155 1210 1265 1320 1375 1430 1485 1540 1595 1(550 1705 1760 1815 1870 1925 1980 2035 2090 2145 2200 2255 2310 2365 2420 2475 2530 2585 2640 2695 2750 2805 2860 2915 2970 3025 3080 3135 3190 3245 3300 3355 3410 3465 3520 3575 3630 3685 3740 3795 3850 3905 3960 4015 4070 4125 4180 4235 4290 4345 4400 4455 4510 4565 4620 4675 4730 4785 4840 4895 4950 5005 5060 5115 5170 5225 5280 5335 5390 5445 The column at the left gives the first figure of the number of dollars of valuation, and the row at the top the second figure. A decimal point is understood before each of the other numbers. E.g., the tax on $10 is $0.055, " $87 " $0.4785, " $5900 " $32.45. To find the tax on $9805, the collector's commission being 1%, the actual computation of a collector would be as follows : Tax on $9800 = $53.90 " $5 - .03 $53.93 Commission 1% = .54 $54.47 Exercises. 1. From the preceding table find the tax at 5-J- mills on $1 on each of the following valuations, the collector's commission being 1% : (a) $1750, (b) $2500, (c) $5475, (d) $17,645, (e) $18,750, (/) $9250, (g) $7625. 2. Prepare the first two rows of a tax table (opposite and 1) at the rate of 8| mills on $1. GOVERNMENT REVENUES. 167 3. From Ex. 2, compute the tax on each of the following valua- tions at 8-J- mills on $1, collector's commission being 1%: (a) $1200, (6) $19,150, (c) $15,175, (d) $1750, (e) $17,150, (/) $1825, (g) $825, (h) $500. 4. Taxes are levied in a certain village as follows : for streets" $2000, for fire apparatus, etc. $1500, for school purposes $6000, for salaries and office rent $3400, for repair of bridge $500, for general purposes $500 ; the total valuation of property is $1,950,000 ; what is the rate of taxation ? 5. In Ex. 4, what would be the taxes of a man whose property is valued by the assessors at $2500 ? 6. The rate of taxation being 5 mills on $1, what are the taxes on property valued by the assessors at $9500, collector's commission 1%? Suppose the owner does not pay promptly and is fined 5%, what is his tax? (Use the table.) 7. Find the tax on (a) $18,500 at 4 mills on a dollar, (b) $6000 "3.8 " (c) $3500 "8.4 " (d) $21,400 " 7.2 " (e) $5500 "5 (/) $6750 " 4.8 " (g) $1800 " 54- 8. The assessed valuation of a district being $950,725, what is the rate of taxation necessary to raise $8000 ? 9. To raise $2000, a tax of 1 mills on a dollar was levied ; what was the assessed valuation ? 10. What would be the various taxes levied on a man whose property is valued by the assessors at $12,800, if the rates were as follows : state tax 1-J mills, county 2 mills, town 0.8 mill, school 1.4 mills ? 11. At 7 mills on a dollar, how much is the tax of a man who owns a farm of 250 acres, worth $70 an acre, but assessed for only f of its value ? 12. The rate of taxation in a certain town is 5 mills on a dollar, and the amount to be raised is $4783.87 ; what is the assessed valua- tion ? 13. At 6 mills on a dollar, how much is the tax of a man who owns a farm of 300 acres assessed at $10 an acre, and who is assessed on $2000 of personal property, and who pays a poll tax of $1 ? CHAPTER XIX. Commission and Brokerage. PRODUCE bought in quantities or sent to cities for sale is usually bought or sold through a commission merchant or a broker. A commission merchant usually has the goods consigned to him and sells them in his own name, remitting the net proceeds (the sum realized less the commission) to the con- signor. If he is buying for a customer, he charges the sum paid plus his commission. A broker does not receive the goods, but sells them for the consignor in advance or buys them for his customer, and they are shipped directly to the buyer. His commis- sions, called brokerage, are therefore less than those of the commission merchant. Commission and brokerage are reckoned as a certain per cent of the amount paid in buying or realized in selling, but more often as a certain amount for a given transaction. Thus, it is more common to pay 50 cts. a ton for selling hay than to pay a commission of 4% or 5%. Stocks are bought and sold on a brokerage of 12 cts. for each share, as explained in Chap. XX. There are numerous other cases involving commission and brokerage, as the buying and selling of securities. Some of these are mentioned in the exercises. Since no new principles are involved, illustrative problems are unnecessary. COMMISSION AND BROKERAGE. 169 Exercises. 1. What is the commission for buying a carload of 400 bu. of grain at of a cent a bushel ? for selling 4 carloads of hay at $5 a car ? for selling 500 bu. of beans at 95 cts. a bushel, commis- sion 5% ? 2. How much does a broker receive for selling 1200 bales of cotton, brokerage 25 cts. a bale ? 500 bbls. of rye flour at $2.95, brokerage 2i% ? 10,000 bu. wheat, brokerage (of a cent a bushel) ? 3. A commercial traveler sells goods at a commission of 3% ; to how much must his sales amount that he may have an income of $4500 a year ? 4. A commission merchant receives 100 boxes of Mexican oranges which he sells at $3.50 a box and remits $323.80 net proceeds; what is the rate of his commission ? 5. What are the net proceeds of a sale of 8750 Ibs. of leather at 25 cts. a pound, commission 2% ? 6. A speculator buys 1000 bbls. of May pork (ie., to be delivered the following May) at $7.82^-, and sells it at $7.90; he pays a bro- kerage of 2 cts. (on each barrel) for buying and the same for selling ; does he gain or lose, and how much ? 7. A speculator buys 10,000 bu. of May wheat at 83 (cts. a bushel) and sells it at 82 ; the brokerage is (of a cent a bushel) for buying and the same for selling ; how much does he lose ? 8. An auctioneer offers his services at $8 a day or 2% of amount sold ; a merchant accepts the latter offer and the stock is disposed of in 4 da., realizing $1875.50 ; how much less would he have paid if he had taken the first offer ? 9. A collector has a $500 note placed in his hands with power to compromise ; he accepts 75 cts. on a dollar and charges 5% of the sum collected, and 25 cts. for a draft ; what are the net proceeds ? 10. A broker buys flour for a customer at $3.30 a barrel, charging 2% ; the bill, including commissions, is $4039.20 ; how many barrels are bought ? 11. A dealer buys 1000 doz. eggs at an average price of 16 cts. a dozen and sends them to a commission merchant who sells them at an advance of 4 cts. a dozen, charging 10% commission ; the express was $7.50 ; did the dealer gain or lose, and how much ? 12. At 5%, what is the brokerage for selling 1000 bu. of potatoes at 38 cts. a bushel ? 13. A commission merchant sells 275 bu. of onions at 60 cts. a bushel, and remits the proceeds after deducting his commission of 7i what is the amount remitted ? 