SlLVBfl,BUHDETT AMD mill MiiiiMlMtiiBiiiMiiii m MEMOIRIAM Irving Str Ingham Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/arithnewadvancedOOcookrich Cl)e iljotmal Cour0e m iQumber THE NEW ADVANCED ARITHMETIC BY JOHN W. COOK PRESIDENT ILLINOIS STATE NORMAL UNIVERSITY ANT) MISS N. CROPSEY ASSISTANT SUPERINTENDENT CITY SCHOOL* INDIANAPOLIS, INDIANA SILVER, BURDETT & COMPANY New York . . . BOSTON . . . Chicago 1902 t^ V Copyright, 189S, 1896, Bv Silver, Burdett and Company. H. M. PLIMPION i. CO.. PRINTERS 4 BINDERS, NORWOOD, MASS., USA. PREFACE. IT has seemed to the authors of the J^grmal Course IN Number that there is room for another series uf Arithmetics, notwithstanding the fact that there are many admirable books on the subject already in the field. The Elementary Arithmetic is the result of the ex perience of a supervisor of primary schools in a leading American city. Finding it quite impossible to secure satisfactory results by the use of such elementary arith- metics as were available, she began the experiment of supplying supplementary material. An effort was made to prepare problems that should be in the highest degree practical, that shonld develop the subject systematically, and that should appeal constantly to the child's ability to think. Believing that abundant practice is a prime necessity to the acquisition of skill, the number of prob- lems was made unusually large. The accumulations of several years have been carefully re-examined, re-arranged, and supplemented, and are now presented to the public for its candid consideration. Not the least valuable fea- ture of this book is the careful gradation of the examples, securing thereby a natural and logical development of number work. No space is occupied with the presenta- tion of theory, — that side of the subject being left to the succeeding book. The first thoughts are what and Jww, — these so presented that the processes shall be 800574 It prefa ce. easily comprehended and mastered. Subsequently, the why may be intelligently considered and readily under- stood. The Advanckd Arithmetic is the outgrowth of a somewhat similar experience in the class-room of a teach- ers' training-school. For many years an opportunity was afforded to study the effects upon large numbers of pu- pils of the current methods of instruction in arithmetic. The result of such observation was the conviction that the rational side of the subject is seriously neglected. An effort was made to supplement the ordinary text-book by a study of principles and by explanations of pi'ocesses. The accumulations of fifteen years have been edited with all of the discrimination of which the authors were capa- ble. Great care has been exercised in the presentation of principles and in the formulation of processes, to the end that the learner shall have every facility for the use of his reasoning powers, and at no point be relieved from the proper exercise of his mental activity and acumen. It is hoped that tlie book may contribute somewhat to the movement, now so happily going oil, that looks toward the disestablishment of the method of pure au- thority, and the establishment of a method that makes its appeal to intelligence and reason. The authors desire to express tlieir appreciation of the excellent suggestions offered by many friends; but espe- cial tlianks are due Professor David Felmley, of the Illinois State Normal School, for his discriminating criticisms and valuable assistance. THE AUTHORS. SUGGESTIOIS^S. METHOD is determined chiefly by aim. The answer whicli the teacher makes to the question, " Why should boys and girls study arithmetic?" will guide him in the details of instruction. Arithmetic is one of the traditional "three R's." Some knowledge of its processes is necessary to any degree of intelligence. Its highly practical character is conceded by every one. The arithmetical operations employed in ordinary busi- ness affairs are simple, but they must be performed with absolute accuracy and with great rapidity. They are based, primarily, upon the memory. The necessity for perfect familiarity with the fundamental facts of number becomes apparent. Neither accuracy nor rapidity is poS' sible without a thorough mastery of the primary work. This mastery is acquired through constant repetition of the old in connection with the acquisition of the new. One of the teacher's maxims, constantly, must be "Ee- viewf Review!! REVIEW!!! But arithmetic has another and a higher function. It must cultivate t^at quick intelligence which is able to analyze given conditions and determine what should b& done in the particular case. A true problem in arithmetic is a statement or seriea of statements in which something is told and somethiag vi aUGGESTIONS. is asked ; the answer to what is asked being implied iv what is told. The chief activity of the child, in the solu- tion of true i)robleius, is, first, tlie analyzing of given con ditions, and second, the performing of certain operations. The former of these activities should never become me- chanical; the latter should pass into the mechanical stage as early as possible. The study of advanced arithmetic differs in certain essential particulars from all other studies in the common- school course. The thought lacks the continuity of his- tory or geography. The work consists of a series of efforts that are more or less distinct. Each problem stands alone in its statement; but the operations involved in it fall under certain general types. Arithmetic, consequently, requires a constant dealing with both particular and gen- eral notions. These general processes may be discovered in illustrative problems, and the possible number of them be absolutely exhausted. When this has been done, tlie "case" arrangement of problems should cease, and the pupils should be permitted to perform that analytic ac- tivity that discovers conditions in a problem, and that synthetic activity that unites it to its proper class. The problems in this book have been prepared with this thought in mind. Another phase ot arithmetic work should be clearly appreciated. It surpasses all other studies in the low;er grades in the number of its generalizations and the ease with which they are made. Tlie generally accepted prin- ciples tnat instruction should proceed from the concrete to the abstract, and from the individual to the general, have constant application. But the abstract should be the result of a conscious abstraction, and the gwieraL SUGGESTIONS. vu should be a conscious generalization. As illustrating this tliought, and showing how the general is reached from the individual, particular attention is called to the method of deriving rules from processes. The formulation and statement of a rule is a generalization. It is a fundamental principle of true teaching, that whatever is done by the pupil shall be accomplished, through his conscious, personal effort, and that whatever his acquisition, it shall be consciously his own, — not his in the memory -alone, which is the same as his in the book, but his vitally and substantially, as the blood in his veins or the innate ideas of right and wrong. Such knowledge is of a rooted and growing order that gives satisfaction and power to its possessor. To suggest and stimulate such teaching, and to secure such growth, many questions are asked and many direc- tions are given, in this book, which are designed to throw the pupil upon his own resources. The questions cannot be answered by any statements found on the pages, — no need to tax the memory for words and phrases ; yet all the questions and directions are simple and easily to be answered by the pupil, if he has thought his way clearly. Too great emphasis cannot be given to the statement already made, that the work of analyzing should never become mechanical. While there is value in concise and definite formulas, there is infinitely greater value in free- dom. The mind should be free to discover conditions, relations, and sequences; it should be as free in stating conclusions and results ; but this freedom cannot exist if the reasoning is compelled, per force, to follow a memo- rized formula. The forms of analyses given in the fol- viii SUGGESTIONS. lowing pages are presented as models to be studied and mastered, but not to be memorized. If the pupil, in his own way and in different words, shall clearly present the steps of reasoning and draw the correct conclusion, his work should be approved. An additional suggestion must suffice. The teacher's "knowledge of the subject should be organic. Arithmetic should be recognized as a science that is deduced from the idea of addition. When the so-called fundamental processes have been mastered, little remains but repeti- tion. The fraction differs from the integer in that it in- troduces a double unity. The decimal fraction differs from the common fraction in the method of expressing its denominator Percentage is "a case" in decimal frac- tions. Compound numbers differ from simple numbers because of the introduction of variable scales, etc., etc. As the power to generalize relieves the mind from the overwhelming burden of a countless multitude of indi- viduals, so each new step in advance is easily held if correllated with the fundamental ideas. J. W. C. CONTENTS. PAKT I. Section I. Definitions . , Notation . . , Numeration . Roman Notation Reduction . . Section II. Addition . Section III. Subtraction Section IV, Multiplication .... Multiplication by Factors Surface Measure . . . Section V. Division Long Division .... Division by Aliquol; Parts Division by Factors . . Law of Signs .... Properties of Numbers . Tests of Divisibility , . Factoring ..... Cancellation 11 22 45 49 54 55 59 65 67 72 73 FAoa Section VI. Fractions . . . . . 76 Reduction 78 Addition of Fractions . . 83 Least Common Multiple 83 Subtraction of Fractions . 90 Multiplication of Fractions 96 Division of Fractions . . 103 Complex Fractions . . . 108 Decimal Fractions . . 125 Numeration of Fractions 126 Notation of Fractions 127 Reduction of Decimals • 129 Addition of Decimals 132 Subtraction of Decimals 133 Multiplication of Decimals 135 Division of Decimals . . 138 Measurement of the Circle 143 Federal Money .... . 147 Bills and Statements . . 148 Denominate Numbers . 151 Measures of Length . . . 151 Surface Measure . . . . 156 Surveyor's Measure . . 159 United States Surveys . . 160 Measures of Volume . . . 162 Wood Measure . . . . 163 Lumber Measure . . . . 165 Measures of Capacity . . 167 Weight . . . 168 CONTENTS. English Money French Money . . German Money Circular Measure . Longitude aud Time The International Date Lite Page 172 172 173 173 174 '80 Calendar 182 Miscellaneous Tables . . 183 PART IL Section VII. Percentage 203 Section VIII. Applications of Percentage 224 Profit and Loss .... 224 Commission 231 Commercial Discount . . 234 Stocks, Bonds, and Broker- age 236 Taxes 242 United States Revenue . . 245 Insurance 248 Property Insurance . . . 249 Life Insurance 250 Interest 255 Accurate Interest .... 266 Partial Payments .... 273 Compound Interest . . . 274 General Problems in Simple Interest 278 Present Worth and True Discount 2d-2 Bank Discount 285 Exchange 289 Foreign Exchange . . . 293 Equation of Pa\'ments . . 295 Section' IX. I^atio 306 Proportion 307 Compound Ratio . . . . 311 Partnership 314 Section X. Involution . . ... 317 Evolution ... . . 318 The Right Triangle . . . 324 Cube Root ..... 328 Cube Roof of Decimal Frac- tions 331 Cube Root of Common Frac- tions 332 The Cone 341 Prisms and Pyramids . . S42 General Reviews .... 344 Algebraic Questions . . . 356 The Literal Notation . . 356 Evaluation of Algebraic [ex- pressions 358 Axioms 361 Equations containing Frac- tions 364 Positive and Negative Quan- tities 369 Law of Signs in Division . 375 Appendix. Greatest Common Divisor . 379 Least Common Multiple . 384 Mariner's Measure . . 386 Average of Accounts . . 386 Origin of Units .... 387 Metric System of Weights and Mea.«iirps .... 388 THE NORMAL COURSE IN NUMBER THE NEW ADVANCED ARITHMETIC. ^att I. SECTION L DEFINITIONS. 1. Measuring is the process of finding how many times a quantity contains a part of itself which is taken as a stand- ard. lUustrate. 2. That portion of a measured quantity which is used as a standard is called a Unit. When we count a basket of eggs, we measure the quantity of eggs by using one of them as a standard. Such a unit is a natural unit. When we measure a quantity of cloth, we use a portion of itself, called a yard, as a standard. Such a unit is an artificial unit. 3. From the repetitions of the unit in counting or measur- ing, the successive numbers, one, two, three, etc, arise ; thus, Number is that which answers the question " How many ? " 4. Arithmetic is the science which treats of number and the methods of employing it in computation. 5. To simplify counting or measuring, units are gathered into equal groups, each of which forms a new unit. Thus, in measuring a basket of eggs, they are grouped into dozens. The quantity may be expressed as a number of single eggs or as a number of dozens. 1 2 NEW ADVANCED ARITHMETIC. A Decimal System of numbers is a system in which ones are grouped into tens ; tens into tens of tens, or hundreds ; hundreds into tens of hundreds, or thousands ; thousands into ten-thousands, etc. Ones are units of the first rank, or order ; tens are units of the second order ; hundreds, of the third order, etc. Illustrate these groupings with bundles of splints. The scale in any system of numbers is the numbei of units in each order required to form one of the next higher. NOTATION. 6. The art of expressing numbers by means of characters is Notation. 7. A system of Notation which will express all nuipbers must include a set of characters to represent numbers and the laws for using them. 8. The Arabic System of Notation employs ten charac- ters called figures. They are 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven). 8 (eight), 9 (nme), (cipher). 9. Any given figure always expresses th^ same number of units, but the order or kind of units is expressed by the place in w^hich the figure is written. Ones stand in the first place, tens in the next place to the left, hundreds in the third place, etc. ; thus, A figure standing in any place expresses units ten times as large as if standing one place to the right The cipher is used to fill vacant orders. NUMERATION. 10. The art of reading numbers expressed by figures ki Numeration. 11. Three orders form a period. The name of the lowest order in each period is ones ; of the next higher is tens ; of the third is hundreds. NUMERATWX. 8 The names of the first twelve periods in their order are as follows : 1. Units. 2. Thousands. 3. Millions. 4. Billions. 5. Tril- lions. 6. Quadrillions. 7. Quintillions. 8. Sextillions. 9. Sep- tillions. 10. Octillions. 11. Nonillions. 12. Decillions. Note. — The meaniug of-the prefix iu the word biUlon is two; iu tlie word trillion is three ; in quadrillion is four, aud so on. Observe that the number of auy period above millions is two more than the meaning of the prefix in the uame of that period. 12. Arraugement of orders aud periods. Trillions. Billions. Millions. Thousands. Units. 865,406,38 2,104,579 Note. — Learn the names of the periods in their order from left to right. 13. To read a number, group the figures into periods, be- ginuing at the right, and separating tlie periods by commas. Beginning at the left, read the number in each period as if it stood alone ; then add the name of the period. Note. — The English system of Numeration is in use in England and upon tlie continent of Europe, except in France. It employs six orders for a period. In studying this system the meaning of the names of the periods is made plain. A million is a thousand thousand. A billion is the square of a million ; a trillion, the third power of a million; a quadrillion, the fourth power of a million, etc. 14. Eead the following numbers : 1. 2345. 6. 250849. 11. 683471. 2. 4638. 7. 381307. 12. 829406. 3. 7912. 8. 408391. 13. 200619. 4. 3105. 9. 716004. 14. 100054. 5. 26853. 10. 5Q0836. 15. 973070. 16. 253087G. 23. 17. 3890432. 24. 18. 470G3502. 25. 19. 50780439. 26. 20. 480983048. 27. 21. 379068452016. 28. 22. 690750142953. 29. A^£;H'' .4Z)r.4.VC£i) ARITHMETIC. 85968345620961. 6390086133859016. 7000492587291563295. 8230075913748426950. 27. 2005300861943186627. 10030006729062127390037. 300750916400853269057040. ]S;oTE. — Freciueiit dictatiou exercises should be given with successively larger numbers until pupils have acquired proficiency in writing numbers. 15. Before writing the following number's, tell how each will appear when written. Illustration. Three thousand eight hundred seven is ex- pressed by writing the following : three, comma, eight, cipher, seven. 1. Forty thousand six. 2. Ninety-seven thousand five hundred twelve. 3. Three hundred sixty-nine thousand twenty- four. 4. Four million eight thousand two. 5. Fifty-six million nineteen thousand thirty-three. 6 Eighty-one million five hundred thirteen thousand two hundred fifty-one. 7o Three hundred million ninety thousand four. 8 Five billion six million seven thousand eight. 9. Seventy -two billion six hundred thirty-five thousand two hundred fifty-one. 10. One hundred three billion two nuUion seventeen thousand one hundred four. 11 Two trillion three billion four million five thousand six. 12 Nmety-one trillion two hundred seven billion sixty- nine million four thousand three 13 Eighty-six trillion one million twenty-three. NUMERA TION. 5 14. Two hundred sixteen trillion five hundred thousand. 15. Nineteen trillion four. XuTE. 1. Teachers should supply dictation exercises in writing nam- bers until a good degree of proficiency is acquired. 2. Observe which of the number names are eompound words. 3 Note that the word " aud " is not used in these exercises. 16. THE ROMAN NOTATION. This method expresses number by the use of certain print letters. They are I, V, X, L, C, D, M. 1 = 1, V = 5, X=10, L = 50, C = 100, D = 500, M = 1000. Their use is determined by the following PRINCIPLES. 1. Repeating a letter repeats its value. II = 2, XX = 20. 2, When a letter is placed after one of greater value, the tTso express a number equal to the sum of their values. XV =: 15, CI= 101. 3 When a letter is placed before one of greater value, the two express a number equal to the difference of their values. IX = 0, XC = 90. (Limited to IV, IX, XL, and XC.) 4. When a letter is placed between two, each of greater value, its value is taken from the sum of their values. XIX r= 19, XIV = 14. 5. Placing a dash over a letter multiplies its value by a thousand X = 10,000, M = 1,000,000. EXERCISES 1 Express by the Roman characters all numbers from one to one hundred 2, 125. 5 419c 8. 3, 263c 6. 599o 9. 4, 379 7. 648. 10. Note Use dictation exercises freely. 2A 752 11. 1776. 1066. 12. 1799. 1492. 13. 1896. 6 NEW ADVANCED ARITHMETIC. 17. REDUCTION. 1. Reduction is the process of changing the unit of a number without changing its value. 2. Express 6 pints in quarts ; 8 quarts in gallons ; 6 feet in yards ; 32 ounces in pounds ; 20 mills in cents ; 300 cents in dollars. 3. Have tliese numbers been changed to a larger or to a smaller unit? The process of reducing a number to larger units is called Reduction Ascending. 4. How are pints reduced to quarts? quarts to gallons? feet to yards? ounces to pounds? mills to cents? ones to tens? tens to hundreds? tens to thousands? cents to dol- lars? Give many similar examples. DIRECTION. 5. To reduce a ntitnber to a tiu/tnber of larger miits, divide it by the number of the given units which -makes one of the larger units. 6. Express 3 quarts in pints; 3 gallons in quarts; 4 yards in feet; 3 dimes in cents; 5 tens in ones; 6 hundreds in tens; 7 thousands in hundreds, in tens. 7. Have these numbers been changed to a larger or to a smaller unit? The process of reducing a number to smaller units is called Reduction Descending. 8. How are quarts reduced to pints? gallons to quarts? yards to feet? dimes to cents? tens to ones? hundreds to tens? thousands to luuidreds? to tens? DIRECTION. 9. To reduce a number to a number of smaller unitSf multiithi it bi/ the number of tlie smallet units to which one of the larger units is equal. REDUCTION. 7 Note. — A knowledge of the fundamental nature of the decimal STStem is of the utmost importance in aritlniietical operations. This is obtained through ]iractice in the two forms of Keduction. Multiply ex- amjiles like the following 18. 1. In 50,000 there are how many tens? hundreds? thousands ? ten-thousands ? 2. In 17,000,000 there are how many thousands? hun- dred-thousands? tens? hundreds? ones? ten-thousands? 3. In 38,000 mills there are how many cents? dimes? dollars ? 4. In 65 dollars there are how man}' dimes? cents? mills? 19. 1 is what part of 10? One ten is what part of 100? One hundred is what part of 1000? In 1,111 each unit is what part of the unit standing in the first place at its left^ 20. It is customary to fix the place of ones by placing a period at its right, thus : 1 . When the period is thus used it is called the decimal point. From what has been observed what kind of unit will the right-hand figure in 1.1 express? What is r)x\Q tenth of one tenth? What, then, is the kind of unit expressed by the right-hand figure in 1.11? What is one tenth of one hundredth? What kind of unit is ex- pressed by the right-hand figure in 1,111? Similarly, show what kind of units is expressed by a figure in the fourth place at the right of tile decimal point; in the fifth place; in the sixth place. 21. Table of six places at the right of the decimal point. c § o c 5 a H W H H W S .236418 t .6 11. 2. .8 12. 3. .2 13. 4. .24 14. 5. .37 15. 6. .04 16. 7. ol27 17. 8. .209 18. 9. .842 19. 10. .094 20. 31. 16.07 (Read : 32. 25.375 i-LlALi 33. 239.004 34. 508.0089 35. 2851.3675 36. 3750.1049 21. .00836 22, .04826 23. .39627 24. .30861 25. .000003 26. .000072 27. .000739 28. 004936 29. .003972 30. .386492 8 NEW ADVANCED ARITHMETIC. 22. The name of any number is the same as the place in which its right-hand figure stands. Read the follov/ing numbers : .007 .491 .0005 o0036 .0683 .2758 .3085 .4902 .00006 . .00034 16 and 7 hundredths or as 1607 hundredths.) 37. 6.00039 38. 28643907502 39. 10000.807501 40. .197527 23. Perfect familiarity with the names and numbers of the places at the right of the decimal point is necessary for accurate and rapid writing. The name of a number is de- termined by the place of its right-hand figure. In writing numbers like the foregoing correctly the first time two things must be known : the number of figures required to express the number; the place in which the right-hand figure must stand to express the kind of units. 24. Write the following numbers : 1. Twenty-three hundredths. 2. Seven tenths. 3. Sixty-nine hundredths. NOTATION. 9 4. One hundred forty-eight thousandths. 5. Two hundred eleven thousandths. 6. Six hundred ninetj'-three thousandths. 7. Nine ten-thousandths. 8. Four ten-thousandths. 9. Thirt3'-two ten-thousandths. 10. One hundred eighty-three ten-thousandths. 11. Nine hundred seven ten-thousandths 12. One thousand five hundred seventy=six ten thou- sandths. 13. Nine thousand four hundred three ten-thousandths. 14. Eight hundred-tliousandths. 15. Five hundred- thousandths. 16. Forty-five hundred-thousandths. 17. Ninety-four hundred-thousandths. 18. Three hundred fifty-two hundred-thousandths. 19. Eight hundred ten hundred-thousandths. 20. Four thousand seven hundred nineteen hundi-ed- thou- sandths. 21. Seven thousand twenty-three hundred-thousandths. 22. Two thousand sis hundred-thousandths. 23. Thirty-four thousand six hundred twentj^-one hundred- thousandths. 24. Fifty-nine thousand one hundred six hundred-thou- sandths. 25. Nineteen thousand three hundred-thousandths. 26. Fifty thousand one hundred-thousandths. 27. 25 millionths. 32. 5008 millionths. 28. 8 millionths. 33. 37592 millionths. 29. 63 millionths. 34. 8090G6 millionths. 30. 478 millionths. 35. 26 and 68 hundredths. 31. 2895 millionths. 36. 94 and 39 thousandths. 10 NEW ADVANCED ARITHMETIC, 37. 290 aud 463 hundred-thousandths. 38. 40073 and 50093 millionths. 39. 61 aud 13 millionths. 40. Two thousand five and three thousand forty-six mil- lionths. 25. Numbers like exercises 1 to 30, Art. 22, are called Decimal Fractions. Numbers 31 to 39 are called Mixed Decimals. Name the Decimal Fractions in Art. 24. Name the Mixed Decimals in the same section. 26. REDUCTIONS. 1. In .6 there are how many hundredths? How many thousandths? Write each of the numbers. How many ten- thousandths? Write the number. How many millionths? Write the number. Read 3.o as hundredths, as thousandths, ten-thousandths, hundred-thousandths, millionths. Write each of the numbers. In .600000 there are how many tenths? ten-thousandths? hundredths? hundred-thou- sandths ? millionths ? 2. In 3.2 there are how many hundredths? How many ten-thousandths? How many millionths? Write each of the numbers. In 5.000000 how many hundred-thousandths? hundredths? thousandths? tenths? Write each of the numbers. 3. In 400000, there are how many tens? How many hundreds? How many ten-thousands? How many thou- sands? ADDITION. 11 SECTIOX 11. ADDITION. 27. Like numbers are numbers that are made by repeating the same unit. 28. The Sum of two or more like numbers is a number which contains all of their units. 29. The sign + Cpl^s) between two numbers shows that their sum is to be found. 30. At first, to unite two numbers, we count the units of the second upon the first, thus : In uniting four blocks and three blocks, we think (or say) four, five, six, seven, as we place the three blocks one at a time with the four. In this way, hy counting objects, we learn and commit to memory the following forty-five sums: 12 3 2 4 3 5 4 3 1112 12 12 3 6 5 4 7 G 5 4 8 7 1 2 3 1 2 3 4 1 2 — — - — — — ■— 6 5 9 8 7 6 5 9 8 3 4 1 o 3 4 5 2 3 — - — — — — ~™ "" 7 6 9 8 7 6 9 8 7 4 5 3 4 5 6 4 5 6 — — — — — — ~ — — 9 8 7 9 8 9 8 9 9 5 6 7 n 7 7 8 8 9 31. This series of forty-five sums of the nine primary num- bers taken in twos is called the Addition Table. 12 NEW ADVANCED ARITHMETIC. The last twenty sums in the addition table may be learned without objects by uniting with the first number enough ot the second to make ten ; thus, 8 + 5 = 8 + 2 + 3 = 10 + 3 = 13. 32. Addition is the process of finding the sum of two or more like numbers. Since 7 + 5 = 12, 17 + 5 = 22, 27 + 5 = 32, etc. Practice sim- ilarly with all of the " endings." 33. As a preparation for " column addition" give frequent exercises in adding by twos, threes, fours, etc. , starting with one, two, three, etc., and carrying the work to 50. 34. EXERCISES. The following problems are for seat, board, or home work, and for oral practice in recitation. Practice upon them until they can be performed with great rapidity. They contain frequent repetitions of the work in the addition table. Add from the bottom, naming results only, thus: (first problem) 7, 13, 21, 24, 26, etc. To test the accui-acy of the w^ork add downward. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 957 5 968649 5678887985 7565778 2 99 685998 2 763 274 3 676972 3 897963786 8436356839 69669 5 7499 (11) (12) (13) (14) (1.5) (16) (17) (18) 89 79 38 93 37 43 93 98 29 63 34 84 28 92 89 79 93 94 34 68 22 99 83 78 26 33 68 89 21 99 83 89 97 29 93 62 88 78 29 35 96 98 82 99 83 87 39 73 ADDITION. m (20) (21) (22) 324 752 945 8685 561 433 887 4944 872 512 654 5636 324 311 472 3768 607 721 541 9483 830 869 635 7521 962 863 796 2754 574 127 4856 385 7039 (23) (24) (25) (20) 3580 2816 77281 497529 6295 6389 94969 678315 3782 3528 8S799 855274 9061 4893 68698 998865 3648 4678 89769 677345 7296 8253 84858 499586 6802 7594 48979 394759 3690 6031 87857 862979 3285 8539 36686 371789 9735 7426 76978 542963 3869 752784 (27) (28) (29) (30) 476583 9759869 898488 5492284 369868 8387698 764257 9785746 878725 4555966 701480 9637478 766958 1804795 925895 4476204 832777 8468872 699054 6544058 794489 6804187 854606 9669733 468985 7557688 988987 1368572 739849 5687899 504469 2943489 747494 8848786 566587 6918394 836366 9579665 677876 31647o8 669118 3695467 757769 5819404 577669 6086164 515987 8578308 8402326 492834 7533579 13 14 NEW ADVANCED ARITHMETIC. How were these numbers written for addition? Wliy? With what cohimn was the addition begun in each case? "Why? W^hen the sum of the numbers expressed in any column exceeded nine, what was done? 35. FORMAL STATEMENT. Write the numbers to be added so that units of the same order shall stand in the satne eolumum Jieo inning tvith the lowest order, find the stim of the numbers expressed in each column* If this sum exceeds nine, in any ease, reduce it to the next higher denomina- tion, placing the remainder, if any, under the column added, and adding the rediicetl number to the first term in the next higher order. 36. TEST OF ACCURACY. 1. Add the numbers expressed in each column, in the re- verse order. 2, Separate the problem into two or more problems and then unite the several results. What must be true of numbers in order that they may be united? What is the denomination of the sum? 37. How many mills are there in one cent? cents in one dime? dimes in one dollar? 38. TABLE OF FEDERAL MONEY. 10 mills make 1 cent. 10 cents make 1 dime. 10 (limes make 1 dollar. 39. In writing Federal Money, separate the order of dimes from the order of dollars by a decimal point. 126 dollars, 4 dimes, 7 cents, and 6 mills is written, $126,476. It is usually read, 126 dollars, 47 cents, 6 mills. ADDITION. 15 40. Write the following: 1. 83 dollars, 27 cents, 4 mills. 2. 59 dollars, 20 cents, 1 mill. 3. 73 dollars, 5 cents, 2 mills. 4. 148 dollars, 2 dimes, 9 mills. 5. 300 dollars, 7 mills. Read the above numbers, (1) as dollars, dimes, cents, and mills. (2) as dollars, cents, and mills. 41. Find the sum of the following numbers: 125 dollars, 26 cents, 8 mills. 64 dollars, 33 cents, 4 mills. 278 dollars, 5 cents. 471 dollars, 6 mills. 312 dollars, 59 cents, 7 mills. 42. Tell how the numbers are written for addition. Tell where the addition should begin. Describe the process, using a form similar to that on page 14. 43. Write, read, and add the following : 1. $48,041, 863.247, $146.28, $276,007, $160,406. 2. $361.79, $483,062, $583,802, $1272.84, $2169.176. 3. $2678.145, $5684.297, $462.01, $5000.36, $790.46, $579,614. 4. $2638.95, $5406.63, $2384.25, 376.52, $857.35, $96834.67, $3790.48. 5. $3762.05, $67452.84, $3568.90, $3.5.70, $49.32, $5.83, $23.71. 6. $4.21, $3.85, $9.63, $85.16, $128.95, $673.70, $2895..30, $853.60. 7. $893.40, $6.87, $3708.90, $7570.00, $0.75, 54603.55, $3780.25. 16 NEW ADVANCED ARITHMETIC. 8. S5.25, SG0.70, $375.08, 895.80, §3617.50, 8817.46, ?3064.38, 80.85. 9. 83768.60, 8479.00, 86328.50, 82.91, 8325.75, 89.00, $754.08, 835.10. 10. 87000.00, 8215.80, 8725.60, 850.50, 81.87, 86384.50, $4536.40. 44. How many gills are there in one pint? pints in one quart? quarts in one gallon? These are used in measuring what? Write the table of Liquid Measure. The abbreviation for gallon is gal. ; for quart is qt.; for pint is pt. ; for gill is gi. 1. Add : 5 gal. 2 qt. 1 pt. 3 gi. 4 " 3 " " 2 " 7 " 1 " 1 " 2 " 12 " 2 " 1 " 3 " 14 " " " 1 " How are these numbers wi-itten for addition? Where does the addition begin ? Describe the process as you did in Federal Money. 2. Add: 6 gal. 1 qt, 1 pt. 3 gi. 4 " " 1 " 2 '* 12 " 3 " " 1 " 27 '^ 2 " 1 " 2 '* 36 " 3 " " " 3. Add 8 gal. 3 qt. 1 pt. 3 gi. ; 8 gal. 2 qt. 1 pt. 2 gi. ; 4 gal. 1 qt. 1 pt. ; 1 gal. 1 pt. 1 gi. ; 6 gal. 3 qt. 3 gi. ; 6 gal. 2 gi. 45. How many quarts are there in one peek? How many pecks in one bushel? Write the table of Dry Measure. ADDITION. 17 1. Add; 5 bu. 3 pk. 6 qt, 6 '^ 2 '^ 4 9 " 1 " 6 28 '' " 7 16 " 2 " 5 4 " 3 " 2 2. Add 4 bu. 1 pk. ; 7 bu. 3 pk. 2 qt. ; 12 bu. 2 pk. 6 qt. ; 19 bu. 1 pk.; 26 bu. 3 pk. ; 14 bu. ; 18 bu. 1 pk. ; 2 pk. 3. Add 21 bu. 1 pk. ; 36 bu. 2 pk. ; 19 bu. 3 pk. ; 2 pk. ; 12 bu. 3 pk. ; 6 bu. 4 qt. 46. Write the following problems very carefully on paper, slate, or blackboard, and solve : 1. 824 + 69 + 703 + 9208 + 29607 = ? 2. 65 + 20007 + 893 + 566 + 15869 + 587 = ? 3. 5607 + 20189 + 46827 + 463912 + 51872 + 56928 + 324501 + 873 = ? 4. 70128 4- 58694 + 79106 + 436912 + 586107 + 371009 + 400106 =3 ? 5. 4007 + 93281 + 56185 + 47594 + 508069 + 724378 + 563128 = ? 6. Paid $2480 for a farm, S268 for a span of horses, |65 for a wagon, $32 for a set of harness, S18 for a plough, $124 for a mowing-machine, and $384 for other utensils. What was the entire cost? 7. The area of Maine is 33,040 square miles ; of New Hampshire, 9,305; of Vermont, 9,565; of Massachusetts, 8,315 : of Rhode Island, 1 ,250 ; of Connecticut, 4,990. AVhat is the entire area of the New England States ? 8. The area of New York is 49,170 square miles ; of New Jersey, 7,815; of Pennsylvania, 45,215; of Delaware, 2,050; of Maryland, 12,210; of District of Columbia, 70; of Vir- ginia, 42,450; of AVest Virginia, 24,780. What is the area of the Middle States? 18' NEW ADVANCED ARITHMETIC. 9. The area of North Carolina is 52,250 square miles; of South Carolina, 30,570; of Georgia, 59,475; of Florida, 68,680; of Tennessee, 42,050; of Alabama, 52,250; of Mississippi, 46,810; of Louisiana, 48,720; of Texas, 265,780; of Arkansas, 53,850; of Indian Territory, 64,690. What is the area of the Southern States? 10. What is the united area of the New England^ Middle, and Southern States? 11. On Monday, a merchant sold 3 gal. 2 qt. 1 pt. of molasses ; on Tuesday, 5 gal. 3 qt. ; on Wednesday, 4 gal. 1 pt. ; on Thursday, 6 gal. 1 qt. 1 pt. ; on Priday, 2 gal. 2 qt. 1 pt. ; on Saturday, 7 gal. 3 qt. 1 pt. How much did he sell in the entire week? 12. A dealer sold to A, 426 bu. 3 pk. 2 qt. of oats ; to B, 329 bu. 3 pk. 5 qt. ; to C, 189 bu. 1 pk. 7 qt. ; to D, 426 bu. 2 pk. 5 qt. ; to E, 562 bu. 3 pk. 6 qt. How much did he sell to all? 13. A farmer values his horses at $350, his cows at $275, his sheep at $411.75, his hogs at $129.25, and his poultry at $27.25. What is his value of all? 14. From New York to Albany is 143 miles; Albany to Suspension Bridge, 304 miles ; Suspension Bridge to Detroit 230 miles ; Detroit to Chicago, 284 miles. What distance from New York to Chicago? 15. A merchant bought at one time 224 barrels of flour for $1,344 ; at another, 217 barrels for $1,193.50; at another, 192 barrels for $1,056; at another, 486 barrels for $2,916. How many barrels did he buy? What was the cost of all? 16. A steamship sailed 239 miles each day for three days, and 227 miles each day for the next three days. How far did she sail in the six days? 17. Add .26.3, .187, 2.08, 4.019, 16.008, .563, .472, .008. 18. Add 26.0029, .0086, .0278, 43.0196, 217. .3059, .0062, .0487. ADDITION. 19 19. Add .00869, .0571, 36.426, 68.07956, .00964, .473, .86o2, .17, .0080. 