170 HIGHER ARITHMETIC. 14. A commission merchant remits $266 as the proceeds of a sale of 200 bbls. of apples, his commission being 5% ; at what price per barrel did he sell them ? 15. A man sends a carload of 13 tons of hay to Boston where it sells for $14 a ton, and receives $175.50 after paying his broker ; how much was the brokerage a ton ? 16. In Ex. 15, if the hay cost the man $8.50 a ton, and the freight cost 21 cts. a 100 (Ibs.), did he make or lose by the transaction, and how much ? 17. A commission merchant sells 4000 heads of cabbage at $3.50 a hundred, and remits $126 ; what was his rate of commission ? 18. 400 bu. of beans at 62 Ibs. to the bushel are shipped to Boston, the freight being 28 cts. a 100 (Ibs.) ; the beans cost the shipper 70 cts. a bushel and were sold through a broker at 95 cts., brokerage 5% ; how much did the shipper gain ? 19. A commission merchant sold 600 Ibs. of butter at 24 cts., 480 doz. eggs at 20 cts., 1200 Ibs. poultry at 7 cts.; what are the net proceeds after deducting $14 for freight and cartage and 2-$-% com- mission ? 20. A broker remits $1706 after deducting 2% for brokerage and 25 cts. for the draft ; how much was his brokerage ? 21. A salesman received $6782.88 in one year, this representing his commissions at 1J% ; find the amount of his sales. 22. A lawyer having a debt of $3250 to collect, compromises for 97^- cts. on a dollar ; his commissions are 2 J% ; how much does he remit to his client ? 23. A lawyer collects a debt for a client, takes 3J% for his pay, and remits the balance, $1935 ; what was the debt and the fee ? 24. An agent buys goods on commission at 2|%, and pays $40 for freight ; the whole amount was $1628.73 j what was the sum expended for goods ? 25. A real estate agent sold some western land for a man and, after retaining $23.40 as his commission, remitted $2116.60 ; what rate of commission did he charge ? 26. An agent sells some property for s dollars on a commission of r% ; what are the net proceeds ? 27. The net proceeds from the sale of some property is p dollars, and the rate of commission is r% ; at what price was it sold ? 28. An agent sells some property for s dollars and remits p dollars as the net proceeds ; what was the rate of commission ? CHAPTER XX. Stocks and Bonds. WHEN a number of persons wish to engage in business the law allows them to form a corporation usually known as a stock company with a certain capital stock, each person owning a certain number of shares of that stock, each share being allowed one vote at the meetings for the election of directors. The business of these companies is managed by officers, usually elected by the directors. If the company makes more than its expenses, part or all of the surplus is divided among the stockholders in the form of dividends. If a stock is paying a good rate of dividend, that is, a higher rate than can be received from ordinary invest- ments, a $100 share will cost more than $100, and the stock is said to be above par. If it is paying about the same rate that ordinary investments bring, a $100 share may be bought for $100, and the stock is said to be at par. If it is paying low dividends, or none, it will be below par. The dividends are expressed either as a certain per cent of the par value or as a certain number of dollars per share. E.g., a 5% stock is one which is paying 5% on the par value ; and if stock is paying a dividend of $3 it pays $3 on each share. If the par value of one share is $100, then a stock paying $3 a share is the same as a 3% stock. But if, as is often the case with mining stocks, the par value is $25, a dividend of $3 a share is at the rate of 12%. 172 HIGHER ARITHMETIC. Sometimes a company issues two kinds of stock, pre- ferred, which is entitled to the dividends to a certain amount (e.g., to 5% of the par value), and the common, which is entitled to part or all of the balance. On Jan. 1, 1897, the total capital stock of all steam railways in North America was $5,008,352,237, of which $3,986,753,937 was common stock and $1,021,598,300 was preferred. When a company needs more money than has been paid in by the stockholders it often borrows money and issues bonds payable at a certain time and bearing a certain rate of interest. These bonds are usually secured by a mortgage on the property of the company, taken in the name of trustees for the bondholders. Similarly, when a national, state, county, or city government wishes to borrow money it issues bonds, but without mortgages. Bonds either have coupons annexed, which are cut off as interest becomes due and are collected for the owner by the bank where he keeps his account, or are registered, that is, bear the name and address of the owner, the interest being sent when due. Bonds are spoken of as "4's reg.," " 5's coup.," etc., meaning that they draw 4% of their par value and are registered, or draw 5% of their par value and have coupons annexed. Exercises. 1. "Which would you prefer to own, common stock or preferred stock ? Why ? Suppose the preferred stock paid 5% and the common 7% ? the common 3% ? 2. Suppose the capital stock of a company is $100,000, half being preferred and half common, the former being entitled to 5% and the latter to the balance ; suppose $6000 to be distributed in dividends, what rate of dividend would be received by the common stock ? If the dividends remain the same from year to year, which kind of stock would you prefer to have at the same price ? 3. A company having $100,000 capital, of which $30,000 is pre- ferred stock entitled to 5%, the balance going to the common stock, has $6000 available for dividends ; what is the rate of dividend of the common stock ? 4. Which would you prefer to own, $1000 of stock in a certain railway, or one of its $1000 bonds ? Suppose it was a 5% bond, while the stock paid 7% ? 5% ? 4% ? STOCKS AND BONDS. 173 5. Which would you prefer to own, a coupon bond or a registered bond ? Why ? Which is the safer against loss by theft ? Which is the more easily transferred in case you wish to sell ? 6. A certain railway stock is paying 9% dividends annually, and another is paying 2% ; are they above or below par ? Why ? 7. United States 4% bonds are sold at 117, that is, a $100 bond costs $117, while Atchison railway 4% bonds are sold at 80 ; what is the reason for this difference in price ? 