20. Add .000524, .00524, .0524, .068397, 6.080576, 12.008642, 99.0864, .000748, .429783. 21. A man gave each of his four sons S375 ; he gave his daughter as much as he gave to two sons ; and to his wife as much as he gave to three sons. How much did he give to all? 22. A man left to his heirs a certain quantity of land. To the first he gave 320 acres ; to the second as much as to the first, and 80 acres more ; to the third as much as to the second, and 96 acres more; to the fourth as much as to all of the others, and to his wife as much as to the first three. How many acres did he bequeath? 23. In 1893, the enrollment in the public schools of Illi- nois was as follows : Number of male pupils enrolled in graded schools, 228,412; number of females, 235,885; number of male pupils enrolled in ungraded schools, 189,851 ; number of females, 171,937. What was the total enrollment for the year? 24. For the same year the number of male teachers in graded schools w^as 1,692; of female teachers, was 8,410; the number of male teachers in ungraded schools was 4,861 ; the number of female teachers was 9,277. What was the whole number of teachers employed? 25- For the same year the amount paid to male teachers in graded schools was $1,337,360.50; to female teachers, $1,168,903.82; to male teachers in ungraded schools, $4,272,782.46; to female teachers, $1,641,281.79. What was the whole amount paid to teachers? 26. The Permanent School Fund of Illinois is as follows : School Fund Proper, $613,362.96; the Surplus Revenue Fund, $335,592.32; the College Fund, $156,613.32; the Seminary Fund, $59,838.72 ; the County Funds, $158,616.63 ; 20 NEW ADVANCED ARITHMETIC. the Township Funds, $12,220,722.14; the University Fund, $006,207.6-1. What is the entire fund? 27. The following form is called a "■ Bill," or " Statement of account." Columbus, Ohio, May 15, 1896. James Watson To Robert S. William*, Dr. 1 10 72 60 • 1, 25 1 10 4 1 Fifth Reader, 1 Advanced Arithmetic, 1 English Grammar, 1 Geography, 1 History United States, Received payment, Robert S. Williams. 28. John Jones, of Utica, N. Y., bought of Pixley & Co., September 12, 1895, the following articles. Prepare the bill. 1 suit of clothes, $16.50; 3 pair hose, 90- ; 1 light over- coat, $12; 6 pair cuffs, $1.35; 1 doz. collars, $2.30; 1 hat, $3.25; 1 umbrella, $1.85. Receipt the bill. 29. On January 24, 1896, Frank Walton bought of William Snow of Louisville, Ky., 1 barrel flour, $3,25; 3 hams, $3.83; 30 lbs. sugar, $1.45; 5 lbs. coffee, $1.85; 2 dozen oranges, $0.55 : 6 cans tomatoes, $1.15. Bill not paid. Make the statement. 30. Bill of Robert Thompson, Bloomington, 111., rendered to James Dixon, May 12, 1896. Items: Turkish lounge, $25.00; center table, $14.50; 6 chairs, $13.35; rocker, $4.75; chiffonier, $18.75; hall tree, $17.50. 31. Bill of Joseph Stoner, blacksmith, rendered to Charles Smith, December 10, 1895, Albany, N. Y. Items: sharpen- ing 3 plows, $1.20; shoeing team, $2.40; setting tires, $1.65; repairing buggy, $1.50; babbitting harvester, $5.75. Receipt the bill. ADDITION. 21 47. The following device is often used bj accountants; 83457 26 29063 21 840521 25 364307 35 428630 32 976538 27 2725516 For the result read all of the last result and the right-hand figures of the previous results. 1. Find the value of the U. S. coinage of 1891 from the following statement: double eagles, 825,891,340; eagles, $1,956,000; half eagles, 81,347,065 ; quarter eagles, 827,600; silver dollars, 823,562,735; half dollars, 8100,300; quarter dollars, 81,551,150; dunes, 82,304,671.60; nickels, 8841,- 715.50; cents, 8470,723.50. 2. Money in circulation in United States, Dec. 1, 1894, was as follows : gold coin, 8465,789,187; gold certificates, 858,925,899; silver dollars, 857,449,865; minor coins, 861,606,967; silver certificates, 8332,317,084 ; "Sherman" notes, 8124,574,906; United States notes, 8276,910,489; currency certificates, 857,135,000; National Bank notes, $202,517,054. What was the total amount in circulation? 3. Distances along the Chicago and Alton Railroad — Chicago to Joliet, 37 miles; Joliet to Bloomington, 89 miles; Bloomington to Springfield, 59 miles ; Springfield to Alton, 72 miles ; Alton to St. Louis, 26 miles. What is the total length of the road? 4. Two ships meet in mid-ocean. One sails 416 mUes eastward the first day, 386 miles the second day, and 369 miles the third day. The other sails 396 miles westward the first day, 278 the second day, and 339 the third day. How far apart are they at the end of the third day? 3A 22 NEW ADVANCED ARITHMETIC. SECTIOIS^ III. SUBTRACTION. 48. As Addition is the process of uniting two or more like numbers, Subtraction is the process of separating a number into two smaller numbers. As such facts as 2 + = 2 are not counted in the addition table, so facts like 2 — = 2 are not counted in the subtraction table. 49. The following are the 81 primary problems in subtrac- tion. Neither accuracy nor rapidity is possible uutU the difference between the numbers in each pair can be given with readiness. 2 3 3 4 4 4 5 12 13 2 14 5 5 5 6 6 6 6 3 2 15 4 3 2 6 7 7 7 7 7 7 16 5 4 3 2 1 8 8 8 8 8 8 8 7 6 5 4 3 2 1 9 9 9 9 9 9 9 8 7 6 5 4 3 2 9 10 10 10 10 10 10 1 9 8 7 6 5 4 10 10 10 11 11 11 11 3 2 1 9 8 7 6 SUBTRACTION. 23 11 11 11 11 12 12 12 5 4 3 2 9 8 7 12 12 12 12 13 13 13 6 5 4 3 9 8 7 13 13 13 14 14 14 14 6 5 4 9 8 7 6 14 15 15 15 15 16 16 5 9 8 7 6 9 8 16 17 17 18 7 9 8 _9 50. John had 17 marbles and lost 8, How many had he left? How many objects would be needed to illustrate this problem ? Use the problem to test the accuracy of the fol lowing definitions : 51. Subtraction is the process of separating a number into two parts, one of which is given for the purpose of finding the other. 52. The Minuend is a number that is to be separated into two parts, one of which is given for the purpose of finding the other. 53. The Subtrahend is the given part of the minuend. 54. The Remainder or Difference is the required part of the minuend. 55. The sign — (minus) when placed between two num- bers shoves that their difference is to be found. If the minuend is dollars, what will the subtrahend be? Why? What will the remainder be? Why? 56- The minuend, subtrahend, and remainder are like numbers. 24 NEW ADVANCED ARITHMETIC. John had 17 marbles and James had 12. How many more had John than James? How many marbles would be needed to illustrate this problem? Will the definition given include this problem? 57. Problems in subtraction assume these two forms. When objects are employed they are reducible to the first form (see Art. 50), since the 12 marbles that James had indicate the size of one of the two parts into which John's are to be separated. 58. A problem in subtraction is in its simplest form when each term in the minuend equals or exceeds the cor- responding term in the subtrahend. Illustration. 8462 Minuend. 5341 Subtrahend. 59. Problems are not generally in this form, but they must be made to assume it before the subtraction can be performed. Problem. 721 564 Is this problem in its simplest form ? If not, show why it is not. 60. EXPLANATION. - 1. Since the ones' term of the minuend is less than the corresponding term of the subtrahend, one of the tens in the minuend may be reduced to ones and added to the ones' term. 10 ones plus 1 equal 11 ones. 11 minus 4 equals 7. 2. Since the tens' term of the minuend is less than the corresponding term of the subtrahend, one of the hundreds in the minuend may be reduced to tens and added to the tens' term. 10 tens plus 1 ten equal 11 tens. 11 tens minus 6 tens eoual 5 tens. SUBTRACTION. 25 3. 6 hundreds minus 5 hundreds equal 1 hundred. 61. Since the minuend has been separated into the sub- trahend and remainder their sum must equal the minuend. Hence, To prove a probletn in subtraction, find the sum of the subtraJiend and remainder; if it equals the minuendf the ivork is correct, 62. Since the problem given is like all problems in sub- traction, all may be solved as this has been. Hence, we may make the following RULE FOR SUBTRACTION. Write the subtrahend below the minuend, tvith units of the same order in the same column. Beginning tvith the lowest denomination, subtract each tenn of the subtraJiend from the corresponding term of the minuend. If any term of the subtrahend exceeds the corresponding term of the minuend, increase the smaller term by one of the next higher, and proceed as before. 63. EXAMPLES FOR PRACTICE. 1. $24,685 6. 5444454 13.574 4567956 2. $369.72 126.41 3. $48,567 26.785 4. $584.32 296.48 5. $3557.888 1899.899 7. 4222332 2789679 8. 26689888 16789899 9. 92820 63574 10. 102875 63986 26 NEW ADVANCED ARITHMETIC. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Practice with great 31. 32. 33. 34. 35. 36. 146238 87159 196205 159438 2666776 1789789 611345566 235845897 833344565 436789768 746643455 366799687 82345533 49455769 923334454 583584889 3445444554 1575678567 5112556 2686897 21. 14455677 4857898 22. 84557666 59698979 23. 423445575 259576976 24. 21234566 17457' 25. 170308444 24154456 26. 8800337.005 347942.462 27. 40080222 28848567 28. 847000221 512356345 29. $85200620.05 40065010.23 30. 403201058 250042006 Remainder. 9 on these 30 problems until you can read the results rapidity. Minuend. Subtrahend. 824 63 978 ? 467 ? 826 279 1276 ? 638 ? 745 12864 24671 ? 18983 SUBTRACTION. 27 How find subtrahend when fninuend and remainder are given? Wliy? Make a rule. How find minuend when subtrahend and remainder are given ? Why ? Make a rule. 37. 60241—42374 + 26082 = ? 38. 280621 + 460082-31892 — 42671=? 39. 529706-31793-64802 + 5729=? 40. A merchant bought 324 bu. 2 pk. of oats, and sold to A 59 bu. 3 pk. and to B 123 bu. 1 pk. How much did he have left? 41. A farmer had 861 acres of land, and gave 294 acres to his son. How many acres did he have left? 42. From a barrel containing 37 gal. 3 qt. 1 pt. and 3 gi. of vinegar, a merchant sold to A, 7 gal. 2 qt. 2 gi. ; and to B, 4 gal. 3 qt. 1 pt. 2 gi. more than he did to A. How much was left in the barrel ? 43. C and D were 350 miles apart. They traveled toward each other, C at the rate of 46 miles a day, and D at the rate of 37 miles a day. How far apart were they at the end of the second day? Which had traveled farther? How much ? 44. Four men together owned 786 cattle. E owned 227, and F, 339. G owned as many less than E as F did more than E. How many did H own? 45. A merchant paid $2,500 for a quantity of silk. For other dry goods he paid $265 more than he did for the silk. For groceries he paid $683 less than what he had before expended. To one customer he sold goods amounting to $5,280 ; to another, $325 less than half as much ; to another, $2,895 less than to the first, thus disposing of all of his stock. Did he gain or lose, and how much? 46. One cask contains 39 gal. 1 qt. 1 pt. 2 gi. of alcohol, and another 28 gal. 3 qt. 1 pt. 3 gi. How much more does the first contain than the second ? 28 NEW ADVANCED ARITHMETIC. 47. In one bin there are 873 bu. 2 pk. 5 qt. of com ; in another, 698 bu, 3 pk. 7 qt. How much less in the second than in the first? 48. A has $650; B has 825 less than half as much. C has Si 25 less than both A and B. D has §275 less than A, B, and C. How much have all of them? 49. A and B start from the same place at the same time and travel on the same road. A travels at the rate of 26 miles an hour, and B at the rate of 39 miles an hour. How far apart will they be at the end of 6 hours? If B should then stop, and A should travel at B's rate, how long would it take him to overtake B? 50. From A to B is 492 miles; from B to C is half as many miles, plus 42. How find the difference in the two distances ? How many miles from A to C ? 51. Three persons bought a mill valued at $25642. The first paid S6743.25; the second t«'ice as much; and the third, the remainder. How much did the third pay? 52. At one time I spent ^ of a dollar ; at another, | of a dollar ; and at another, | of a dollar. If I had but a dollar at the beginning, what part of it had I left? How many cents ? 53. How many more days in the months of March, April, May, and June, counted together, than in the months of September and October? 54. "When shopping, a lady bought ribbon for 36 cents, lace for $1.48, gloves for 81.75, and velvet for 81.27. She gave in payment a five-dollar bill, and received her change in nickels and cents. How many nickels did she receive? 55. 8.27-6.52 = ? 56. 264.008 - 79.169 = ? 57. .80641 - .27835 = ? 58. .01803 — .00657 = ? 59. 2.061 - .8934 = ? SUBTRACTION. 29 60. 7.006521 - .009730 = ? 61. 4.069 + 72.0083 - 10.15328 = ? 62. .08031 + .2483 + .005687 - .0148 - 00693 = ? 63. 263.094 - 172.86 + 52.0048 - ? 64. $821,054 + $63,006 - $279,838 - $346,765 = ? 65. Bought several articles costing respectively 63;^, 89)?!, 48?', $1.38, $2.76, $4.75. Gave merchaut a $20 bill. What ohaoge should I receive ? 66. How many years after the discovery of America was Washington born? How old was Washington at the time of the Declaration of Independence ? When he was first inau- gurated as president? (Omit months and days.) 67. Lincoln was born how many years after Washington? How old was he at the time of his death? If still living, how old would he be? (Omit months and days.) 68. How many years from the battle of Bunker Hill to the attack on Fort Sumter? From the surrender at Yorktown to Lee's surrender? From the battle of Waterloo to the battle of Gettysburg? 69. The National Debt of U. S., on July 1, 1870, was $2480672427.81. On July 1, 1894, it was $1632253636.68. How much had it diminished in 24 years ? 70. The area of France is 204092 square miles. That of Great Britain and Ireland is 120979 square miles. What is the difference of their areas? Which of the States of the American Union is larger than either? It is how much larger than France ? Than Great Britain and Ireland? Which is larger, France or California? How much? Great Britain and Ireland or California? How much? 71. How many years ago was the Declaration of Inde- pendence signed ? How many years from the signing of the Declaration of Independence to the close of the Civil War? 80 NEW ADVANCED ARITHMETIC. AREAS OF STATES AND TERRITORIES. NEW ENGLAND. Maine, 33,040. Vt., 9,565. R. I., 1,2.50. N. H., 9,305. Mass., 8,315. Conn., 4,990. MIDDLE ATLANTIC GROUP. N.Y., 49,170. Del., 2,050. Va., 42,450. N. J., 7,815. Md., 12,210. W. Va. , 24,780. Penn., 45,215. D. C, 70. COTTON STATES. / N. C, 52,250. Ala., 52,250. Texas, 265,780. S. C, 30,570. Miss., 46,810. Ark., 53,850. Ga. 59,475. La., 48,720. Teun., 42,050. Fla. 58,680. CENTRAL STATES. Ky., 40,400. 111., 56,650. Wis., 56,040. Ohio, 41,060. Mo, 69,415. Minn., 83,365. lud., 36,350. Mich., 58,915. THE GREAT PLAIN. Iowa, 56,025. N. D., 70,795. Neb., 77,510. Okla., 39,030. S. D., 77,650. Kan., 82,080. MOUNTAIN STATES. Ind. T. , 31,400. Mont., 146,080. Colo., 103,925. N. M., 122,580. Idaho, 84,800. Utah, 84,970. Ariz., 113,020. Wy., 97,890. Nev., 110,700. Cal., 158,360. PACIFIC STATES. Oreg. 96,030. Alaska, 531,409. Wash., 69,180. SUBTRACTION. 31 Find difference in area between — 71. Maine and the rest of New England. 72. Texas and the other Gulf States. 73. New England and Illinois. 74. The Central States and the Cotton States. 75. The Pacific States and the Middle Atlantic Group. 76. The Central States and the Great Plain. ^ 77. The five States on the east bank of the Mississippi and the three States on the west bank. 78. How many States each equal to Illinois can be cut out of Texas ? 79. Find the total area of all the States east of the Mis- sissippi River, excluding Minnesota and Louisiana. 80. Find the total area of the original thirteen States. 81. A man expended 810,564 for 4 tracts of land. For the first he paid Sl,9G8.50; for the second, $2,680; for the thu-d, $3,127.50. What did he pay for the fourth? 82. The sum of the areas of Maine, Ky., Md., Penn., and a fifth State is 189,072 square miles. What is the area of the fifth State? Which is it? 83. A man bought three buildings. He paid for the first $7,846; for the second, 82,875 more; for the thu-d, 83,182 less than for the second. He put in as part payment a farm for 82,125, and paid the remainder in cash. What was his cash payment? Latitude and longitude are measured in degrees, minutes, and seconds. Sixty seconds (marked") make a minute. Sixty min- utes (marked ' ) make a degree (marked ° ). 84. New York is in 74 ° 3 " west longitude and Boston 71° 3' 30" west longitude. Boston is how far east of New York? 85. Chicago is in 87° 35' west longitude. How far is it west of N. Y. ? Of Boston ? 32 NEW ADVANCED ARITHMETIC. 86. Albany is 298 miles east of Buffalo aud Chicago is 589 west of Buffalo. AVhat is the distance from Albany to Chicago ? 87. Berlin is 13° 23' 43" E. and New Orleans 90° 3' 28" W. ; what is the difference of their longitudes? 88. Boston is in 42° 21' 24" N. latitude. The latitude of New York is 40° 42' 43" N. Boston is how much farther north than New York? 89. London is in latitude 51° 30' 48" N. It is how much farther north than Boston? 90. New Orleans is 29° 57' N. and Rio Janeiro 22° 54' S. What is their difference in latitude? 91. Find the time from July 6, 1888, to Sept. 10, 1896. 92. Find the time from March 12, 1889, to Oct. 18, 1895. 93. Find the time from June 15, 1887, to April 5, 1897. Note. How many jears from June 15, 1887, to June 15, 1896? How many months from June In, 1896, to March 15, 189/ ? How many days lu March after the 15th ? To these add the 5 days in April. 94. Find the time from Aug. 21, 1890, to May 16, 1897, 95. Find the time from Sept. 12, 1891, to Dec. 25, 1897. 96. Find the time from Oct. 28, 1886, to June 19, 1895. MUL TIP Lie A TION. 33 SECTioivr ly. MULTIPLICATION. 64. What is the sum of 5 and 5 ? "What is the smu of two 5's? What is the sum of 8 and 8 and 8? What is the sum of three 8's? AVhat is the sum of five 6's? of four 9's? of seven lO's? This will suggest the manner in which the mul- tiplication table is built up. As commonly used it consists of sums formed by repeating the nine primary numbers up to nine of each. 65. Multiplication is a short method of finding the sum of tV70 or more equal numbers. 66. The Multiplicand is one of the two or more equal num- bers that are to be united. 67. The Multiplier is the number of equal numbers that are to be united. 68. The Product is the sum of two or more equal numbers that have been united by multiplication. 69. The sign of multiplication is an oblique cross. When the multiplier comes first, the sign is read times When the multiplicand precedes, the sign is read multiplied by. 70. Numbers are spoken of as abstract or concrete. This distinction is of little value except in Multiplication and Division. A number of named objects, as 6 books, is called a concrete number. A number whose unit is not named, as 7, or a number of numbers, as 5 nines, is called abstract. Accordingly, 1. The Multiplicand may be abstract or concrete. 2. The Multiplier is abstract. 3. The Product is like the Multiplicand. 84 NEW ADVANCED ARITHMETIC. 71. Multiplication Table. 1 2 3 4 5 6,7 8 9 10 11 12 2 4 6 8 10 12 ; 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 23 32 36 40 44 48 6 10 15 20 Z5 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 ■ 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 108 110 121 132 12 24 3G 48 CO 72 84 90 120 132 144 72. EXAMPLES FOR PRACTICE. These problems and others like them should be used until the results can be spoken with great rapidity. 1. 1348G9725 2 2. 843512769 3 3. 4G9 1058327 4 5. 7964031825 6 6. 6324507198 7 7. 31924065708 4. 9601475328 5 8. 2573960814 9 MULTIPLICATIOX. .^ 35 Go through these problems, giving the product of the multiplier and each term of the multiplicand "v\'ithout regard to the denomination of the product. Repeat the operations, naming the denomination of each product. Repeat the operations, naming the denomination of each product, reducing it to the next higher order when it exceeds nine, and giving the denomination of the remainder and of the reduced number in each case. 73. "When the product in any case does not exceed nme, where should it be -written ? "When the product exceeds nine, where should the remainder be written? What should be done with the reduced number? 74. Since the method employed in these cases is applica- ble iu all similar cases, we may make the following rule for multiplication when the multiplier is less than 10. RULE. Write the multiplier under the lotvest term of the tniiU tiplicand. MuUiply this term of the imtltiplicand by the multiplier. If the product is less than lO, j^lace it tinder the term multiplied. If the product is more than lO, reduce it to the next higher denomination, placing the re- mainder, if there should be one, under the term mtilti- plied, and adding the reduced number to the product of the multiplier and the next higher term of the multiplicand, 75. ]Multiplication by tens, hundreds, etc. Illustrative Example. 8364 22 16728 16728 184008 76. In this problem the tens' term of the multiplier is how many times the units' term? The product of each tenu of the multiplicand and the tens' term of the multiplier is how 36 NEW ADVANCED ARITHMETIC. many times the product of the same terms and the units* term of the multiplier? Where, then, should these results be written? 77. Since a figure standing in tens' order expresses a number ten times as great as if standing in units' order, the products obtained by using the tens' term of the multiplier will belong one order higher than the terms multiplied. Hence, To multiply by tens, proceed as before, placing the results one order to the left of the term multiplied. 78. Apply the same reasoning to the hundreds', thou- sands', etc., terms of the multiplier, and make a statement for each. 79. GENERAL RULE. Write the tniiltiplier tinder the multiplicand. Beginning with the units' term of the multiplier, mul- tiply the multiplicand by each term of the multiplier, successively, placing the right-hand figtire of each partial product under the term by which it was obtained. Unite the several partial products. 80. PROOFS. 1. Use the multiplicand as the tmiltiplier. 2. Divide the product by either of its factors. The qug^ tient will be the other factor, if the work is correct. Note. Proof 2 cannot be used until after a study of division. 81. Illustrative Analysis. Multiply 826 by 352. 1. We are to unite 352 826's. "We first unite 2 826's, then 50 826's, then 300 826's, and then unite these several products. Note. 826's is read eight hundred twenty-sixes. 2. 2 times 826 one." are 1652 ones. 3. 50 times 826 are 5 times ten times 826. Ten times 826 are 826 tens. Since 5 times 826 tons are tens, the ric;ht-hand figure of tliis partial pro- duct will be written under the tens' figure of the multiplicand. MULTIPLICATION. 37 4. 300 times 826 are 3 times 100 times 826. 100 times 826 are 826 hun- dreds. Siuce 3 times 826 Imndreds are hundreds, the right-hand figure of this partial product will be written under the hundreds' figure of the mul- tiplicand. 5- Uniting the partial products gives the complete product. 82. EXAMPLES FOR PRACTICE. 1. Multiply 864 by 24; by 35, by o8 ; by 79 2. Multiply 978 by 61 ; by 83 ; by 96 ; by 89. 3. Multiply 4625 by 189 ; by 265 ; by 374. 4. Multiply 3718 by 264 ; by 357; by 819. 5. Multiply 24689 by 345 ; by 678 ; by 921, 6. Multiply 13576 by 987; by 654; by 321. 7. Multiply 90876 by 234 ; by 567 ; by 892. 8. Multiply 34005 by 306 ; by 508 : by 809. 9. 874 X 81326. 10. 596 X 493725. 11. 2086 X 321068. 12. 34708 X 97540016. 13. 65413 X 2496038. 83. To multiply by 10, 100, 1000. etc., annex as many zeros to the right of the multiplicand as there are in the multiplier. Note. If the decimal point is used, it must be moved as many orders to the right as there are zeros in the multiplier, and the vacant orders m*st be filled with the zeros. 84. PROBLEMS. 1. Multiply 7642 by 10; by 100; by 10000; by 1000; by 100000 2. Multiply 246092 by 1000; by 10; by 10000; by 100. 3. Multiply 9284 by 200 : by 3000; by 60 ; by 70000. 4. Multiply 5681 by 2600 ; by 87000 ; by 24600 ; by 360. 4A 38 NEW ADVANCED ARITHMETIC. 85. GENERAL PROBLEMS. 1. What is the cost of 5 cords of wood at 86 a cord? Analysis. Since each cord cost $6, 5 cords cost 5 sixes of dollars (or 5 times S6), which are $30. 2. What is the cost of 236 acres of land at $72 an acre? Analyze as above. Note. In such problems as number 2, where the multiplicand is smaller than the multiplier, the order of factors may be changed by the following analysis : If the land cost $1 an acre, 236 acres would cost $23G. Since the laud cost $72 an acre, 236 acres cost 72 times S236. Use this form of analysis until it cau be employed easily and rapidly. 3. If a man travel at the rate of 46 miles a day, how many miles will he travel in 27 days? 4. James earned S264; John, $432; William, twice as much as both James and John; Henry, three times as much as the difference between William's earnings and John's earnings. What did all earn? 5. Thomas bought 4 gal. 3 qt. 1 pt. 2 gi= of milk , Eeuben bought 5 times as much. How much did both buy? 6. If 16 men can do a piece of work in 12 days, in how many days can one man do it? 7c If a railway train run 42 miles an hour, how many miles will it run in 678 hours? 8. A locomotive division is 126 miles long. How many miles does an engineer ride in July, if he makes a trip every day? 9. A merchant bought 7 loads of oats, each containing 63 bu. 2 pk. o qt. How many did he buy? What did they cost him, at 30 cents a bushel ? 10. The distance from Bloomington to Chicago is 126 miles. How many feet of wire are there in 7 telegraph lines connecting the two cities? MUL TIP Lie A TION. 39 U. A man had $3,000. He bought 6 horses at $125 each, 5 cows at $57 each, 2 wagons at $48 each, and 40 acres of land at $40 an acre. How much money had he left? 12. Multiply the sum of 826 and 439 by twice their dif- ference. 13. Multiply the sum of I X 729 and 8 X 563 by 9 times the difference between 23 X 48 and 65 x 76. 86. MULTIPLICATION BY FACTORS. Since 6 = 2 x 3, 6 times any number = 2 x S times, or 3x2 times, that number ^ hence, if I wish to multiply a number by 6, I may multiply it by 3 and that product by 2, or multiply it by 2 and that product by 3. The same plan may be followed with any number that can be factored. RULE. Separate the multiplier into two or more factors, mul- tiply hy one of them, the resulting product by a second^ and so continue until all of the factors have been used. The last product is the reQtiired result. 87. PROBLEMS. 1. Multiply 456 by 15, using factors of 15. Analysis. Since 15 = 3 X 5, 15 times 456 = 3 times 5 times 456. 5 times 456 = 2280; 3 times 2280 == 6840. FORM. 2. 5246X18=? ^246 3. 6792x25 = ? 15738 4. 24680 X 48 = ? 6 5. 56072X64 = ? ^^^-^ 88. Multiplication by numbers slightly less than a power of 10. 1. 869 X 99 = ? Analysis. If the multiplier were 100, the product would be 86900. Since the multiplier is one less than 100, the product is once 869 less than 86900. 86900 - 869 = 86031. 40 NEW ADVANCED ARITHMETIC. 2. Multiply 7824 by 999 ; by 9999. 3. Multiply 862534 by 98; by 998; by 97; by 997; by 9997. , 89. Learn the following table of aliquot parts ; 16|= J of 100. 1G6| = J of 1000. 2| r= i of 10. 25 -lot 100. 250 = 1 of 1000. 5 = J of 10. 50 = ^ of 100. 500 = ^ of 1000. 7^ = 1 of 10. 75 = 1 of 100. 750 = f of 1000. 3i = ^ of 10. 33i = iof 100. 333i = i of 1000. 6| = f of 10. 66f = f of 100. 666| = 1 of 1000. 12^ -^oi 100. 125 = i of 1000. 371 = 1 of 100. 375 = 1 of 1000. &2h = 1 of 100. 625 — 1 of 1000. 8^= 1 of 10. 83i = 1 of 100. 833i = f of 1000. 8| = |of 10. 87^ = 1 of 100. 875 = 1 of 1000. The twelfths and ; sixteenths of 100 and 1000 may be added for more extended work. Illustrative Example. FORM. 1-284 1284 X 37^=? 128400 1(3050 48150 Analysis. 37^^=1 of 100. If the multiplier were 100, the product would be 128400. If the nmltiplier were | of 100, tlie product would be i of 128400, which equals 16050. Since the multiplier is | of 100, the product is 3 times 16050, which equals 48150. 90. PROBLEMS. 1. Multiply 48464 by 62i; by 12|; by 625; by 375; by 75. 2. Multiply 58647 by 33,\; by 66§; by 333^; by 666§. 3. Multiply 86484 by 16|; by 166§; by 831^; by 833^. 4. What is the cost of 24 dozens of eggs, at 12^ cents a dozen ? Analysis. At one dollar a dozen, 24 dozens cost $24. Since the price 13 12i cents, or | of a dollar a dozen, 24 dozens cost \ of $24, which equals $3. MUL :iPLICATION. 41 5. What will 16§ lbs. of sugar cost at 6 cents a pound? 25 lbs.? 33^ lbs.? G6§ lbs.? 83^ lbs.? Analysis. 100 lbs. cost $6. i of 100 lbs. cost \ of S6. 6. What is the cost of 324 objects at 8^ cts. each? at in cts.? at 16§ cts.? at 25 cts.? at 33^ cts.? at 37i cts.? at 50 cts. ? at 75 cts. ? 7. Find the cost of 625 objects at 16| cts.; at 25 cts.; at 50 cts. ; at 66^ cts. ; at 40 cts. 8. Find the cost of 824 objects at 33^ cts. ; at 62^ cts. ; at 37i cts. ; at 83^ cts. 9. Find the cost of Q2b horses at $83^- each; at §87^ each ; at $62i each. 10. Find the cost of 824 cattle at $37i each; at §33^ each ; at $50 each. 91. MISCELLANEOUS PROBLEMS. 1. 8 X .024 =? 9 X .0073 =? 27 X -1694 =? 2. 36x3.056=? 48X17.0835=? 59x24.16947=? 3. 83 X .24356 =? 92 x .3607 =? 98x7.00869=? 4. Multiply .086794 by 8 ; by 9 ; by 26 ; by 37. 5. Multiply 365.432 by 45; by 69; by 138; by 246. 6. Q>Q''^ X 7086957. 7. 230000 x 569842. 8. 998 X 47952. 9. (84625 + 53796) X (63824 — 21706). 10. (821 — 463 + 279) X (425 + 872 - 328). 11. Multiply 3 gal. 2 qt. 1 pt. by 5 ; by 8 ; by 9 ; by 12. 12. Multiply 7 yd. 2 ft. 8 in. by 7 ; by 10; by 11. 13. Multiply 46 bu. 3 pk. by 15 ; by 24 ; by 38 ; by 49. 14. Bought 1 60 acres of land at $65 an acre, and 80 acres at S75 an acre. Sold 120 acres at $80 an acre and the remainder at $66.50 an acre. What was the gain or loss? 42 NEW ADVANCED ARITHMETIC. 15. A dealer bought 3 horses at S85 each ; 5 at $96 each ; 7 at $124.50 each. He shipped them to the city, the freight averaging $12.50 each. He sold the cheapest at $110.50 each; the second lot at $128 each, and the third lot at $164 each. If his personal expenses and the care of the horses amounted to $32, what did he gain by the transaction? 16. Put the following problems in the form of bills with the teacher as the seller and yourself as purchaser : 1. 12 yds. calico at 6]- f ; 6 doz. eggs at 12^(/ ; 3i lbs. coffee at 42;*; 15 yds. muslin at 11 <^; 9 yds. summer silk at 86/ ; 3 pairs shoes at $1.45 ; 2 bu. potatoes at 65;. 2. 4 yds. linen at 40/; 2 pairs gloves at $1.75; 6 yds. silk at $1.25; 15 yds. sheeting at 25^ ; 6 yds. pillow-casing at 12i^'; 8 yds. towelling at 24 ^ ; ^ yd. velvet at $1.25; 6 handkerchiefs at 25/ ; 2 waists at $1.25. 17. A stone falls 16 feet the first second, (16 +32^ feet the second second, (16 + 32 + 32) feet the third second, and so on. How many feet will it fall in 8 seconds? 18. Light travels 186000 miles a second. It takes 498 seconds for a light wave to pass from the sun to the earth. What is the distance ? 19. A bushel of corn in the ear weighs 70 pounds ; shelled, 56 pounds. How many pounds of cobs in a crib containing 1800 bushels of ears? 20. Make a pendulum by fastening a split bullet to a thread 30 inches long. Count the vibrations in a second. How many vibrations will it make in a minute? in an hour? in a day ? in a week ? 21. A car is loaded with 49 steel rails 32 feet long and weighing 78 pounds to the yard. The weight of the car and its load is 62592 pounds. AVhat is the weight of the car? 22. If the above car be loaded with 512 bushels of shelled corn, what will the car and its load weigh? 23. What is the estimated number of words in a book containing 240 pages, each page averaging 350 words? MULTIPLICA TIOX. 92. SURFACE MEASURE. 43 1. This figure is 3 inches long and 2 inches wide. How many square inches in each horizontal row? In the whole figure ? 2. Draw a surface 5 inches long and 3 inches wide. How many square inches in each row ? in the whole figure ? 3. This page is 5 inches by 7 inches. How many square inches in a single row along the right-hand margin? How many such rows? How many square inches in the page? Note. Id the following problems the measurements should be made by the pupils, with tape-line, yard-stick, or foot-rule. The nearest integral number of units should be taken. Thus, if the desk-top in the uext problem is 20f inches long, it should be taken as 21 inches. 4. How many square inches in your desk-top? in all the desk-tops in the school-room? 5. How many square inches in a pane in your nearest window? in all the panes? in all the windows? 6. How many square feet in your school- room floor? square inches? 44 NEW ADVANCED ARITHMETIC. 7. In a well-lighted school-room the floor-surface is not more than six times the window surface. Is your room "well-lighted"? 8. Would the leaves in this book cover all the glass in the school-room windows ? 9. How many square feet of blackboard in your school- room? What did it cost? (Reckon natural slate @ 25(* per sq. ft. ; artificial slate at 1 If per sq. ft. ) 10. Draw on blackboard a diagram of the school-room floor, scale one inch to the foot. 11. Draw the same on paper, scale one quarter-ioch to the foot. 12. Draw a diagram of the north wall using same scale. 13. Calculate separately the number of feet of wainscot- ing, blackboard, plaster, doors, windows. 14. How many square feet in a base-ball diamond? Note. The teacher may increa.se this list indefiuitely. The best prob- lems are those drawn from the pupil's surroundings or from his studies. 15. How many square feet are there in a lot 99 feet wide and 208 feet deep? 16. What is the value of the above lot at 9i cents a square foot? 17. If flooring costs 3 cents a square foot, what did the lumber in the floor of your school- room cost ? 18. If it costs 3 cents a square foot to lath and plaster a ^all, what was the cost of plastering the ceiling of your bchool-room ? 19. If matting sells for 4 cents a square foot, what will it cost to cover the floor of your school-room with it? DIVISION. 45 SECTIOT^ Y. DIVISION. 93. Separate 12 crayons into groups of 4 crayons each. Separate 12 crayons into 4 equal groups. In each of these processes the 12 has been separated into equal numbers; either of the processes is called Division. In the first pro- cess we are given the number to be separated and the size of the equal groups, — that is, the 12 are to be measured off into 4's. This kind of division is called Measurement. In the second process we are given the number to be separated and the number of equal groups to be made of it. This kind of division is called Partition. 94. DEFINITIONS. 1. Division is the process of separating a number into equal numbers. 2. Measurement is the process of separating a number into equal numbers of a given size. 3. The Dividend in Measurement is the number that is to be separated into equal numbers of a given size. 4. The Divisor in Measurement is one of the equal num- bers into which the dividend is to be separated. 5. The Quotient in Measurement is the number of equal numbers into which the dividend has been separated. 6. Partition is the process of separating a number into a given number of equal numbers. 7. The Dividend in Partition is the number to be sepa- rated into a given number of equal numbers. 8. The Divisor in Partition is the number of equal num- bers into which the dividend is to be separated. 46 XEW ADVAXCED ARITHMETIC. 9. The Quotient in Partition is one of the equal numbers into which the dividend has been separated. 10. The Remainder in each case is the undivided part of the dividend. 95 1. In measurement the divisor and dividend have the same unit. The quotient, being a number of numbers, is abstract. The remainder is like the dividend. 2. In partition the dividend and quotient are alike and the divisor is abstract. The remainder is like the di\idend. 93. "Which of the following problems Ulustrate measure- ment and which partition? Name divisor, dividend, and quotient in each case and show how the definitions apply. PROBLEMS. 1. An 80-acre field was divided into 10-acre lots. How many did it make ? 2. A stage coach went 6 miles an hour ; how many hours were required to go 30 miles ? 3. A school-room contains 54 seats arranged in 6 equal rows ; how many seats are there in each row? 4. At 3 cents each, how many oranges can be bought for 30 cents? 5 If 5 barrels of flour cost $20. what is the price per barrel? 6. If a school of 42 pupils were divided into 6 equal classes, how many pupils would there be in each class? 7. "With divisor and quotient in each of above problems, make a problem in multiplication; with dividend and quo- tient, a problem in division, and tell kind. 97. There are four signs of division. They are : ) , -i- , : , DIVISION. 47 The divisor is placed at the left of the first sign, at the right of the second and third, and below the fourth. Illustrations, 4)12(3. 12-^4 = 3. 12:4 = 3. ^^ = 3. 98. By reversing the order of the multiplication table the measmement and partition tables are formed. Show how this is done with 6x8 = 48. 99. PROBLEMS IN MEASUREMENT. 1. Use 2 as a divisor and give the quotients for all num- bers from 2 to 19. Illustration. In 2 there is one 2. In 3 there is one 2 and half of another, etc. 2. Using 3 as a divisor, do the same with numbers from 3 to 29. 3. With 4 from 4 to 39. 8. With 9 from 9 to 89. 4. With 5 from 5 to 49. 9. AVith 10 from 10 to 99. 5. With 6 from 6 to 59. 10. With 11 from 11 to 109. 6. With 7 from 7 to 69. 11. With 12 from 12 to 119. 7. With 8 from 8 to 79. 12. With 16 from 16 to 144. 100. PROBLEMS IN PARTITION. 1, Use 2 as a di\'isor, and give quotients for all numbers from 2 to 19. Illustration. One half of 2 is 1. One half of 3 is \\. 2. With 3 to 29. Continue these exercises through the same numbers as in division. 101. EXAMPLES FOR PRACTICE IN MEASUREMENT- 1. Divide 8648 by 3. FORM. .3 ) 8648 ^8S2f 48 NEW ADVANCED ARITHMETIC. 2. 7463-^4=? 7. 3. 9608 -^ 5 =? 8. 4. 21279 4- 6 =? 9. 5. 39852 ^ 7 =? 10. 6. 463024 -^ 8 =? 11. )2. Explain the eleven precedi FORM. 3) 8048 Analysis. There are two 3's in 8 with a remainder of 2. Since the 8 is thousands, the quotient and remainder are thousands. 2 thousands = 20 hundreds. 20 hundreds and 6 hundreds are 26 hundreds. There are eight 3's in 26, with a remainder of 2. Since the 26 is hundreds, the quotient and remainder are hundreds. 2 hundreds = 20 tens. 20 tens and 4 tens are 24 tens. There are eight 3's in 24. Since the 24 is tens, the quotient is tens. There are two 3's in 8, with a remainder of 2. 2 is §^ of 3. Hence in 8648 there are 2882| threes. 560074-^9 =? 730620^ 10=? 582763 -^ 11 =? 9806436 -^ 12 =? 2668937 -^ 16 =? Analysis. One third of 8 thousands is 2 thousands, with a remainder of 2 thou.sands 2 thousands = 20 hundreds. 20 hundreds and 6 hundreils are 26 hundreds. One third of 26 hundreds is 8 hundreds, with a remainder of 2 hundreds. 2 hundreds = 20 tens. 20 tens and 4 tens are 24 tens. One third of 24 tens is 8 tens. One third of 8 is 2, with a remainder of 2. J of 2 is |. Hence one third of 8648 is 2882|. 103. In problems like the preceding, in which the product of divisor and quotient, the remainder, and the new partial dividend are remembered and not written, the work is said to be by Short Division. The products are not written be- cause they are included in the ordinary multiplication table. Solve and analyze the following rapidly by both methods : 24937068 -Ml = ? 569478370 4- 16 = ? 439450682 ^ 12 = ? 71.3069458 -^ 9 = ? 864037926 4- 11 = ? 1. 289634 -;- 9 = ? 6. 2. 5879639 -4-11 = ? 7. 3. 608579031 -^ 12 = ? 8. 4k 2.50769037 -^ 16 = ? 9 5. 508763018 ^ 12 = ? 10. DIVISICN. 49 104. LONG DIVISION. Illustrative Problem. 98684 ~ i2 = ? FORM. Divisor. Dividend. Quotient. 42 ) 98684 ( 2349 84 146 126 208 188 "404 378 2G Remainder. 105. EXPLANATION. There are two 42's in 98. Multiplying the divisor by the first term of the quotient, we find their product to be 84, which we write beneath the partial dividend. The remainder is found to be 14. Since the 98 is thousands, the quotient and remainder are thousands. 14 thousands = 140 hundreds. To this number is added the 6 hundreds of the dividend. 140 hundreds and 6 hundreds = 146 hundreds. In 146 there are three 42's. Performing the multiplication, to find how much 146 exceeds three 42's, the remainder is found to be 20. Since the 146 is hundreds, the quotient and remainder are hundreds. 20 hundreds = 200 tens. 260 tens + 8 tens = 208 tens. In 208 there are four 42's, with a remainder of 40. Since the 208 is tens, the quotient and remainder are tens. 40 tens = 400 ones. 400 ones + 4 ones = 404 ones. In 404 there are nine 42's, with a remainder of 26. The 26 may be left as a remainder, or we may indicate the part of 42 that it is and join it to the quotient. It is |§ of 42. Hence in 98684 there are 2349 42's and If of another, or simply 2349 1| 42's. 50 XEW ADVANCED ARITHMETIC. To test the work find the product of the divisor and quo- tient, and to it add the remainder. The result should equal the dividend. 106. Analyze the same problem by partition, usmg the analysis previously given. 107. REMAINDERS. Note that in many problems we have an integral quotient with a remainder, as, 25 -^ 4 = G with a remainder of 1 ; or we may complete the division obtaining a fractional quotient, 25 -^ 4 = 6^. AVhether we shall say, 25 divided into 4's are six 4's with a remainder of 1, or 25 divided into 4's are six and one fourth 4's, depends upon the nature of the problem in which these numerical relations are found. 25 cents will buy how many tin cups at 4 cents each? 25 cents will buy how many pounds of sugar at 4 cents each? In which of these problems is there an undivided remainder? Make five concrete problems in measurement involving remainders; five in partition. Make five concrete problems in measurement involving fractional quotients ; five in partition. 108. RULE FOR LONG DIVISION. At the left of the dividend take a partial dividend that win contain the divisor, and place the first term of the quotient at the right. Multiply the divisor by the term of the quotient thus obtained, u-rite the product beneath the partial dividend, and subtract. To the remainder thus obtained annejr the ne.rt term o/ the dividend for a second partial dividend, and proceed as before. Note 1. If a new partial dividend ■will not contain the divisor, place a cipher in the quotient, annex the next term of the dividend, and proceed as before. DIVISION. 51. NoTs 2. In obtaining any term of the quotient, compare the first term of the divisor with the first part of the partial dividend. Note 3. I'rove the problems as in short division. 109. EXAMPLES FOR PRACTICE. Note. Long Division usually begins with 13 as a divisor. In solviug the following problems let the pupils make and refer to the multiplication tables for 13, 14, etc., to 19, until they can find readily any term of the quotient. Thus? 2 X 13 — 26; 3 X 13 =: 39, etc. I. Divide by 13: 31668; 56641; 49784; 73433; 987205; 1266238. 2o Divide by 14 3430; 4536; 6490; 80132; 96039; 1219b53; 1285310. 3. Divide by 15: 3705; 5777; 10269; 91185; 59947; 131549. 4. Divide by 16: 3952; 97264; 140318; 942448; 1376048; 892751; 394887; 1427759; 8895779; 1086331. 5. Divide by 17: 228225; 453151; 821993; 998746; 1201847; 15943688; 1482275; 6293332; 1172540. 6. Divide by 18: 244153; 438753; 644508; 988513; 877858; 1313982; 1743954; 1507995. 7. Divide by 19: 6574; 32908; 268669; 586915 •, 1290105; 1078000; 1798712; 1515876; 1311040; 1410003. 8. Divide 68324 by 20; 30; 40; 50; 60 9. Divide 47906 by 21 ; 31 ; 41 ; 51 ; 61 10. Divide 74583 by 22 ; 32 ; 42 ; 52 II. Divide 194873 by 23; 33; 43; 53 12. Divide 108460 by 24 ; 34 ; 44 ; 54 13. Divide 298765 by 25 ; 35 ; 45 ; 55 14 Divide 370625 by 26 ; 36 ; 46 ; 56 15. Divide 872056 by 27; 37; 47; 57 16. Divide 476921 by 28; 38; 48; 58 60; 70; 80; 90, 61; 71; 81; 91. 62 ; 72; 82; 92. 63 ; 73; 83; 93. 64 ; 7-^ 84; 94. 65 ; 75; 85; 95. 66 ; 76; 86; 96. 67 ; 77; 87; 97. 68 • 78; 88; 98. 52 NEW ADVANCED ARITHMETIC. 17. Divide 572183 by 29; 39; 49; 59; 69; 79; 89; 99. 18. 109278^234; 109278 -^ 467o 19. 934605-^567; 934605-^1667. 20. 27732494 -^ 4658 ; 27732494 -^ 5943. 21. 794006387 h- 568946. 22. 74320876 -i- 6958. 23. 14173345 -f- 2005. 24. 14173345 -^ 7069. 25. 602305812 -^ 70003. 26. 602305812 -^ 8604. 27. A bought 37 acres of land, for which he paid S2,664. "What was the price per acre? Measurement or partition? Why? Analysis. Since 37 acres cost $2,664, each acre cost one thirty-seventh of $2,664. 28. At $72 an acre, how many acres of land can b3 bought for S2,664? Measurement or partition ? Why ? Analysis. Since each acre cost $72, as many acres can be bought for $2,664 as there are 72's in 2,664. There are thirty -seven 72's in 2,664. Hence, for $2,664, 37 acres can be bought, at $72 an acre. Analyze each of the following : 29. At 43 miles an hour, how long will it take a train to run 1,677 miles? 30. If a train run 1,677 miles in 39 hours, what is the rate per hour? 31. 38 pieces of cloth cost ^1,786. What was the average price ? 32 At $47 a piece, how many pieces of cloth can be bought for $1,786? 33. A farmer sold his wheat at 97 cents a bushel, receiv- ing $353.08. How many bushels did he sell? DIVISION. 53 34. A farmer sold 364 bushels of wheat for $353.08. What was the price per bushel? 35. 15 bu. 3 pk. 4 qt. of oats were divided into 4 equal [Ales. What amount was there in each pile? 36. Divide 24 gal. 3 qt. 1 pt. 2 gi. of vinegar into 7 equal parts. What is the amount in each part? 37. The divisor is 328, the quotient 407, and the remain- der 279. What is the dividend? 38. Make a rule for finding the dividend when the divisor, quotient, and remainder are given. 39. The dividend is 364,280, the quotient 877, and the remainder 325. What is the divisor? 40. Make a rule for finding the divisor when the dividend, remainder, and quotient are given. 110. DiTision by 10, 100, etc. 1. 8640 -MO. 2. 4900-^100. 3. 596000 -MOOD. 4. Make a rule based on the three preceding problems. 5. 58764 -^ 100. What is the quotient in Problem 5? the remainder? Make a rule for such cases. Does this include every case? RULE FOR DIVIDING BY A POWER OF 10. Cut off fyoui the right of the iliviaend as many fignres as there are ciphers in the dirisor. The part thus cut off is the remainder, and. the rest of the dii'idend is the Qtiotient. 6. Divide 79640 by 10; 100; 1000^ 10000. 7. Make and solve six problems Hire the above. 111. Changing problems from measurement to partition, or from partition to measurement, 1. Paid 75 cents for oranges at 5 cents each. How many did 1 buy? 5A 54 NEW ADVANCED ARITHMETIC. Give analysis like that of Problem 28, Art. 109. Change to partition. Analysis. If the oranges had cost cue cent each, 75 cents would have bou"-ht 75 oranges. Since they cost 5 cents each, 75 cents bought one fifth oi 75 oranges. One fifth of 75 oranges is 15 oranges. 2. 25 acres of land cost $6.30. What was the average price per acre? Give analysis like that of Problem 27, Art. 109. Change to measurement. Analysis. If each acre had cost $1, 25 acres would have cost $25. Since 25 acres cost $650, each acre cost as many times |l as there are 25's a 650. There are 26 25's in 650. Hence, each acre cost $26. 3. Review Problems 29-34, inclusive. Art. 109, making the changes as indicated above. 112. Division by Aliquot Parts. Review the aliquot parts of 100 and 1000 given in multi- plication. FORM. 73800 1. Divide 73800 by 37^. 738 5904 1968 Analysis. 37iis|oflOO. If the divisor were 100, the quotient would be 7.38. If the divisor were \ of 100, the quotient would he eiglit 738's, which are 5904. Since the divLsor is | of 100, the (luotient is i of 5904, which is 1968. 2. Divide 465500 by 25 ; 37^; 62^; 87i. 3. Divide 39683000 by 125 ; 375; 625; 875. 4. Divide 269500 by 8JL ; 16^; 33^; U?,] 58 J., ; 91§. 5. Divide 5005000 by 162 ; 331 ; 41f,; 58'; 91|. Dl VISION. 55 6. How many dozens of eggs, at 12^ cents a dozen, can be bought for $2 ? $5? $3.75? $6.50? Analysks. (a) If the egi;s were $1 a dozeu, $2 would bu\- 2 dozens. Since the eggs are ^ of $1 a dozen, $2 will buy 8X2 dozens, which ~ 16 dozeus. Or (b) At 12.^ cents a dozen, $1 will buy 8 dozens, and $2 will buy twice 8 dozens, which equals 16 dozens. 7. How many pounds of butter can be bought for $7.50 at 25 cents a pound? at 33^ cents? at 16| ceuts? at 50 cents ? 8. How many yards of cloth can be bought for $10.50 at 37 J cents a yard? at -if' i of h i' o- ' 4 5 15. A fraction is in its lowest terms when the numerator and denominator are prune to each other. 130. PRINCIPLES. I. Multiplying the numerator of a fraction by an integer multiplies the fraction by the integer. Since the number of fractional units in the fractional number is multiplied by the integer, while their size is un- changed, the fraction is multiplied by the integer. Illustrate. II. Dividing the numerator of a fraction by an integer divides the fraction by the integer. Since the number of fractional units in the fractional number is divided by the integer, while their size is un- changed, the fraction is divided by the integer. Illustrate. Ill Multiplying the denominator of a fraction by an integer divides the fraction by the integer. 78 NEW ADVANCED ARITHMETIC. If the number of equal parts into which a unit has been separated be doubled, each part will be one half as large as before. If the denominator of a fraction be multiplied by any integer, the unit will l)e divided into as many times the number of parts that it was before as the integer is times one. The resulting fractional units will be the same part of the former fractional units that one is of the integer. Since the number of fractional units is unchanged, the frac- tion must be divided by the integer. Illustrate. IV. Dividmg the denominator of a fraction by an integer multiplies the fraction by the integer. If the number of equal parts into which a unit has been separated be divided by an integer, the resulting fractional units will be as many times the former fractional units as the integer is times one. Since the numerator is unchanged, the fraction is multiplied by the integer. Illustrate. V. Multiplying both terms of a fraction by the same number does not change its value. Illustration. ^ = *. There are 4 times as many^ frac- tional units in | as in i, but each is only \ as large. VI. Dividing both terms of a fraction by the same num- ber does not change its value. Illustration. * = i. There are only \ as many frac- tional units in -^ as in |, but each is 4 times as large. 131. REDUCTION. Reduction of fractions is the process of changing their denomination without changing their value. Review Reduction, page 6. 132. Illustrative Example. In So there are how many quarters ? i^NALYSis. Since in $1 there are 4 (jiiarters, in $5 there are 5 fours of quarters, which are 20 quarters. FRACTIONS. 79 1. In $7 there are how many tenths of $1 ? in $8? in $9? in $16? in $86? 2. How many quarter-yards in 5 yards ? in 12 yards? in 15 yards? in 25 yards? in 63 yards? 3. Change 7 to fifths; 8 to thirds; 12 to eighths; 15 to sixteenths. 4. Tell how to reduce any integer to an equivalent frac- tion having any denominator. 133. Illustrative Example. In $7| there are how many quarters ? Analysis. Since in $1 there are 4 quarters, in $7 there are 7 fours of quarters, which are 28 quarters. 28 quarters aud 3 quarters are 31 quarters. PROBLEMS. Change the following to improper fractions and anah^ze the process. Note. Illustrate Problems 1-3 with paper circles. 1. 5^. 6. 8§. 11. 64 f. 16. H- 21. 33i. 2. 2|. 7. 12^. 12. 85ig. 17. m. 22. 561. 3. 3|. 8. 15|. 13. 91^. 18. 41§. 23. 66§. 4. H- 9. 201*. 14. 34/^. 19. 26|. 24. 81i-. 5. 5/.. 10. 25|. 15. 512^^-r- 20. 62i. 25. 83|. 134. Solve the above problems, giving the following Analysis. 5^ = ? Since in one there are 2 halves, in any integer there are twice as many halves as ones ; hence in 5 there are two times 5 halves, which are ^-, ^^- + ^ —_^- Show how the following rule may be made from the above analysis. RULE. ]For reducing a tnioced nuniber to an improper fraction. Multiply the integer by the denominator of the fraction. To this restilt add the numerator, and write the sum over the denominator. 80 NEW ADVANCED ARITHMETIC. 135. Change: 1. 8 to llths. 7. 9 to 15ths. 13. 18f to 7ths. 2. 6f to 5ths. 8. 10 to 18ths. 14. 34^7 to 17ths. 3. 12 to lOths. 9. llf to6tbs. 15. 46/^ to 25ths. 4. 12^ to 3ds. 10. 15 to 21sts. 16. 503?o to 30tlis. 5. 7 to 8ths. 11. Uj\ to lOths. 17. 65H to 12ths. 6. 7| to Sths. 12. 17VV to leths. 18. 72/^ to 18ths. 136. Illustrative Example. Change 24 quarters of a dollar to dollars. Analysis. Since in $1 there are 4 quarters, in 24 quarters of a dollar there are as many dollars as there are fours in 24. There are 6 fours iu 24 ; hence in 24 quarters of a dollar there are $6. Similarly reduce the following fractions to whole numbers : 1. f, f, ¥, Y- 6. ih M, ¥, if- 2. hS ¥, -VS ¥• 7. ¥, ¥, H, V. 3. V, ¥, -¥, ¥- 8. If, -¥, f^, H- 4. n, n, n, n- 9. H^ ¥eS ^1^ W 5. ff, 11, ih u- 10. 2|5 W, ¥o^ W 137. Illustrative Example. Reduce ^"- to a mixed number. Analysis. Since in one there are f , in ^ there are as many ones as tliere are fives in 27. There are 54 fives in 27 ; hence there are 5f oaes in y. Change the following fractions to whole or mixed numbers : 1. ¥, ¥-, ¥• 9. ¥/, W, ¥^- 2. -¥, ¥, V- 10. 'W, ¥4S W- 3. ¥, ¥-, H- 11. w, W, %¥• 4. n, ff, ¥• 12. !M, *s^ n§- 5. ih ff, !^ 13. Ml, fH, -VW 6. ?i, H. ?t- 14. %h ' II, W- 7. W» W, -W- 15. W, Vif, W- 8. V^ W> W- 16. ^ 3 , ^H=^- FRACTIONS. 81 RULE. To reduce an improper fraction to a whole or mixed number, divide the numerator by the denouilnator. 17. 7 5 feet are how many thirds of a foot? ^^ of a foot equal how many feet? 18. 12| yards equal how many fifths of a yard? - •*6 't n 13. In a box weighing 12i pounds, a grocer packed for shipment ISJ pounds of ham, |f of a pound of tea, 3| pounds of coffee. 6iV pounds of sugar. What was the total weight of the paclvage ? 14. The United States coins weigh, — cent 48 grains, 5-cent piece 73^ grains, dime SSyV grains, quarter-dollar 96^% grains, half-dollar 192^^ grains, dollar 412i grains, quarter- eagle 64^ grains, half-eagle 129 grains, eagle 258 grains, double-eagle 516 grains. What is the eiitire weight of the series ? 15. A man has in his purse 2 silver dollars, 3 half-dollars, 6 dimes, and 7 5-cent pieces. AVhat is the weight of the whole? (Addition.) 90 NEW ADVANCED ARITHMETIC. 16. How many acres are there in 4 tracts of land, the first containing 885, tlie second, 112§ acres; the third, 146^ acres; and the fourth, 39j'\i acres? 17. P'ind the sum of 12i.\ pounds, 31 GJ pounds, 518| pounds, 2005^ pounds, and 17^ pounds. 18. Find the amount of coal in 5 car-loads weighing as follows: 28| tons, '29J tons, 'dO^Q tons, 27^§ tons, and 31| tons. 19. A mail carrier traveled 12^% miles on Monday, 11§ miles on Tuesday, 13| miles on Wednesday, 10| miles on Thursday, 9{'^ miles on Friday, and 14|f miles on Saturday. Find the whole distance traveled in the 6 days. 148. SUBTRACTION OF FRACTIONS. 1. Define subtraction, minuend, subtrahend, remainder. 2. Illustrative Example. 3 — 1 = ? Analysis. If | be separated iuto 2 parts, one of which is \, the other will be f or |. 7 3 _ -2 11 7_ — ? 13 8 — ? II \ — 1 i — » — ■ T2 12 — • lo T5 — ■ 15 18 — ■ When the denominators are alike, how is the subtraction performed ? 3. I — I = ? How many eighths are needed to make ^? to make ^? How many eighths remain? 4 11 2 . 11 3 . 19 3 • 3 — _7 *• 1 a 5120 4 ' 20 5)5 12' 5. Illustrative Problem. Separate | of a sheet of paper into two parts, one of which shall be | of a sheet. This 2 of a sheet of paper is to be separated into two such parts that one of them shall be I of a sheet. Thirds are not readily formed from fourths ; hence, I change the fourths into something from which thirds may be made. Fold the fourths together. Fold what you now have into three equal parts. Open the sheet. It is now folded into FRACTIONS. 91 Fold the heet into four equal parts ; thus, Fold down one of the fourths ; thus, how many equal parts? Show ^ of the sheet. How many twelfths does it contain ? Show | of the sheet. Now tear off the fourth first folded down. What part of the sheet is left? How many twelfths are there in it? How many are needed to make § of a sheet? How many, then, should you tear off? What is left? What, then, does | - § equal? 6. Application to an abstract problem. i — 1 = ? I am to separate * into two such parts tuat one of them shall be i. Since i is not easily formed from fifths, I change i to twentieths, from which ^ may be made. J = ^B. ^^ are needed to make l- If ig be separated into two parts, one of which is /„, the other is ^i ; hence, * — i — 11 — 2 0' Explain the following in the same manner : 1. 2. 3. 4. 3 1 — 5 G — 3 2 ^: 5. 6. 7. 8. 5 I — 7 4 — 6 _ 2 — 7 _ 1 — V 9. 10. 11. 12. to G — • 7 _ 4 _ ■? T n 7 ij _ a — V 1 J 5 — • 14 _ a — ? 7. From ' the foregoing the following analysis may be derived : if — f = ? Since these fractions are unlike, I change them to equivalent fractions having the I. c. d. Explain the following problems as above 1. i - S = ? 4. f 2. A-^-? 5- i 3. 5 - I = ? 6 I :! — _4 — 1 1 — 1^ — # = ? 13 7 — • 92 NEW ADVANCED ARITHMETIC. 7. H - /2 = ? 13. n - /3 = ? 8. -B - 1 = ? 14. u - ^■'o = ? 9. ^-^ = ? 15. t§ - /o = ? 10. hi - 1^3 = ? 16. M - H = ? 11. §1 - H = ? 17. M-3I = ? 12. ii-U = ? 18. gB-ie = ? Form a rule from the analysis just given. 149. Review Art. 145. Make a rule for problems like the following. Give results rapidly. 1. i -h- 2. h 1 4- 3. 1 4 -h 4. 1 2 - I. 5. h _ 1. 6. i -h 7. i -tV 8. h -h 9. 1 7 -tV 10. } -t\ 11. I -h- 12. I 13. 1 1 " f — T 14. Va-^'o 15. V2-l\l 16. \ - iV. 17. I'o - -I'g 18. 5 — J- 19. 2 _ 2 :5 4" 20. n-?. 21. 5 — ? 22. 1 — I- 23. .? — rr- 24. 1 — t- 25. 1 — I- 26. 2 _ 2 i; s» • 27. 5 — f. 28. i- |. 29. l- T^(T- 30. 3. _ 5 A- 31. 3 7 A- 32. t- 1 7* 33. 4 t\- 34. f- A- 35. (T f- 36. t- t\. 37. f- T%. 38. g- A- 39. H -H 150. Illustrative Problem. 7^ — 4| = what? () IG n ^ 4| 9 12 2/2 " Analysis. J = ^*2 : 1=^X2- Since j*^ is less than ^, I take one of the 7 ones, leaving 6 ones, and reduce it to twelfths. 1 = ff ; H + T2 — rl > H - f2 = A ; 6-4 = 2. Hence, 7^ - 4| = 2^2- Note. That instead of adding \\ to /j I may subtract the /j from the H and add the remainder, -^, to the i*2> •'hu^ obtaining ^, as before FRACTIONS. 93 Observe that the process is identical with that of subtrac- tion of whole numbers. Explain the following in the same way : 1. 17i-8i = ? 7. 83^-59| = ? 2. 26a-l9f = ? 8. 92H - 46f ^ = ? 3. 124^0 - 98i = ? 9. I26i| - 97^ = ? 4. 317/g - 2681 = ? 10. 532i| - 48329 = ? 5. 91/5 - 48a - ? 11- 624if - 279|5 = ? 6. 46Uf-178|^ = ? 12. 1217^1 - 968i| = ? 151. ADDITION AND SUBTRACTION. I. From the sum of f and /^ take their difference. 3. The remainder is /^ and the subtrahend |. What is the minuend? 4. What must be added to Gi^ to produce lljf? 5. A man received $4| for butter, $.5* for cheese, and an amount equal to their sum for vegetables. He paid $3^ for sugar, $4* for coffee, and an amount equal to their differ- ence for tea. What amount was left? 6. 85 - (4Vt - Vo) =? 7. 4|+ (6ii-2f)=? 8. b\l - {21 + IH) =? 9. A man bought 160 acres of land. To A, he sold 24^ acres; to B, ^1^^ acres; to C, as much as to A and B; to D, as much as the difference between A's and B's. How many acres were left? 10. (8}f + 41) -(2^^.4-34)=? II. 3 barrels contain 156^ gallons of oil. In the first are 51f gallons; in the second 49f gallons. How many gallons in the third ? 12. Monday night the Mississippi River at St. Louis stood at 15 feet above low-water mark. Tuesday it rose 1^ feet, 94 NEW ADVANCED ARITHMETIC. Wednesday 1^ feet; Thursday g feet; Friday it fell H feet-, Saturday 2 J feet. What was its height Saturday night? 13. John is 5§ years older than Thomas. Thomas is 2f years younger than Harry and 1 ^ years older than Richard. John is how much older than Harry? 14. A pole 22| feet long is broken in two. One piece is 2| feet longer than the other. What is the length of each ! piece ? Query. How much added to the shorter would make it equal to the longer ? Adding this to the pole, would give what total length ? 15. From Bloomington to Decatur on the Illinois Central railroad the distance is 4o§ miles. Clinton is 1^^ miles nearer to Decatur than to Bloomington. How far is Clinton from each? 16. A owns 12y'^ acres of land; B owns 7§ acres more than A. - C owns as much as both A and B, and D owns as much as the difference between A's and B's; find B's, C's, and D's. Find the whole amount owned. 17. 3 of A's money increased by * of his money lacks ^ of his money of being $955. How much mone}^ had he? 18. The distance from Albany to Syracuse is 148 miles. A starts from Albany for Syracuse and B from Syracuse for Albany at the same time. A walks 23| miles the first day, 18§ miles the second day, 24j\ miles the third, and 29}J miles the fourth. B travels in the same time 15 3 miles, 19.^4 miles, 26^ miles, and 31^ miles. Make diagrams and show : a. How far apart they were at the end of each day. b. How far each is from the starting-point at the end of each day. c. How far each is from his destination at the end oi eacn day. FRACTIOXS. 95 19. A piece of cloth contains TyV yards. What will be left after using | of a j-ard for a vest, 2^ j-ards for a coat, and 2^ yards for a pair of pantaloons ? 20. A farmer having 7 apple-trees gathered from them as follows: 21 barrels, 3| barrels, 4f barrels, 5/^ barrels, 2;* barrels, 3§ barrels, 5i barrels. He sold to one man 123 barrels; to a second, 2^ barrels; to a third 4| barrels. How many barrels were left? 21. In 1834 the amount of gold in the eagle was reduced from 247^ grains to 232i grains. How much was taken out? 22 From an ounce (480 grains) of standard gold were minted an eagle, a half-eagle, and a quarter-eagle. How many grains remained? (See Art. 147, Problem 14.) 23. In 1853 the weight of the dime was reduced from 41^ grains to 38i'2 grains. How many grains were taken out? 24. From a standard silver dollar, 41 2i grains, 10 dimes are coined? How many grains of silver remain? 152. MULTIPLICATION OF FRACTIONS. 1. Define multiplication, multiplicand, multiplier, prod- uct. What does the numerator show? What is the effect pro- duced by multiplying the numerator? Why? Unite five 2's of thirds; six 3's of fifths; seven 5's of eighths. What did you do in each of these cases? How, then, may you multipl}' a fraction b}" an integer? 2 Since the numerator of a fraction expresses the numWr of fractional units in the fractional number, it is evident that a fraction is multiplied by multiplying its numerator. Note. The sis^n X was first introduced by William Oughtred iu 1631. At first the multiplier was uniformly placed after the sign. Now the mul- tiplier frequently precedes it. The sign X is read " multiplied by " when the multiplier follows ; as 7 lbs. X 5, 9A. X |, f oz. X 8 (" three fourths of an ounce multiplied by eight "). 96 NEW ADVANCED ARITHMETIC. The sign is read " times " when the multiplier preceding it is an integer or a mixed number, as 5 X 7 lbs., 6| X 8 ft. (" six and three fourths times eight feet"). The sign is read " of " when the multiplier before it is a simple frac- tion; as I X $20 ("three fourths of twenty dollars"). The sign is read " b}' " when the factors are dimensions; as, a pane 14" X 32" ("a pane fourteen inches by thirty-two inches"). A door -3' — 8" XT- 6" (" a door three feet eight by seven feet six "). 153. THE MULTIPLIER AN INTEGER. PROBLEMS. 1. Multiply ^- by 7c Analysis. Seven 4's of fifths are %'. In \^ there are as many ones as there are 5's in 28. There are 5| fives in 28 ; lience -/ = 5|. 2. 5 X 4. 9. T^o X y. 16. 8jo X 21. 3. 1 X 5. 10. i\ X 11. 17. ,15| X 32. 4. 1 X5. 11. 1 X 10. 18. 26^1 X 13 5. i X 7. 12 A X 6. 19. 9f X 14. 6. 1 X 9. 13. ,S X 9. 20. 35H X 43 7. ^x 4. 14. H X 8. 21. 52| X 49. 8. fi X 8. 15. 7^ X 12. 22. 69S X 56. Note. Observe that in these problems the number of fractional units ifl multiplied in each case. 154. PROBLEMS. 1. Multiply t X 4. What fractional unit is four times as large as an eighth? Wliat, then, is 4 times §? Here we multiply the size of the fractional units, by dividing the denominator by the integer, hence we say " 4 times ^ is ^ ; " g is clearly 4 times |, since it has the same number of fractional units and they are 4 times as large. 2 FRACTIONS. 97 5?T X 7. 14. ]| X 14. 26. Hi X 72. 3. tV X 8. 15. i* X 19. 27. m X 64. 4. H X 12. 16. II X 34. 28. ^\\ X 48. 5. M X 13. 17. H X 10. 29. yi X 38. 6. ^i X 7. 18. ^i X 18. 30. 2V5 X 75. 7. ff X 19. 19. ^5 X 37. 31. ^^^ X 75. 8. H X 21. 20. t*3 X 25. 32. fH X 64. 9. If X 17. 21. U X 19. 33. 3V3 X 49. 10. fi X 26. 22. \l X 13. 34. ||i X 25. 11. H X 11. 23. |a X 27. 35. //g X 24. 12. 43^ X 8. 24. il X 29. 36. 78/4 X 28. 13. ^^ X 20. 25. ^1 X 28. 37. ^^%%^ X 33. 155. PROBLEMS. Ii has been shown in Art. 86 that the continued product of a multiplicand and the factors of a multiplier is the same as the product of the multiplicand and the multiplier itself ; thus: 25 X 3 X 2 =1 25 X 6. In such problems as j% x 10, we multiply the size of the fractional units by 5, obtaining §, then the number of frac- tional units by 2, obtaining ^3^. 1. If X 12=? s If = 4 times 3 times Jf 3 times |f = i^. 4 12. II X 56. 22. /4L X 180. 13. ^7 X 63. 23. \l^ X 48. 14. |§ X 75. 21. Ill X 72. 15. ii X 65. 25. 3V5 X 125. 16. |i X 28. 26. 1-12 X 120. 17. li X 65. 27. 4^5 X 218. 18. fi X 87. 28. Ill X 240. 19. H X 100. 29. f|a X 250. 20. t¥s X 144. 30. fig X 144. 21. T%\ X 51. 31. tVo% X 245. An. 4.LYSIS. 12 times V^- = '^=8 2. f X 12. 3. t\ X 9. 4. H X 28 5. H X 24 6. 1 .T T5 X 20 7. u X 35 8. 4? X 24 9 hi X 33 10. l| X 40 11. U X 45 98 NEW ADVANCED ARITHMETIC. "WTiich of these three methods may always be employed? Which is the most convenient? RULE. To tnultiply a fraction by an integer, divide the denomi- nator by the integer, ff possible ; if not possible, divide the denominator by the largest possible factor of the integer; if the denoniiutitor is not so divisible, nuiltiply the nume- rator by the integer. Since dividing the denominator by a factor of the integer is the same as omitting that factor from the denominator and the integer, the rule may be shortened : Omit all factors common to the integer and the denomi- nator, and mtiltiply the numerator by the remaining factor of the integer. 156. THE MULTIPLIER A FRACTION. PROBLEMS. 1. Multiply 8 by I . This means find f of 8. Whenever the multiplier is a fraction, the problem may be read in the same way. There are two processes involved: finding ^ of 8 (partition), and uniting three such parts (multiplication). Analysis. ^ of 8 is 2. | of 8 are 3 twos, which are 6 Find: 2. I of 12 ; I of 15 ; f of 21 ; la of 28. 3. § of 6 bushels; § of f ; | of 10 cents ; f of |§. 4. ^ of $24 ; f of V- ; t'u of 40 acres; -^^ of y>. In the preceding examples the multiplicand is first divided by the denominator of the multiplier, and the quotient is multiplied by the numerator. Since the order of the opera- tions is immaterial, we may first multiply the muUiplicand by the numerator and divide this product by the denomina- tor. Generally this will be more convenient; hence, the FRACTIONS. 99 RULE, To tnnltiply by a fraction, tnultiply by the numerator and tliiide the product by the denominator. 5 Multiply 8 by f ; 9 by § ; 12 by f ; 15 by f. 6. Multiply 21 by f ; 24 by | ; 30 by | ; 32 by §. 7. Multiply f by f ; | by 3 ; j% by § ; ii by g. 8. Multiply ^i by f ; |-| by fg ; if by | ; H by §. 9. Multiply U by If; U by H; tVo by H; HI by #f. 10. Find j'o of f. How find t^ of §? How y'o of §? 11. Find g of Vi ; f of j% ; § of ^ ; § of f . 12. Find f of i*i; f of |; f of A; i of |. 13. Find I ox t ; f of ^ ; i% of {2 ; § of §§. In these problems, how is the numerator of the product formed? the denominator? Where cancellation is possible, what should first be done? RULE. To }nultij>Iy a fraction by a fraction, cancel all factors common to a numerator and a denominator, and miiltiply together the reinaininrj factors of the numerators for the numerator of the product, and the remaining factors of the denominators for the denominator of the product. 157. PROBLEMS, Multiply : 1. H by 15. 4. 129 by fi. 7. il by f . 2. i| by 48. 5. 15 by 32. 8. 1? by §f . 3. 54 by hh 6. ^5 by |. 9. f of /, by 1 of ^rs- 10. § of 3 of f by H of If of A. Find : 11. y- of 560 12. §§ of 784. 13. f§ of 1872. Multiply : 14. f Jf ^' by t of f of j%. 15. 8§ by 6. Analysis. 6 times | is 4. 6 times 8 is 48. 48 + 4 = 52. 100 NEW ADVANCED ARITHMETIC. Multiply : 16. 12| by 10. 17. 155 by 24. FORM. Ill 18. 125^2 by 48. 19. 584^1 by 50. 20. G24^V by 86, 21. Ill by 81. 5i Use this method wheu the fractions are small. Analysis. J of 11=2 and a remainder of 3. 3 = |. I + I = ¥• i of i = A- 8 X I = 5^. 8 X 11 = 88, etc. 96i Multiply : 22. 15|byl2^. 23. 86|by27l. 24. 1248/o by 492|. Reduce the mixed numbers to improper fractions. 25. What is the cost of ^^ of an acre of land at 888 an acre? 26. What is the cost of /- of a quire of paper at 23 cents a quire? 27. What is the cost of 4g cords of wood at 84.75 a cord? 28. What is the cost of 23J yards of broadcloth at 82J a yard ? 29. What is the cost of ^^^s tons of coal at 82.57 a ton? 30. What is f of I ? t of /^ of \\ ? 31. What is f of ^2 of \\ of 50 ? 32. What is /^ of 2^ of \\ of 5 J of 12? 33. Find the cost of 3^ yards of cloth at 83.75; of 4| yards at 82.40; of 15f yards at 83.20. 