8. In 1896 $68,981,244 of dividends was paid on $3,986,753,937 of common stock in the North American railway companies, and $16,533,019 on $1,021,598,300 of preferred stock; what was the average dividend in each case ? Purchasing stocks and bonds. Since one usually does not know who has stock for sale he applies (directly or through a bank) to a stock broker who belongs to some stock exchange, where stocks and bonds are bought and sold. The leading stock exchange of the United States is in New York. The broker charges brokerage, usually -J% of the par value. This is charged for buying and also for selling. A newspaper quotation of 122 means that $100 of stock, which we shall always take as representing the par value of one share, as is usually the case with railway stocks, is selling for $122. But the seller would receive only $122 $&, or $121, for each share, because he must pay his broker ; and the buyer must pay $122'+ $, or $122$-, for each share, because he too must pay his broker. In stock quotations, fractions are expressed in eighths, quarters, or halves. E-g>, stock is often quoted at 97-f, but never at 62f. Fractions of a share are not usually sold ; if a person has $1000 to invest in Canada Southern Railway stock, quoted at 48|-, he would pay $49 a share, and purchase 20 shares and have $20 left. The purchaser receives a certificate of stock, signed by the proper officers of the company, stating that he owns so many shares. When he sells his stock he sends this certificate, properly indorsed, to his broker ; it is delivered to the company and another certificate is made out for the new purchaser. 174 HIGHER ARITHMETIC. Newspaper quotations of the prices of stocks and bonds are given in the daily papers and form the best basis for a series of problems. The brokerage must be considered in each case. In the absence of a daily paper the following quotations may be used, and on them are based the prob- lems on pp. 175, 176. STOCKS. BONDS. Atchison 16* U. S. 4's reg. 116| " prefd. 26f U. S. 4's coup. 117 C. B. & Q. 79f U. S. 2's 95 C. & N. W. 104J Atchison 4's 80 Canada South. 51 Bait. & Ohio 5's 99 N. J. Central 107i Erie 7's 137 Canadian Pacif. 57 North. Pac. 6's 114* D. L. & W. 160} Wabash 5's 107* Lake Shore 148* N. J. Central 5's 118i N. Y. Central 97 111. Central 4*'s llOi Pullman Car Co. 158 C. B. & Q. 5's 100 Illustrative problems. 1. Suppose a man buys 10 shares of Atchison as quoted above and sells them 6 mo. later when quoted at 18, having received no divi- dends ; does he gain or lose, and how much, money being worth at the rate of 4% a year to him ? 1. He buys for 16* + fc, and sells for 18 fc, .-. he gains 1, that is, $1.25 on a share. 2. .-. " 10 -$1.25, or $12.50. 3. But he loses * of 4% of 10 ($16* + $i), or $3.33, interest. 4. .-. his net gain is $12.50 $3.33, or $9.17. 2. Suppose a man buys 50 shares of C. & N. W. as quoted above and sells them 6 mo. later when quoted at 102*, meanwhile receiving a 3% dividend ; does he gain or lose, and how much, money being worth at the rate of 5% a year to him ? 1. He loses ($104* + $i) ($102* $*), or $2, on a share. 2. .-. " 50 -$2= $100. 3. He also loses 2*% of 50 ($104* + $i) = $130.47, interest. 4. .-. his total loss is $100 + $130.47 = $230.47. 5. He gains 3% of 50 $100 = $150. 6. .-. his net loss is $230.47 $150 = $80.47. STOCKS AND BONDS. 175 3. Not considering the length of time the bond runs, what rate of income does a purchaser receive from investing in U. S. 4's reg. as quoted on p. 174 ? 1. He receives $4 on every ($116 + $i) invested. 2. Let r% stand for the rate. 3. .-. r% of $116|- = $4. 4 - '** =&=**>* Exercises. Unless otherwise directed, use the quotations given on p. 174, remembering the brokerage in each case. In finding the rate of income on bonds the time of maturity is not considered in these exercises. 1. What will 20 shares of Pullman Car Co. stock cost ? 2. Also 125 shares of N. Y. Central ? 3. Also 75 shares of Lake Shore ? 4. What sum will be received from the sale of 10 shares of C. B. & Q. ? 5. Also from the sale of 40 shares of D. L. & W. ? 6. Suppose a man buys 100 shares of N. J. Central as quoted and sells it when quoted at 115f, what is the gain, not considering divi- dends or interest ? 7. Solve Ex. 6, supposing the stock had paid a 2% dividend mean- while, and that 8 mo. had elapsed and that money was worth at the rate of 6% a year to the investor. 8. Suppose a man sells 100 shares of Canada Southern as quoted, this stock paying 2|% dividends annually, and invests the proceeds in 31 shares of I). L. & W. which pays 9% dividends annually, putting the balance in a savings bank where it draws 4% ; find the alteration in income. 9. Suppose a man sells 50 shares of C. B. & Q., which pays 4% dividends, and invests the proceeds in Lake Shore, which pays 8%, buying as many shares as possible, and placing the balance in a savings bank where it draws 4% ; find the alteration in income. 10. Suppose a man has $2500 to invest ; what is the greatest num- ber of shares of Pullman Car Co. that he can buy, and how much will he have left ? 11. Which investment pays the better, a 5% bond and mortgage or Erie 7's as quoted, the interest being paid promptly ? 12. Also Erie 7's or North. Pac. 6's ? 13. Also U. S. 2's or U. S. 4's coup. ? 176 HIGHER ARITHMETIC. 14. Also C. B. & Q. 5's or North. Pac. 6's ? 15. A man's income in Erie 7's is $245 ; how much has he invested, at par value ? How much did the bonds cost him, as quoted ? 16. A man's income in D. L. & W. stock, while it pays 9% annu- ally, is 6% on the sum invested ; what was the quotation when he made the investment ? 17. A man's income is 5f% on the sum invested in C. & N. W. which he purchased when quoted at 104 ; find the rate of dividends. 18. A broker bought on his own account 50 shares of Atchison prefd. as quoted and sold it at 2?i ; how much did he gain ? 19. Tamarack Mining Co. stock pays a semi-annual dividend of 3% ; how much will the holder of 50 $100-shares receive ? 20. A bank with a capital of $150,000 declares a quarterly dividend of 2% ; what is the total amount of this dividend and how much will the owner of $1200 of stock receive ? 21. To raise more money a company sometimes assesses its stock- holders. If a certain mining company levies an assessment of 10%, how much must be paid by the holder of 50 $100-shares ? 22. How much must be invested in Wabash 5's as quoted to bring an annual income of $1000 ? 23. The common stock of a certain railway company is $20,000,000, and the preferred stock (which in this case is entitled to 6% annually) is $4,000,000. The company declares a semi-annual dividend, paying the usual amount to the preferred stockholders and 2 \% to the others. How much money was distributed in dividends ? 24. How much must be invested in 111. Central 4's as quoted to bring an annual income of $1350 ? 25. How much income will be derived from an investment of $991.25 in Bait. & Ohio 5's as quoted ? 26. A certain stock is quoted at 260 ; a broker is instructed to buy a certain number of shares at this price ; his bill including brokerage is $2081; how many shares did he buy ? 27. The average rate of dividends paid to stockholders in national banks in 1872 was 10.19%, the dividends amounting to $46,687,115; in 1895, 6.96%, amounting to $45,969,663 ; what was the total capital for each of these years ? 28. In Ex. 27, find the rate of increase of capitals and the rate of decrease of dividends. 29. Of the bonds quoted on p. 174, which yields the highest rate of income on the investment ? 30. Of the same bonds, which yields the lowest rate of income ? CHAPTER XXI. Insurance. FOB the majority of citizens insurance business is con- fined to three general lines, the practical problem being substantially the same in all cases. The three lines are 1. Fire insurance, 2. Life insurance, 3. Accident insurance, and the practical problem is, Given the face of the policy and the rate to find the premium. Less common are such special forms as tornado, plate glass, and steam-boiler insurance, insurance against loss by theft, marine insurance, etc. The technical features of insurance are so constantly changing that it is inexpedient to enter into the subject with any detail. The premium is computed either as a certain per cent of the face of the policy, or, what is analogous to it, as a certain sum on each $100 of insurance. The latter is the usual form. Both this certain per cent and this certain sum go by the name rate of insurance. E.g., the rate for insuring the life of a man 30 yrs. old in a certain company, the policy to mature at death, is $22.85 annually on $1000, although it might be stated as 2.285%. The rate for insuring a business block against fire for 1 yr. (the usual time for insuring places of business) may be $1.10 on $100. The rate for insuring a house against fire for 3 yrs. (the usual time for insuring dwelling houses) may be $0.95 (for the 3 yrs.) on $100. 178 HIGHER ARITHMETIC. Exercises. 1. What are the premiums for insuring business prop- erty against loss by fire for 1 yr. for the following amounts at the specified rates ? (a) $2000 at $0.90 per $100, contents for $5000 at $0.95 per $100. (6) $3500 " $1.10 " " $10,000 " $1.25 (c) $8000 " $1.35 " " $50,000 ' $1.40 " (d) $7500 " $1.20 " " $35,000 " $1.30 " 2. What are the premiums for insuring dwelling property against loss by fire for 3 yrs. for the following amounts at the specified rates ? (a) $1000 at $0.90 per $100, contents for $1500 at same rate. (6) $7000 " $1.10 " " $5000 " (c) $6500 " $0.95 " $4000 (d) $4000 " $1.05 " " $3750 " 3. What are the premiums for insuring manufacturing establish- ments against loss by fire for 1 yr. for the following amounts at the specified rates ? (a) $10,000 at $2.25 per $100. (6) $50,000 " $1.75 " (c) $25,000 " $1.95 (d) $15,000 " $2.10 " 4. What is the annual premium for insuring against loss by fire a business block for $8000 at $1.10 per $100, its ground and first floor contents for $10,000 at $1.20 per $100, its other contents for $8000 at $1.35 per $100, and two plate glass windows against damage from other causes than fire at $2.70 per window ? 5. What is the annual premium for insuring a leaded glass window in a church for $1250 at 2% ? 6. How much would be the annual premiums paid by a man 30 yrs. old for $5000 of life insurance at $22.85 per $1000 ? on the 10-pay- ment plan (of paying only ten times, the policy maturing at death) at $54.65 per $1000 ? on the 25-payment plan at $28.46 per $1000 ? on . the single-payment plan at $428.14 per $1000 ? 7. As in Ex. 6, on a 10-year endowment policy (one in which ten payments are made, the policy then maturing, or maturing at death if before 10 yrs.) for $5000 at $106.75 per $1000 ? on a 25-year endow- ment policy for $5000 at $38.85 per $1000 ? 8. What is the premium on a $1000 tornado insurance policy for 1 yr. at 20 cts. per $100 ? for 3 yrs. at 50 cts. per $100 ? for 5 yrs. at 80 cts. per $100 ? CHAPTER XXII. Miscellaneous Exercises. 1. Multiply 1854.362 by 0.000087931, correct to 0.000001. 2. Multiply 162.5473 by 8726.47231, correct to 0.0001. 3. The distance of the moon from the earth is 59.97 times the earth's radius ; if this radius is 3962.824 mi., find the distance to the moon, correct to 1 mi. 4. Divide 634.7538292 by 0.0657391, correct to 0.001. 5. Divide 15.63214725 by 0.0057123, correct to 0.001. 6. How many days, hours, minutes, and seconds in a year of 365. 24226 da.? 7. How often does the heart beat in a life of 75 yrs. of 365 da. each, supposing that the number of beats is 140 per min. during the first 3 yrs. of life, 120 for the next 3, 100 for the next 6, 90 for the next 10, 75 for the next 28, 70 for the next 20, and 80 for the last 5 ? 8. Knowing that 1,040,318,228,677 = 2,870,564 X 362,407 + 5,741,129, state the quotient and the remainder from dividing 1,040,318,228,677 by 2,870,564 ; also by 362,407. 9. Supposing a person can count one hundred in 30 sees. , and that after counting incessantly for 30 yrs. he dies, and his son goes on counting for 30 yrs. and then dies, and so on ; how many generations must elapse before one trillion is counted ? 10. A person loses T ^ of his fortune and then T ^ of the remainder ; would the result have been the same if he had first lost ^ and then T ^ of the remainder ? Generalize for - and - 11. How much is the fraction -^ 3 - increased or diminished when 5 is added to each term ? 12. In any year show that the same days of the month in March and November fall on the same day of the week. 180 HIGHER ARITHMETIC. 1H - 7, 13. Reduce to simplest form 14. Reduce the fraction g ' , 2 ^ 1X *> J^ J 7 to its simplest form. l(|-^i-AJ 15. Simplify the expression T a of - i+=4 16. Simplify the expression 17. Divide 3 - f of T % by 21 J + & + 4 X 5. 18. The first of a series of cog-wheels, working into one another in a straight line, has 7 n teeth ; the second has 6 w, the third 5 n, and the number in the fourth is to that in the third as 2 to 3. If the wheels are set in motion, how many revolutions must each make before they are simultaneously in their original positions ? 19. Show that with a 1-ct. piece, two 2-ct. pieces, a 5-ct. piece, four dimes, a half-dollar, and nine silver dollars one can pay any sum less than or equal to $ 10. 20. The moon revolves about the earth in 27 da. 7 hrs. 43 mins. 11.5 sees. ; what is the average angle passed over in a day ? 