34. Bought ^ cords of wood at 83.50 ; 5^ cords at 84.40 ; 6| cords at 84.56. Sold the wood at an average price of 85.00 a cord. What was the gain? 35. A farmer gathered 42^ loads of corn averaging 33§ bushels from his 25-acre fiekL What was the total yield? 36. Multiply 17| by 14J; 26g by 293; 132^^ by 68|; 694|by87^. FRACTIONS. lOi 37. A steamer ran at an average rate of 386f miles for 65 successive days. What distance was covered? 38. Find tlie cost of 469| bushels of wheat at 603- cents. £9. Find the cost of 35| pounds of coffee at 28f cents. 40. A father worked 6 days at S2J per day, his son 5 days at Sl|, his daughter 4 days at $| : what were their total earnings for the week 'i 41. What is the area of a door oi feet X 7^ feet? 42. Draw a square b\ inches by 5i inches. Draw lines dividing it into square inches. Multiply 5^ by 5^ and point out each partial product in the diagram. 43. The sheet on which I now write is 7|" X lOf"'. How many square inches ? Rhomboid. 44. A rhomboid is a four-sided figure with parallel sides and oblique angles. It was shown in Art. 92 that the area of a rectangle is the number of square units in a row the length of the rectangle multiplied by the number of such rows ; or, briefly, area = base X altitude. In the rhomboid the perpendicular distance from the base to the side opposite is the altitude. 8A 102 XEW ADVANCED ARITHMETIC. 45. Cut a rhomboid out of paper. Fold the base AB upou itself, ereasiug it along the altitude PK. Cut along P K ; arrange the parts as in the sec- ond figure. The base, altitude, and area of this rectan- gle are the same as in the rhomboid ; hence area of rhom- boid tude. base X alti- 46. Cut out two equal paper triangles, M and N. Place a pair of equal sides together thus, making the rhomboid B S. Since the base and altitude of triangle M and of the rhom- . ^ . , base X altitude bold are equal, area of triangle = ■ 47 Cut out five paper triangles, measure accurately their bases and altitudes, and determine their areas. 48. Determine the area of any triangular spaces in the school-yard. Draw each on some convenient scale. 49. Draw and calculate the area of the following triangles. Use the scale of one inch to the foot in the first set, one quarter-inch to the foot in the second set. FRA CTIONS. l03 Base. 50. 4 ft. 51. 2 yd. 52. 3f ft. 53. 4tV ft 54. 3ift. Altitude. 2\ ft. \\ ft. 43 ft. 2§ ft. 1] y^i- Base. Altitude. 55. 20 ft 171 ft. 56. 18| ft 61 ft 57. 93 ft 14| ft 58. 6 ft 14^ ft 59. 4§ yd. Z-\ yd. 60. A cubic foot of water weighs 62 J pounds. Ice is |f as heavy as water. What is the weight of a cubic foot of ice? 61. "What is the weight of a cubic foot of Joliet limestone which is 2 J times as heavy as water? of dry pine \l as heavy ? 62. A gallon of water weighs 8 J pounds. What is the weight of a gallon of mercury 13| times as heavy? of a gal- lon of milk which is \%% as heavy as water? 63. Standard silver is ^^ pure. How many grains of silver in the standard dollar of 412^ grains? 64. The rear wheel of a bicycle is 7^ feet in circumfer- ence, and revolves 2A times as often as the pedals. How many miles are traveled in 1,000 revolutions of the pedals? (1 mile = 5,280 feet) 65. From what number can 5^| be taken nine times with no remainder? 158. DIVISION OF FRACTIONS. 1. Define Measurement, Divisor, Dividend, Quotient. 2. Define Partition, Divisor, Dividend, Quotient. 3. Illustrate a problem in measurement and one in parti- tion by using objects. 159. Divisor an Integer. Illustrative Problem. 1. Divide \^ into 5 equal parts. Is this a problem in measurement or partition? Why? How can it be performed with objects? 104 XEW ADVANCED ARITHMETIC. According to the definitions heretofore given, this is a problem in partition. It ma}' be read : P'ind I or \^ ; \ of 2. Divide if by6; 1^ by 3 ; f i by 7 ; i4byl2; §lby9; U by 13. 3. Divide II by 16; i\ by 17; |i by 19; g| by 17; A\ by 29. 4. Divide fQ by 15; fi by 13; ff by 19 ; ^i^ by 24 ; 45 by 7; 8? by 14 ; 5j3^ by 19. How may all of these divisions be performed? Make a nile based upon the solution of these problems.. Jllastrative Problem. 5. Divide | by 5. How does this problem differ from the preceding? In what other way ma}' a fraction be divided by an integer? ^ of \^= -^-Q, obtained by multiplying the denominator by 5, which di^ades each fractional unit by 5. Explain fully the eSect of multiplying 'he denominator by an integer. 6. Divide ^% by 4 ; f by 7 ; \\ by 9 ; 2i by 10 ; 4f by 13. 7. Divide I by 8; § by 11 ; f^ by 12; ^^ by 16. 8. Divide 3^ by 7; 5? by 8; 7f by 12; ^ by 9. 9. Divide ^5 by 12. An-altsis. r2 = 3 of i- Hence r2 of ^ = i of i of j^. i of ^^ = ^. 1 r>f 2 — 2 3 01 15 — ^• 10. Divide j§ by 15; \1 by 20; fo by 25; |i by 14; §Jbyl8; Mby 14; B by 26. 11. Di\ide |i by 34 ; f § by 39 ; ^4 by 36 ; 4^ by 38 ; ft by 85. How was the division performed in the first set o. prob- lems? How in the second set? How in the third? ]\Iake a rule based on these solutions. "What cancellations should be performed? FRACTIONS. 105 160. Divisor a Fraction. Illustrative Problem. 1. Divide 4 by 5. Analysis. To divide 4 by | is to separate 4 into equal parts, each of which is f . 4 = ^/. lu \^- there are six 2's of tliirds (or 6 times |). Illustrate this problem by folding 4 paper squares into thirds, and then separating the thirds into groups of two each. 2. Divide 8 by f . 8 = 4^". In %Q there are 20 times f. What is done with the integer? How is the division per- fonned? Make a rule. Divide : 3. 7 by |. 4. 12 by g. 5. 10 by ^. RULE. To divide a tvhole ntnnher by ti fraction, reduce tJie whole tiuniber to the same deitoniinfifion as the fraction, and divide the numerator of the dividend by tlia mime-' fator of the divisor. Divide : 6. 18 by ^^. 8. 32 by if. 10. 63 by U- 7. 24 by If 9. 45 by Jg. 11. 36 by ||. 12. How many boxes each holding § of a quart can be filled from 8 quarts of berries? 13. How many yards of cloth at f of a dollar a j'ard can be bought for $12? 14. If a man can dig a ditch -^-^ of a rod in length in an hour, how many rods can he dig in 3| days of 9 hours each? 15. At 2^ cents each, how many apples can be bought for 52 cents? Reduce the mixed number to an improper fraction 2^ -~ ^. 106 NEW ADVANCED ARITHMETIC. 16. At $21 a yard, how many yards of cloth can be bought for 863 ? 17. At 83^ a day, how many days must a man work to earn $144? 18. At S24| an acre, how many acres can be bought for $724? 19. If one horse cost $124[^, how many horses can be bought for 83,990? 161. Illustratice Problem. 1. Divide 1 by §. Analysis. I = |- Iu f there are as many f as there are 2's in 5. The quotient of 5 by 2 may be expressed thus : |. Hence 1 -^ | = |. Explain the following in the same way : Divide 1 by § ; by f ; by f ; by ^ ; by U ; by ii ; by if ; byH; byf-^ byn- Note. The denominator is the number of fractional units in 1. The numerator is the number of fractional units in the fraction ; hence the rule. RULE. To divide 1 by a /faction, divide the deuoininator of the fraction by its niinierator. Note. This is usually called " inverting the divisor." 2. Divide 15 by f. Analysis. If the dividend were 1, the quotient would be |. Since the dividend is 15, the quotient is 15 times I, which ecjuals, etc. Analyze Problems 2 to 11 (Art. 160) by this method. 162. Division of a Fraction by a Fraction. Illustrative Problem. Divide ^ by ^. Fold a paper square into fourths. Tear out one of the fourths. We are now to see how many pieces, each of which is one third of the paper square, can be made from the three fourths. Since thirds are not easily made from fourths, we change the three fourths to nine twelfths. Four twelfths FRA CTIOXS. 107 make one third. Nine twelfths make two thirds with one twelfth left, which is one fourth of another third. Hence, in i^ of a sheet of paper there are 2\ thirds of a sheet. Explain the following problems by the same method. Divide : 1. |byi. 3. gby|. 5. § by ^. 2. §byi. 4. Aby|. 6. fbyi. Now omit the reference to objects. What do you do with divisor and dividend? RULE. To divide a fraction by a fraction, change them to equivalent fractions having the I. c. d., and divide the numerator of the dividend by the numerator of the divisor. 7. Divide A by i. Analysis. ^ = H- ^ - f§. ?§ 4- f§ = 63 - 10 = f§ = 6^^. 8. Divide i by t- Analysis. ^ = 3^. f = ff . 5% ^ |§ = 6 - 25 = ^%. 9. Di^^de ^\ by f ; ^% by f ; % by ^ ; Yi by |. 10. Divide 21 by If; 3} by 2^ ; 1^ by 3|. 163. Shorter Analysi 1. Divide 5 by |. Analysis. 1-1 = 1- i -f = 1 of 1 = i|. 2. Divide S byf; 1 byl f by 1; by a ; f by v 5 I by|i; HbySI. Cancel common factors. RULE. To divide a fraction by a fraction, invert the divisor and proceed as in multiplication. Use the following form, ^ -H /tt = M X ¥ = if • 3. Divide U by /o ; I? by ^\ ; H by il- 108 NEW ADVANCED ARITHMETIC. 164. PROBLEMS. 1. At $f a pound, how many pounds of coffee can be bought for Soi? 2. If a man travel 43 miles an hour, in how many hours will he travel 23 1 mUes? 3. At 82-^ a yard, how manv vards of cloth can be bousht forSGi? 4. If each bag hold If bushels, how many bags will be needed to hold 40 ^ bushels of oats? 5. At Sfy a yard, how much cloth will 8f buy? 6. If an acre of land will yield 23/^ bushels of wheat, how many acres are necessary to yield 1,874* bushels? '• a - 1) - (f X I) = ? 8- (! + *)-^(t-|) = ? 9- § of f of f\ -^ i of i of 185 = ? 10. (^ X if) - m - i; = ? 11. Divide il by 9 ; if by 25 ; /^ by 10. 12. Divide 6 by * ; 12 by f ; 15 by |. 165. COMPLEX FRACTIONS. Problems in division of fractions are sometimes written in the form of a fraction ; thus, 5 -f- J may be written -|- 4 Such expressions are called Complex Fractions. 1. The following method of reading complex fractions is recommended. The complex fraction whose numerator is | and denominator §. Read : 2^ § 5. 2_ .|_ f 5" J^ 4 V r 7' V 8^' 4i' 6- 2. The longest straight line used separates the numerator from the denominator. This line mav be regarded as a sign FRACTIONS. 109 of division.. The expression above it is the dividend, and that below it the divisor. Solve the following as problems in division. Reduce 1. 3 4 5. .5 2. 91 -3 6. 8-§ 3. % 12' 7. 8 X J ^ X 8 4. 1 of fof 9 TO 8. 10. 11. 12. -A f of 2i 91 - -^3 ^f §x 6i H- -1 IX 9§ 92 — J 166. To Find the Part which One Number is of Another. 1. 3 is what part of 7 ? Analysis. 1 is ^ of 7, hence 3 is f of 7. 2. 5 is what part of 12? 6, of 17? 11, of 22? 5, of 20? 8, of 24? 9, of 36? 17, of 12? 19, of 7? The part which one number is of another is alwa^'s ex- pressed by a fraction, of which the number that is the part is the numerator, and the other the denominator. Make a rule for finding the part that one integer is of another. 3. § is what part of .5? Analysis. 5 = ^f. | is the same part of ^ that 2 is of 15. 2 is ^ of 15 ; hence f is ^ of 5. 4. I is what part of 8? | is what part of 12? | is what part of 10? 1 is what part of 9? j^ is what part of 8? j\ is what part of 6 ? 5. 1^ is what part of 4? -J-§ is what part of 14? \l is what part of 126? 6. 2^ is what part of 6 ? 3 j is what part of 4 ? 5| is what part of 10? 110 NEW ADVANCED ARITHMETIC. Change the mixed numbers to improper fractions. Make a rule for finding the part that a fraction is of ao integer. 7. I is what part of | ? Analysis. | = f^. f - |§. f^ is the same part of f§ that 24 is of 25. 24 is II of 25 ; hence -f is f | of f . What was done in the above problem ? Mrke a rule for such cases. 8. f is? what part of f ? 9. I is what part of § ? 10. i\ is what part of i ? 11. With each of the following pairs of numbers, find the part which the first is of the second, and give the results rapidly. (1) h h (9) h h (17) 7t3„ 8f. (2) h h (10) h h (18) 8^, lOf (3) 1, i. (11) 4|, 51. (19) 12^3^, 15 J. (4) h h (12) J, 81. (20) 19i, 14|. (5) h 6. (13) i, 9. (21) 211, 25|. (6) 1, 8. (14) 9, i. (22) H, T^o- (7) 8, 61. (15) f,|. (23) H, H- (8) 6^, 8. (16) 5§, 4|. (24) 22^1^, 455»j Show how the following general rule is derived : To find the part that one nnniher is of another, divide the number ex-pressing the part by the number of which it is a part. 167. To Find a Number when a Specified Part of it is Given. Illustrative Problem. 15 is ^ of what number? Since 15 is I of the required number, ^ of that number is J^ of 15. ^ of 15 is 5. I of the required number is 8 fives, which are 40. Heuce 15 is § of 40. FRACTIONS. 1 Find the number of w lich 1. 45 is f . 8. 125 is U. ls. 18| is ^. 2. 48 is 3. 9. 324 is ^|. 16. 413 is ^^. 3. 36 is j%. 10. 441 is |i. 17. 462 is ^5. 4. 72 is if. 11. |is|. 18. TT is 1%' 5. 75 is If. 12. 2 ia •*> 3 1° jr. 19. 91§ is H- 6. 84 is ^^. 13. 3| is /5. 20. 23| is -If. 7. 90 is \l. 14. 7| is il- 21. 34A is U 111 22. A has §3^ and B $7*. A's money is the same part of B's that B's is of C's. How mucn has C? 23. A farmer has 15f acres of meadow and 40 acres of oats. The part which the meadow is of the oats-field is the same that the latter is of the corn-field. How many acres of corn has he? 24. A house cost § as much as the lot. Both cost $896. Find the cost of each. 25. The area of North America is 6,446,000 square miles. This is about § of the area of Africa. Whi.t is the approxi- mate area of the latter? 26. The area of Australasia is 3,288,000 square miles. It is about what part as large as North America? as Africa (11,514,000)? as South America (6,837,000)? 27. The annual expenditure of England per capita for military purposes is S3. 72, and for education is 70 cents per capita. The latter is what part of the former? (Approxi- mate.) 28. The distance from one corner of a field to another corner is 770 yards; this is y^^ of a mile. How many feet are there in a mile? 29. A certain farm contains 340 acres which are ^| of a section. How many acres are there in a section? 30. A merchant sold goods to the amount of $316.80 on Monday; this was ^^ of his sales on Tuesday, which were 112 NEW ADVANCED ARITHMETIC. * of his sales on Wednesday. What was the aggregate of his sales for the three days? 31. In a school-room there are three windows each 9 feet X 3i. If the lighting area is 2^^ of the floor area, what is the latter? If the room is 30 feet wide, what is its length? 168. MISCELLANEOUS PROBLEMS. Oral Exercises. (Use no written work in the solution of these problems.) 1. Find the 1. c. m. of 4, 5, 6 ; of 8, 12, 20; of 5, 15, 30, 40; of 4, 6, 8, 10, 12, 15; of 12, 15, 18, 20, 30. 2. Find the 1. c. m. of 12, 18, 24; of 15, 30, 40, 45; of 18, 24, 30, 36; of 60, 90, 105, 120; of 80, 120, 160, 240. 3. Change to whole or mixed numbers ->/, ^, ^^a, f|, L25 JL60 19 » 24 • 4. Change to improper fractions 9f, lO^Sj-, I2f , 15f, 18§, 21i. 5. Add i and 1 ; f and f ; | and | ; | and ^ ; §, |, and |; I, ^, i^o, and tV- 6. |-i=? f-|=? |-f=? A- + =? 7. Multiply f by 4 ; /^ by 24 ; ^^ by 45 ; ^6^ by 44 ; ^^ by 8. 8. Find f of 16 ; ^^ of 36 ; ^\ of 10: ^^ of 13 ; jf of 7. 9. Find § of I; t of if ; /^ of T-V; A of ^^. 10. Multiply t by f ; 2^ by 3^ ; 6^ by 2i ; f by ^^ ; H by?,. 11. Divide 4 by ^; 7 by i; lObyi; 12byTir; 6 by §; 7 by 1% ; 8 by | ; 6 by H; 15 by ^^ ; 14 by § ; 18 by f ; 20 by 5; lOby f; BbyU; 8 by H; 7 by H- 12. Divide i¥ by 5 ; H by 6 ; /^ by 16 ; i\ by 18 ; j§ by 36; H by 48; H by 51 ; ^ by 87; ff by 39; f| by 72. FRACTIONS. 113 13. Divide 1 by i; 1 by i; 1 by ^V ; 1 by §; 1 by t\; Ibyfg; 1 byH; 1 by A^o- If one be divided by any fraction, what will tlie quotient be? 14. Divide § by t ; t by f ; | by ^V ; H by f § ; ^ by 6f . 15. t-ft=? t-f-? tXf=.? 1^1-? f is what part of | '^ | is what part of ^ ? 16. 7 is what part of 13? 8, of 19? 11, of 44? 18, of 27? § is what part of 6 ? of 9 ? of 15 ? ^ is what part of 4 ? of 7 ? of 10? of 12? of 15? 17. I is what part of * ? of -i% ? of f ? of |§ ? /„ is what part of I'i of t^^? of \V of 2^? of 3J? 18. 14f is f of what number? 19. I of 15 is f of wliat number? 20. I of I of 15 is I of i of 9 times what number? 21. f of 21 is I of what number? 22. I? of m is ^Q of 5 times what number? 23. John has ^§ of a dollar; William has f as much, and this is 16 cents more than | of Henry's money. How much has Henry? i24. John lost f of his marbles, and has 15 left. How many had he at first? 25. Mary's money is f of Laura's, and both have 90 cents. How much has each? 26. A's farm is | of f of B's. A and B together own 81 acres. How many acres has each? 27. John is 7; Fred 9 : (1) The difference in their ages is what part of John's age? (2) Of Fred's age? (3) John's age is what part of Fred's age? (4) John's age is what part of Fred's age less than Fred's age? (5) Fred's age is what part of John's age more than John's age? Note. In concrete problem.s, or where a fraction expresses the relation between two numbers, " of " follows the fraction. If one number is a frao- 114 NEW ADVANCED ARITHMETIC. tion of a second number more or less than the second number, the " of" phrase is frequently omitted ; thus Jane's age is \ less than Mary's means, "Jane's age is \ of Mary's less than Mary's." 28. A raised 150 bushels of potatoes, which was \ less than what B raised. How many did B raise? 29. James has 884, which is \ less than B's money. How much has B ? 30. A walked 120 miles, which is i more than B walked, and i less than C walked. How far did B and C walk? 31. 63 is 1 more than what number? It is \ less than what number? 32. The time past noon is ^ of the time till midnight. What o'clock is it ? 33. The time till midnight is V of the time past noon. Wliat o'clock is it? 34. The time till midnight is f of the time past noon. What o'clock is it? At what hour is the time till midnight | of the time past 3 A. M.? 35. How many cubic inches in a brick 8" X 4" x 2\ "? How many half-inch cubes in a two-inch cube? 36. Arrange the fractions f, \\^ j\, ^|, in order of magnitude. 37. I of water is oxygen. What weight of oxygen in a gallon of water weighing 8 J pounds? 38. ^ of air is oxygen. How much oxygen in a cubic foot of air weigMng 52.5 grains? 39. If 15 gold pens cost S25. what is the cost of 3 gold pens? 40 If 21 sheep are worth 856, what are 3 sheep worth? Query. Is it necessary to find the cost of one shfep? 41. If 24 men consume a barrel of flour (196 pounds) in 2 weeks, how many pounds do 3 men consume ? FRACTIONS. 115 42. If 32 horses in 3 days consume 60 bushels of oats, how many bushels do 20 horses consume in one day ? 43. If 12 bushels of oats are worth 9f bushels of corn, 10 bushels of oats are worth how much corn? 44 If 8 barrels of flour cost $25^, what do 11 barrels cost? 45. If at noon on a certain day a 15-foot pole casts a 12- foot shadow, what length of shadow is cast by a pole 32 feet long ? 46. A pine block 3 " X 3 " X 3 " weighs 5 ounces. What is the weight of a similar block 3 " X 4 " X 6 " ? 47. A can dig a ditch in 6 days. What part of it can he dig in one day? B can do the same work in 4 days ; what part can both dig in one day working together? 48 A and B can trim 2% of a hedge in one day. In what time can they trim .^l of the hedge? fg of the hedge? 49. A can do a piece of work in 7 days. A and B can do the same work in 4 days. In what time can B do the work alone ? 50. A can mow a field in 5 days, and B can mow it in 6 days. In how many days can they mow it, working together ? 51. A can perform a certain work in 8 days, B in 10 days, and C in 12 days. In what time can the three perform it, work i ng together ? 52. Three pipes. A, B. and C, fill a cistern in 3 hours. A alone fills it in 8 hours; B in 12 hours. In what time can C alone fill it? 53. Pipe A can fill a cistern in 10 hours; pipe B in 12 hours; pipe C empties it in 1.5 hours. When all three pipes are open, how long is the cistern in filling? 54 A can saw a cord of wood in ^ a day, B in ^ of a day. In what time can both saw it working together? Query. How much does each saw in a day i 116 NEW ADVANCED ARITHMETIC. 55. John can spade a garden in J of a day, Charles in * of a day. In what time can they do it, working together? 56. If I of a yard of cloth cost 81*, what will j\ of a yard cost? 57. If to I of the cost of an article $2 be added, the result will be f of the cost; what is the cost? 58. J of the distance from Chicago to Elgin is -f^ of the distance from Chicago to Aurora. Elgin is 4 miles farther from Chicago. How far is each from Chicago? Analysis. f of dist. to E = jSg of dist. to A. ^ of dist. to E = 1% of dist. to A. Dist. to E = f^ of dist. to A. Dist. to A = i| of dist. to A. Difference of di.^tances = ^ of dist. to A. Difference of distances = 4 miles. i\ of dist. to A = 4 miles. ^ of dist. to A = 2 miles, f^ of dist. to A = 42 miles, dist to E. jl of dist. to A = 3S miles, dist, to A. 59. If f of the distance from C to D is | of the distance from E to F, and if the sum of the two distances is 82 miles, what is the distance from C to D? from E to F? 60. A house and lot cost 85,200. The lot cost yV as much as the house. What did each cost? 61. IJ] is f of what number? 62. What is the cost of 6^ yards of cloth at 82i a yard? of 4 1 yards at $2^ a yard? of 75 yards at 81-4^ a yard? 63. How many pounds of material can be bought for 85^ at 8^ a pound? for 84f at 8v\ a pound? for 88f at 8/^ a pound? for 810^ at 82 a pound? 64. If A had 84 and lost 8|, what part of his money did he lose? if he had 85 and lost ^? if he had 86^ and lost 84? if he had 812^ and lost 87? if he had 810i and lost S6i? FRACTIONS. 117 65. What is the cost of i- a yard and \ of a yard at $f a yard? of § + § at $§ ? of | + f at %,\ ? of \ + | at $§ ? 66. A grocer bought equal lots of eggs at 9 cents per dozen and 10 cents per dozen. He sold them at 12 cents per dozen, clearing 60 cents. How many dozen did he buy? 67. Sarah's age is § of Mary's and % of Ruth's. The sum of their ages is 46 years. How old is each? IsOTE. 1 iud au expressiou for each iu sixths of iSarah's age. WRITTEN PROBLEMS. 1. The product of three numbers is 124:2. Xwo of the numbers are 7 J and 8]. What is the third? 2. What number divided by % of ^2 equals 132§? 3. What nimiber diminished by 2.53y'3 leaves 84y ? 4. What is the 1. c. m. of 60, 125, 180, 225, 250? 5. Bought 8^ yards of cloth at S3.\ a yard, 12-; yards at $2 1, and 18^ yards at $4^. How many bushels of corn at 26| cents a bushel will pay the bill? 6. 15 X A X ^* X 1068=? 7. What is the cost of 28 pounds of butter at 37 ^ cents a pound, 84 bushels of corn at 43^^ cents a bushel, 135 bushels of oats at 25 cents a bushel, 160 bushels of rye at 62^ cents a bushel ? 8. A man performed y^s of ^I's journey the first day, jAj of it the second day. § of it the third day, and had 17g miles left. What was the length of his journey? 9. A man left f of his estate to his eldest son, ^-^ of it to his second son, and the remainder to his third son. The share of the second was S520 less than that of the third. What was the value of the estate ? What was each son's share ? 9A 118 NEW ADVANCED ARITHMETIC. 10. Bought a farm of 3244 acres of land for $15,646. If the house and a house-lot of 12 acres were counted at $3,150, what price per acre was paid for the rest ? 11. What number is that } of which exceeds ^ of it by 112? 12. What number is that § of j of which is 291 less than I of I of it? Owning ^j of a farm, I sold ^ of my share. If what I had left was worth $211), what would the whole farm be worth at the same rate? 14. I bought a house and lot for $5,784,. paying j^^ as much for the lot as for the house. How much did I pay for each? 15. I of A's farm equals f of B's. Together they have 551 acres. How many has eacliT 16. A and B can do a certain piece of work in 12| days. They worked togeMier 6| days ; A then left, and B finished the work in 8J days. In how many days could each do the work alone? 17. How many bushels of grain can be bought for $l,260f at 31^ cents a bushel? at 56^? at 66f ? at 87i? 18. If 56| acres of land cost $1,200, what is the price per acre? 19. How long will 225 pounds of meat last a crew of 5 men, at the rate of If pounds per day for each man? 20. A man's crop of oats weighed 66,677 pounds. He sold it for 3^ cents a bushel. Counting 32 pounds for a bushel, what was the value of his crop? 21. A crop of wheat weighing 90,144 pounds brought $1,252. Counting 60 pounds to a bushel, what was the price per bushel for wheat sold? 22. f + A + i^rr - t of ^^^ = ? FRA CTIONS. 119 23. Bought 8J yards of broadcloth at $4^ a yard, 5J yards of cassimere at $2h a yard, 5? yards of silk at $2^ a yard, and paid for all with corn at 41| cents a bushel. How many bushels were required ? 24. What is the 1. c. m. of 824, 936, 1020? 25. A man owned f of a factory. He sold | of his share. He gave ^ of the remainder to his daughter, ^ of what then remained to hi^ son, and sold J of the remainder for §14,000. What was the value of the factory? What was the daugh- ters share ? the son's share ? What was the value of what he had left ? 26. Find the sum, difference, and product of 4£ and 6-j\. 27 Find the quotient arising from dividing the sum of 8/3 and 5f by their difference. 28. A can do a piece of work in 15 days, B in 12 days, and C in 10 days. A works 2 days, B 3 days, and C 3 days. In what time can A and B finish the job by working together? 29. Thomas can dig a ditch in 3| days, Richard in 6f days, Harry in 4* days. In what time can all complete it working together? 30. Divide f of 6§ by § of y^ of 12|. 31. X f X -f X 16| X ^0. 32. A cistern having a capacity of 88|^ barrels contained 63/o barrels. After 15^^ barrels were pumped out, the cis- tern was what part full ? 33. A had a journey of 43^2 miles to perform. After traveling 36^ miles, what part of the journey remained ? 34. 23 is what part of 48i? of 614? of 58§? 35. T^ is what part of 87? of 169? of 4:6^? of 83tV? 36. 7| is what part of lof? of 28|? of 72|? 120 NEW ADVANCED ARITHMETIC. 37. Divide 76 into two such parts that f of the first shall equal ^ of the second. 38. Divide 417 into three such parts that the first shall be j\ of the second, and the third shall be \^ of the second. 39. How is the value of a proper fraction affected by adding the same number to both terms? Why? 40. How is the value of an improper fraction affected by adding the same number to both terms? Why? 41. How is the value of an improper fraction affected by subtracting the same number from both terms? Why? Of a proper fraction ? Wh}-? 42. What is the circumference of a wheel that makes 24 1 revolutions in 400 feet? 43. If another wheel have a tire two feet shorter, it makes how many more revolutions in the same distance ? 44. It takes six hours to complete a certain journey at the ordinary rate of travel. How many hours are required to complete three fourths of the journey if the rate be in- creased by one half of itself? 45. An adult inspires 30 cubic inches of air at an ordinary inspiration. If they breathe 1 5 325 tii' 2 -^ 7 1 1 _1_ 1 7 5 a 8^' 12^' 16f' 6i' 33i' 6§' 2>\ RULE. Multiply or divide both terms of the fraction frti some number that uill make tlie denominator a itower of ten. Then ea-press the denominator by the position of the riyht-hand figure of the numerator with respect to the decimal point* 182. ADDITION OF DECIMAL FRACTIONS. Define addition, sum. Numbers are "written how? Why? Addition begins where? Why? Make a rule. Analyze each problem as in simple addition. 183. PROBLEMS. 1. 2 3 4. .625 4.073 126.0009 4006.092 .0984 26.0084 482.1872 9.059701 .4907 59.00462 600.50983 683.086409 .00864 83.01879 17 008459 34.189357 .09769 50.00043 6.098725 1086.049078 \ 5. Add .00862, 4.04378, 73.096, 168.00097, 49.287005, 83.460037. 6. Add 77.02081, 94.09069, 88.00799, 686.060098, 897.0609, .084858, .087857, .3060686, .76978. 7. Add .04069, .008972, .0934, .0083462, .027309, .5302681, .05003701. DECIMAL FRACTIONS. 133 8. Add 7 tenths, 56 hundredths, 93 thousandths, 329 hundred-thousandths, 8052 millionths, 42067 ten-thou- sandths, 43 hundredths, 98 ten-millionths. 184. ADDITION OF COMPLEX DECIMALS. Illustrative Example. Add 3^ tenths, 7f thousandths, 17-i2r hundredths, 86 2| ten-thousandths. Analysis. Ten-thousandths being the lowest denomination, all num- bers of higher denomination are to be reduced to teu-thousaudths, unless they become pure decimals before such reduction is completed. .3^ = .3 + .^i-^ = .3125 .007f = .007 + .^5?a = .0074 .17t\ = .17 + .-4f^ = .17l8fV .0862| = .08621 Note. If all of the mixed decimals can be reduced to pure decimals, reduce them before addition. 1. Add 3J tenths, 29^ thousandths, 56] thousandths, 24^ hundredths, 183^ thousandths, 86^ hundredths. 2. Add 15f thousandths, 38^ hundredths, 409 .\ ten- thousandths, 3^ tenths, 9^ hundredths, 7f thousands, 5-^j tenths, 2^ hundredths. 3. Add 64| hundred- thousandths, 53* hundredths, 78.^ thousandths, 86| ten-thousandths, 4920/^ hundred-thou- sandths, 6/2 tenths, 9/5 hundredths. 4. Add 423W millionths, 29^^ hundredths, 46J units, 126§ tenths, 479^V hundredths. 185. SUBTRACTION OP DECIMAL FRACTIONS. Define subtraction, minuend, subtrahend, remainder. Perform and explain a problem in subtraction of simple numbers. Give the rule. Apply the same to the fol- lowing problems. Note. Make minuend and subtrahend of the same denomination be>- fore subtracting. lOA 134 NEW ADVANCED ARITHMETIC. 186. PROBLEMS. 1. .0861 - .0295 = ? 2. .7043 — .4805 = ? 3. .0461 - .00356 = ? 4. 4.02603 - .9078 = ? 5. 26.1059 - 19.74308 = ? 6. 461.083024 — 86.59260834 = ? 7. .023 — .000465 = ? 8. 92. — .06479 = ? 9. 400. — .00004 = ? 10. .00583 - .0000583 = ? 11. .7^ - .0081 ^ ? 12. .43J — .0047^ = ? 13. .0084} — . 00023 1^^- = ? Note. .0084} = .00841^ = .00840^. 14. 42.08^ - 34.0574H = ? 15. 83.50703xV — 59.02 = ? 16. .80352 _ .0454 = ? 17. 84.0431 + 56.057 — 16.059 - 23.00845 = ? 18. 625 — .0748 + 29.0536 - 439.00596 - .089037 = ? 19. .08^ + .0043^ — .0061 = ? 20. 5^ — .061 + .009^^ + -OO-tV =' 21. 28 thousandths — 46 ten-thousandths =? 22. 423 millionths — 17 hundred-thousandths =? 23. 46 tenths — 46 thousandths =? 24. 3824 hundredths — 3824 ten-millionths =? 25. \ of a tenth — | of a thousandth =? 26. ^ of a thousandth — ^ of a millionth — ? 27. 426| ten-thousandths — 38^ hundred-thousandths =? 28. 9251 hundredths — 4659 hundred-millionths —? DE CI MA L FRAC TIOXS. 135 29. 94f thousandths — 53}| ten-millionths =? 30. \ of ten — ] of a hundredth =? 31. ^ of one hundred — ^ of a hundredth =? 187. MULTIPLICATION OF DECIMAL FRACTIONS. 1. Define all terms in multiplication. 2. "What is the effect of removing the decimal point one place to the right? t'^\o places? four places? How multiply by 10? by 100? by 10000? by 1000000? 3. "What is the effect of removing the decimal point one place to the left? three places? six places? How divide by 10? by 1000? by 1000000? 4. Multiply .008764 by 10, and read the result; by 100; by 1000; by'lOOOOOO; by 10000000. 5. Divide 4968.307 by 10, and read the result; by 100; by 1000; by 10000. 6. Mate general rules for multiplying and dividing by powers of ten. 188. 1. Multiply .0536 by 28. Note. Since tlie multiplicand is ten-tliousandtlis, the product is ten- thousandths; hence, 28 times .0536 = 1.5008. 2. Multiply .824 by .01 ; by .001 ; by .1 ; by .0001, and read the results. Multiplying a number by .1 is equivalent to dividing it by what divisor? 3. Multiply .497 by .39. This problem means: Find .39 of .497. How do you find .01 of .497? What will the result be? How many decimal places will it have? "What do you do with this result? 4. Explanation. I first find .01 of .497. .01 of .497 is .00497, which is found by removing the decimal point two orders to the left, and filling the vacant orders with ciphers. .39 of .497 is 39 times .00497. 136 NEW ADVANCED ARITHMETIC. (1) (2) .497 } S .00497 197 > .39 J 39 .04473 .1491 .19383 5. How many decimal places are there in the product? Why? Make a rule for "pointing" the product. RULE. In Multiplication of Decimal Fractions, multiply as in simple numbers, and, point off as many decimal places in the product as there are in multiplicand and multiplier. 189. PROBLEMS. Multiply : 1. .542 by 58. 3. .00436 by .8. 5. 6.0281 by .072. 2. .0693 by 324. 4. .7093 by .49. 6. 28.0563 by .057^. Note. Simplify the decimal fractions. 7. .800694 by .17^- 8. .006294^ by .00863f. 9. .0976^ by 24^^. Note. Change ^s to a decimal fraction. 10. 25864 by .03972. 11. 30.6895 by 4.300906. 12. 50638 thousandths by 9026 hundredths. 13. 49060037 millionths by 207003 ten-thousandths. 14. 409 billionths by 36 millionths. 15. 5063087 teu-millionths by 6204 thousandths. 16. 29 ten-thousandths by 29 ten-millionths. 17. 48 tenths by 48 ten-thousandths. Give the results rapidly in the following problems : 18. 9 hundredths by 7 thousandths. 19. 15 ten-thousandths by 6 thousandths. 20. 8 thousands by 9 thousandths. DECIMAL FRACTIONS. 137 21. 24 ten-thousands by 24 ten-thousandths. 22. 17 hundreds by 4 hundred- thousandths. 23. 23 tenths by 4. 24. 11 milliouths by 11 thousandths. 25. 18 ten-thousandths by 4 hundredths. 26. 7 ten-milliouths by 14 hundred-thousandths. 27. What is the product of tenths and thousandths? of thousandths and hundredths? of ten-thousandths and ten- thousandths? of hundreds and thousandths? of thousands and hundredths? of millions and millionths? of hundred- thousands and hundredths ? of millionths and hundreds ? of ten-thousandthfe'. and thousands? 28. .125 is what part of 1? of 1 ten? 29. .375 is what part of 1 ? .0375 is what part of .1 ? of 1 ? 30. .0025 is what part of .01 ? 31. .0625 is what part of . 1 ? 32. .000875 is what part of .001 ? 33. .006| is what part of .01? Read each of the following as a part of 1 standing in the first order at the left of its significant figures : 34. .00075, .0025, .00125, .0000875, .033.^, .0000662. 35. What is the cost of 864 bushels of oats at $0.41 per "bushel ? 36. What is the cost of 17 horses at $112,375 each? 37. What is the cost of 384 acres oi land at $67,065 each? 38. What is the cost of 18.56 yards of cloth at $2.5625 a yard? 39. What is the cost of 29 books at $0.0625 each? 40. What is the cost of 465.375 bushels of wheat at $0.9175 per bushel? 138 NEW ADVANCED ARITHMETIC. 190. DIVISION OF DECIMAL FRACTIONS. 1. Define partition, dividend, divisor, quotient, remainder. .97254-^8=:':' Explain by partition. Follow the form in simple numbers. 2. Explain the following in the same way. 191. The divisor a whole number. 1. .7658^7 = ? 6. .042864 -f- 24 = ? 2. .04536 -^ 6 = ? 7. .008399 -^ i}7 = ? 3. .157032 -^ 9 = ? 8. .010867 -^ 46 = ? 4. 71.40636 ~ 12 = ? 9. 4.20638 4- 55 = ? 5. 146.0736 -^ 16 = ? 10. 436.0095 h- 77 = ? 11. What is the denomination of the quotient in partition? In the preceding problems the quotients are like what? Hcjw many decimal places are there in each quotient? 12. Make a rule for pointing the quotient when the divisor is a whole number. 13. Explain the same problems by division, using the form given in simple numbers. 14. The quotient in each case is like what ? RULE. In dirision of decimal fractions, ulien the divisor is a whole tiutnber, divide as in simple nufiibers, atid point off aa niatiy decimal places in the Quotient as there are in the dividend, 192. The divisor a decimal. 