21. The length of an arc of 97 21' 47.2" is 23 in.; find, correct to 0.1", the arc of the same circle 1 in. long. 22. What fraction of the circumference is an arc of 27 17' 30" ? (Answer correct to 0.0001.) 23. Prove that the sum of a common fraction and its reciprocal is greater than 2. Is there any exception ? 24. Prove that if the same number is added to both terms of a fraction the new fraction is nearer unity than the old. 25. Prove that, of three consecutive numbers, the difference between the squares of the first and third is four times the second. 26. Prove that the difference between two numbers composed of the same digits, as 937 and 793, is a multiple of 9. 27. Show that the integral part of a quotient is not changed by adding to the dividend a number less than the difference between the divisor and the remainder. 28. Given the sum and the difference of two numbers, show how to find each. Prove your statement. 29. Is the product of two square numbers always a square ? 30. Prove that no number ending in 5 and not in 25 can be square. MISCELLANEOUS EXERCISES. 181 31. Prove that a fraction whose terms are composed of the same number of digits is not altered in value by repeating the same number of times the figures of both terms. E.g., f| = ||f| = fHHI 32. What is the common fraction which, reduced to a decimal fraction, equals 0.4275275 ? 33. Prove that an integer cannot have for a square root a fractional number. 34. Prove that any odd square number diminished by 1 is a mul- tiple of 8. 35. State the test of divisibility of a number by 33. 36. Show that if two numbers are prime to one another, any powers to which they may be raised are also prime to one another. 37. Extract the 32d root of 429,497,296. 38. Show that a factor of each of two numbers is also a factor of their greatest common divisor. 39. If your school building is heated by a furnace, compare the area of a cross section of the cold-air pipe with the sum of the areas of cross sections of the hot-air pipes, and determine the ratio. 40. What is the ratio of an arc of 321 22' to one of 37 21' 1" ? 41. Divide the arc of 88 27' 33" into three parts proportional to the numbers 3.2, 5.6, 8.5. 42. Divide the length of 28.75 in. into three parts proportional to the numbers f , f , T V 43. How many pounds each of nickel and lead must be added to an alloy weighing 10 Ibs. and consisting of 11 parts (by weight) nickel, 7 parts tin, and 5 parts lead, so that the new alloy shall consist of 19 parts nickel, 41 parts tin, and 17 parts lead ? 44. It takes a letter 43 da. to go from New York to Siam, a dis- tance of 12,990 mi., and 34 da. to go to Adelaide, Australia, a distance of 12,845 mi. What is the ratio of the average rate on the latter route to that on the former ? 45. The area of Lake Superior is 32,000 sq. mi. and it drains an area of 85,000 sq. mi. ; the area of Lake Erie is 10,000 sq. mi. and its drainage is 39,680 sq. mi. Are the areas proportional to the drainage ? If not, what would be the drainage of Lake Superior to make them so? 46. What force can a man weighing 165 Ibs. exert on a stone by pressing on a horizontal crowbar 6 ft. long, propped at a distance of 5 in. from the point of contact with the stone, not considering the weight of the bar ? 182 HIGHER ARITHMETIC. 47. A uniform rod 2 ft. long weighs 1 Ib. ; what weight must be hung at one end in order that the rod may balance on a point 3 in. from that end ? 48. Two men carry a weight of 20 Ibs. on a pole, one end being held by each ; the weight is 2 ft. from one end and 5 ft. from the other ; how many pounds does each support ? 49. In a pair of nut crackers the nut is placed 1 in. from the hinge, and the hand is applied at a distance of 6 in. from the hinge ; if the nut requires a force of 22.5 Ibs. to break it, how much pressure must be exerted by the hand ? 50. Three persons are associated in a common enterprise, the first having invested $4000, the second $7000, and the third $9000. At the end of a year their gains amount to $7340, out of which they pay the first $2000 for managing the business and divide the balance among the three in proportion to their investments. How much did each receive ? 51. A country is 600 mi. long and 320 mi. wide ; find the dimen- sions of the paper on which a map of the country might be drawn, the scale being fa in. to the mile. 52. The average number of deaths in the world each minute is estimated at 67, and the average number of births at 70 ; how many of each in a year of 365 da. ? 53. By what fractional part of an inch should the highest mountain in Alaska, 19,500 ft., be represented on a globe 16 in. in diameter, the earth's radius being taken as 4000 mi. ? 54. In a certain enterprise in which three persons are engaged, A puts in $3500 for 25 mo., B $2400 for 15 mo., C $4500 for 12 mo.; they gain $5000 ; what is the share of each ? 55. A, B, and C rent a pasture for 6 mo. for $100 ; A puts in 25 cattle for the whole time, B 30 for 4- mo., C 45 for 3 mo. ; find the rent paid by each. 56. The streets of a certain city have an area of 8 km 2 . In a cer- tain storm the average depth of snow was 25 cm. Assuming 12 cm 3 of snow to produce 1 cm 3 of water, find the volume of water produced by this snow, and the weight in metric tons. 57. The wheels of a bicycle are 28 in. in diameter. The sprocket wheel connected with the pedals has 18 sprockets ; the other, 8. How many miles an hour does the rider make for one revolution of the pedals per sec.? If he travels 15 mi. per hr., how many revolutions of the pedals per min. ? MISCELLANEOUS EXERCISES. 183 58. Milk gives about 20% in weight of cream, and cream gives about 30% in weight of butter. How many liters of milk will produce 100 kg of butter, and how many kilograms of butter from 100 1 of milk ? The density of milk is 1.03. 59. At a certain school rain fell one day to the depth of 36 mm. Calculate the volume of water which fell upon the school yard, a hectare in area ; also the weight of this volume of water ; also the respective weights of the oxygen and hydrogen contained, knowing that water is formed of eight parts in weight of oxygen to one of hydrogen. 60. A cubic foot of water weighs 1000 oz., and in freezing expands ^-Q of itself in length, breadth, and thickness ; find the weight of a cubic foot of ice, correct to 0.1. 61. A liter of good milk weighs 1.030 kg. A milkman furnished 4.5 1 of milk weighing 4.59 kg. Was there any water in it, and if so, how much ? 62. How long will it take a man to walk around a square field whose area is 6f- acres, at the rate of a mile in 10|- mins. ? 