1. Illustrative Problem. .Sib -f- .5. The dividend is the product of the divisor and quotient. The number of decimal places in the product equals the num- ber in both multiplicand and multiplier. Hence the numl)er of decimal places in the quotient is equal to the number iu DECIMA L FRA CTIONS. 139 the dividend minus the number in the divisor. In this case the number of decimal phices in the quotient is 3 — 1, or 2. .5) .875 1775 2. 87.5^.005 = ? We annex ciphers until the dividend contains as many decimal places as the divisor. The quotient contains 3—8 decimal place's. .005) 87.500 17500 193. PROBLEMS. 3. .6241 -^ .79 11. 4. 1.0276 -^ .028. 12. 5. 44.814 -^ .;>7. 13. 6. .39071 -^ .0089. 14. 7. .091512 -~ .0124. 15. 8. 7.5522 -^ 2.46o 16. 9. .153032 ^ 00376. 17. 10. .02336081 ^ .00583. 18. 5.66703747 -^ 70.83. .052629096 H- 5470.8. 183.057 -h .379. 39.3888 -^ .0528. 5.81715 -:- .00695. .633447 -^ .000783. .7737013 -^ .0000859. 112.1021 -^ .02453. How does the rule apply to this problem ? 19 20. 21. 22 23 24. 25. 33. 34. 35c 36. 37. 2099.274 ^ .3607. 26. 26624.32 -^ .4379. 27. 5481 ^ ,063. 28. 48760 ^ .0092. 29. 766300 ^ .00079. 30. 133574 — .000329. 31. 24980020 -^ .0406. 32. 8.6947 -^ 28. .46083 -^- .37. .070685 ^ .000056. 860.025 90.3864 .0378. 4.77. .016458 -^ .000963. .023907 ^ .0001839. .7 -^ 43, carry the quotient to 5 decimal places. 4.6-^58, " " 6 " " 63. ~ 97, " " 4 " " ,1 _i- 329 " " 6 ' " 4. -^ 586, " " 8 " 140 NEW ADVANCED ARITHMETIC. 194. To divide by a power of 10, remove the decimal point as many places to tlie left as there are zeros in the divisor. 1. Divide 8493.7 by 10; by 100; by 10000; by 1000000. 2. Divide 49.683 by 100; by 10000; by 1000; by 10; by 1000000. 195. Division by the factors of a number. 1. Divide 8.75 by 50. 8.75 h- 10= .875. .875 ^5 = .175. 2. Divide 12.25 by 500; by 5000; by 250; by 25000. 3. Divide 75 by 750 ; by 7500 ; by 75000. 4. If 24 boxes of fruit cost $52.32, what does each box cost ? 196. REVIEW PROBLEMS. 1. Change .00875 to a common fraction. 2. Change ^2 ^ ^ decimal fraction. 3. How many square feet in a piece of ground 86.48 feet long and 39.6 feet wide? 4. If a man travel 29.6 miles a day, in how many days will he travel 1,016.088 miles? 5. Change ^^ to a decimal fraction. 6. What is the cost of 473.5 bushels of corn at .47 of a dollar a bushel? 7. At .47 of a dollar a bushel, how many bushels of corn can be bought for S222..545? 8. Add 342 thousandths, 568i hundredths, 634^- tenths, I of a hundredth, and J of a ten-thousandth. 9. From } of a tenth take ^ of a thousandth. 10. Change .00^ to a common fraction. 11. Change .04 to a common fraction. 12. Change y^j to a decimal fraction of three orders. 13. If 97 books cost $317,675, what will each cost? DECIMAL FRACTIONS. 141 14. If S8.5 bales of cloth cost $3,048.43, what is the price of each bale ? 15. If a pair of shoes cost $2,625, how many pairs can be bought for $49,875? 16. At 83 cents a yard, how many yards of cloth can be purchased for $61,005 ? 17. At $.045 per pound, how many pounds of sugar can be purchased for $1.89? 18. Define a decimal fraction. How does it differ from a common fraction ? 19. Find 1. c. m. of 18, 24, 36, 40, 180. 20. Define a pure decimal ; a mixed decimal. 21. 23^ X 4f =? 831 X 25H =? 3^ X 24.08 =? 22. 693|-f-7; 826i-^9; 938^^ -^ 15. 23. Tell how to change a decimal fraction to a common fraction. Give two ways of changing a common fraction to a decimal fraction, H X 4f ^ .08;V X 1 . 24. — ? 25 What common fractions can be changed to pure deci- mals ? Explain in two ways why this is so. 26. 2^ is what part of 15? of 20? of 31? of 6§? of 8|? of 18|? of 50? of 100? 27. How many twos are there as factors in the denomina- tor of f ? How many successive one-place reductions of the numerator must be made to introduce these twos? How many ciphers, then, should be annexed at once? Same questions for ^^ ; for ^| ? How about y^ ? ^^? 28. Study the following fractions in a similar way. fl S .39 » 25' T^5' 50" JL9_ 29. In addition of mixed decimals, what should be done before beginning to add? Answer a similar question for subtraction. 142 NEW ADVANCED ARITHMETIC. 30. Loaned $824.40 with the uuderstanding that .06^ of the amount should be paid me for its use for one year. At this rate, what sum should be paid for its use for 18 months? for 30 months? for 3 years and 5 months? 31. AV^hat is the rule for " pointing" the product in multi- plication of decimals? Give the reason. 32. Sold a house and lot for $1,824.60, losing 16^ per cent of the cost; what was the cost? 33. What shall goods costing $624.80 be sold for to gain S~h per cent of their cost? 34. Define measurement, and all terms. Define partition, and all terms. 35. Give and explain the rule for "pointing" the quo- tient when the divisor is a decimal fraction. How multiply by a power of 10? How divide by a power of 10? 36. Multiply 324.086 by 2.5648. Divide 831.2157728 by 324.086. 37. There are 2,150.42 cubic inches in a bushel. What is the capacity in bushels of a bin 48 feet long, 8 feet wide, and 12 feet high? 38. There are 231 cubic inches in a gallon. What is the capacity in 40-gallon barrels of a cubical cistern 6x8 feet on the bottom and 10 feet deep? 39. A brick is 8 X 4 X 2 inches. What is its volume? This is what part of a cubic foot? ^ of a brick wall is usually filled with mortar. How many bricks will be required to build a wall 40 feet long, 8 feet high, and 8 inches thick? 40. Find the cost of an 8-inch foundation for a building 18 feet X 24 feet, the wall to be 7 feet high, and brick $8 a thousand in the wall. 41. Two men start from the same place and travel in opposite directions, one at the rate of 3.85 miles per hour, and the other at the rate of 4.12i miles per hour. How far apart will they be at the end of 13 hours? MEASUREMENT OF THE CIRCLE. 143 42. A freight train runninG; at an average rate of 16| miles au hour starts from Albany at 6 a, m. At 9 a. m. an express train starts from the same point and runs iu the same direction at an average rate of 42^ miles an hour. At what time will it overtake the freight train ? How far from Albany ? 197. MEASUREMENT OF THE CIRCLE. Measure accurately with a tape-line the circumferences and diameters of five circles, such as the bottom of a pail or the head of a barrel. If the figure measured is not a true circle, take half the sum of the longest and shortest diame- ters as the true diameter. Divide each circumference by its diameter, getting the quotient true to hundredths, thus : 14.5, ^ 4^ ^ 14.1S75 ^ 4.5 = 3.15+ These quotients differ because of inaccuracies in measure- ments. Find their average. This is a close approximation circumference to the true quotient. This quotient, -, is repre- diameter seuted by the character -n- (called j*;/). Hence circumference = tt X diameter. PROBLEMS. 1. Measure the circumferences of 5 trees and calculate the diameter of each. 2. Measure the circumferences of 5 other round objects, — ball, stove, apple, etc., — and calculate their diameters. 3. How long is the tire on a 4-foot wheel? 4. How far is a wheelman advanced by each revolution of his 28-inch wheel? 5. A bicycle is said to be geared to 70 inches when each revolution of its pedals propels it a distance equal to the circumference of a 70-inch wheel. How many revolutions of the pedals will move the wheel one mile? 144 NEW ADVANCED ARITHMETIC. Note. By more accurate methods the value of ir has been ivund to 200 decimal places. Hereafter call its value 3^ or 3.1416—. 6. My bicycle bas 28-incb wbeels, 7 sprockets on the rear bub, 17 sprockets at the pedal. How many turns of the pedals will carry me two miles ? 7. The circumference of the earth at the equator is 24,897 miles. What is its equatorial diameter? 8. What is the diameter of a circular mUe race- track ? 9. The wheels of a wagon are respectively 42 inches and 49 inches. In going what distance does the fore wheel make one more turn than the hind wheel? 10. How is a circle drawn ? 11. Draw a cu'cle whose diameter is 6 inches ; one whose radius is 8 inches. (The radius is h of the diameter.) Cal- culate the circumference of each. Cover the circumference of each with a string. Measure it. 198. DEFINITIONS. 1. A Plane is a surface against which a straight edge will fit in all directions, as the desk-top, the blackboard. Point out surfaces in the school-room that are not planes nor made of planes. Such are called curved surfaces. 2. A circle is a plane figure bounded by a line all points of which are equally distant from a point within called the centre. 3. The bounding line is called the circumference. Any straight line from centre to circumference is called a radius. A line passing through the centre and terminating in the circumference is called the diameter. FORMUL/E. D = 2R. C = 7rxDor7rD. C = 7rX2Ror27rR. NoTK. When letters rei)reseiit numbers, the sign of multiplication may be omitted. MEASUREMENT OF THE CIRCLE. 145 199. Cut from a potato or turaip a thin circular slice. Cut it iu ti^-o, and divide each semicircle into 8 equal wedges. Be careful not to cut the rind. Straighten the rind of each semicircle and fit the wedges together. The circle is now a rhombus. Its base is half the circum- 27rR ^ ference, — - — , or - K. Its altitude is R ; hence its area is - R X R = tt R'-. NoTK. An exponent is a figure written above and to the right of a figure or letter to show liow many times the number represented by the latter is to be used as a factor ; thus, 3- = 3 X 3 ; 4^ = 4 X 4 X 4. PROBLEMS. 1. If R in the above diagram is 7 inches, what is the length of AB ? What is the area of the circle ? 2. TMiat is the area of one face of a silver dollar? 3. The face of a watch is IJ inches in diameter. "WTiat is its area? Solution, it /?2 = 31 x (\"T- = -V- X J" X f" = 2l| square inches. 4. By how many acres does a square mile exceed a mile circie ? 1 mile = 320 rods- 160 square rods = 1 acre. 146 NEW ADVANCED ARITHMETIC. 5. "WTiat is the length of the sweat-band in a 6| hat? 6. How high must a 3-iuch tin-cup be made to hold one pint? One gallon = 231 cubic inches. 7. Measure pails, tin-cups, coffee-barrels, and other cylin- ders and determine their capacity in cubic inches. 8. What is the volume of a new lead pencil \" X 7 " ? 9. Around a circular pond 500 feet in diameter is a gravel walk 30 feet wide. What is the area of the walk? Ql-ery. "What is the area of the entire circle, including walk and pond ? the area of the pond only ? 10. How many square inches of tin are needed to make a quart cup 4 inches in diameter, making no allowance for seams? 11. How many barrels of 3U- gallons each will a cylindri- cal water-tank hold if 14 feet in diameter and 12 ft., 10 in. deep, inside measure? 12. Mercury is shipped from the mines in cylindrical steel bottles holding 100 pounds each. If these bottles are 4 inches in diameter, what must be their depth? Mercury is 13.6 times as heavy as water. 13. What is the weight of a dry pine log 12 feet long and 30 inches in diameter? Specific gravity of dry pine = .48. 14. How many cubic inches in a cylindrical tile 12 inches long? Outside diameter 8 inches, inside diameter 6 inches? 15. How many square inches in the entire surface of a cylindrical block 6 inches in diameter and 6 inches high? How many cubic inches in its volume? Which is the larger, surface or volume in a 4 '^ X 4" cylinder? in an 8" X 8" cylinder ? FEDERAL MONEY. 147 200. FEDERAL MONEY. 1. Money is that medium by which exchanges of property are ordinarUy effected. 2. Federal Money is that sj'stem of mouey established by the Congress of the United States of America. 3. The unit of value is the dollar. 4. AVhat are the denominations? Give the table. "What is the scale? 5. A coin is a piece of metal on which certain characters are stamped by government authority, making it legally current as money. 6. The coins are as follows : WEIGHT. One cent . . Bronze, 48 grains Troy. 3-ceut piece . Copper and nickel, 30 (( 5-ceut piece . Copper and nickel, 73.16 " (I SILVER. Dime . Silver and copper, 38 iV " (( Quarter-dollar > . it 96^0 " (( Half-dollar • i; u 192.9 " li Dollar . . . Ik a (( 412.5 " ti Gold and copper. GOLD. Quarter-eagle Three-dollar . " " " Half-eagle . " " " Eagle = SIO . " " " Double-eagle. " '• " 7. The silver and gold coins are one tenth copper. 8. The gold dollar weighs 25.8 grains Troy. To find the weights of the remaining coins, multiply this number by the number of dollars expressing the value of the coin. 9. The government also provides a currency made of paper. It consists of treasury notes, national-bank notes, gold certiflcates, and silver certificates. 148 NEW ADVANCED ARITHMETIC. 201. BILLS AND STATEMENTS. 1. A bill is an itemized statement of indebtedness. "When it includes items purcliased at different times, it is usually called "a statement of account." When payment is made, the statement is " receipted." 202. Arrange the following in bill forms : 1. On February 10, 1892, D. C. Smith bought of R. C. Rogers & Co., Rochester, N. Y., 47 rolls wall-paper, at 27 cents; 60 yards border, at 3 cents; 13 shades, at Si. 72; 112 feet moulding, at 17 cents; 13 sets curtain fixtures, at 79 cents. 2. The following is a statement of W. P. Johnson's account with A. B. Cole & Co., Salem, Mass., made June 30, 1891: 1891, June 15, 2 gallons molasses, at 64 cents; June 16, 1 sack flour, Si. 65; 10 pounds starch, at 7 cents; 9 pounds turkey, at 12 cents; June 21, 5 gallons oil, at 15 cents; 2 loaves bread, at 5 cents; 2 dozen eggs, at 13 cents, June 23, 24 pounds sugar, at 5 cents; June 24, h bushel potatoes, 40 cents; June 26, 2 pounds cheese, at 20 cents. Receipt this statement. 3. R. P. Young, in account with G. G. Johnson, Normal, 111. 1891, December 1, 1 dozen cakes, 10 cents; 2 loaves bread, at 5 cents ; 1 pound halibut, 20 cents ; December 3, 25 pounds sugar, at 4 cents; 1 peck sweet potatoes, 25 cents ; ^ bushel apples, 50 cents ; December 4, 2 barrels kindling, at 25 cents ; December 7. 3 chimneys, at 8 cents ; 1 quart oysters, 35 cents ; December 8, 1 pound tea, 75 cents; 1 dozen cakes, 10 cents; December 10, 2 dozen eggs, at 16 cents; December 11, 3 lemons, at 4 cents; 1 sack salt, 35 cents ; December 14, 4 cans plums, at 20 cents ; 1 bushel apples, $1.25; December 17. 1 peck onions, 28 cents; December 20, 3 chickens, at 25 cents; 5 gallons oil, at 15 cents; December 24, 12 pounds turkey, at 11 cents; 1 quart oysters, 35 cents ; 5 bunches celery, at 4 cents. FEDERAL MONEY. 149 203. ALIQUOT PARTS. 6^ cents is fj of $ H " iV " ^ 12^ " i " $ 16| « i " $ 25 " i " $ 33^ ceuts is i of $1. 66| cents is | of $1. 37^ " i " f I. 75 "I " $1. 50 '■ ^ " $1. 83^ " I " $1. 62^ " I " $1. 87^ " I " $1. PROBLEMS. 1. What is the cost of 384 pounds of sugar at 6| cents per pound ? Analysis. If the sugar were $1 a pound, 384 pounds would cost $384. Since the sugar costs jg of a dollar a pound, 384 pounds will cost ^^ of $384, which equals $24. 2. What is the cost of 600 articles at 8^ cents each? 3. Of 688 articles at 12| cents each? 4. Of 1,272 articles at 25 cents each? 5. Of 5,865 articles at 33| cents each? 6. Of 575 articles at 50 cents each? 7. Of 473 articles at 10 cents each? 8. Of 972 articles at 20 cents each? Continue this exercise until great facility is acquired. 9. What is the cost of 568 yards of cloth at 37^ cents a yard? Analysis. If the cloth were $1 a yard. 568 yards would cost $568. If the cloth were | of $1 a yard, 568 yards would cost ^ of $568. which is $71. Since the cloth is f of $1 a yard, 568 yards will cost 3 X $71, which is $213. 10. Of 480 pounds of tea at 62^ cents per pound? 11. Of 560 articles at 87^ cents each? 12. Of 1,272 articles at 66jf cents each? 13. Of 2,480 articles at 75 cents each? 14. Of 42,684 articles at 83^ cents each? IIA 150 NEW ADVAXCED ARITHMETIC. 15. At 12?, cents a dozen, bow many dozens of eggs can be purchased for §7o? Analysis. At $1 a dozen, $75 will buy 75 dozens. At ^ of SI a dozen, $75 will buy 8 X 75 dozens, -which equals GOO dozens. 16. At 25 cents each, how many articles can be bought for $84? for 8125? for8M4.oO? for 8328.75? for S875.25? 17. At 33 i cents each, how many articles can be bought for$4U> for $i)5? for6875> for 8'J75.33i ? for 61, 275. 66§ ? 18. At 16 5 cents each, how many articles can be bought for 818? for $'J6? for 8324.1G2? for $425.33^? for $585.50? for 8728.66:^ ? for 82,548.831, ? 19. At 8^ cents each, how many articles can be pur- chased for 25 cents? for 50 cents? for 83' cents? for 815? for $85? for $125.16,2? for 8250.25? for 8324.33.1? for $354,415? for 8681.50? for 8724.58J-? for 81,242.66^-? for $1,461.75? for 8l,5'J5.83i? for $2,568,915? 20. At 50 cents each, how many articles can be bought for $50? for $125? for 81,250? for $864.50? for $965.25? for 81,386.75? 21. At 37^ cents a pound, how many pounds of butter can be bought for $15? AxALTSis. At $1 a pound, SI.t will l)uy 15 pounds. At |- of a dollar a pound, $15 will buy 8 X 15 pounds, which equals 120 pounds. At | of a dollar a pound, $15 will buy ^ of 120 pounds, which is 40 pounds. 22. At 62i cents each, how many articles will $125 buy? 8200? 8645?" 8874.25? 81,025.50? $2,550? 23. At 87 h cents each, how many articles will $28 buy? $42? $91? $126; $357? $826.42? 24. At 66;| cents each, how many articles will $8 buy? $28? $64? $186? $432.50? $786.48? 25. Make 5 problems in which the cost of each article is 75 cents; 83 J cents; 91§ cents. DENOMINATE NUMBERS. 151 204. DENOMINATE NUMBERS. 1. Measuring is the process of finding how many times a given quantity contains another quantity called the unit of measure. 2. The unit of measure is called the Standard Unit, and is usually defined by law. 3. A number composed of standard or derived units employed in the measuring of magnitudes is a Denominate Number, as 3 bushels, 5 pounds. 4. A denominate number composed of Ijut one kind of unit is a Simple Denominate Number, as 3 pecks. 5. A denominate number composed of more than one kind of unit, but which is reducible t > a simjjle denominate number, is a Compound Denominate Number, as 4 bushels, 3 pecks, which is reducible either to bushels or pecks. 6. A scale is the statement of the number of units of each kind required to form one of the next higher kind. Note. We have seen tliat the decimal scale is uniform. Nearly all of the scales in Compound Denominate Numbers are not uniform. While the scale in Federal Money is 10, in English Money it i.s 4, 12, 20. 205. MEASURES OF LENGTH. 1. A Line is that which has extension in only one dir'^ction. 2. Measures of length are called Linear Measures. 206. LINEAR MEASURE. Table. 12 inches (in.) = 1 foot ^ft.). 3 ft. =: 1 yard (yd.). bl yd. = 1 rod ^-d.). 320 rd. = 1 mile. 152 XEW ADVAXCED ARITHMETIC. Ho"w many yards in a mile ? How many feet ? Remember these numbers. (Perform without analysis.) 1. How many inches in 5 feet ? 7 feet? 8 feet? 15 feet? ie\ feet? 18^ feet? 7.8 feet? 5.16 feet? 2. How mauy feet in 5 yards? 12 yards? 16^ yards? 18} yards? 24 yards? 33. V yards? 42.6 yards? 3. How many yards in 4 rods? 10 rods? 32 rods? 33} rods? 37i^ rods? 9.52 rods? 4. How many rods in 6 miles? 9 miles? 16.V nailes? 24^ miles ? 5. How mauy inches in 4 feet, 5 inches? 16 feet, 6 inches? 31 feet, 8 inches? 6. How many inches in 5 yd. 1 ft. 6 in. ? 8 yd. 2 ft. 8 in. ? 19 yd. 2 ft. 3.V in.? 4 rd. 2 yd. 3 in.? 22 rd. 2 ft. 11 in.? 7. How many feet in 36 inches? 60 inches? 84 inches? 100 inches? 115 inches? 8. How many yards in 12 feet? 18 feet? 22 feet? 26 feet? 82 feet? 9. How many rods in 11 yards? 33 yards? 44 yards? 38^ yards? 49^ yards? 35| yards? 10. How many miles in 640 rods? 960 rods? 1,280 rods? 1,350 rods? 2,080 rods? 11. How many feet and inches in 65 inches? 88 inches? 131 inches? 164 inches? 236 inches? 12. How many yards, feet, and inches in 62 inches? 93 inches? 167 inches? 328 inches? 434 inches? 13. How many rods, yards, and feet in 37 feet? 62 feet? 69 feet? 86 feet? 14. How many miles, rods, and yards in 1,797 yards? 3,569 yards? 19,540 yards? ■ DENOMINATE NUMBERS. 153 207. Define reduction. Define each kind. What is the form employed in Problem 1, above? In Problem 7? What is the general rule for reduction ascending? for reduction descending? (See Art. 17.) Illustrative Example. 1. Reduce 2 mi. 46 rd. 3 yd. 2 ft. 8 in. Analysis. Since iu 1 mile there are 320 rods, iu any number of miles there are 320 times as many rods; hence, in 2 miles there are 320 times 2 rods, which are 640 rods. 640 rods + 46 rods = 686 rods. Since in 1 rod there are 5i yards, in any number of rods there are 5i times as many yards; hence, in 686 rods there are bh times 686 yards, etc. 2. Reduce 3 yd. 2 ft. 10 in. to inches. 3. Reduce 21 rd. 4 yd. 1 ft. to feet. Reduce to feet : 4 5 rd. 3 yd. 2 ft. - 5. 8 rd. 4 yd. 1 ft. 6. 13 rd. 5 yd. 1 ft. 7. 38 rd. 2 yd. 8. 2 mi. 124 rd. 4 yd. 9. 5 mi. 312 rd. 2 ft. 10 6 mi. 196 rd. 3 yd. 1 ft. Reduce to inches : 11. 3 yd. 2 ft. 12. 4 yd. 1 fto 8 in. 13. 5 yd. 2 ft. 10 in. 14. 4 yd. 2 ft. 5 in. 15. 2 rd. 4 yd. 7 in. 16. 1 mile. 17. \ mile. 208 Reduce to units of lower denominations ? 1. f of a mUe. 154 NEW ADVANCED ARITHMETIC. Analysis. Since in 1 mile tiieie are 320 rods, in f of a mile there aw f of 320 ruds = ^^^ rods = 274f rods, f of a rod ~ f of V- yard = V ^^ a yard = 1 f yards. 4 of a yard = f of 3 feet = ^ feet = 1 j feet. ^ of a foot = ^ of 12 inches = y iuclies = 8* inches; hence, etc. KUUM. f nil. = f X -^--'O rd. = LM^-2-^ rd. = 2742 rd. I rd. = f X V y'l- = ¥ yd. = H yd. ^yd. = f X ;3 ft. = Y ft. = If ft. I ft. = f X 12 ill- = Y = ^f i"- 2 § of a mile. 7. .625 of a rod. 3. ^ of a rod. 8. ,86 of a mile. 4. § of a mile. 9. .047 of a mile. 5. ^\ of a yard. 10. .253 of a rod. 6. .375 of a yanl. 11. .08^ of 4 miles. KoTK. .375 X 3 = l.l 125 feet. .125 X 12 = 1.5 inches. 209. Reduce to units of higher denominations : 1. 80526 inches. Analysis. Since there are 12 isclies in 1 foot, in 80526 inches there are as many feet as there are I2's in 80526. There are 6710 12's in 80526, with a remainder of 6; hence, in 80526 inches there are 6710 feet and 6 inches. Since there are 3 feet in one yard, in 6710 feet there are as many yards as tliere are 3's in 6710. There are 2236 3's in 6710, with a remainder of 2; hence, in 6710 feet there are 2236 yards and 2 feet. Since in T rod there are 5^ yards, in 2236 yards tliere are as many rods as there are times 5^ yards in 2236 yards, or as there are times 11 lialf-yards in 4472 half- yards. There are 406 times 11 in 4472. with a remainder of 6; lience. in 4472 half-yards there are 406 rods, with a remainder of 6 half-yard^, or 3 yards. Since there are 320 rods in a mile, etc. 6. 317 feet. 7. 461 yards. 8- 2893 inches. 9- 26459 inches. 210. 1. Reduce f of a foot to <^he fraction of a rod. 2. 1253 inches. 3. 1367 inches. 4. 1598 inches. 5. 2291 inches. 10. 17891 feet. DENOMINATE NUMBERS. 155 Analysis. 1 foot is ^ of a yard. | of a foot is | of ^ of a yard, which is 2% of a yard. Since tliere are Y yards in a rod, ^ of a yard is f\ of a rod, and 1 yard is y\ of a rod. ^\ of a yard is ^^ of ^j of a rod, which is yfj '^f a rod. Short form. | X J X jx- 2. 6 inches are what part of a rod? 3. f of a yard are what part of a mile? 4. 6 feet are what part of a rod? of a mile? 5. ^ of an inch is what part of a yard? of 2 yards? of 2 J yards ? 6. 2 feet 3 inches are what part of a yard ? of a rod ? 7. 1 foot 9 inches are what part of a yard? of a rod? 8. 4 yards 2 feet are what part of a rod ? of a mile ? 9. I of a foot are what part of a rod ? 10. 2\ feet are what part of a rod? 3 J feet? 4|^ feet? 11. 1% of a rod are what part of a mile? 2\ rods? 3^ rods ? 12. Reduce to the fraction of a rod .12 of a foot; .015 of a foot? .08 J of a yard ; .7 of an inch. 13. Reduce .25 of a foot to the fraction of a mile; .375 of a yard; .875 of a rod. 14. Reduce /^ of a foot to the fraction of 2 miles ; of 3^ miles. 211. 1. Change 3 yd. 2 ft. 3 in. to the decimal of a rod. Analysis. 3 in. are J of a foot. \ = .25. 2.25 ft. = \ as many yds. = .75 yd. 3 75 yd. = -^j as many rd. = .68+ rd. 2. Change 4 yd. 2 ft. 6 in. to the decimal of a rod. (Re- ject all terms below fourth place.) 3. Change 3 rd. 5 yd. 1 ft. 8 in. to the decimal of a mile. 4. Change to the decimal of a mile : 4 rods ; 6 rods ; 20 rods; 10 rods, 3 yards; 50 rods, 5 yards, 2 feet; 80 rods, 2 yards, 2 feet, 9 inches. 156 NEW ADVANCED ARITHMETIC. 212. SURFACE MEASURE. 1. A surface is that which has length and breadth only, 2. A surface is measured by finding how many surface units it contains? 3. The surface unit is usually a square whose side is a linear unit. 4. A square has the following properties : (a) It is a plane. (b) It is bounded by four equal straight lines. (c) Its angles are all right angles. 5. How many square inches in a square foot ? square feet in a square yard ? square yards in a square rod ? 6. Draw a square rod on the scale of 1 inch to the yard. 7. Complete and learn the following table. square inches = 1 square foot. square feet = 1 square yard. square yards = 1 square rod. 160 square rods — 1 acre (A). 640 acres = 1 square mile. In land surveys a square mile is called a section. 213. PROBLEMS. 1. Reduce 2 sq. rd. 7 sq. yd. 5 sq. ft. 86 sq. in. to square inches. 2. Reduce 3 A. 84 sq. rd. 10 sq. yd. 2 sq. ft. to a simple number. 3. Reduce 3 sections, 480 A. 125 sq. rd. to square yards. 4. Reduce 12,652 square inches to a compound number. 5. Reduce 224,725 square rods to square rods, acres, and square miles. Note. The only troublesome divisor in square mejisure is 30j. Ob- serve the method in the following problem. DENOMINATE NUMBERS 157 €. Reduce 2,480 square feet to square rods, etc METHOD. 2480 square feet = 275 sq. yd. 5 sq. ft. 275 square yards =1100 fourths of a square yard. 30^ square yards = 121 fourths of a square yard. , 1100 fourths -^ 121 fourths = 9, with a remainder of 11 j fourths. 11 fourths of a sq. yard = 2 sq. yd. 6 sq. ft. 108 sq. in. Add first rem. 5 3 sq. yd. 2 sq. ft. 108 sq. in. Hence, 2480 square feet = 9 sq. rd. 3 sq. yd. 2 sq. ft. 108 sq. in. 7. Reduce 327 square yards to square rods, etc. 8. Reduce 5,873 square yards to a compound number. 9. Reduce -j^f of a square mile to lower denominations. 10. Reduce f of a square rod to lower denominations. 11. Reduce .372 of an acre to lower denominations. 12. Reduce 3.75 square yards to lower denominations. 13. 36 square inches is what part of a square yard? 14. 6 A. 64 sq. rd. is what part of a section? 15. Reduce 19 A. 32 sq. rd. to the decimal of a square mile. 16. The base of the Great Pyramid of Gizeh, which is a square 764 feet X 764 feet, is what decimal fraction of a square mile? 214. THE AREAS OP RECTANGULAR SURFACES. 1. What is the area of a rectangular field 40 rods wide and 92 rods long? What is its value at $63 an acre? 2. What Is the cost of plastering the walls and ceiling of a room 36 feet by 48 feet, and 12 feet high, at 27 cents a square yard, no deductions being made for openings ? 158 NEW ADVANCED ARITHMETIC. 3. " Develop " the several plastered surfaces in Problem i- on the scale of 6 feet to 1 inch on the blackboard, or 6 feet to \ incli on your tablet. Make a similar diagram for each problem in this set. End, 36'. .Side, 48'. End, 3"^'. I Side, 48', 12' Ceiling, 36' X 48'. 4. What is the cost, at 26 cents a square yard, of plaster- ing a cottage containing 6 rooms, 2 of which are 14 feet by 15 feet, 2 are 10 feet by 12 feet, and 2 are 13 feet by 15 feet, the ceilings being 10 feet high, and no allowance being made for openings ? 5. What will it cost to paper the walls and ceilings of this cottage, with paper at 20 cents a roll, deducting 400- square feet for base-board, allowing for 14 windows, each 3 feet by 6 feet, and for 7 doors, each 3 feet by 8i feet, the paperer's charge being 20 cents a roll, and the border costing 5 cents a yard, no deductions being made for border? Note. Wall-paper is 18 inches wide. A roll is 8 yards long. 6. Find cost of papering the four walls and ceiling of your school -room at 5 cents a roll, and 3 cents per yard for border for walls. 7. What is the cost of carpeting a room that is 16 feet by 19 feet, the carpet being a yard wide, and costing Si. 12^ a yard, if there is a loss of H yards in matching, the carpet running the longer dimension of the room? Note. How many breadths are needed for this room ? How mncli is to he turned nnder at one side ? Make a plan of the room, using a scale of an inch for a foot, and mark the breadths. DENOMINATE NUMBERS. 159 8. How many yards of carpet, each strip being | of a yard wide, the loss in matching being 6 inches in each strip except tlie first, Avill be needed for a room 22 feet long and 18 feet wide, the strips running the long way? How mucli if the strips ran crosswise? Make a plan of the room for each case. Which plan is more economical? "Why? 9. Cost of covering your school-room with cocoa matting at 30 cents per square yard ? 10. Cost of carpeting it with ingrain carpet 36 inches wide, if the " design" in the carpet is 26 inches long? Note. How many times is the design repeated in one strip "] How much must be cut off or turned under at the end ? at the side ■? 11. If corn is planted in hills 4 feet apart each way, how many hills in an acre field 9 rods wide? 12. If corn is planted in hills 3'-8" apart each way, and yields 3 ears to the hill, 100 ears to the bushel, what is tlie yield in the above field? 13. In a shower an inch of rain fell. How many tons t® the acre? 14. The water from a roof 37^' X 56' is gathered mto a cistern holding 200 barrels of 31J gallons. What depth of rainfall on the roof will fill the cistern ? 15. If the cistern is a cylinder 10 feet in diameter, an inch of rain from the above roof will fill it to what depth ? 16 The pressure of the atmosphere in Central Illinois averages 14.4 pounds to the square inch. What is the down- ward pressure on the lid of a trunk 38 '' X 21 "? 215. SURVEYORS' MEASURE. The surveyors' chain, invented by Edmund Gunter, about 1620, consists of 100 links. Its length is 4 rods. 1. How many feet in a chain ? 2. How many inches in a link? 160 NEW ADVANCED ARITHMETIC. 3. How many chains in a mile? 4. Make a table embodying all these facts and label it surveyors' long measure. 5. How many square links in a square chain? square chains in a square mile? square rods in a square chain? square chains in an acre? 6. Arrange these facts in a table and label it surreyors' square measure. 7. How many acres in a rectangular field 22.67 chains X 9.26 chains? 8. How many acres in a rectangular field &2 rd. 3 yd. 2 ft. 3 in. by 37 rd. 8 in. ? Note in the above problems tlie relative simplicity of surveyors' 216. UNITED STATES SURVEYS. 1. Most of the lands of the United States west of the orig- inal thirteen States, except the tract between the Ohio and Tennessee rivers, are surveyed in accordance with the fol- lowing system : 2. In each great survey district there is run a Principal Meridian and an east and west line called a Base Line. On each side of the Principal Me- ridian at distances of six miles are north and south lines called Range Lines, which divide the land into strips six miles wide called Ranges. 3. By east and west lines paral- lel to the Base Line the ranges are divided into townships six miles square. A Township is designated by giving its number and direction from the Base Line, the number and position of its range, and the name or number of the Principal Meridian. 6 7 18 19 30 31 5 8 17 20 29 32 4 9 16 21 28 33 3 10 15 22 27 34 2 11 14 23 26 35 1 12 13 24 25 36 DENOMINA TE NUMBERS. 161 Sec. 28. \^ Section 28, etc. N. E. i, Section Sec- 3 1 4 2 Thus the writer is in Township 24 North, Range 2 East of the 3d Principal Meridian. 4. Townships are subdivided into 36 sections numbered thus: The sections are divided into halves and quarters ; the quarters into halves and quarters, and so on. Tracts are described thus : 1. W. \ Sec. 28, T. 24 N. 2 E., 3d P. M. 2. S. ^S.E. 3. N. W. 1 28, etc. 4. W. i N. E. I S. E. tion 28, etc. How many acres in each of these tracts? What fraction of a section is each ? 217. PROBLEMS. Draw a 6-inch square representing a section. Mark in it each of the following tracts and tell how many acres each contains. 1. The N. E. i of the N. E. \. 2. The S. 1 of the N. W. \. 3. The S. I of the N. W. ^ of the S. W. \. 4. The E. i of the N. \ of S. E. \ of the S. E. \. 5. The N. i of the S. E. \ of the S. E. \ of the N. W. \. 6. The N. W. 1 of the S. W. \ of the N. E. \ of the N. E. 1. The N. W. \ of the N. W. ^ of the N. ^Y. \. The N. W. 1 of the N. W. \ of the N. E. \. The N. ^ of the S. W. \ of the N. W. \ of the 7. 8. 9. S. F 10 The S. i of the S. h of the S. W. I of the N. W. J. 162 NEW ADVANCED ARITHMETIC. 218. MEASURES OF VOLUME. I. A Solid is a figure having length, breadth, and thick- ness. 2. A Rectangular Solid is a solid bounded by six rectangles. 3. The rectangles are called the faces of the solid. 4. The intersections of the faces are called the edges. 5. Name five rectangular solids. Bring a rectangular solid to the class. Show the faces and the edges. 6. A cube is a rectangular solid whose faces are squares. 7. A cubic inch is a cube whose edges are each one inch. 8. Solids are measured by finding how many units of volume they contain. 9. The units of volume are usually cubes whose edges are linear units. 10. Complete this table : Table of Cubic Measure. cubic inches = 1 cubic foot. cubic feet = 1 cubic yard. 128 cubic feet = 1 cord (woOd). 100 cubic feet = 1 cord (stone). II. Wood that is to be used for fuel is measured by the cord. The sticks are usually cut 4 feet long, and are then called " cord-wood." For convenience in measuring they are usually " corded," that is, piled 4 feet high, the length of the stick making the width of the pile. In such a pile, for every 8 feet of length there is a cord of wood. Prove that it contains 128 cubic feet. DENOMINATE NUMBERS. 163 A pile 4 feet wide, 4 feet high, and 1 foot long contains 1 cord foot. How many cubic feet does it contain? 219. PROBLEMS. 1. Reduce 4 cubic yards 15 cubic feet 964 cubic inches to cubic inches. 2. Reduce 8 cords 7 cord feet to cubic feet. 3. Reduce 864,952 cubic inches to a compound number. 4. Reduce 1,264 cubic feet to cords, etc. 5. Reduce ^ of a cubic yard to cubic feet and cubic inches. 6. Reduce ^^ of a cord to integers of lower denominations. 7. Reduce .36 of a cord. 8. 27 cubic inches is what part of a cubic yard? 9. Change 864 cubic inches to the decimal of a cord. k 220. WOOD MEASURE. PROBLEMS. 1. How many cords of cord-wood in 3 piles of wood, each being 4 feet high, the first being 36 feet long, the second 42 feet long, and the third 73 feet long? Find its cost at $4.75 per cord. Note. Where cord-wood is piled 4 feet high, there is cue cord for every 8 feet of length. 