63. Having found the average number of inches of rainfall per yr. in your vicinity, determine the average number of gallons of water that fall upon the roof of your school building in one year. 64. Of the world's supply of wool in a certain year, 2,456,733,600 Ibs., Great Britain produced 147 Ibs. to the continent of Europe's 640 and North America's 319 ; North America produced 6 Ibs. to Austral- asia's 11 ; Great Britain produced 15 Ibs. to the Cape of Good Hope's 13 ; Australasia produced 22 Ibs. to the River Plate's 15 ; Australasia produces 577,500,000 Ibs ; how many pounds (correct to 100,000) were produced by all other countries together ? 65. The value of the food-fishing industry in Alaska and Massa- chusetts together in a certain year was $8,149,987, Massachusetts exceeding Alaska by $3,547,877 ; what was the value in each ? 66. A gas holder is to be constructed in the form of a circular cylinder, such that the radius of the circle shall be equal to the height ; find its dimensions to contain 100,000 cu. ft. of gas. 67. If a terrestrial globe is constructed 36 in. in diameter, find the size on its surface of the United States, whose area is 3,500,000 sq. mi., the earth's diameter being 8000 mi. 68. If 100 Ibs. of copper are drawn into 1 mi. of wire, find the diameter, copper being 8.9 times as heavy as water and 1 cu. ft. of water weighing 1000 oz. avoirdupois. 184 HIGHER ARITHMETIC. 69. An India rubber band 8 in. long, in. wide, T ^ in. thick, is stretched until it is 18 in. long and J in. wide ; what is then its thick- ness, assuming the volume to remain constant ? 70. What is the amount of pressure exerted against one side of the upright gate of a canal, the gate being 24 ft. wide and submerged to the depth of 10 ft. ? 71. A locomotive traveled for 32 sees, on a certain railroad at the rate of 112 mi. per hr.; how many revolutions were made in this time by the driving wheels, which were 78 in. in diameter ? 72. If a man whose body has a surface of 15 sq. ft. dives in fresh water to the depth of 70 ft., what pressure does his body sustain ? 73. Sea- water weighing 64.05 Ibs. per cu. ft., what is the pressure per sq. in. at the depth of 4655 fathoms of 6 ft. ? 74. What length of paper yd. wide will be required to cover a wall 15 ft. 8 in. long by 11 ft. 3 in. high, no allowance being made for matching ? 75. The diagonals of a quadrilateral field are 24 chains and 35 chains in length respectively, and are perpendicular to one another ; how many acres in the field ? 76. How many degrees in an arc equal in length to the radius of the circle ? 77. A circle is inscribed in a square, the radius of the circle being 8.5 in.; find the area between the sides of the square and the circum- ference of the circle. 78. Find the difference between the area of a circle 15.4 in. in diameter and that of a regular inscribed hexagon. (The side of a regular inscribed hexagon equals the radius of the circle.) 79. Three circles each 4 ft. in diameter touch one another ; find the area of the triangular figure enclosed by them. 80. What is the length of the edge of the largest cube that can be cut out of a sphere 1 ft. in diameter ? 81. What is the length of the edge of a cube whose surface is 9 sq. ft. 54 sq. in. ? 82. A bar of metal 9 in. wide, 2 in. thick, and 8 ft. long weighs 1 Ib. per cu. in. ; find the length and thickness of another bar of the same metal, and of the same width and volume, if 2 in. cut off from the end weighs 27 Ibs. 83. For a period of a week note carefully the number of minutes spent in the preparation of each of your various lessons. Represent the averages for the various subjects graphically. MISCELLANEOUS EXERCISES. 185 84. Gunpowder being composed of sulphur, 75% niter, and the balance charcoal, how many pounds of each in 200 Ibs. of powder ? 85. Three persons contribute $1000, $1200, $1780 respectively, and after trading 15 yrs. dissolve partnership ; the firm then being worth $18,000, what did each man receive ? 86. A piece of wood which weighs 70 oz. in air has attached to it a piece of copper which weighs 36 oz. in air and 31.5 oz. in water ; the united mass weighs 11.7 oz. in water ; what is the specific gravity of the wood ? 87. Find the weight of 10 mi. of steel wire 0.147 in. in diameter, the specific gravity being 7.872. 88. Find the weight of a cast iron cylinder 8 ft. long, with a radius of 3.5 in., assuming the specific gravity to be 7.108. 89. What is the weight of a circular plate of copper 11 in. in diameter and f in. thick, copper weighing 549 Ibs. per cu. ft.? 90. What is the weight of a slate blackboard 19.5 ft. long, 3.5 ft. wide, and f- in. thick, the specific gravity of the slate being 2.848 ? 91. If 31 cm 3 of gold weighs 599 g, find the specific gravity of gold. 92. Find the time between 3 and 4 o'clock when the hour and minute hands of a watch are (1) together, (2) opposite, (3) at right angles to one another. 93. A locomotive is going at the rate of 45 mi. per hr. ; how many revolutions does the drive-wheel, 22 ft. in circumference, make in a second ? 94. How many seconds will a train 184 ft. long, traveling at the rate of 21 mi. per hr., take in passing another train 223 ft. long, going in the same direction at the rate of 16 mi. per hr.? how many seconds if they are going in opposite directions ? 95. How many telegraph poles 58 ft. apart will a traveler by train going at the rate of 48 mi. per hr. pass in a minute ? 96. A train leaves C for M at 9 A.M., traveling at a uniform rate of 15 mi. per hr. ; an express train leaves M for C at 10 A.M. at 40 mi. per hr. ; at what time will they meet, and at what distance from C, the distance from C to M being 50 mi. ? 97. A sledge party travels northward on an ice-floe at the rate of 12 mi. per da. ; the floe is itself drifting eastward at the rate of 100 rods per hr. ; at what rate is the sledge really moving ? 98. A starts out on a bicycle at the rate of 8 mi. per hr. ; after he has gone 3 mi., B follows at the rate of 10 mi. per hr. ; after how many hours will B overtake A ? 186 HIGHER ARITHMETIC. 99. Out of a circle 18 in. in diameter there is cut a circle 13.5 in. in diameter ; what per cent of the original circle is left ? 100. A laborer asks to have his time changed from 10 hrs. to 8 hrs. a day without decrease of daily pay ; by what per cent of his hourly wages does he ask them to be increased ? 101. There was formerly in use a discount known as " true dis- count," which was the interest on the present worth of a given amount, that is, on such a sum as placed at interest for the given time should equal the given amount ; show that the bank discount equals the true discount plus the interest on the true discount. 102. What is the present worth of $1356.80 due in 1 yr. 4 mo., the rate being 4|% ? 103. What is the present worth and true discount of $1120 due in 2 yrs. at 6% ? 104. What is the present worth and true discount of $1000 due in 11 mo. at 5% ? 105. What is the present worth and true discount of $1430.40 due in 16 mo., the rate being 3% ? 106. A dealer sells a machine for $80, taking a note to be paid without grace in 8 equal monthly payments without interest ; after two payments he takes the note to a bank and discounts it at 6% ; find the proceeds. 107. A dealer sells a bicycle for $50, taking a note to be paid without grace in 10 equal monthly payments without interest ; after 4 payments he discounts the note at 5% ; find the proceeds. 108. A man sells a lot for $500, taking a note to be paid without grace in 10 equal monthly payments with interest at 5% ; after half of the payments have been made he discounts the note at 6% ; find the proceeds. 109. A bookseller agrees to furnish a certain number of books for $66.30, after giving a discount of 15% upon the list prices. He him- self gets a discount of 25%. What is his gain ? 110. A house cost $5000 and rents for $25 a month, with $25 to pay annually for repairs and $50 for taxes ; what is the difference in the income from this and from the same money invested in 6% stock at 96? 111. A certain set of books in 10 volumes is offered for $66.60 cash, or $69 payable $5 cash and $5 each succeeding month until the total amount of $69 has been paid. Which is the cheaper arrange- ment, money being worth MISCELLANEOUS EXERCISES. 187 112. A man invests f of his money at 4% and the rest at 4|%. His annual income is $997.60. What is the ratio between the two parts of his income ? What are the two amounts invested ? What is the average interest upon his capital ? 113. On a mortgage for $1700 dated May 28, 1900, there was paid Nov. 12, 1900, $80; Sept. 20, 1901, $314; Jan. 2, 1902, $50; Apr. 17, 1902, $160 ; what was due Dec. 12, 1902, at 6% ? 114. Which investment yields the better rate of income, one of $4200, yielding $168 semi-annually, or one of $7500, producing $712.50 annually ? 115. If an agent's commission is $290.40 when he sells $11,606 worth of goods, how much would it be when he seljs $7416 worth ? 116. What is the difference on a bill of $1750 between a discount of 40% and a discount of 30% 10% ? 117. What per cent on the cost is gained by selling goods at the list price, they having been purchased at "a quarter off " ? 118. How long will it take a sum of money to double itself at 6% simple interest ? at 6%, compounded semi-annually ? at 6%, com- pounded quarterly ? (Answer correct to 0.001.) 119. The cost of maintaining the life-saving service of the United States for a certain year was $1,345,324 ; the value of the property saved was $9,145,000, and the value of the property involved was $10,647,000 ; these values were what per cent of the cost of maintain- ing the service ? 120. The average number of hours which it takes for a letter to go from New York to London is 162.5 by one route and 176.7 by another ; what per cent is gained by taking the shorter passage ? 121. The total valuation of farms in the United States in a certain year was $13,279,252,649, of the implements $494,247,467, and of the live stock $2,208,767,573; the value of farm products was $2,460,107,454. The value of the products was what per cent of the total valuation of farms, implements, and live stock ? 122. After decreasing 13% from the acreage in a certain year, the number of acres in corn in the United States the following year was 62,671,724 ; what was the acreage in the former year, correct to 1000 ? 123. The United States produced 4,019,995 tons of steel in a cer- tain year, and that produced by other countries constituted 65.26% of the total production of the world ; required the total production. 124. In a certain year the combined capital of all the fire insurance companies of the United States was $70,225,220 ; the total income for 188 HIGHER ARITHMETIC. that year was $175,749,635, the amount paid for losses $89,212,971, for expenses and surplus $54,203,408, the balance going to dividends ; what was the average rate of dividend ? 125. 59.9% of the total production of copper in the world comes from outside the United States ; how many tons does this represent, the production of the United States being 105,774 tons annually ? 126. The dividends paid by the National Banks in a certain year were $45,333,270, the capital being $672,951,450; what was the average rate of dividend ? (Answer to 0.1%.) 127. Find z, given (a) 5* = 20 ; (6) 100* = 2 ; (c) 8* = 100 ; (d) 4*+! = 50. 128. Show that log (1 + 2 + 3) = log 1 + log 2 + log 3. Is this true f or 1 + 2 ? f or 1 + 2 + 3 + 4 ? 129. Why is 10 the most practical base for a system of logarithms ? Why can 1 not be the base of a system ? 130. What is the logarithm of 125 to the base 6 ? of 729 to the base 3 ? of 64 to the base 4 ? to the base 2 ? 131. Show that if 3 is the base of the system of logarithms, log 81 + log 243 = log 19683, by finding each logarithm. 132. We write numbers on a scale of 10 ; that is, we write up to 10, then to 2 10, then to 3 10, 10-10 Show that we might write on a scale of 9 using one less digit ; or on a scale of 8 using two less digits, and so on to a scale of 2. 133. How many characters are necessary for the scale of 10 ? of 8 ? of 12 ? of 100 ? of n ? 134. What characters are needed in the scale of 2 ? Write 15 on the scale of 2. 135. Write the numbers 12, 144, 145, 155, 1728, 1738 on the scale of 12. (The letters , e may be taken to stand for 10 and 11.) 136. Add 1164 and 2345 on the scale of 7. 137. From 3542 take 1164 on the scale of 7. 138. Multiply 3542 by 4 on the scale of 7. 139. Write 25 and its nth power, on the scale of 5. 140. Write the numbers 15 and 21 on the scale of 2 ; multiply one by the other and hence show that if we used the scale of 2 it would not be necessary to learn the multiplication table. 141. On the scale of 10 one-half is written 0.5 ; on the scale of 12 it is written 0.6. Write the fractions , |, , $ as decimals; also write them on the scale of 12, and hence show that the scale of 12 is adapted to computation better than the scale of 10. -APPENDIX. NOTE I, to p. 36. THE THEORY OF SQUARE BOOT CONTINUED. Trial divisor. The expression 2f is often called the trial divisor, 2f+n being called the complete divisor. It will be noticed in the example on p. 36 that the second trial divisor, 2/ 2 , equals the sum of the first complete divisor, 2/ x + n i> and n^ In other words, the new trial divisor can always be found by adding n to the last com- plete divisor. For 2/i + HI, added to m, equals 2 (/i + m). But /i + m= / 2 , .