164 NEW ADVANCED ARITHMETIC. 2. Each of the following piles is 4 feet high. Find how many cords each contains. (a) 83 feet long. (d) 47^ feet long. (b) 69 feet long. (e) 61 feet 5 inches long. (c) 35^ feet long. (/) 93 feet 10 inches long. Find the cost of each pile at $5.25 a cord. 3. How much cord-wood in each of the following piles? What is the cost at $4.75 a cord? (a) 26 feet long, 6 feet high. 13 3 g^X0X$4.75 $185.25 p| ~ 8 $23.15f. Explanation. The number of cubic feet in this pile is 26 X 6 X 4. Since there are 128 cubic feet in a cord, the number of cords is Since cord- wood is 4 feet 128 long, the four may be omitted from dividend and divisor, thus leaving 32 as a divisor. Note that we then divide the area of the side of the pile of 4-foot wood by the area of the side of a cord. (6) 54 feet long, 7 feet high. (c) 61 feet long, 7 feet high. (d) 42 feet 8 inches long, 6 feet 4 inches high. Note. Call inches twelfths of a foot, and change mixed nnmbers to improper fractions, thus : 128 X 19 X $4.75 42 ft. 8 in. = 42§ ft. r= 4^. 6 ft. 4 in. = 6^ = J^. 3 ^ 3 v^ 32 (e) 86 feet 3 inches long, 7 feet 6 inches high. (/) 124 feet 5 inches long, 8 feet 6 inches high. (g) 97 feet 6 inches long, 5 feet 9 inches high. (h) 224 feet long, 12 feet high. DENOMINATE NUMBERS, 165 321 LUMBER MEASURE. 1. The unit of lumber measure is the board foot, one foot long, one foot wide, one inch thick. 2. Lumber that is less than one inch thick is counted as if an inch thick. If lumber is more than inch thick, the excess is taken into account. 3. How many cubic inches in a board foot? How many board feet in a cubic foot? 4. What is the width of an inch board that contains as many board feet as it is feet long? 5. Of a two-inch plank? of a three-inch stud? ' 6. "What is the end-area of each of the above pieces? 7. What is the end-area of a 6 " X 8 " sill ? How many board feet in each foot of its length? What di\isor have you employed? 8. How many board feet in a beam 10 " x 12 ", 24 feet long? 9. Show the truth of the following : thickness X •width X length Number of board feet = — • 12 In what units must thickness, width, and length be expressed ? Bills of lumber are regularly made out in these units. 222. PROBLEMS. 1. Wliat is the cost of 16 sills 6" X 8" X 18' (a $18 per thousand feet? FORM. 16 X 6 X8 X 18 X SIS 12 X 1000 ANALYSIS. (1) Since a number of dollars is required, I write ?18, the cost of 1000 board feet. I express the cost of one board foot by -writing 1000 as a divisor. I express the cost of a stick 1 foot long, 1 inch wide, 12A 166 NEW ADVANCED ARITHMETIC. and 1 inch thick by writing 12 as a divisor. I multiply by 18 because the sill is 18 feet long; by 8 because it iii 8 inches wide; by 6 because it is 6 inches thick ; by 1 6 because there are 1 6 sills. 6 X 8 X 18 (2) The number of board feet in each sill is expressed by r^ -. 1 write 16 as a multiplier because there are 16 sills, 1000 as a divisor to get number of thousand feet, then multiply by 18 because the number of ■dollars paid must be 18 times the number of thousand feet. 2. What is the cost of the following bill of lumber at $21 •a thousand (M) ? c 8 sUls, 8 X 10, 16 feet long; 48 studs, 2x4, 18 feet long; 22 joists, 2 X 10, 16 feet long; 50 rafters, 2 X 4, 14 feet long; 24 joists, 2 X 8, 16 feet long. 3. Find the cost of the following bill of lumber, at $16.50 per M : 32 common boards, 8 inches wide, 14 feet long ; 65 fence boards, 6 inches wide, 16 feet long; 16 corner posts, 4x4, 18 feet long ; 7 sills, 6 X 8, 14 feet long; 46 rafters, 2 X 6, 16 feet long; 36 joists, 2 X 8, 18 feet long. 4. Find the cost of the following bill of lumber : 86 pieces maple flooring 1x3 x 16 @ $40 per M. 86 " ash " 1 X 3^ X 16 @ $36 per M. 72 >' clear pine boards 1 X 10 x 16 @ $32 per M. 24 rafters 2x6 X 16 @ $18 per M. 36 joists 2 X 10 X 22 @ $21 per M. 5. What is the cost of 46 planks, 2 inches thick, 10 inches wide, and 18 feet long, at $22 per M.? 6. What is the cost of 65 2^-inch oak planks, 12 inches wide and 16 feet long, at $36 per M. ? 7. How many posts set 8 feet apart are required for wire fencing around a field 800 feet square and for two cross fences dividing the field into four equal squares ? Show that the cost of these posts at 8\ cents each is $49.75. DENOMINATE NUMBERS. 167 8. Find the cost of the lumber and posts to fence a field AO rods by 60 rods with a 5-board fence ; the posts costing 22 cents each, and placed 8 feet apart; the boards being 16 feet long and 6 inches wide, and costing 818.50 per M. Make a plan of the field. Cut boards so as to have as little waste as possible. 9. With lumber and posts as in the preceding problem, find the cost of material, except the nails, to fence the N. ^ of N. W. \ of S. E. J of a section. 10. Find the cost of lumber and posts to put a 4-board fence around a section of land and to make division fences separating it into quarter-sections, boards, and posts as in Problem 8. 223. MEASURES OF CAPACITY. 1. The units of Liqmd Measure are the gallon, the quart, the pint, and the gill. The primary unit is the gallon, and contains 231 cubic inches. The other u-nits are divisions of the gallon. TABLE. 4 gUls (gi.) = 1 pint (pt). 2 pt. =1 quart (qt.) 4 qt. =1 gallon (gal.). Note. The pint is divided iDto 16 eqnal parts, each of which is called an ounce. This mea,«ure is used by apothecaries. The teacher should show a 1 -ounce, a 2-ounce, and a 4-onnce bottle. In computing the capacity of tanks and cisterns, the barrel of 31 § g^- lons is tlie unit. 2 The units of Dry Measure are the bushel, the peck, the quart, and the pint. The primary unit is the bushel, which contains 2,150| cubic inches. The other units are divisions of the bushel. Note. The standard bushel is 18i inches in diameter and 8 inchefl deep. It contains 2,1501 cubic inches, or nearly 1^ cubic feet. 168 NEW ADVANCED ARITHMETIC. TABLE. 2 pints = 1 quart. 8 quarts = 1 peck (pk.). 4 pk. = 1 bushel (bu.) 3. How many cubic inches are there in the liquid quart? in the dry quart? About how many gallons are there in a cubic foot of water? PROBLEMS. 1. Reduce 3 gal. 3 qt. 1 pt. 2 gi. to gills. 2. Reduce 17 bu. 3 pk. 5 qt. 1 pt. to pints. 3. Reduce 587 gills to a compound number. 4. Reduce 1,267 pints to a compound number. 5. Reduce to integers of lower denominations ^-| of a gallon. 6. Reduce || of a bushel to integers of lower denomina- tions. 7. ^ of a pint is what part of a bushel? 8. § of a gill is what part of a gallon? 9. Reduce .625 of a bushel to integers of lower denomina- tions. 10. Reduce 1 gill to the decimal of a gallon. 224. WEIGHT. 1. Weight is the measure of the downward pressure of bodies at or near the surface of the earth. 2. There are four systems of weight. These are Avoir- dupois, Troy, Apothecaries', and Metric. 3. The standard from which the units of the first three are derived is the Troy [)ound of the mint. Note 1. For the metric system, see Appendix. I DENOMINATE NUMBERS. 169 4. The Troy pound is divided into 5,760 equal parts called grains. 7,000 grains equal a pound avoirdupois, and 5,760 the pound apothecaries'. 5. Troy weight is used in measuring gold, silver, precious stones, jewels, etc. 6. Apothecaries' weight is used in mixing medicines. 7. Avoirdupois weight is used in measuring ordinary arti- cles of merchandise. 8. Avoirdupois Weight. TABLE. 16 oz. =■ 1 lb. 2©00 lb. = 1 ton (T.). Note 1 . The long ton is used in the United States custom-houses and in the Eastern States in weighing coal and iron. 28 lb. — 1 quarter (qr.). 4 qr. = 1 hundred weight (cwt.). 20 cwt. = 1 tju. Note 2. The relation of Avoirdupois weight to Troy weight may be seen by comparing the following table with the Troy table : y\ of 7000 grains = 437j grains = 1 oz. av. jig of 437^ grains = 2'\\ grains = 1 dram av. Note 3. 62^ lb. Avoirdupois = 1000 oz. = the weight of a cubic foot of distilled water. 9. Troy Weight. TABLE. 24 gr. = 1 pennyweight (pwt.). 20 pwt. =^ 1 oz. 12 oz. = 1 lb. 10. Apothecaries' Weight. TABLE. 20 gr. = 1 scruple (9). " 3 scruples = 1 dram (3). 8 drams = 1 ounce (§). 12 ounces = 1 pound (ib). 170 NEW ADVANCED ARITHMETIC. Note. The Troy pound is little used. Gold and silver buUion are sold by the ounce; gold ornaments by the pennyweight; jewels by the caraZ (3.2 grain.s). The word carat is also used in the sense of twenti/-fourths in stating the purity of gold. Grold 14 carats fine is ^| gold, ^| alloy. The metric system is rapidly displacing Apothecaries' weight in phar- macy. 11. Comparison of Weights. 1 lb. Troy = f J§g = i^i of 1 lb. Avoirdupois. 1 oz. Troy = ^§^^ = ff § of 1 oz. " The ounce and pound Apothecaries' equal the ounce and pound Troy, respectively. 225. PROBLEMS. 1. How many ounces in 8 lb. 7 oz. av. ? in 37 lb. 13 oz.? in 548 lb. 15 oz. ? in J of 1 lb. ? in f of 1 T. ? in .325 of a long ton? 2. How many grains in 2 lb. 5 oz. 11 pwt. 16 gr. Troy? in 3 lb. 8 pwt. ? in H of 1 lb. ? in .0875 of 1 lb. ? 3. How many grains in 2 lb. 3 oz. I sc. 17 gr. Apotheca- ries'? in -1^^ of 1 lb.? in .28 of 1 oz.? 4. Change the following simple numbers to compound numbers: 568 oz. av. ; 2825 gr. Troy; 6827 gr. Apotheca- ries' ; 174 sc. ; 869 pwt. 5. What is the cost of 2 tons of sugar at 3J cents a pound? of a 3-ounce silver watch-case at 8.6 cents a penny- weight? of 2 oz. 5 gr. of quinine at J of a cent a grain? 6. 2 lb. 12 oz. is what part of 1 T. ? 7. Reduce 6 grains Troy to the decimal of a pound. 8. 2 sc. 12 gr. is what part of 2 lb. 5 dr. ? 9. Reduce 4.28 lb. Troy to numbers of lower denomina- tions. 10. /j of 1 lb. Apothecaries' =? U. 12 oz. is what part of 1 T. ? DENOMINATE NUMBERS. XII 12. Change 6 gr. to the decimal of a Troy pound. 13. Change 15 lb. Avoirdupois to Troy weight. 14. Change 24 lb. Troy to Avoirdupois weight. 15. Change 22 lb. 13 oz. Avoirdupois to Troy weight. 16. Change 18 lb. 7 oz. Troy to Avoirdupois weight. 17. What are you worth if you are worth your weight in gold coin ? in silver dollars ? 18. "A pint is a pound the world around." Is this statement exactly true for water? 19. The Orloflf diamond (194| carats) weighs how many ounces Avoirdupois? 20. What is the value of the gold in an 18-carat watch- case weighing 3 ounces Troy? 1 oz. gold = $20.67. Note. The grains are usually measured by weight. The weight of the bushel is determiaed by law. These laws are not absolutely uniform in the several States, except in the case of wheat. The following is the number of Avoirdupois pounds for a bushel in the great majority of cases: Indian corn, 56 ; oats, 32 ; rye, 56 ; wheat, 60. 21. How many bushels in 8,640 lbs. of wheat? In 7,924 lbs. of corn? In 5,872 lbs. of oats? In 10,240 lbs. of rye? 226. Apothecaries' Fluid Measure. 60 minims (m.) = 1 fluid drachm (fl. 3). 8 fluid drachms = 1 fluid ounce (fl. §). 16 fluid ounces = 1 pint (O.). 8 pints = 1 gallon (cong.). Compare this table with Apothecaries' Weight. 227. PROBLEMS. 1. How many 20-minim doses of laudanum in a fluid ounce ? 2. How many 3-ounce bottles of perfumery may be filled from one gallon ? 172 NEW ADVANCED ARITHMETIC. 3. A stationer bought 5 gallons of ink for $3.00. He bought -i-ounce bottles ai 10 cents per dozen, filled them with ink, and sold them at 5 cents each. What was his gain? 228. ENGLISH MONEY. 4 farthings (qr.) = 1 penny (d.). 12 pence = 1 shilling (s.). 20 shillings = 1 pound (£). £= Zi'ftra = pound, d. — denai'ius^ Latin for "penny." qr. = quadrans = fourth. Note. The Troy pound of silver was originally coined int.o 240 silver pennies of 24 grains (1 pennyweight) eacli. A cross was stamped so deep that the penny was readily broken into fourths (farthings). The present value of the pound sterling is $4.8665. The gold coin of this value is called the sovereign. The shilling is coined of silver ; the penny and half-penny of copper. The guinea (21 s.) and crown (5 s.) are no longer coined. English gold coins are 22 carats fine. PROBLEMS. 1. Reduce £7 15 s. 7 d. 1 qr. to farthings. 2. Reduce £28 9 d. to pence. 3. Reduce 586 d. to a compound number. 4. tS^ of £1 = what? 5. £ 7.048 = what? 6. 7 d. 2 qr. is what part of £1 ? 7. Change 15 s. 9d. to the decimal of a pound. 8. By how much does the quarter dollar exceed the shilling in value ? 229. FRENCH MONEY. The franc (worth 19.3 cents in U. S. money) is the unit. The scale is decimal. TABLE. lO millimes (m.) = 1 centime (c.) 10 centimes = 1 decime. 10 decimes = 1 franc. DENOMINATE NUMBERS. 173 230. GERMAN MONEY. The unit is the inark, or reichsmark^ worth 23.8 cents in U. S. money. TABLE. 100 pfeunige (pf.) = 1 mark (RM.). The 5-franc piece has been proposed as an international coin nearly equal to the dollar, to 4 shillings, and to 4 marks. 1. What part of the dollar must be takeu out to make it conform to the 5-franc piece ? 2. What part of 4 shillings ? 3. The weight of the 4 marks must be increased by what part of itself? 231. CIRCULAR MEASURE. 1. An arc is any portion of a circumference. 2. The unit of arc measurement is the degree (°). A degree is ^^^ of a circumference. What part of the circum- ference is an arc of 90= ? 60= ? 45°? 270=? 30? 15°? 22i=? Note. The arc and the degree that measures it must be portions of the same circumference. 3. An arc of 90° is called a quadrant ; an arc of 60= a sextant ; an arc of 30= a sign. 4. The degree is divided into 60 equal parts called min- utes (') ; the minute into 60 equal parts called seconds ("). 5. Make a table setting forth these facts and label it Table of Circular Measure. 6. An angle ifi the difference in direction between two lines proceeding from the same point, called the vertex. 7. If the four angles formed by two intersecting straight lines are equal, each angle is called a right angle. r 8. An angle larger than a right angle is an obtuse angle, an angle smaller than a right angle is an acute angle. 174 NEW ADVANCED ARITHMETIC, 9. The unit of angular measurement is called a degree. This degree is one ninetieth of a right angle. Note. If, with the vertex of a right angle as a center, and any radius whatever, a circumference be described intersecting the sides of the angle, the intercepted arc is a quadrant (90°). Hence a right angle is called an angle of 90°. It is evident that any acute angle is as many ninetieths of a right angle as its intercepted arc is of a quadrant. For this reason the term " degree " is applied both to the ninetieth of the ri<5ht '(ngle and the ninetieth of the quadrant. PROBLEMS. 1. Reduce 68° 45' 36" to seconds. 2. Reduce 46824" to a compound number. «. Add 42° 17' 26", 51^ 48' 51", 7° 56' 48", 12° 46' 28'\ 4. What is the difference between 29° 12' 42" and 16° 46' 25" ? 5. Multiply 12° 7' 18" by 16. 6. Divide 49° 17' 24" by 12. 7. 5" is what part of a degree ? 8. T^^of 1°=? 232. LONGITUDE AND TIME. 1. Find in your atlas a map of the world in hemispheres. What are the lines called that extend from the top of the map to the bottom? What are they for? What are the cross lines called? What are they for? AYhat is a prime meridian ? AVhat two prime meridians are used in your geography? 2. Find a map of the United States. Find the longitude of the following places with reference to Greenwich and to Washington City : (1) Cape Cod, Mass. (4) Denver, Col. (2) Erie, Pa. (5) Leavenworth, Kan, (3) Washington, D. C. (6) Memphis, Tenn. DENOMINATE NUMBERS. 175 3. The longitude of a place is its distance in degrees, minutes, and seconds, east or west of a prime meridian. It is measured on the arc of a parallel, or of the equator. 4. The difference of longitude of two places on opposite sides of the prime meridian is the sum of their respective longitudes. 5. If A is 10 miles east of B, and C is 12 miles west of B, then A is how many miles east of C? How found? If A is 5 miles west of B, and C is 21 miles west of B, then C is how many miles west of A ? How found ? Apply these illustrations in the following problems? The Greenwich meridian is referred to in the following : 233. TABLE. Paris 2° 20' 22" E. London 0° 5' 38" W. New York 71° 0' 3" W. Boston , 71° 3' 30" W. Chicago o 87° 35' 0" W. New Orleans . . . . • . 90° 3' 28" W. San Francisco . . . , . 122° 26' 15" W. Berlin 13° 23' 43" E. St. Petersburg . . . . . 30° 16' 0" E. Pekin 116° 26' 0" E. Calcutta . , 88° 19' 2" E. Pittsburg ....... 80° 2' 0" W. St. Louis 90° 12' 11" W. Cincinnati 84° 26' 0" W. Rome 12° 27' 14" E. Honolulu 157° 52' 0" W. Sydney 151° 11' 0" E. PROBLEMS. 1. What is the difference between the longitude of Paris and that of New York? 176 NEW ADVANCED ARITHMETIC. 2. Of Berlin and of London? 3. Of Chicago and Calcutta? 4. Of Sydney and Honolulu? 5. Of St. Petersburg and of St. Louis? 6. Of Rome and of Cincinnati? How find the difference of longitude of two places on opposite sides of the prime meridian ? Why ? 7. Of Boston and of Chicago? 8. Of London and of New Orleans? 9. Of Pekin and of Calcutta? 10. Of Pitts'ourg and of San Francisco? How find the difference of longitude of two places on the same side of the prime meridian ? Why ? Longitude to Time. 234. It is noon at any place when the sun is on its meridian. Before it is noon again at that place the earth must make about one revolution on its axis. The time occupied in making this revolution is divided into 24 hours, and is called one day. During this time the entire circum- ference of each parallel has passed under the sun. Since each circumference contains 360°, ^ of 360°, or 15°, passes under the sun each hour, j.}j^ of 15°, or 15', each minute, and b\j of 15', or 15", each second. When it is noon at any place, what time is it 15° east of that place? 30° E.? 45° E.? 90° E.? 180° E.? 15° we^t? 30° W.? 45° AV.? 90° W.? 120° W.? 180° W.? 15' E.? 15' W. ? 30° 30' E. ? 45° 45' W. ? 60° 30' 15" E. ? 00° 45' 30" W. ? 235. Prove that the following statements are true: A difference of 15° in the longitude of two places makes a difference of one hour in their time. A difference of 15' in the longitudes of two places makes a difference of one minute in their time. A difference of 15" in the longitude of two places makes a difference of one second in tlieir time. DENOMINATE NUMBERS. Yll PROBLEMS. 1. "Wliat is the difference in time of two places whose difference of longitude is 36° 42' 30" ? Analysis. Since a difference of 15° in the longitude of two places makes a difference of 1 hour in their time, a difference of 36° of longi- tude makes a difference of 2 hours in their time, with a remainder of 6°. 6° = 360'. 360' + 42' = 402.' Since a difference of 15' in the longi- tude of two places makes a difference of 1 minute in their time, a dif- ference of 402' of longitude makes a difference of 26 minutes of time, with a remainder of 12' of longitude. 12' = 720". 720" + 30" = 750". Since a difference of 15" of longitude makes a difference of 1 second of time, a difference of 750" of longitude makes a difference of 50 seconds of time. Therefore, a difference of 36° 42' 30" of longitude makes a difference of 2 hr. 26 min. 50 sec. in time. Find the difference of time when the difference of longi- tude is : 2. 94° 17' 45". 5. 64° 0' 50". 8. 82° 31' 30". 3. 112° 48' 15". 6. 48' 45". 9. 59° 59' 48". 4. 6° 56' 46". 7. 150° 12' 42". 10. 128° 19' 18". Time to Longitude. 236. If the time at A is an hour later than at B, what is their difference of longitude? if 2 hours later? if 5 hours later? if 1 minute later? 5 minutes? 10 minutes? 1 second? 5 seconds? 20 seconds? A is east or west of B? How do you kno^v? Change the word later to earlier, and ask the same questions. 237. Prove the truth of the following statements : A difference of an hour in the times of two places shows a difference of 15° in their longitudes. A difference of a minute in the times of two places shows a difference of 15' in their longitudes. A difference of a second in the times of two places shows a difference of 15" in their longitudes. 12 178 NEW ADVANCED ARITHMETIC. PROBLEMS. 1. The time in one town is 2 hr. 35 min. 22 sec. earlier than in another. Which is farther east ? How many degrees, etc. ? Analysis. Since the time is earlier in the first town than in the sec ond, the sun will not reach its meridian until it has passed the meridian ot the other; it is, consequently, farther west. Since a difference of 1 sec- ond in the time of two places shows a difference of 15" in their longitude, a difference of 22 seconds in their times shows a difference of 330" of longitude, which equals 5' 30". Since a difference of 1 minute in the times of two places shows a difference of 15' in their longitude, a differ- ence of 35 minutes in their times shows a difference of 525 in their longi- tude. 525' + 5' = 530' = 8° 50'. Since a difference of 1 hour in the times of two places shows a difference of 15° in their longitude, a differ- ence of 2 hours in their times sliows a difference of 30° in their longi- tude. 30° + 8° = 38°. Their difference in longitude is 38° 50' 30". Why begin with the lowest denomination ? In the following problems A and B represent places the difference of whose times is given. Find their difference of longitude, and tell which is farther east. 2. 4 hr. 25 min. 15 sec. A's later. 3. 8 hr. 6 min. 20 sec. B's later. 4. 6 hr. 40 min. 18 sec. A's earlier. 5. 9 hr. 52 min. 3 sec. B's earlier. 6. 1 hr. 59 min. 59 sec. A's later. 7. 12 hr. B's earlier. n 8. 2 min. 3t sec. L"? later. 9. 11 hr. 24 sec. A's earlier. 10. 10 hr. 31 min. 29 sec. A's later. Make a rule for each of the two general processes. 11. When it is noon at Paris, what is the time at St. Petersburg? at San Francisco? 12. When it is 6 A. m. at London, what is the time at New York? at Cincinnati? at Rome? at Honolulu? at Sydney? DENOMINATE NUMBERS. 179 13. "When it is 35 minutes past 3 p. m. at Berlin it is 34 min. 41 1'^ sec. past 8 p. m. at a second city. Find the name of the city in the table. 14. When it is 4 p. m. at Chicago it is 40 min. 35 sec. past 1 Po M. at a second city. Find its name in the table. 15. A ship's chronometer indicates that the time at Green- wich is 25 minutes past 3 p. m. By observations the captain ascertains that it is noon where the ship is. What is the longitude of the ship? Note. The teacher may form inauy problems from the table of longitudes. 238. A Shorter Method. Since a difference of 15° in the longitude of two places makes a difference of one hour in their times, a difference of 1° in their longitude makes a difference of 4 minutes, and a difference of 1' a difference of 4 seconds in their times. The two sets of facts may be combined for rapid oral work. Illustration. Difference of longitude 49° 36' 21". Analysis. A difference of 45° makes a difference of 3 hours. A dif- ference of 4° makes a difference of 16 minutes. A difference of 30' makes a difference of 2 minutes. A difference of 6' makes a difference of 24 seconds. A difference of 21" makes a difference of If seconds. Com- bining results, the difference of time is found to be 3 hr. 18 min. 25| sec. Solve Problems 2-10 (page 177) by this method. FORM. 49° 36' 21" 3 16 24 2 If 3 hr. 18 min. 251 sec. 239. Apply the same facts to the method of finding differ- ence of longitude when difference of time is given. Illustration. Difference of time 5 hr. 39 min. 50 sec. A difference of 4 seconds shows a difference of 1', and a differ- ence of 4 minutes a difference of 1°. 180 NEW ADVANCED ARITHMETIC. Analysis. A difference of 5 hours shows a difference of 75°. A dif- ference of 36 minutes (9 fours of minutes) shows a difference of 9°, and of 3 minutes a difference of 45'. A difference of 48 seconds shows a differ- ence of 12' and of 2 seconds a difference of 30". Combining results, the difference of longitude is 84° 57' 30". Solve Problems 2-15 in the last set by this method. FORM. 5hr. 39 min. 50 sec. 75° 9° 45' 12' 30" 84° 57' 30' 240. THE INTERNATIONAL DATE LINE. Travellers across the Pacific Ocean westward set their time forward a dav on crossing the 180th 'r.eridian. Islands in the equatorial portion of the Pacific were colonized by Europeans coming from the east with the trade winds, and have the same reckoning as the American continent. Australia, New Zealand, and the neighboring islands originally colonized by the Dutch have the time of Asia, one day in advance. On many charts is shown the International Date Line separating these lands. It passes through Behring's Strait, thence southwest east of Japan, but west of the Philippines, thence east, southeast, and south to the east of New Zealand- Prior to 1867 this line passed east of Alaska. 241. TIME MEASURES. 60 seconds (sec.) make 1 minute (min.). 60 minutes 24 hours 7 days 365 days 366 days 100 years 1 hour. 1 day. 1 week. 1 common year. 1 leap year. 1 century. 1. If the year is not the last in the century, its number must be divisible by 4 to make it a leap year. If it is the closing year of a century, it is not a leap year unless its number is divisible by 400. DENOMINATE NUMBERS. 181 2. The months containing 30 days are April, June, Sep- tember, and November. 3. The months containing 31 days are January, March, May, July, August, October, and December. 4. February contains 28 days in a common year, and 29 days in a leap year. PROBLEMS. 1. How many seconds are there in a day? 2. How many hours are there in a common year? in a leap year? 3. Change f of a common year to days, hours, minutes, and seconds. 4. 15 minutes is what part of September? 5. Which of the following are leap years? 1866, 1880, 1500, 2000, 1894, 1892. 6. Find the time from June 12, 1881, to March 5, 1884. Method. Count by years a.s far as possible, then by calendar months, then the remaining days; thus, from June 12, 1881, to June 12, 1883, is 2 years. From June 12, 1883, to February 12, 1884, is 8 months. There are 1 7 days left in February and 5 in March, hence the time is 2 years, 8 months, and 22 days. Note. A calendar month beginning with the first day of the month completes the month ; a calendar month beginning with any other day ends with the next preceding day in the following month. The periods Oct. 1-31, June 10-July 9, Jan. 31-Feb. 28, are calendar months. 7. Find the time : (1) From December 5, 1881, to June 16, 1886.- (2; From May 5, 1879, to September 20, 1883. (3) From August 16, 1888, to June 24, 1892. (4) From January 21, 1886, to November 7, 1893. 8. How many days from January 23 to July 29, common year? Note. 8 + 28 + 31, etc. 9. How many days from October 16, 1890, to June 3, 182 NEW ADVANCED ARITHMETIC. 242. THE CALENDAR. 1. There are three natural time uuits : the year, the month, and the day. 2. The natural day from midnight till midnight is not of uniform length. The mean solar day is the average of all the days in the year. 3. The month, the period from one new moon to the next, equals 29h days nearly. 4. The year, the period between two successive vernal equinoxes, equals 365.2422 days. 5. A calendar is an adjustment of tliese natural time uuits for civil purposes. A lunar calendar makes the mouth a leading luiit. Mouths are alternately of 29 and 30 days. The year of 12 months or 354 days is called a lunar year. Since the lunar year is 11 days too short, extra mouths (7 in 19 years) must be added from time to time. 6. The lunar year is still used iu Mohammedan countries. 7. The Juliau calendar, established by Julius Ca;sar in 46 b. c, pro- vided a common year of 3G5 days and, every fourth year, a leap year of 366 days. The mouths beginning with Jlarch were alternately of 31 aud 30 days. Later August was given an additional day at the expense ol February, and Octolier and December were made months of 31 days instead of September aud November. 8. The average Julian year of 365.25 days exceeds the true solar year by .0078 day or j^-g day. Gregory XIII. in 1582 corrected as much of tlie excess as had accumulated since 325 a. d. by decreeing that the 5th of October should be the 15th. He provided that every year that is divisible by 4, and not by 100, is a leap year. Of the century years, only those divisible by 400 are leap years. The Gregorian calendar thus provides 'j7 leap years in 400 years. 9. The Gregorian calendar was not adopted by the nations of northern Europe until 1700. It was adopted in England in 1752. Russia still uses the Julian calendar. 243. Questions. 1. What years since 1582 have been leap years in Russia and not in Italy 1 2. Christmas in Russia comes how many days later than with us ? 3. By what fraction of a day does tlie average Gregorian year exceed the true year 1 4. In how many years will the excess in the Gregorian calendar amount $0 one day 1 5. wiiat is the length of the true year in days, hours, minutes, and seconds 7 DENOMINATE NUMBERS. 183 244. MISCELLANEOUS TABLES. 1, 12 oues — 1 dozen. 12 doz. — 1 gross. 12 gross r= 1 great gross. 20 ones = 1 score. 2. 24 sheets of paper ~ 1 quire. 20 quires = 1 ream. 2 reams = 1 bundle. 5 bundles = bale. Mariners' Measure. 6 feet = 1 fathom. 120 fathoms = 1 cable-length, 80 cable-lenoths = 1 mile. 245. ADDITION. Define addition, sum. Tell how the numbers are written for addition. 1. "^hat is the sum of the followino; numbers : 1 mi. 1.S2 rd. 4 yd. 1 ft. 7 in. 2 309 .") 2 9 5 169 3 4 8 274 4 2 11 15 3 2 6 2. A man travelled 23 mi. 186 rd. on Monday. 19 mi. 29.') rd. 4 yd. on Tuesday, 36 mi. 83 rd. 5 yd. on Wednes- day, 19 mi. 317 rd. 2 yd. on Thursday, 28 mi. 297 rd. on Friday, and 34 mi. 168 rd. on Saturday; how far did he travel in the six days? 184 NEW ADVANCED ARITHMETIC. 3. Add: 4. Add: 5. Add: 6. Add 7. Add; 3 sq. yd. 5 sq. ft. 68 sq. in . 5 7 124 7 6 99 2 8 136 4 2 79 2 A. 121 sq. rd . 17 sq. yd. 3 139 24 7 86 19 12 117 28 1 110 sq. rd. 26 sq. yd. 4 sq. ft. 83 sq. in. j 129 14 7 132 1 147 29 6 59 ! 153 17 2 126 88 21 8 94 i 2 cu. yd. 19 cu. ft. 824 cu . in. [ 5 24 1232 6 16 714 12 21 936 17 26 1532 j 2 15 1129 1 5 cd» 6 cd. ft. 14 cu. fto ! 12 4 9 6 7 12 8 2 11 27 5 15 8. Find the sum of 5 cd. 7 cd. ft., 6 cd. 9 cd. ft., 12 cd. 4 cd. ft., 9 cd. 4 cd. ft. 9. Sold 3 cd. 5 cd. ft. 12 cu. ft , 12 cd. 14 cu. ft., 9 cd. 5 cd. ft. 15 cu. ft. How much in all? 10. Find the sum of 2 gal. 3 qt. 1 pt., 5 gal. 2 qt 1 pt., 7 gal. 1 qt., 10 gal. 1 pt., 6 gal. 2 qt. 1 pt. 11. Find the sum of 4 bu. 3 pk. 5 qt., 6 bu. 2 pk. 7 qt., 12 bu. 1 pk. 6 qt. DENOMINATE NUMBERS. 185 12. Add 2 lb. 8 oz. 14 pwt. 18 gr., 3 lb. 7 oz. 12 pwt. 10 gr., 5 lb. 10 oz. 8 pwt. 17 gr., 12 lb. 15 pwt. 21 gr. Troy. 13. Add 1 lb. 3 oz. 5 dr. 2 sc. 16 gr., 2 lb. 7 oz. 6 dr. 14 gr., 8 oz. 2 sc. 18 gr., 5 dr. 2 sc. 9 gr. Apothecaries'. 14. Made 4 purchases in London, costing respectively £4 8 s. 7 d., £7 15 s. 9 d., £5 18 s. 8 d., and £10 7 s. 6 d. What was the amount expended? 15. What is the difference in latitude of Boston (42° 21' 24" N.) and Rio de Janeiro (22° 54' S.) ? 16. A man bought a quantity of broadcloth for £ 17 9 s. ; of silk for £23 11 d.; of cotton goods for 18 s. 9^ d. ; of linen goods for £29 15 s. 11| d. ; of groceries for 17 s. 81 d. ; of boots and shoes for £31 19 s. b\ d. What did he pay for all? 17. Add } of a mile and f of 8 rods. 18. Add ^^ of a square yard and .37 of a square rod. 19. Add 2f weeks, 5.33i days, 6.375 hours. 20. Add £ 21, f of £ 1, .27 of £ 1. 246. SUBTRACTION. Define subtraction, minuend, subtrahend, remainder. Tell how the numbers are written for subtraction. Give the rule. What is the proof ? 1. The distance from A to B is 12 mi. 83 rd. 3 yd. 2 ft., and from A to C is 8 mi. 117 rd. 5 yd. 1 ft. ; the first dis- tance is how much greater than the second ? 2. From 4 mi. 68 rd. 2 yd. 1 ft. 4 in. take 2 mi. 97 rd. 4 yd. 2 ft. 9 in. 3 From 27 sq. yd. 6 sq. ft. 83 sq. in. take 16 sq. yd. 8 sq. ft. 141 sq. in. 4 From 5 A. 83 sq. rd. 13 sq. yd. 5 sq. ft. 67 sq. in. take 2 A. 98 sq. rd. 29 sq. yd. 7 sq. ft. 110 sq. in. 186 NEW ADVANCED ARITHMETIC. 5. From 2 sec. 512 A. 73 sq. rd. take 1 sec. 538 A. 95 sq. rd. 26 sq. yd. 6. Find the difference of the following : £ 17 lis. 7 d. 2 qr. 12 15 V 3^ 4 15 7 3 7. Find the difference between 15 cu. yd. 18 cu. ft. 1276 cu. in. and 7 cu. yd. 23 cu. ft. and 1528 cu. in. 8. Bought 600 lb. of sugar. Sold 124 lb. 6 oz., 73 lb. 13 oz., 48 1b. 9 oz., 173 lb. 14 oz. How much was left? 9. Bought 624 ed. of wood. Sold 75 cd. 7 cd. ft., 116 cd. M cu. ft., 124 cd. 5 cd. ft. 12 cu. ft., 283 cd. 4 cd. ft. 10 cu. ft. How much was left? 10. What is the difference between 8 lb. Apothecaries' weight, and 5 lb. 7 oz. 4 dr. 2 sc. 16 gr. ? 11. A is in long. 124° 42' 36" E., and B is 67° 49' 24" E. What is their difference in longitude? 12. From a cask containing 38 gallons the following amounts were drawn : 4 gal. 3 qt. 1 pt., 7 gal. 2 qt., 12 gal. 1 qt. 1 pt., 8 gal. 3 qt. 1 pt. How much was left in the cask ? 13. From I of 1 rd. take .63 of 1 rd. 14. From the sum of -{'^ of 3 lb. and .35 of 2 lb., take the sum of 5 of 5 oz. and .64 of 2 lb. Apothecaries'. 15. Started to walk 124 miles. Went the first day 18 miles, 74 rods ; the second day, § of 23 miles ; the third day .28 of 95 miles. What distance remained? 16. Find difference in time between Januar}' 21, 1X95, and July 28, 1848; between August 12, 1876, and May 10, 1890. 17. From 12 lb. 9 oz. 7 pwt. 11 gr. take 7 lb. 10 oz. 15 pwt. 18 gr. DENOMINATE NUMBERS. 187 18. How long a time from the battle of Bunker Hill to the firing on Fort Sumter? 19. A cylindrical cistern is 10 feet deep and has a diame- ter of 8 feet. What is its capacity in gallons? (tt = 2^.) In barrels? (3H gallons.) Being | full, the following amounts were drawn out : 1^ barrels, 2\ barrels, 3.25 barrels, 6.325 barrels. How many barrels were left in the cistern? 1728 X 22 X 16 X 10 X 2 Suggestion. z „„, „„ = barrels. 20. From I of 1 cord take .16 of 3 cords. 21. A man paid £11 12 ?. 8^ d. for a wagon. He gave the merchant a £ 20 note. What change should he receive ? Give the rule for subtraction of simple numbers. 22. A man having 34 cords of wood sold to one man 5 cd. 7 cd. ft. 12 cu. ft., to another 15 cd. 14 cu. ft., and to a third 8 cd. 5 cd. ft. How much had he left? 247. Multiplication. Define multiplication, multiplicand, multiplier, product. Give the rule for multiplication of simple numbers. AVhat is the denomination of the product? Be able to give the analysis of reduction at each step. 1. Multiply £7 12 s. 9 d. 3 qr. by 9. FOKM. £7 12s. 9d. 3qr. 9 63 1U8 81 4)27(6 5 7 6 24 68 20)115(5 12)~sf(7 3 100 84 15 ~3~ 2. Multiply 3 mi. 25 rd. 4 yd. 2 ft. 8 in. by 12. 3. Multiply 8 cu. yd. 13 cu. ft. 124 cu. in. by 24. 188 NEW ADVANCED ARITHMETIC. 4. A pile of wood is 4 feet wide, 4 feet high, and 27 feet long. How many cords in 15 S'leh piles? What is it worth at 84.50 a cord? 5. A ship sails from N. Y., 'ongitude 74° 0' 3" W., and makes an average dail}^ easting of 'J° 24' 3G." What is her longitude at the end of 7 days? 6. 27 cans hold an average of 10 gal. 3 qt. 1 pt. How much do they all contain? 7. A bin is 8 feet wide, 12 feet long, and 7 feet high. How many bushels of shelled corn will it hold ? How much will 18 such bins hold? 8. 36 men worked an average of ''2 d. 7 hr. and 30 min. How much money will pay them, at 81.25 a day, counting 10 hours to the day? 9. Find the cost to the druggist of 36 prescriptions of quinine, each containing 24 grains, if quinine cost 42 cents for an avoirdupois ounce (437i grains). 10. Sold 28 loads of oats, each containing 74 bu. 3 pk., at IGj cents a bushel. What was the amount received? 11. Bought a city lot containing | of an acre at 82.25 a square yard. What did it cost? 12. Multiply 15 hr. 24 min. 38 sec. by 42. 13. Multiply 16° 17' 22" by 76. 14. Multiply 128 lbs. 7 oz. by 56. Division. 248. Define partition, divisor, dividend, quotient, re- m.ainder. What terms are alike ? What kind of a number is ihe di\T[sor? 249. Define measurement, divisor, dividend, quotient, re- mainder. Which terms are alike? What kind of a number is the quotient? DENOMINATE NUMBERS. 189 1. If £18 12 s. 3 d. 3 qr. be divided equally among 7 persons, what will each receive? FORM. £ s. d. qr. 7) 18 (£2 12 3 3 14 80 12 4 ~T 7)'92"(13s. 7)'T5~(2d. lyTilc^x. 91 U 7 2. How many articles at £2 7 s. 8 d. 3 qr. each can be purchased for £40 11 s. 4 d. 3 qr.? Analysis. To simplify this problem, divisor and dividend should be reduced to the lowest deuoniiuation found in the numbers. The divisor equals 2291 qr. The dividend equals 38947 qr. There are 17 229rs in 38947; hence, 17 such articles can be purchased. 3. Divide 8 mi. 186 rd. 4 3'd. by 7. 4. Divide 8 mi. 186 rd. 4 3'd. by 7 3'd. 5. If a field containing 71 A. 82 sq. rd. be divided into 8 equal parts, what will each part contain ? 6. Bought 36 2 X 4 16-foot studs, 48 2 X 8 18-foot joists, 4 sills 8 X 8 16 feet long, at $18.50 per thousand; 1,250 feet flooring at $32; 1,460 feet sheathing at 820, and 1,480 feet siding at 833.50. If the bill were divided into 4 equal pay- ments, what would each amount to? 7. How many paving-stones 4 ft. 4 in. long and 3 ft. wide will be needed to make a 3-foot walk 186 ft. 8 in. long? 8. How many bricks, of ordinary size, will be required to pave a court 16 feet wide and 80 ft. 8 in. long? 9. DiWde 4 lb. 2 oz. 13 pwt. by 1 lb. 4 oz. 17 pwt. 16 gr. 10. Divide £19 17 s. Od. 2 qr. by £1 11 d. 11. If a man travel at an average rate of 4 mi. 25 rd. 5 yd. an hour, how many hours wiU be required to travel 175 miles ? 190 NEW ADVANCED ARITHMETIC. 12 Divide 24 T. 826 lbs. by 724 lbs. 8 oz. 13 Bought 15 cd. 7 cd. ft. 12 cu. ft, which was dehvered in 16 loads. What was the average load? 14. If 7 hr. 15 mill. 40 sec. is the average time required for a man to produce a certain article, how many such arti- cles can be produced in 348 working hours? 15. How many cases, each holding 2 gal. 3 qt. 1 pt., will be needed to hold 48 gal. 3 qt. 1 pt. ? 16. A box has a capacity of 1 bu. 3 pk. 5 qt. How many times must a laborer fill it to remove 324 bu. 2 pk. of oats? 17. A cellar is 18 ft. 6 in. by 24 ft. 4 in. and 5 ft. deep. How many loads J of a cu. yd. each will the excavated dirt make? 18. A man, having 44 A. 96 sq. rd. of land, sold 5 A. 92 sq. rd. What part of the land does he still own? 19. A cubical tank, 10 feet square at the base, has a capacity of 8,000 gallons. What is its height? 20. A cubic foot of pure gold may be coined into how many dollars? Gold is 19] times as heavy as water. 250. MISCELLANEOUS PROBLEMS, REVIEW. 1. Find the sum of $83.2, $632.7, $504.9, $473.3, $712.5, S 190.04. 2. Find the sum of $6041.072, $4003.926, $9621.863, $7028.414, $8631.372, $36027.496, $48971.022. 3. What is the cost of 7 articles at $8,464 each? Of 36 articles at $15,842 each? Of 329 articles at $76,575 each? Of 974 articles at $83,125 each? Of 87 articles at $479,375 each? 4. If 23 barrels of flour cost $155.25, what is the price per barrel? . DENOMINATE NUMBERS. 191 5. If 725 acres of land cost $49,571,875, -what is the price per acre ? 6. What is the difference between 87000 and 82874.6G4? 7. A man received 86,126.82 for his farm, 82,579.12 for his stock, and 81,966.47 for his grain. He bought a house for 83,582.96; furuitui-e for 81,391.65 ; a horse for 8164.25 ; a carriage for 8164.28 ; and harness for 836.80. How much money did he have left? 8. A man purchased a library for 87 63. 65 J, paj'ing an average price of 82.34^ per volume. How many volumes did he buy? 9. Find the entire cost of the following articles : 1 desk, S28.50; 1 bookcase, 868.30; 1 half dozen chairs, 818.25; 1 rocker, 812.70; 1 bedstead, 829.50; 1 bureau, 829.58; I washstand, 811. "6; one stove, 837.49; 1 table, 824.76; 1 lounge, 819.46. 10. At 47:^ cents each, how many bushels of corn cau be purchased for 8343.98? 11. A paid 8491.76 for a pair of horses, and 8278.97 for a carriage. How much more did he pay for the horses than tor the carriage? 12. 86215.824 is how much more than 81987.948? 13. A street-car company bought 864 mules, paying $79,200 for them; what was the average price? 14. At 817. 58^- each, how many calves can be bought for 85,697? 15. A man bought six farms. For the first he paid 86,012.07; for the second, 84,631.26; for the third, $3,712.84 ; for the fourth, 88,067.53 ; for the fifth, 87,824.86; for the sixth, 6,098.94. What did he pav for aU? 16. A merchant sold a man the following articles : sugar, $1.40; coffee, 97 cents; tea, 83 cents; salt, 48 cents; flour, $1.85; apples, 82.38; potatoes, 86 cents; molasses, 85 192 NEW ADVANCED ARITHMETIC. cents. He received in payment a 820-bill. "What amount of money should he return? 17. What is the entire cost of the following articles : one horse, 8116.87; one buggy, Si 29. 40; a set of harness, 828.90; a whip, §2.55; one wagon, 865.75; one blanket, 83.78; one sleigh, 836.47? 18. Change 873 yards to a compound number. 19. Change 2 ft. 3 in. to a fraction of a mile. 20. Add : 4 bu. 3 pk. 5 qt. 1 pt. ; 6 bu. 2 pk. 7 qt. ; 12 bu. 1 pk. 6 qt. 1 pt. ; and 23 bu. 3 qt. 21. A railway train, running at the average rate- of 34 mi. 68 rd. 4 yd. 2 ft. per hour, went from A to B in 9 hours. What is the distance between the two places? 22. Multiply 217 rd. 4 yd. 2 ft. 7 in. by 23. 23. The area of a floor is 25 sq. yd. 6 sq. ft. 83 sq. in. What is the entire area of 1 2 such floors ? 24. How much laud is there in 9 fields, if each contain 53 A. 47 sq. rd. 26 sq. yd. ? 25. How many 4-ouuce vials can be filled from 5 gal. 3 qt. 1 pt. 3 gi. of alcohol? 26. Divide 583 bu. 3 pk. 7 qt. of corn into 16 equal parts. 27. How many 40-gallon barrels of water will a cubical cistern contain that is 10 feet deep? 28. Multiply 45 A. 24 sq, rd. 18 sq. yd. by 38. 29. 37 equal quantities of land contain 37 sec. 201 A. 88 sq. rd. 23 sq. yd. 2 sq. ft. 72 sq. in. What does each contain? 30. A railway train, moving at a uniform rate, ran 307 mL 299 rd. 3|^ yd. in 9 hours. What was the rate per hour? 31. How many revolutions will a carriage wheel, whose circumference is 11 ft. 4 in., make in describing a distance of 1 mi. 125 rd. 4 yd. 10 in.? 32. Divide 41 rd. 4 yd. 10 in. by 4 yd. 2 ft. 8 in. DENOMINATE NUMBERS. . 193 33. If 80 cu. 3'd. 4 cu. ft. 848 cu. in. of earth were re- moved iu 28 equal loads, how much did each load contain? S*. How many piles of wood, each containing 2 cd. 75 cu. ft., can be made from 93 cd. 12 cu. ft. ? 35. 3.46 miles = ? 36. Change 4 inches to the decimal of a mile. 37. 3 yd 1 ft. 8 in. is what part of 225 rd. 4 yd.? 38. What is the sum of the following numbers? 1 mi. 182 rd. 4 yd. 1 ft. 7 in. 2 309 5 2 9 5 169 3 4 8 274 2 2 11 15 1 2 6 39. Divide 66 sq. yd. 6 sq. ft. by 2 sq. yd. 7 sq. ft. 40. How many lots, each containing 4 A. 36 sq. rd., can be formed from 50 A. 112 sq. rd. ? 41. From a 40-gal. barrel of vinegar a merchant sold to one man 4 gal. 3 qt. 1 pt. 1 gi. ; to a second 5 gal. 2 qt. 3 gi. ; and to a third 13 gal. 1 qt. 1 pt. 2 gi. What amount was left in the barrel? 42. What quantity of oats will 15 bins contain if the capacitj^ of each be 186 bu, 3 pk. 7 qt. ? 43. Change 2 yd. 1 ft. 11 in. to a fraction of a rod. 44. 4 of a mile = ? 45. § of a rod = ? 46. How much wood is there in 24 piles of wood, each containing 9 cd. 86 cu. ft.? 47. Multiply 18 cu. ft. 724 cu. in. by 46. 48. Reduce 2 mi. 180 rd. 3 yd. 2 ft. 10 in. to inches. 49. How many feet in 321 rd. 4 yd. 1 ft.? 50. Reduce 87889 inches to integers of higher denomi- nations. 194 NEW ADVANCED ARITHMETIC. 51. Reduce i\ of a square mile to integers of lower denominations. 52. Reduce .028 of an acre. 53. 108 square inches is what part of a square rod? 54. 2 sq. yd. 5 sq. ft. 64 sq. in. is what part of 16 sq. yd. 2 sq. ft. 76 sq. in.? Express the result as a decimal fraction. 55. Reduce 4813 feet to integers of higher denominations. 56. Change 429 yards to a compound number. 57. An English officer bought 65 horses for his company at an average price of £21 12 s. 6 d. ; what was the aggre- gate cost? 58. How many bushels of oats will a rectangular bin con- tain that is 6 ft. long, 4 ft. wide, and 5 ft. 8 in. high? 59. A farmer bought the following tracts of land, all lying in the same section : The N. J- of the S. E. I of the S. W. -}. The S. J- of the N. E. ^ of the S. W. i. The N. J of the S. W. i of the S. E. i. Draw a diagram of the section and show his purchase. What did it cost at $62.50 an acre? 60. Find the cost of the posts and fencing necessary to build a four-board fence around the land just described : posts being worth 22 cents each, and placed 8 feet apart, and the fencing being 1 in. thick, 6 in. wide, 16 ft. long, and costing $18.25 per M.? 61. Reduce }^ of an acre to square rods, etc. 62. 2^ qr. is what part of £l ? 63. How many revolutions will a carriage wheel, whose circumference is 14 ft. 8 in., make in going 2 mi. 84 rd.? 64. Change 6 grains Troy to the decimal of a pound. 65. What are 7 loads of hay, each weighing 2,460 pounds, worth at S8.25 a ton? DENOMINATE NUMBERS. 195 66. Add § ci an acre, 5 of a square rod, | of a square yard, and -^^ cf a square foot. 67. What is the value of 3 oz. 5 dr. of quinine at 3 cents ji grain? 68. Reduce /^ of a common year to days, hours, etc. 69. A man sets his watch at Chicago local time. After travelling for some tune he finds that it is 1 hr. 24 min. 30 sec. faster than the local time where he is. What is his longitude ? 70. 12" of arc is what part of a circumference? 71. What is the cost of the following piles of cord- wood at $4.75 a cord? 1 pile, 8 ft. high, 22 ft. long. 2 piles, each 6^ ft. high, 31 ft. long. 1 pile, 9^ ft. high and 32 i ft. long. 72. A square cistern, whose bottom is 8 feet on a side, is 12 feet deep. How many gallons of water are there in it when it is I full? 73. Change \\ oi o. cubic yard to cubic feet and cubic inches. 74. What is the cost of the Brussels carpet, at Si. 6 3 a yard, to cover the floor of a room that is 22 feet long and 19 feet wide, the strips to run the long way? 75. What change shall be made in a watch that is set to New York local tune to make it agree with Chicago local time? 76. When eggs are sold at the rate of 2 for 3| cents, wh.at is the cost >f 4^ gross? 77. Find the cost of the following bill of lumber at $21.50 per M. ; 5 sills, 8 X 10, 18 ft. long. 36 joists, 2 X 10, 16 ft. long. 42 studs, 2x4, 22 ft. long. 70 boards, averaging 9 in. wide and 14 ft. long. 196 NEW ADVANCED ARITHMETIC. 78. If 17 T. 15 cwt. 3 qr. 12 lb. of coal be divided equally among 12 bins, how much will each contain? 79. How many quarts of milk will a vessel hold whose capacity is 1 peck? 80. Add .44 of a common year to 29 days, 19 hr. 18 min. 81. Divide an angle of 100° 10' 1" into 13 equal parts. 82. Reduce 8975 grains Troy to a compound number. 83. Reduce 2 lb., 9 oz. 5 dr. 2 sc. 15 gr. Apothecaries' to grains. 84. A room is 15 feet by 20 feet wjth walls 12 feet high. There are 3 windows 2 J feet by 6 feet, and 2 doors 2 feet 8 inches by 8 feet 4 inches. If wall-paper cost 22 cents a roll, and border 5 cents a yard, what is the whole cost for walls and ceiling? 85- Make a bill of the following items, and receipt it : 32 pounds of sugar, at 6^- cents. 48 yards of calico, at 8^ cents. 28 bushels of potatoes, at 37^ cents. 18 bushels of apples, at 87J cents. 32 yards of cloth, at 75 cents. 3 dozen plates, at 50 cents. 86. What is the cost of 3^ reams of letter-paper, at 12^ cents a quire ? 87. "What is the time from March 11, 1883, to January 19, 1889? 88. Change 88537 square feet to square yards, etc. 89. How many days from January 25, 1892, to December 11? 90. A steamboat going down stream is propelled 12 miles an hour by steam, and 320 feet a minute by the current ; in what time can she go 175 miles? In what time wUl she go the same distance up stream? 91. Reduce 25" to the decimal of a degree. DENOMINATE NUMBERS. 197 92. To ^ of a mile add 7569 inches, and multiply the result by 7. 93. From 8.36 bushels take 3f pecks, and divide the remainder by 12. 94. 2\ quarts is what part of 3 gal. 1 pt? 95. 1 gill is what part of 2 gallons? Change the result to a decimal fraction. 96. In a pacing race two horses started together and went a mile. The time of the faster was 2 min. 6 sec, and of the other 2 min. 6| sec. How much was the winning horse ahead at the finish? 97. When a locomotive, having a driving wheel 5 feet and 10 inches in diameter, is running at the rate of a mile in a minute, how many revolutions do the " drivers" make in a second ? 98. Sound travels at the rate of 1,120 feet a second under ordinary conditions. If the report of a gun is heard 3|^ seconds after the flash of the discharge is seen, what part of a mile is the observer from the gun ? 99. Over what area may a horse graze if tied to a stake by a 50-foot rope ? 100. Over what area may he graze if tied to the corner of a barn 30' X 40' by a 50-foot rope? Draw diagram. 101. An experiment showed that a current of electricity passed over 7,200 miles in f of a second. In how long a time would it describe the equatorial circuit of the earth? 102. Out of a sheet of paper 8" X 10" cut a circle 6 inches in diameter. What part of the paper is cut away ? 103. What is the longitude of Quebec if it is 5 minutes and 42 seconds past 1 p. m., when it is noon at Chicago? 104. Water flows into a tank through three pipes. The first would fill it in 3J hours, the second in 4i hours, and the third in 5^ hours ; in what time will the tliree pipes fill it ? 14A 198 NEW ADVANCED ARITHMETIC. 105. How many 2-iuch circles equal in area 1 6-inch circle ? 106, A borrower paid 8347 for the use of $2,400, paying 7% of the amount loaned for its use for a year. How many years, months, and days should he keep it, counting 30 days for a month? - 107 A 52-gallon oil barrel was J full. 13 gallons were drawn out. What fraction of its capacity did it then con- tain? Change this fraction to a decimal. 108. At the time of her marriage, 8 years ago, Mrs. S. was ten years younger thfin her husband. Her age is now J of his. What was her age at the time of her marriage? 109. In a school of 57.5 pupils the number of boys is /^ of the number of girls. How many are there of each? 110. What is the cost of a city lot 80' X IGO' if sold for as many silver dollars as can be laid upon it in a single layer and placed side by side in rows parallel to the sides of the lot? 111. What is the area of a 4-inch circle? of an 8-inch circle? Divide the latter area by the former. The first is what part of the second? 112. What is the area of a 5-inch circle? of a 10-inch circle? The second is how many times the first? 113. If the radius (R) of one cu'cle is twice the radius (r) of a smaller circle, ttR- is how many times 7:r'-? 114. How large a water-pipe is needed to carry 4 times as much water as a o-iuch pipe ? 115. What is the cost of the gold in a 14-carat chain weighing 12 pennyweight at 820.67 per ounce? 116. A cubic inch of gold has been beaten so thin as to t'over ^'4 of an acre, \^'hat was its thickness? 117. How many layers of such gold would o(iual the thick- ness of a leaf of this book? DENOMINATE NUMBERS. 199 Note. Measure thickness of the book exclusive of covers, and divide by the number of leaves. 118. Simplify: ^^ - H + 3} 119. An iron beam 16 ft. loag, 2\ in. wide, and 8 in. deep weighs 900 lbs. This specimen of irou is how many times as heavy as an equal volume of water? 120. What part of § of % is | of -^-? Change the result to a decimal fraction. 121. Shingles are sold in bundles, each containing the equivalent of 250 shingles 4 inches wide. If shingles are laid 4i inches to the weather, how many bunches must be bought for a roof 20' X 30'? What is the cost of laying them at 80 cents per square ? Note. A " square" contains 100 square feet. 122. Add f of an acre and 217^ sq. rd. 123. A rug 16' X 12' is placed in the middle of a floor 19' X 15'. AVhat is the width and area of the uncovered strip ? 124. Change the following to decimal fractions of five places: ^\, yf^, ^\.. 125. 1,000,000 American eagles will coin into how many sovereigns? 126. An English immigrant changes his money, £248, into federal money. What does he receive for it? 127. Gold is 19;^ times as heavy as water. What is the weight of a cubic foot? What is its value? 128. If the top of 3'our desk were covered with gold two feet deep, what would it weigh? What would be its value? 129. How high is a rectangular block of gold one foot square at the base and weighing one ton ? What is its value ? 130. The length of one degree of longitude at the 40th parallel is 53.063 miles. How far apart do two men live od this parallel whose noons are just one minute apart? 200 NEW ADVANCED ARITHMETIC. 131. How long is the 40th parallel? 132. The length of a degree of longitude at latitude 45 ^ is 48.995 miles. Calculate as accurately as you can the dis- tance on this parallel from the Connecticut River to St. Paul, Minn. 133. Add 86512, 43972, 64829, 93517, 48695, 82724, 60982, 93728, 46479, 794736, and get a correct result in 20 seconds or less. 134. Multiply 96534 by 48967, and obtain a correct result in 50 seconds or less. 135. Divide 8497068314 by 59637, and obtain a correct result in 80 seconds or less. 136. Write the rule for the multiplication of a fraction, and illustrate it by an example, telling how you know that you have a correct result. 137. Write the rule for the division of a fraction by a fraction, illustrate it by an example, and explain in writing how you know that you have a correct result. 138. If 8 men can do a piece of work in 12 J days of 8 hours each, in how many 10-hour days can 5 men do the same work? The following 27 problems were used in examinations for State certificates in Indiana and Illinois : 139. At 90 cents per yard how much will it cost to carpet a room 20 by 27 feet with carpet 2^ feet wide, allowing one foot waste on each cut for matching? 140. If 12i yards of dress goods will make a dress, how many yards of cambric If yards wide will be required to line one half of it? If the goods are 1 yard wide ? 141. If one bushel of wheat will make 40 pounds of Hour, how many barrels of flour can be made from the contents of a bin full of wheat, the dimensions of the bin being 10' X 5' X 4'? DENOMINATE NUMBERS. 201 142. A can do a piece of -svork in 8^ hours, A aud B together can do it in 4-^ hours, and A and C can do it together in 4 hours. How many hours will it take B and C to do the work? 143. How much will it cost to plaster the walls and ceiling of a room 27 ft. long, 15 ft. wide, and 12 ft. high, at 25 cents a square yard, allowing 432 sq. ft. for doors and windows? 144. Find the circumference and area of a circle whose diameter is 2 ft. 4 in. 145. Divide 125.37 by 15.75. Solve by analysis, and show why the rule for pointing is correct. 146. A vessel sailed from a port directly on a line of lati- tude a certain distance, then sailed due north a certain other distance, when the captain found his chronometer forty min- utes slow. In what direction had he first sailed and how many degrees? 147. A gold mine produces $420,000 in a single year. How many pounds av. did the output weigh, 23.22 grains be- ino- worth 81? Its volume? 148. Find the value of 1 — -x-z + o, y - - , • 149.' Five men in a factory accomplish as much as eight boys. What part of a man's work does a boy do? Change this result to per cent. What per cent of a boy's work does a man do ? 150. The diameter of a cylindrical tank is 10^ feet, and its length is 30^ feet. How many gallons will it hold? 151. (4.4 — .00027) X 2.1 X .005 -^ .000005. 152. After paying § of a debt aud I of the remainder, I owe S430.371 less than at fii-st. What was the debt at first? 153. Reduce 57 A. 96 sq. rd. to the decimal of a square mile. 202 NEW ADVAXCED ARITHMETIC. 154. A man walks a certain distance at the rate of 4^ miles an hour, and rides back at the rate of 7.^ miles an hour. If it takes him 8 hours to go both ways, what is the distance ? 155. Find the cost of 25 pieces of scantling b" X 3^", 10 ft. long, at $10.25 i?er M. 156. When it is 4 hr. 20 min. p. m. , 65° 25' west longi- tude, what is the time 17° 20' east longitude? 157. The Capitol at Washington is 751 feet long and 384 feet wide. How many acres does it cover? 158. If it cost $120 to build a wall 40 ft. long, 14 ft. high, 1 ft. 6 in. thick, what will it cost, at the same rate, to build a wall 180 ft. long, 21 ft. high, and 1 ft. 3 in. thick? 159. A lake whose area is 45 acres is covered with ice an inch thick. Find the weight of the ice in tons, if a cubic foot weighs 920 ounces avoirdupois. 160. A can hoe a row of corn in a certain field in 30 min- utes, B in 20 minutes, and C in 35 minutes. What is the least number of rows that each can hoe that all may finish at the same time? 161. A owns y\ of a ship's cargo, valued at $493.000 ; B owns ^1 of the remainder ; C owns y\ as much as A and B, and D owns the remainder. How much does each own? 162. How many square rods in a piece of laud % of a mile long and ^ of a mile wide? 163. Light occupies 16 minutes and 36 seconds in cross- ing the earth's orbit. If the earth is 95 millions of miles from the sun what is the velocity of light? 164. .0001 -^ .00000001 =? 165. A man bought a horse and a carriage for $280. § of the cost of the carriage was 3 of the cost of the horse. What was the cost of each? PER CENT A GE. 20,'^ j^art II. SECTION YII. 251. PERCENTAGE. 1. Percentage is a system of calculations by hundredths. 2. Per cent means hundredth or hundredths. 1 per cent is iJ-^; 7 per cent is -i^^; f per cent is f of ^l^. 3. Any per cent is a decimal fraction having 100 for its denominator. It may be expressed in the form of a com- 7 ^ 81 . mon fraction, as j^^' j^' j^' in the form of a decimal fraction, as .07, .00§, .08^, or with the per cent symbol, as 7%, 1%, 8^-%. 4. Per cent differs from decimal fractions in general iu two ways : (1) Its denominator is always 100. (2) This denominator may be expressed by the sign %. 252. Express the following as common fractions, and re- duce to lowest terms. 1. 7% 10. 83 i% 19. A% 28. n% 2. 15% 11. 225% 20. 12% 29. .ih% 3. 21% 12. 411% 21. 31% 30. .7% 4. 36% 13. 561% 22. 1% 31. .08^% 5. 25% 14. 831% 23. A% 32. 2.25% 6. 50% 15. 1000% 24. i?% 33. .01% 7. 75% 16. 465% 25. f% 34. .001% 8. 381% 17. 116-1% 26. i% 35. .00^% 9. 621% 18. 1% 27. i% 36. 2H%. 204 NEW ADVANCED ARITHMETIC. ' RULE. To change any per cent to a common fraction, erase the per cent sign and tvrite lOO for a denominator, 253. Write each of the expressions in Art. 253 as a deci- mal fraction. 254. Express the following common fractions with the per cent sign, and also as decimal fractions: 3 100 6. 39 100 11. 48 100 16. 1.25 100 17 100 7. 12i 100 12. .02 100 17. .25 100 125 100 8. 100 13. 100 18. 79 100 62^ 100 9. 100 14. 75 100 19. 87^ 100 250 100 10. .7 100 15. .08J 100 20. .01 lUO 255. There is no problem in percentage that will not fall into one of three general problems. A mastery of these general problems gives the technique of the subject. 256. FIRST GENERAL PROBLEM. To find any per cent of any number. RULE. Find one per cent of the number and multiply the result by the nutnber of per cent. 257. ILLUSTRATIVE PROBLEM. Find 18% of (521. Analysis. 18% of a nnmher is -jiffj of that niiinber. ^-^j of G24 ig 6.24, wliic'h is found by making the 024 stand two orders farther to thft right, jij^g of 624 is 18 times 6.24, etc. PERCENTAGE. 205 This method can always be employed with integers or. decimal fractious. Find: 1. 7% of 824 20. 1?% of 867 2. 19% of 916 21. .2% of 163 3. 26% of 589 22. .25% of 7826 4. 35% of 1230 23. 1% of 7826 5. 431% of 1584 ,24. .125% of 5624 6. 52 J % of 6825 25. \% of 5624 7. S&% of 42563 26. ^5% of 3162 8. 117% of 324^ 27. ^^% of 4563 9. 125% of 861| 28. .06|% of 58635 10. 250% of 936.8 29. ^^% of 58635 11. 1000% of 78.32 30. .061% of 32064 12. 17% of .4 31. ^5% of 32064 13. 23% of .625 32. .0625% of 24638 14. 31% of 3 J 33. iV% of 24638 15. 1% of 125 34. f % of 896.24 16. 1% of 324 35. .625% of 896.24 17. 1% of 762 36. T%% of 756 18. ^%% of 1284 37. 41|% of 756 19. hl% of 825 38. 39 i% of 7824 258. Illustrative Example. Fine 7 % of |. Analysis. ^^ of | = ^ff^ ; ^^ of | = ^Vo- Find : 1. 15% off. 8. 76% of f|. 14. 1% of ^a. 2. 16% off. 9. 72% of 21. 15. 31% of ^. 3. 24 % of tV • N<^™. 21 = |. 16. 4^% of If. 4. 30%ofTSj.. 10. 95 % of 3i. . Note. \^- X ^Uxp 5. 42% of if. 11. 124% of 15 17. 81% offf. 6. 56 % of \^. 12. 500%ofl5| -|. 18. 131% of fa. 7. 63% off. 13. -1% off. 19. 182% of IH- 206 NEW ADVAXCED ARITHMETIC. ORAL PROBLEMS. 20. What is A% of 60? 7% of 80? 5% of 90? 12% of 400? 11% of 900? 21. What is 6% of 25? of 12? of 120? of 1200? of 15? of 150? of 1500? 22. What is 1% of 24? of 36? of 480? of 4800? of 72? of 7200? 23. What is -^^ of 1200? of 120? of 12? off? of §? off? 24. What is 1% of 75? |% of 640? ^7o of 3300? ^% of 1? 1% off? 1% of i§? 25. AYhat is 10% of 2,500 pounds? 16% of $4,000? 7% of 71 miles? §% cf 120 acres? J% of 2,500 bushels? 259. Problems are often simplified by changing per cent to a common fraction in its lowest terms. Illustrative Problem, l. Find 37J% of 96. -1 « 374 75 75 3 3 , Analysis. 371%= — - = — = = -. of 96 = 36. ^"' 100 2 X 100 200 8 8 2. Findl2i% of 72; of 144: of 60 ; of 240 ; off; oi ^%. 3. Find 621% of 2400; of 320; of .048 ; of^^^'; of i»5 ; of 84000. 4. Find 40% of 250; 75% of f; 87^% of ^; 66.2% of .081; 25% of 16; 6]- % of .32; 8J% of 132; 50% of 1; 60% of 2. 5. Find 37A% of 96; of 120; of 144; of 4; of .64. 6. Find 33^% of 27 ; of 81 ; of 122 ; of 650 ; of f ; of .018. 7. Find 16§% of 84 : of 120; of 135; of 225 ; of i ; of f^; of .0144. 8. Find 6§% of 45; of 80 ; of 140; of 328; of ^ ; of |f ; of .18. 9. Find 18|% of 160; of 324; 31^% of 256, of 320; 433% of 180, of j|. PERCENTAGE. 207 10. Find 56|% of .0288; 68^% of /a- 11. Find 20% of 165 ; of f ; of .72. 12. Find 40% of 821 ; 80% of .096. 260. 1. Find 7% of 325. FIRST FORM. 325 .07 For analysis, review Multiplication of Decimals. 22.75 SECOND FORM. 7 1-^ 91 - X 3?P == ^ - "'• 4 Find : 2. 9% of 426. 5. 20% of 630 bushels. 3. 13% of 612. 6. 23% of 1,824 miles. 4. 17%ofS725. 7. 33i% of 756 acres. 8. 125% of 67.2 rods. 9. 37i% of t-t; of .0688; of 432. 10. 2% of 7563; %% of 1200. 11. 37^% of £24 16 s. 8d. 12. 33.\% of 15 lb. 9 oz. 18 pwt. 13. 25% of 10 rd. 2 ft. 4 in. 14. 72% of 75 cwt. 75 lbs. 15. 75% of 440 sheep. 16. A cistern Avith a capacity of 84 barrels is 41? % full. How many barrels does it contain ? 17. How much is 200% of a quantity? 400%? 1000%? 250%? 75%? 37i%? 83J%? 66^%? 41-|%? 18. Find 27% of $864.50. 19. Find 6?% of $965.80. 20. What is f % of $1,286.43? 21. What is 183% of $1,680.48? 208 NEW ADVANCED ARITHMETIC, 22. What is 1% of 8972.84? 23. What is 21% of 7,824 bushels? 24. A merchant bought a stock of goods for $8,324.60. The charge for transportation was 1|% of the cost. What was the entire cost? 25. A owed B a certain sum of money. After paying him 20% of the debt, 25% of the remainder, 50% of what then remained, and 83 J % of the third remainder, what part of the debt was still unpaid? 26. What is the interest on 8468.15 for one 3'ear at 7% ? KoTE. Interest is the amount paid for the use of money, and is com- puted at a given per cent of the amount loaned, called the principal, for one year. 27. What is the interest on 81,236.50 fortwoyearsat 6^% ? 28. What is the interest on 82,580 for 2 years and 6 months at 6| % ? 29. A farmer owns a section of land. 25% of it is meadow, 33^% of the remainder is corn-land, 37| % of the remainder is pasture, 80% of the remainder is wheat-land, and the rest is oat-land. Note. Make a diagram 8 inches on a side, and show the several tracts. 30. A piece of cloth containing 36 yards was found to have lost 3^% of its length by shrinkage after sponging. How much did it lose in length? 31. A man's income is Si ,500. He pays 465 % of it for his household expenses, 20% of it for general expenses, and 13^% of it for personal expenses. What are his expenses for the year? How much does he save? 32. In a school of 650 pupils 52% were girls. How many boys were there? 33. A schoolhouse is 98 feet long. Its width is 83 J % of its length. How wide is it? PERCENTAGE. 209 34. In 1895 there was a shrinkage in the value of farm lands of not less than 22 % . What reduction would this make in the value of a farm of 320 acres formerly worth $85 an acre? 35. A merchant bought an overstock of goods which cost him $12,8G0. He marked them to sell at an advance of 32 per cent. He finally sold them at a discount of 28% of the marked price. Did he gain or lose? How much? 36. The south wall of the room in which I am writing is 15' X 26'. It has three rectangular windows whose aggregate area is 30|g% of the wall. The windows are of uniform size, the width being 40% of the length. What is the area of each window? its width? its length? (Diagram.) 261. The second general problem of percentage is to find the per cent that one number is of another. Illustrative Problem. 8 is what per cent of 15? Two things are to be done in solving this problem. 1. We are to find what part 8 is of 15. 2. The resulting fraction is to be changed to hundredths. The first will require a review of the method of finding the part that one number is of another. The second will require a review of the methods of chang- ing a common fraction to a decimal. Analysis. 8 is yV of 15. -f^ = .531 = 53^ % . Find what per cent the first number is of the second in each of the following pairs : 1. 4 8. 2. 5 20. 3, 6 8. 4. 8 10. 5. 11 20. 6. 18 25. 7. 21 : 30. 8. 5 30. 9. 18 54. 10. 42 63. 11. 7 56. 12. 15 . 24. 13. 21 56. 14. 3 : 36. \ is f off. 1 = 15. 5 35. 16. 5 4. 17. 14 12. 18. 24 . 6. 19. i i- 20. 1 ^^' 21. \ h' Note. 1 = |. J = f . | is f of |. f = 66f % . Hence i is 66|% of J. 22. f : ; 23. t : Note. .4 28. .018 210 NEW ADVANCED ARITHMETIC. 24. Jj : |. 26. i : i- 25. ^ : i. 27. .4 : .25. .40. .40 is 1^ of .20. |4 = f^g = 160%. .2. 30. .007i :' .03. 29. .024 : .1. 31. 2^ : 3^. 32. A boy having 20 marbles lost 3 of them. What per cent of his marbles did he have left? 33. In a school of 42 pupils, 7 were in one class, 14 m a second, 6 in a third, 12 in a fourth, and the remainder in a fifth. Give the per cent of the school in each class. 34. A man owning |- of a mill sold i of it. What per cent of his share did he sell? What per cent of the mill did he still own? 35. A received a salary of $12.5 a month. His board cost him $20, his clothing $5, his other expenses $'30. Each of these items is what per cent of his income ? He saves what per cent? 262. WRITTEN PROBLEMS. Illustrative Problems. 1. $17 is what per cent of $24? Analysis. $17 is ^| of $24. \\ is to be reduced to hundredths. First Method. Reduce the numerator to hundredths, and divide it by the denominator. ix = ^_ofl7. 17 = 17.00. ^:f of 17.00 = 24) 17.00 (.70^. 70§ = 70|fc. 16.8 .20 Second Method. Do anything to the fraction that will make its denominator 100 without changing the value of the /17x4^ fraction 24 X 4J = 100 J PERCENTAGE. 211 70# Multiplying both terms by 4^, |^| is found to equal "Tq^; hence, 17 is 70-^% of 24. 2. f is what per cent of | ? Analysis. f=ii. f = fl- il is Jf of |f. Chauge |f to hua. dredths. 3. 2 is what per cent of 450? Analysis. 2 is :j|^ or ^\^ of 450. 5^3 = V|^ = "^^^ = .OO* = | %. Note. Abundant dictation work is needed to give facility. Jlethod. 18 is what per cent of 37? 37) 18.00 (.4S|4. 118 ^ 3.20 2.96 24 Problems like this should be performed at the rate of two or three a minute. Keep the deci- mal point in its proper place in all of the work. 263. EXAMPLES FOR PRACTICE. Find what per cent the first number is of the second tu. each of the following problems : 1. 18 : 30. 13. 125 : 625. 2. 14:50. 14= 140:720. 3. 23:69. 15. 99:451. 4. 36:81, 16. 328: 1076. 5. 54 : 88. 17. 256 : 72. 6. 69:52. 18. 500: 128. 7. 66:92. 19. 836: 1000. 8. 72-80. 20. f:^^. Form, f x Y- = M = 28) 45.00 9. 84: 64. 21. f : i§. 33. 10. 91:28. 22. f:H- 34- 24 : 70J. 11. 96:124. 23. \:{§. 35. 1% : .5625. 12. 29:37. ?4. .02:. 25. 36. 3.75:71. 25. .15:. 125, 26. 3.5:7.15. 27. 2J^:5f. 28. 7|i : 24i|. 29. .0128:. 456. 7/^:35f. 1 30 31 32 T3- 3i : 28. 212 NEW ADVANCED ARITHMETIC. 37. $24 : $84. 47. 2 yd. 2 ft. 3 iu. : 12 rd. 38. $63: $40. 48. $10.24 : $1280. 39. 125 lbs. : 370 lbs. 49. 375 men : 12000 men. 40. 84 A.: 640 A. 50. fr^. 41. 130 sheep: 1200 sheep. 51. 3i : 7|. 42. 12| days : 19 days. 52. 3 qt. 1 pt. : 5 gal. 2 qt. 43. lOOG rd. : 25 rd. 53. 40 sq. rd. : 8 A. 44. 6 ft. : 324 ft. 54. 3 yd. : 8 rd. 45. 136: 624. 55. 2° 30': 10°. 46. 38: 112. 56. $624: $12. 57. A man bought a farm for $6,250. He paid cash $1,250. What per cent of the purchase price remained unpaid ? 58. A man had 24 cd. 6 cd. ft. of wood. He sold 4 cd. 4 cd. ft. AYhat per cent of his wood was left? 59. 25% of an article is what per cent of | of it? 60. 40% of I of an article is what per cent of all of it? 61. If A's money is 25% of B's more than B's, B's is ■what per cent of A's less than A's ? Note. If A's is 25% of B's more than B's, it is 125% of B's, or f of B's ; hence, B's is f of A's. (Prove this.) If B's is | of A's, it is ^ of A's less than A's ; hence, is 20% of A's less than A's. Observe that " per cent of what 1 " is the important question. 62. If A's money is 25% less than B's, B's is what per cent more than A's? 63. If B's money is 33)l% more than A's, A's is what per cent less than B's ? Note. Form problems until the process is mastered. 64. If A's money is 10% more than B's, B's is what per cent less than A's? 15% more? 20% more? 30% more? 40% more? 50% more? 37i% more? 62^% more? 66^% more? 83i% more? PERCENTAGE. 213 65. The width of the top of your desk is what per cent of its length ? 66. The width of your school- room is what per cent of its length? 67. The length of this book is what per cent of its width ? 68. The thickness of this book is what per cent of its width? of its length? 69. The number of boys in your room is what per cent of the whole number of pupils in the room ? It is what per cent of the number of girls ? 