-. 2 (/i + m) = 2/ 2 . In the extraction of the square root of a long number like 299,066.7969, the ordinary abridged process may be still further shortened. In this example, it will be noticed that after the first three figures of the root have been found the next two can be found by merely dividing the remainder 950.79 by the trial divisor. 5 4 6. 8 7 0.87 29<90'66.79'69 1092 ) 950.79 104 4 90 77.19 1086 7466 .75 1092.8 950.79 1093.67 76.55 69 That this is true in general in the case of a perfect square will now be proved. 190 HIGHER ARITHMETIC. Abridgment theorem. After n -f 1 figures of the square root of a perfect second power (or square) have been found, the next n figures can be found by dividing the next remainder by the next trial divisor. 1. Let Vs =/ + , where s is a perfect second power, / contains n + 1 figures already found, and x contains n figures to be found. 2. v the n + 1 figures of / are followed by the n figures of x, f has the value of a number of 2 n + 1 figures. 3. From 1, s = / 2 + 2/x + x*, or ^^ = x + ^ That is, if */ ^/ the remainder s / 2 is divided by the trial divisor 2/, the quotient x 2 is x plus the fraction x 2 4. .-. if it is shown that is a proper fraction, the integral part of x 2 x + is x, the remaining part of the root, and the theorem is proved. */ 5. v x contains n figures, .-. x < 10", and x 2 < 10 2 . 6. v / has the value of a number of 2 n + 1 figures, .-. / < 10 2n , and 2/ < 2 10 2w . x 2 10 2n 7. .- gy < 2 . 1()2M or i, a proper fraction. Exercise. Show that the abridgment theorem holds only for a perfect second power as stated, by considering the cases of 152,399,000, and 152,399,025, the latter being a perfect second power. NOTE II, to p. 40. THE THEORY OF CUBE BOOT CONTINUED. Trial divisor. The expression 3/ 2 is often called the trial divisor, 3fn -\- n 2 being called the correction, and 3/ 2 + 3/w + n 2 the complete divisor. It will be noticed in the example on p. 39 that the second trial divisor, 3/ 2 (or 780,300), equals the sum of the first complete divisor (3/ 2 + 3fn + n 2 , or 765,100) and the correction (Sfn + n 2 , or 15,100) and the square of n (n 2 , or 100). This is always true in cube root. APPENDIX. 191 For / t =/i H- m. .'. 3/ 2 2 = 3 (/i + m) = 3A 2 + 6/mi + 3 m = 3/i 2 + 3/ini + wi 2 (the preceding complete divisor) + 3/iWi + wi 2 (the correction) + ni 2 (the square of n). And the same reasoning holds for all trial divisors in cube root. This is much shorter than the operation of squaring / and multi- plying by 3 each time. Abridgment theorem. After n + 2 figures of the cube root of a perfect third power have been found, the remain- ing n figures can be found by dividing the next remainder by the next trial divisor. 1. Let vT = /+x, where t is a perfect third power, / contains n + 2 figures already found, and x contains n figures to be found. 2. v the n + 2 figures of / are followed by the n figures of x, / has the value of a number of 2 n + 2 figures. 3. From 1, t =/ + 3/ 2 x + 3/x 2 + x 3 , ... ^^ = x + + -j~ ' That is ' if the remainder t ~f* is divided by the trial divisor 3/ 2 , the quotient is x plus the fractions x 2 . x* 7 and x 2 x s 4. .-. if it is shown that + -^-^ equals a proper fraction, the integral part of the quotient is x and the theorem is proved. 5. v x contains n figures, .-. x < 10 M , x 2 < 10 2 , and x 3 < 10*. 6. v / has the value of a number of 2 n + 2 figures, ... /< iQ2+i, /2 < lQ4+2, and 3/ 2 < 3 10 4 + 2 . x 2 7- x 2 10 2 1 " / K x 3 ,2+l' ' A 10 ' d 10 3 1 3/ 2 '^ 3 . x 2 x . !Q4n+2 ' 01 3 lQn+2 3 1 1 <- 4- '/ ' 3, r 2 ^io 3-io+ 2 ' 8. .'. j + ~ < + 37-2 , a proper fraction. Exercise. Show that with the first three figures of the square root of 14,696,712,600,000,000, obtained by the ordinary process, the next two cannot be correctly found by division. (Similar conditions evi- dently exist in the theory of cube root.) 192 HIGHER ARITHMETIC. Three cube roots. (Omit if the class has not studied quadratic equations and imaginaries.) Just as 4 has two square roots, + 2 and 2, so 8 has three cube roots, 2, 1 + V 3, and 1 V 3, as may easily be proved by cubing. So in general, every number has three and only three cube roots. 1. For let x 3 n ; then if the value of x is found the cube root of n is known. 2. From 1, x 3 n = 0. 3. .-. (x Vn)(z 2 + x Vw + \^ = 0, by factoring (2). 4. This equation is satisfied if either x \ = 0, or x 2 + x tt + Vw 2 = 0. 5. If x Vn = 0, then x = Vn, the ordinary arithmetical cube root. 6. If x 2 + x Vn + Vn 2 = 0, then, by solving the quadratic equation, 7. That is, the three cube roots of n are Exercises on the three cube roots. 1. What are the three cube roots of 1 ? Verify your answer by cubing. 2. Show that each of the imaginary cube roots of 1 is the square of the other. 3. Show that the sum of the three cube roots of any number is zero. 4. Show that the product of the three cube roots of any number is that number. 5. Show that any number has four fourth roots. NOTE III, to p. 146. THE THEORY OF COMPOUND INTEREST CONTINUED. The theory gf compound interest affords an application of logarithms for those who have studied Chap. XI. The work on p. 193, excepting steps 1-4 and Ex. 1, should be omitted by others. APPENDIX. 193 E.g., by logarithms the computation on p. 145 becomes simple : log 1.02 = 0.0086 6 log 1.02 = 0.0516 log 150 = 2.1761 2.2277 = log 168.9, as near as the result can be obtained by the small table on p. 114. The principal formulae are deduced as follows : 1. Let p the number of dollars of principal, r the rate at which the interest is to be compounded annually, t the number of years, and a 1} a 2 , a s , ..... a t the amounts for 1, 2, 3, ..... t years. 2. Then, ai = p-{-rp = (l + r)p, a, = (1 + r)p + r (1 + r)p = (1 + rfp, a 8 = (1 + r) 8 p, and in general 3. a t =(l + ryp. '- 5. From 3, log a, = t log (1 + r) + log p. . log , logy log (! + ) 7. And from 5, r = antilog loga '~ logy - 1. If the interest is compounded semi-annually for t yrs. at r% a year, the amount is evidently the same as if the interest were com- pounded annually for 2 1 yrs. at -% a year. u Exercises. 1. What is the amount of each of the following ? FACE. TIME. RATE. COMPOUNDED. (a) $250 3 yrs. 2 mo. 10 da. 4% semi-annually. (6) $75 1 yr. 4 " 3 " 4% quarterly. 2. Find the principal, given the AMOUNT. TIME. RATE. COMPOUNDED. (a) $384.03 5 yrs. 5% semi-annually. (6) $162.36 2 " 4% 3. In how many years will $200 amount to $310.26 at 5%, com- pounded annually ? 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