70. The percentage of girls in your room is what per cent of the percentage of gMs in the primary room ? 71. What was the percentage of boys in the last gradu- ating class in your high school? of girls? 72. What is the percentage of attendance in your room for one month if all the pupils enrolled except five were present each day, and if each of them was absent three days? 73. What is the population of the town or city in which you live? What is the enrollment in public schools? AYliat per cent of the population is in school ? 264. The third general problem of percentage is to find a number Twhen some per cent of it is given. Illustrative Problem. 24 is 6% of what number? Analysis. Since 24 is 6% of a required number, 1 % of that number is \ of 24, which is 4. 100% of the required number is 100 times 4, which is 400. Method. Find 1% of the required number by dividing the given number by the number of per cent. Multiply this quotient by 100. Such problems may be taken out of percentage by chang- ing the per cent to a common fraction; thus, 6% = ^• 24 is ^^ of what? 214 NEW ADVANCED ARITHMETIC. 265. ORAL PROBLEMS. 1. 48 is 10% of what? 12% of what? 25% of what? S3i% of what? 50% of what? 2» 60 is 1% of what? |% of what? ^7o of what? ^^% of •what? 3. f is 5% of what? g% of what? 25% of what? 100% of what? 621% of what? 4. $21 is 25% less than what? 5. $18 is 91% less tha-u what? 6. $150 is 50% more than what? 7. 60 A. is 30% of what? 8. 75 miles is 25% of what? 9. J of I of a yard is 20% of what? 10. § of I of ^ of a bushel is 16§% of what? 266. WRITTEN PROBLEMS. 1. $6.24 is 18% of what? FORM. $6.24 X 100 Employ eaucellatioa. 2. 750 rd. is 125% of what? 3. 18 gal. 3 qt. 1 pt. is 6|% of what? 4. $62.50 is 15t% of what? 5. 4 sq. rd. 16 sq. yd. is 5^% of what? €. 4 cu. ft. 428 cu. in. is 73f % of what? 7. 67° 30' is 182% of what? 8. A isf% of what? 9. 980 A. is 51% less than what? 10, $6,820 is 241% more than what? 11. 72 men deserted from a regiment, leaving 92f% ot ihe whole number. How many men were there in the rogi- ment before the desertion ? r ER CEN TAGE. 215 12. Paid 663.84 for the use of a certain sum of money for one year at I'/c What was the sum r 13. Paid 690 for the use of a certain sum of money for 2 years at the rate of 6 % for a year. What sum was borrowed? 14. The interest on a certain principal for 4 years and 7 months, at Q'/c •> is $269.94. What is the principsd? 15. 8141,57 is 18% of what number? 16. 84 rods is 125% of what number? 17. 4^^ is 37 i % of what number? 18. 5 lbs. 3 oz. 6 pwt. is 33 i% of what? 19. 35 barrels is 4I5 % of the capacity of a cistern. How many barrels will it hold? 20. 8315.09 is 182% of ^'^^^ number of dollars? 21. The width of a pane of glass in m}^ window is 14 inches, which is 487^^9% of the lengtli. How long are the panes ? 22. AVhat is the area of each pane? This is o\^^^?.2% of what ? 23. Find the luinibcr of Mhich 263 is ] %' ; of which 79 is|%. 24. 826 is i\% of what number? 25. 964 is 500% of what number? 1000% of what number? 1331%. 26. Sold 445.5 pounds of sugar, which was 44% of what I had left. How much had I at first? 27. Sold a farm for 812,860. f of this amount is 50% of the cost of the farm. The gain is what per cent of the cost? 28. What numl)er increased by 20% of itself equals 126? 29. What number diminished by 20% of itself equals 126? 30. A railway train, running at an average rate of 35 miles an hour, for 2f hours, passes over 35% of the conduc- tor's run. ^Miat is the length of his division? 31. What number increased by 25% of itself equals J? 216 XEW ADVAXCED ARITHMETIC. 32. AVhat number diminished by 75% of itself equalSj^? 33. If 340 be added to a number, the result is 117% of the number. What is the number? 34. If 520 be subtracted from a number, the result is 86% of the number. What is the number? 35. A town is found to have gained 824: in population in 5 years. This is an increase of 8% . What was the popula- tion of the town 5 years ago ? 36. A traveler having gone 384 miles has completed 84% of his journey. How much farther has he to go? 37. A farmer having plowed 36] acres finds that he has finished 56% of his field. How large is it? 38. A farmer contracted to deliver to a dealer 1 ,800 bushels of corn. Upon taking his grain to market he found that he had overestimated the capacity of his crib 4%. How much did it contain ? 39. A merchant being obliged to vacate his room sold his stock at a discount of 11 % of the cost and realized 823,568.46. What did the goods cost him? 40. In a certain school there are 168 boys, who form 42% of the whole school. How many girls are there in the school ? 41. A merchant sold a suit of clothes for S28.50, which was 25% less than the marked price. This was 33 ;\''^ more than the cost. What was the cost? 42. $3,825 is 11% more than what? 15% less than what? 43. What number increased by 18% of itself equals 3,379.52? 44. AVhat fraction diminished by 30% of itself equals ^i? 45. 's -^ i equals 24% of what? 4 4- t'- 46. V e ^ " Principal $850 m $08.00 = interest for 1 yr. 10 mo. 23 d. 1 (( 16 « 9 « 24 « 1 « 15" 9" APPLICATIONS OF PERCENTAGE. 269 6 mo. = 4 of 1 yr. ; 4 " = \ of 1 " 20 d. = ^ of 4 mo. 2 " = L of 20 d. 1 " = i of 2 d. 1 mo. = Y*5 of 1 yr. 10 d. = J of 1 mo. 6 " = 1 of 1 " Payment 6 mo. = 4 of 1 yr. 3 " = ^ of 6 mo. 15 d. =iof3 " 9 " = J^ of 3 " 2d payment S72.00 3d «' S153.50 1 mo. = J5 of 1 yr. 15 d. = ^ of 1 mo. $34.00 22.666 3.777 .377 .188 = interest for 6 mo. = " "4 " " 20 d. _ u "2 " -_ « "1 " $61,008 850 = " " 10 mo. 23 d. = first principal. $911,008 86.25 = amount, Jan. 24, 1886. = first payment. $824,758 .08 = new principal, Jan. 24, 1886 $65.9800 5.498 1.832 1.099 $74,409 $72.00 = interest for 1 yr. = " "1 mo. = " " 10 d. = " "6 " = " " 1 yr. 1 mo. 16 d. less than interest. $65.98 = interest for 1 yr. $32.99 16.495 2.749 1.649 = " "6 mo. — a « 3 (( " 15 d. := " "9 " $53,883 74.409 824.758 = " "9 mo. 24 d. = previous interest. = second principal. $953,050 225.50 = amount, Jan. 5, 1888. 2d and 3d payments. $727.55 .08 = 3d principal. $58.2040 4.8503 2.4251 = interest for 1 yr. = " "1 mo. = " " 15 d. $65.4794 727.55 = " " 1 yr. 1 mo. 15 d. 270 NEW ADVANCED ARITHMETIC. $793,029 = amount, Feb. 20, 1889. 265.80 = 4th payment. $527,229 = 4th principal. .08 S42.1776 = interest for 1 yr. $3,514 = interest for 1 mo. 6 d. = ^ of 1 mo. .703 = interest for 6 d. 3 " = ^ of 6 d. .351 r= " "3 " $43,231 = " " 1 yr. 9 d. 527.229 = 4th principal. $570,460 = amount due March 1, 1890. Note. Payments less than the accumulated interest will rarely be made, since they do not diminish the interest-bearing portion of the debt. 1. A note whose principal is $500, dated March 1, 1890, and bearing interest at 6%, has the following indorsements: June 1, 1891, $65. Sept. 16, 1892, $124. What was due Jan. 1, 1894? 2. Principal, $850. Date, May 10, 1891. Rate, 7%. Indorsements : July 15, 1892, $130. June 1, 1893, $46. Dec. 12, 1894, $380. What was due May 10, 1895? 3. Principal, $1,000. Date, Sept. 1, 1892. Rate, 8%. Indorsements : March 12, 1893, $75. June 18, 1894, $275. March 15, 1895, $360. What was due Sept. 1, 1895? 4. Principal, $1,200. Date, July 1, 1890. Rate 5i%. Indorsements : ^^ March 16, 1891, $160. June 12, 1892, $320. Aug. 5, 1893, $500. What was due July 1, 1894? APPLICATIONS OF PERCENTAGE. 271 5. A note of $1,800, bearing interest at 7%, and dated June 12, 1886, was indorsed as follows : March 21, 1887, S183.50. Oct. 12, 1888, $395.75. May 10, 1890, S583.45. What was due July 1, 1892? 6. $1,250. Bloomington, III., Oct. 15, 1886. Four years after date, for value received, we, or either of us, promise to pay to W. O. Davis & Co., or order. Twelve Huntired Fifty Dollars, with interest at 6% per annum. James T. Ronet, John J. Condon. The following statements were written across the back of this note : " Bec'd on the within note, Dec. 1, 1887, $358.80." " Rec'd on the within note, Jan. 21, 1889, $475.00." " Rec'd on the within note, Oct. 25, 1889, $261.50." " Rec'd on the within note, June 15, 1890, $91.40." What was due Oct. 15, 1890? 7. A note for $2,580, dated July 12, 1884, bearing 5% interest, had the following indorsements : Jan. 1, 1885, $75. May 25, 1885, $87.40. Dec. 18, 1885, $260. Oct. 15, 1886, $326.45. June 28, 1887, $752.31. Nov. 12, 1888, $850. What was due July 12, 1889? 8. Principal, $762.84. Date, Sept. 24, 1886. Rate of interest, 10%, Indorsements: Jan. 1, 1887, $51.80. Aug. 23, 1887, $128. May 17, 1888, $125. Oct. 28, 1889, $214.80. March 13, 1890, $306.90. What was due Sept. 24, 1890? 272 NEW ADVANCED ARITHMETIC. 9. Principal, $8,750. Rate, 5%. Date, April 12, 1882. Indorsements : June 20, 1883, $1,250. Aug. 3, 1884, $2,560. Dec. 25, 1885, $3,164.86. July 30, 1886, $1,571.29. Dec. 15, 1887, $1,262.80. What was due Oct. 12, 1888? 10. Principal, $2,350. Date, Aug. 3, 1885. Rate, 71%. Indorsements : Sept. 5, 1886, $250. Jan. 1, 1888, $60. July 25, 1888, $475. March 15, 1889, $560. Aug. 3, 1890, $880. What was stiU due? 11. Principal, $3,000. Date, Jan. 10, 1886. Rate, 6%. Indorsements : March 1, 1887 $260. June 11, 1888, $624. Aug. 25, 1889, $1,030. May 1, 1891, $1,250. Jan. 10, 1892, $280. What was still due ? 302. THE MERCHANTS' RULE. I. Find the amount of the principal for the entire tintCt II. Find, the amount of each payment from the time that it tvas made to the time of settlement. III. From the first amount subtract the sum of the amounts of the serei'al payments. Note. This method allows interest on each payment for all of the time that it is in the creditor's possession. It is a perfectly fair method. 1. A note for $650, dated Jan. 10, 1894, and bearing in- terest ai 5%, has the following indorsements: March 15, APPLICATIONS OF PERCENTAGE. 273 $125; July 12, S240 ; Oct. 5, $85. What was due Jan. 10, 1895 ? 2. A note for $785.40, dated May 1, 1895, and bearing interest at 6%, is indorsed as follows: Aug. 20, $180.20; Nov. 5, $250.80; Feb.. 24, 1896, $236.50. What was due May 1, 1896? 3. Principal, $892.60. Date, July 25, 1895. Rate, 7%. Indorsements: Sept. 1, $325; Nov. 19, $175.50; Jan. 12, 1896, $90; May 10, $300. What was due July 1? 4. Principal, $1,280. Rate, 8%. Date, May 1, 1894. Indorsements: July 25, $300; Sept. 10, $250; Dec. 1, $350; Feb. 10, 1895, $100. What was due April 20, 1895? 303. ANNUAL INTEREST. 1. Annual interest differs from compound interest in one particular : interest does not draw compound interest, but simple interest. A problem will make the difference clear. 2. If a note provides that interest is payable annually, it means that unpaid interest at the end of any period shaU draw simple interest until paid. niustrative Example. A note of $400 is due in 4 yr. 6 mo. The interest at 7% is payable annually. If nothing is paid until the note is due, what will the interest amount to? The interest on $400 for 4 yr. 6 mo. = $126. The $28 due at the end of the first year draws interest for 3J years ; the $28 due the second year, for 2i j-ears ; the third, for 1^ years ; and the fourth for h year. There will then be due, in addition to the $126, the interest on $28 for 3i yr. + 2^ yr. + li yr. + ^ yi'. = the interest for 8 years = $15.68. $126 + $15.68"= $141.68. Make a rule from the above analysis. NEW ADVANCED ARITHMETIC, PROBLEMS. Principal. Time RatA 1. $350 3 yr- 5 mo. 6 d. 6% 2. $400 4 '^ 7 ' ' 9 ' ' 6% 3. 8480 2 8 ' ' 12 ' 7% 4. $510.40 4 10 ' ' 15 ' 7% 5. $560 5 6 ' ' 18 ' 7% 6. $85.50 3 11 ' ' 19 ' 7% 7. $128.20 4 2 ' ' 21 ' 8% 8. $649 3 4 ' ' 24 ' 8% 9. $763.50 6 7 ' ' 20 ' 5% 10. $840 5 1 ' ' 1 ' 5% 11. $24.10 7 7 ' ' 7 ' 5i% 12. $968 6 6 ' ' 6 » 6% 13. $1070 4 2 ' 6% 14 $312.40 5 6 ' ' 20 ' 6% 15. $2060 2 9 ' ' 10 ' 6% 16. $3840 3 4 6% 17. $5625 4 9 ' ' 12 ' 7% 18 $96.90 5 8 ' ' 15 ' 4i% 19. $4500 6 3 ' ' 17 ' 41% 20. $6970 4 5 ' ' 5 ' 5% 21. $386.75 5 10 ' ' 10 ' 5% 22. $193.40 6 6 ' 5i% 23. $4200 4 9 ' 5J% 24 $10000 3 3 ' ' 3 ' 4i% 25. $693.42 4 11 ' ' 6 ' 5% 304. COMPOUND INTEREST. 1. A man borrowed $350 at 7% interest, agreeing that if the interest were not paid at the end of the first year it should be added to the principal to make a new principal for APPLICATIONS OF PERCENTAGE. 275 the second year, and that the interest should be added thus each year until the debt was paid. If he paid nothing until the end of the 3 years and 3 months, how much was then due? 1st principal $350 1st year's interest 24.50 2d principal $374.50 2d year's interest 26.215 3d principal $400,715 3d year's interest 28.049 4th principal $428,764 Interest for 3 mo. 7.502 Amount $436,266 1st principal 350 Interest $86.26 2. Compound interest is interest upon a principal that is increased at regular periods by its accumulated interest. 3. Interest may be compounded at the end of any period agreed upon, instead of annually, as above. RULE To calculate compound interest: 1. At the end of each iteriod increase the principal for that period by the interest accumulated during the period. 2. From the final amount subtract the first principal. 305. PROBLEMS. Find the compound amount and interest : 1. Of $528, for 3 years, at 5%. 2. Of $1200, for 4 years, at 8%. 3. Of $1680.50, for 2 years, at 10%, compounding quarterly. 276 NEW ADVANCED ARITHMETIC. 4. Of $2560, for 3 yr. 8 mo. 25 d., at 6%, compounding semi-annually. 306. TABLE, Showing the amount of^\ at compound interest from 1 year to 10 years, at 3, 4, 4^, 5, 6, ant/ 7 per cent. Years. 1 3 per cent. 4 per cent. 4J per cent. 5 per cent. G per cent. 7 per cent. 1.030000 1.040000 1.045000 1.0.50000 1.060000 1.070000 9 1.060900 1.081600 1.092025 1.102500 1.123600 1.144900 3 1.092727 1.124864 1.141166 1.1,57625 1.191016 1.22.5043 4 1.12,5.509 1.169859 1.192519 1.215506 1.262477 1.310796 5 1.159274 1.216653 1.246182 1.276282 1.338226 1.402552 6 1.194052 1.265319 1.302260 1.340096 1 418519 1.500730 7 1.229874 1.315932 1.360862 1.407100 1.. 503630 1.005781 8 1.266770 1.368.569 1.422101 1.477455 1.593848 1.718186 9 1.304773 1.42.3312 1.486095 1.551328 1.689479 1.8384,59 10 1.343916 1.480244 1.552969 1.628895 1.790848 1.967151 Illustrative Example. Find compound interest of $1UU for 8 yr. 4 mo. 12 d., at 5%. Amount of $1.00 for 8 yr. at 5% $1.4774 100 Amount of .$100 S147.74 Interest of $147.74 for 4 mo. 12 d. 2.70 Amount $150.44 100.00 Compound interest $50.44 With the aid of the table find the amount and compound interest : Principal. Time. Rate. 1. $750 4 yr. 6 mo. i\%. 2. $5,000 9 " 7 " 10 d. 6%. 3. $1,275.45 8 " 2 " 18 " 7%. APPLICATIONS OF PERCENTAGE. 277 Principal. Time. Rate. 4. $1,640.25 4 yr- 5 mo. 24 C I. 5%. 5. $2,850.66 7 1 ' 10" 4%. 6. $834 5 7 " 3%. 7. $796 8 5 ' 10 ' 4%. 8. $1,028.50 9 2 " 12 ' 41%. 9. $1,725.80 3 10 ' 15 ' 5%. 10. $9G0 4 1 ' 21 ' 6%. 11. $2,480 7 11 " 24 ' 7%. 12. $3,812 6 3 ' 28 ' 6%. 13. $86.95 2 7 ' ' 19 ' 5%. 14. $731.25 3 4 ' 17 ' 41%. 15 $5,960 4 5 ' 11 ' 3%. 16. $72.15 5 8 ' 16 ' 4%. 17. $3,824 6 2 ' 18 ' 5%. 18. $7,500 7 10 ' ' 12 ' 3%. 19. $438.90 9 4 ' 15 ' 6%. 20. $1,500 10 10 ' ' 10 ' 7%. 21. $2,500 4 1 ' 1 ' 4i%. Principal. Time. Ra e. 22. $800 4 yr . 4 mo. 6^ 6- comf )ounded semi- Note. Use kalf the rate for double the time. 23. $1,500 3yr. 6mo. 24d. 8% compounded semi-aun'ly. 24. $1,850 4 " 8 " 20 " 9% " " 25. $2,500 2 " 5 " 12 " 12% " quarterly. Note. Use one-fourth the rate for four times the time. 26. $3,150 3jT. 7mo. 18d. 8% compounded semi-ann'ly. 27. $3,675 4 " 2 " 6 " 6% " " 28. $4,180 5 " 3 " 10 " 6% " " 29. $5,000 3 " 7 " 15 " 8% " " 30. $10,000 4 "10 " 18 " 6% " lyA 278 NEW ADVANCED ARITHMETIC. 307. GENERAL PROBLEMS IN SIMPLE INTEREST. 1. "We have seen that five elements have appeared in problems in interest. These are principal, interest, amount, rate per cent, and time. The relations between them are such that if any three of them be given, the other two may be found. Several different problems, consequently, may arise. 308. Problem I. Given the principal, rate per cent, and time, to find the interest. This problem has been discussed sufficiently. 309. Problem II. Given the principal, interest, and time, to find the rate per cent. Illustrative Problem. The interest on $324.00 for o yr. 6 mo. 15 d. is $91. 'J7. What is the rate per cent? AxALYSis. The interest on S324.60 for 3 yr. 6 rno. 15 d., at 1%. is $U.49|. To produce $91.97 in the same time, the rate must be as maiiv times 1% as $91.97 is times $11.49f. $91.97 is 8 times $ll.49|; hence, tlio required rate most be 8%. PROBLEiVIS. Principal, Interest. Time. $66.13 Syr. 5 mo. 10 d. Find rate. $85.36 2 1. 2. 3. 4. 5. 6. 7. 8 9. 10. $650.40 $864.36 $1,275.86 $1,464.29 $242.89 4 $180.04 1 $306.28 2 $2,580.47 $577.02 3 $3,600 $951.75 5 $4,,580 $1,134.94 4 $5,000 $1,6 63.. 38 5 SO. 840. 75 $3,575.72 6 7 8 9 1 2 10 6 3 11 15 6 5 3 10 15 2 1 19 APPLICATIONS OF PERCENTAGE. 279 310. Problem III. Given the principal, interest, and rate per cent, to find the time. Illustrative Problem. The interest on $324.60, at 8%, was $91.97. What was the time? Analysis. The interest on $324.60 for 1 month, at 8%, is $2,164. To produce $91.97 at the same rate, the time must be as many times 1 month as $91.97 is times $2,164. $91.97 is 42i times $2,164; hence, the time was 42| months, which = 3 yr. 6 mo. 15 d. Priuiipal. Interest. Rate. 1. $490.92 849.75 8%. Find time 2. 83,794.08 8803.40 9%. 3. S892.-15 8149.73 5%. 4. 81,231.16 81G9.08 6%. 5. $0,871.48 81,701.10 7%. 6 S89.26 818.04 3%. 7. 813.51 84.37 5i-%. 8. 81.05 80.55 6,\%. 9. 810,874.80 8193.33 4%. 10. 8916.15 858.63 8%. 311. Problem IV. Givsn the interest, rate per cent, and time, to find the principal. Illustrative Problem. What principal will produce 891.97 in 3 yr. 6 mo. 15 d., at 8% ? Analysis. A principal of $1.00, with the above rate and time, will produce $0.28j. To produce $91.97, the principal must he as many times $1.00 as $91.97 is times $0.28i; $91.97 is 324.6 times $0.28^; hence, the required principal is $324.60. Note. Change divisor and dividend to thirds. 280 NEW ADVANCED ARITHMETIC. Interest. Rate. Time. 1. S184.67 7% 3 yr. 6 mo. 9 d. 2. $166.62 8% 3 " 9 ' ' 17 " 3 $35.16 51% 6 " 9 ' ' 24 " 4. $89.68 6% 3 u 2 ' ' 7 " 5. $8.04 7% 2 tt 4 t ' 18 " 6. $150.41 7% 2 4t 4. 4 ' 12 " 7. $233.34 7% 3 " 5 ' ' 13 " 8. $26.67 5% 9 ' ' 15 " 9. $24.21 5% 8 ' ' 15 " 10. $61.74 ^% 5 " 8 ' - 25 " Find principaL 312. Problem V. Given the amount, rate per cent, and time, to find the principal. Illustrative Prohlera. What principal will amount to $416.57 in 3 yr. 6 mo. 15 d. at8%? Analysis. A principal of $1.00 will amount to $1.28^ with the alx)ve time and rate. To amount to $416..57, the principal must be as many times $1.00 as $416..57 is times $1.28^. $416.57 is 324.6 times $1 28^; hence, the reciuired principal is $324.60. Note. Observe that we first found, in Art. 309, what a 1 % rate would produce; in Art. 310, what would be produced in 1 month; in Art. 311, what a $1.00 principal would yield ; and in Art. 311, what a $1.00 prin- cipal would amount to. From these illustrative problems the following general statement may be derived : Perform the problem, assuming one of the required kind to he the required answer ; then compare the result with the given numt)er of the same kind. PROBLEMS. Amount. Rafe. Time. 1. $460 5% 2 yr. 3 mo. 20 d. Find principaL 2. $380.80 6% 3 " 4 " 15 " " " 3. .^524. 10 7% 4 " 6 " 12 " " " 4. $736,50 8% 1 " 9 " 16 " " " APPLICATIONS OF PERCENTAGE. 281 Amount. Rate. Time. 5. $6'Ja.l2 7% 5 mo. 24 d. Find principal 6. $874 H% Syr. 11 " 19 " H 4 t ' 7. 81)26.95 41% 5 " 2 " 21 " 8. $1,284 51% 2 " 7 " 6 " 9. $2,065.48 7% 3 " 8 '> 28 " 10. $3,129.76 6% 1 " 1 " 18 " 313. PROBLEMS. 1. At what rate will $240 gain $8.96 in 6 mouths and 12 days ? 2. In what time will $145.60 gain $42.47 at 5% ? 3. What principal will gain $52.57 in 2 yr. 4 mo., at 6% ? 4. What principal will amount to $132.40 in 5 yr. 7 mo. 20 d., at 10%. 5. At what rate per cent will any principal double itself in 10 years? 8 years? 12 years? 16§ years? 20 years? At what rate will any principal treble itself in the same periods ? 6. In what time will a principal double itself at 3% ? 4% ? 4^% ? 5% ? 51% ? 6% ? 7% ? 8% ? 9% ? 10% ? In what time wUl it treble itself at same rates per cent? 7. What was a man's investment that yielded him an in- come of $1,264.75 in 1 yr. 7 mo. 24 d., at 7% interest? 8. What principal will amount to $5,860.80 in 3 yr. 10 mo. 20 d., at 8% interest? 9. At what price must bonds that bear 4% interest annu- ally be bought, to yield 6% interest on the investment? 10. Bought 4% bonds at 70. What rate per cent of in- terest does the investment yield ? 11. Bought a piece of land for $6,825^ Kept it 2 yr. 6 mo. 18 d., and sold it for $9,841.65. What is the rate per cent of interest that the investment yielded ? 282 NEW ADVANCED ARITHMETIC. 12. How long must $586.40 be on interest at 8% to gain $134.50? 13. AVhat sum of money must be invested at 6% interest, compounded annually, to amount to SlO,000 in 10 years' 14. What investment will yield annually §564.75 at 6% ? 15. In what time will 82,500 amount to S3,650 at U'/c ? 16. What sum put at interest, Jan. 1, 1886, will amount to §343.75, Feb. 1, 1888, at 7% ? Find the lacking numbers in the following problems : 1. Principal. S834.60 Rate. 7% Time. 9 Interest. 848.20 Amount. 2. 6560.40 ? 2 yr. 2 mo. 854.639 ? 3. p 6% 7 mo. 15 d. ? 8350.65 4. 82,500 8% ? 8556.67 ? 5. V 9% 90 d. 872.55 ? 6. 8150.60 ? 72 d. 82.26 7. 8425.75 6% ? 88.52 8. ? 8 7o 93 d. 812.60 9. v 7% 33 d. 8524.12 10. 81,530.38 H% ? 8375.00 314. PRESENT WORTH AND TRUE DISCOUNT. 1. .Tames K. Briggs purchased from Roliert R. Stone a horse and carriage for 8250, with the understanding tliat ho was to pay for them 18 months afterward, without interest. He made the following note : 8250. RocHESTKR, N. Y., May 1, 1890. Eighteen months after date, for value received, I promise to pay to Robert R. Stone, or order. Two Hundred and Fifty Dollars, without interest until due. James K. Briggs. APPLICATIONS OF PERCENTAGE, 283 Ou the 1st day of July, 1890, Stone offered this note for sale. What should a person pay for it so that at its matu- rity he should receive his purchase money and interest on it at 6 % per annum for 1 6 months ? He will receive $250 at the maturity of the note. The question is, What principal, at 6% interest per annum, will amount to $250? A principal of $1.00 will amount to $1.08 in 16 months at 6% per annum. To amount to $250, the principal must be as many times $1.00 as $250 is times $1.08. $250 is 231.48+ times $1.08; hence, the required amount is $231.48+. The problem may be proved by finding the interest on $231.48 for the given time at 6%. 2. The Present "Worth of a debt due at a future time, and not bearing interest, is that sum which, put at interest, at the given rate, for the given time, will amount to the debt. 3. The True Discount of a debt is the difference between the debt and its present worth. RULE. To find the present ivorth of a debt, divide the amount of the debt at its maturity by the amount of $1.00 for the given time at the given rate of interest. To find the true discount, subtract the present toorth from the debt. Note. Observe that true discount is the interest on the present worth for the given time. 315. Another Method, i. Any principal at 6% will amount to 108% of itself in 16 months. In this case the amount is $250. $250, therefore, is 108% of the present worth. 1% of the present worth is ^i^ of $250. 100% of the present worth is jg§ of $250. Which of the general problems of percentage is this? 2. Observe that by this method the discount may be fonnd without finding the present worth. 284 NEW ADVANCED ARITHMETIC. The discount is 8% of the present worth. We have seea that 1 % of the present worth is y J ^ of the amount ; hence, 8% of the present worth is yg^ of the amount. PROBLEMS. 1. "What is the present worth of a non-interest-bearing debt of $1,250, due in 2 yr. 3 mo. 15 d., if money is worth 6% ? What is the true discount? 2. Principal, $875.40; time, 1 yr. 4 mo. 20 d. ; rate, 7%. Find present worth and discount, the principal not bearing interest. In the following problems the principal does not bear interest. Find present worth and discount. Principal. Time. Rate 3. $580.30 3yr. 4 mo. 10 d. 6% 4. $671.95 5 " 7 a 8 u 7% 5. $2,834.25 2 " 5 " 12 " 5% 6. $10,000 1 " 3 " 6 " 4% 7. $383.59 2 " 11 " 21 " 8% 8. $789.13 2 " 9 " 26 " 9% 9. What is the present worth of an interest-bearing debt, if the rate of discount is the same as the rate of interest? 10. A note dated May 12, 1889, for $1,060, due Sept. 21, 1891, and bearing interest at 5%, was discounted Oct. 15, 1890, at 6% . What was its worth? Note. Remember that the amount due at the maturity of the uote is the debt whose present worth is to be found. Find present worth and true discount of the following non-intei'est-bearing notes. Principal. Date. Maturity. D^fcount 11. $84.90 Jan. 1, 1890 March 12, 1892 5% 12. $120 June 12, 1892 Oct. 5, 1894 5% 13. $250.80 March 10, 1888 May 1, 1891 6% APPLICATIONS OF PERCENTAGE. 285 Principal. Date. Maturity. ^^^^[^ V S360 Oct. 15, 1893 June 28, 1896 6% i.5. S480.40 Feb. 20, 1892 Aug. 1, 1895 6|% 16. $560.70 July 28, 1890 Dec. 10, 1894 ^% 17. $1,290 Aug. 3, 1894 May 15, 1897 7% 18. $2,580 Nov. 19, 1893 Oct. 1, 1897 1\% 19. $1,624 Dec. 30, 1895 July 24, 1898 7^% 20. $3,560 April 20, 1896 Dec. 1, 1898 8% 21. $787.50 May 1, 1896 Jan. 19, 1899 S\% 22. $48 Sept. 21, 1896 March 5, 1899 9% 23. $6,000 Jan. 23, 1896 April 21, 1899 9% 24. $980 Dec. 15, 1895 June 12, 1898 10% 25. $25 Oct. 19, 1896 Aug. 1, 1898 10% 316. BANK DISCOUNT. 1. If the note in Art. 314 had been sold to a bank, its value would have been estimated somewhat diflferently from the method there given. The banker would have calculated the interest on $250 for 16 months and 3 days, if days of grace are counted, and would have subtracted the result from $250. The remainder would have been the proceeds, or avails, or cash value, of the note. 2. The Bank Discount of a note not bearing interest is the interest upon the face of the note for the time from the day of discount until its legal maturity. 3. The three days that are sometimes added to the time which a note is to run are called Days of Grace. 4. Several of the States have done away with days of grace. In the following problems take no account of them unless they are mentioned. 286 NEW ADVANCED ARITHMETIC. PROBLEMS. What are the bank discount and avails of the following note, if discounted at 6% on the day that it was made? $600. Chicago, III., May 12, 1891. Ninety days after date, for value received, I promise to pay to the First National Bank of Chicago Six Hundred Dollars, with interest at 6% per annum, after due. Benjamin E. Brown. Analysis. The interest on $600 for 93 days at 6% is 9.30; hence, the bank discount is $9.30, and tlie avails $600 — $9.30 = $590.70. RULE. To find the bank discount of a note tluit does not bear interest : 1. Find the interest on the note for the time that it is to run. 2. To find the avails, subtract the discount from the face of the note. Note. Remember that the sum discounted is the amount to be paid at the maturity of the note ; hence, if the note bears interest, first find the amount of principal and interest. Similarly solve the following : Face of Note. Time Discounted. Rate of Discount. 1. $825.00 60 days 6% 2. $927.18 30 " 5% 3. $264.83 45 " 5% 4. $169.47 90 " 7% 5. $2,968.51 4 months H% 6. $417.80 5 " 8% 7. $361.28 6 7% 8 $248.65 50 days 7% 9. $1,827.90 40 " 6i% 10. $83.70 1 year, 3 months 6% APPLICATIONS OF PERCENTAGE. 287 11. $850. Buffalo, N. Y., July 21, 1896. Three months after date, for value received, 1 promise to pay to Katharine Sedgwick, or order, Eight Hundred Fifty Dollars, at the City Bank. WiLLiSTON Cook. Discounted Aug. 15, 1896, at 7%. 12. $1,200. Philadelphia, Pa., Aug. 1, 1896. Four months after date I promise to pay to Richard M. Johnson, or order. Twelve Hundred Dollars, value received. Thojias R. Williams. Discounted Oct. 15, 1896, at 6%. 13. $128.50. Detroit, Mich., July 10, 1896. Sixty days after date, for value received, we promise to pay to Henry R. Sunderland, or order. One Hundred Twenty- eight and ^o"a Dollars. Arthur G. Hunting. Peter T. Small. Discounted Aug. 1 at 8%. Count days of grace. 14. S480. Boston, Mass., Sept. 1, 1896. Three months after date, for value received, I promise to pay to Samuel S. Huston, or order, Four Hundred Eighty Dollars, with interest at 6 per cent per annum. John L. Whitney. Discounted Sept. 18 at 8%. Count days of grace. 15. $1,540. Aurora, III., Aug. 21, 1896. Two months after date, for value received, I promise to pay to Joseph H. Freeman, or order, Fifteen Hundred Forty Dollars. James S Campbell. Discounted Sept. 1, at 6^%. 288 NEW ADVANCED ARITHMETIC. 16. 8875.00. Chicago, July 10, 1896. Three months after date, for value received, I promise to pay to the Chemical National Bank Eight Hundred Seventy- five Dollars, with interest at 7% per annum after maturity. Robert S. Daniel. Discounted at 7%, July 10. 17. $760. Bloomington, III., Aug. 3, 1896. Four months after date, for value received, I promise to pay to the National State Bank, or order. Seven Hundred Sixty Dollars, with interest at seven per cent per annum after due. James L. Atwood. Discounted at 7 % Aug. 3. 317. Mr. A, desiring to pay a debt of 8460, went to a bank to obtain the monej'. Since he must receive 8460 from the bank, it is evident that he must make his note for more than that amount. He wishes to borrow the money for 4 months. The current rate of interest is 7%. If he should make his note for 81-00, the banker would give him the dif- ference between 81.00 and the interest on it for 4 months and 3 days, if days of grace are counted. The interest on $1.00 at 7% for 4 months and 3 days is 23^- mills. The avails of such a note would be 81.00 — 80.023}i = 80.97(;yi5. In order that he shall receive 8460 his note must be made for as many dollars as there are times 80.976y\ in 460. 460 -i- .976^5 = 471.27; hence, the note must be made for «471.27. Proof. The interest .on $471.27 for 4 months and 3 days at 7% is 811.27+; hence, the avails of such a note would be 8460. Note. In calculating interest on $1.00 for short periods, the second 6% method is most convenient. From the foregoing analysis we may form a APPLICATIONS OF PERCENTAGE. 289 RULE. To find the face of a note that will yield a given amount Jyy bank discount: 1. Find the avails of a note for $1.00 for the given time and at the given rate, and 2. IHvide the given amount by it. Observe that there are two kinds of problems in bank discount. (a) To find the avails of a note when face, time, and rate are given. (h) To find the face when avails, time, and rate are given. PROBLEMS. For what must a note be drav ^ 1 'eld : Amount. Time. Rate of Interest 1. 6724.50 48 days 6% 2. $826.40 60 " 5% 3. $692.24 21 " 8% 4. $5,860 90 " 7% 5. $84 6 mouths 6% 6. $951.28 4 " 9% 7. $10,864.80 5 " 8% 8. $5,712.40 80 days 7% 9. $438.76 50 " 6% 10. $11,-372.12 48 " 5% 318. EXCHANGE. 1. A, in Normal, 111., desired to pay Wm. Dulles, Jr., in New York, $100. It would not be safe to send the money in a letter To send it by express would be expensive. This is what he did. He went to the bank and purchased an order upon another bank in New York to pay the $100 to the order of Wm. Dulles. [See next page.] The Normal bank keeps money on deposit in the New York bank, and 290 NEW ADVANCED ARITHMETIC. S O fca. o __^ o o •p - " X *x ^ ■f. pi:^ '^.j 2 s a; C c; -tj . «*-. c r^ eS ff >— 1 o Sm cc o 03 C c ^^ »^ += ^ Ci^ bC c rS c Cj '•+J c .f^ k—l oJ © F^ ,G 'i r— H X .•*-=" !0.978iV- Note. Change divisor and dividend to twelfths for perfect accuracy. 4. How large a draft, payable in 4 months, exchange ^% premium, interest at 8%, can be bought for §1,250? 5. Find the cost of a 30-day draft for §427.50, exchange at \% discount, iiiterest at 8% . 6. Find tlie cost of a sight draft for $784.90, at \% premium. 7. Exchanged the preceding draft for a 45-day draft, ^% discount, interest 6% . What was the face of the time draft? 8. A commission merchant sold 10,000 bushels of wheat at 87 cents, withdrew his commission of 1^%, and with the remainder bought a 60-day draft, exchange \% premium, interest at 6%. What was its face? 9. A farmer sold 120 acres of land at §65 an acre, re- ceiving from the purchaser his note for 4 months without interest. He at once discounted the note at a bank at 7%, and with the proceeds bought a time draft on New York, due in 4 months, exchange being 1% discount, interest 6%. If he received the face of the draft, did he gain or lose by the deal? How much? When drafts are paid in the country in which they are drawn, the exchange is called Domestic, or Inland. The system is extended also to foreign countries, and constitutes 321. FOREIGN EXCHANGE. 1. Drafts drawn upon banks in foreign countries have their face value expressed in the currency of that country. 294 NEW ADVANCED ARITHMETIC 2. When a system of exchange is established jetween two countries it is necessary to be able to express the value of the currency of each in the currency of the other. Such an expression is called the par of exchange. The legal value of a pound sterling is $4.8665. If exchange varies from this price, it is above or below par. Bills of exchange on England are called Sterling Bills. Their price is quoted at the cost of a pound sterling in United States money. 3. Foreign bills of exchange are sometimes issued in sets of three, called respectively the first, second, and third of exchange. Such bills are transmitted at different times to avoid losses and delays. The one reaching its destination first is paid, and the others become void. 4. The following is the more common form used: Exchange for £250. Chicago, III., June 1, 1892. On demand of this Bill of Exchange pay to the order of Dillon Bros. Two Hundred Fifty Pounds sterling, value re- ceived, and charge the same, as per advice, to BuowN Bros. To Dunn & Co., London. No. 185. 5. Foreign bills may be payable upon presentation, or after a specified time The value of any foreign exchange is now quoted as that of other commodities, at a specified price per pound sterling, per franc, etc. The following is the Chicago quotation for Feb. 15, 18'J2 : For One Pound Sterling $1.!)0 One F"'ranc on France A'i)k One Franc on Belgium .19^ One Franc on Switzerland .19^ One German Mark .24^ One HoUandish Florin .401 APPLICATIONS OF PERCENTAGE. 295 One Austrian Florin .42 One Lira Italian .19^ One Rouble .50 One Finnish Mark .19| One Krona on .Sweden .27 One Krone on Denmark .27 One Krone on Norway .27 PROBLEMS. At the quotations given find the cost of each of the following bills of exchange : 1. For 456 German marks. 3. For 1,283 roubles. 2. For 874 Austrian florins. 4. For 697 lire. 5. For 384 pounds sterling. 6. How large a draft on Paris can be bought for S245.70? 7. . On Brussels for $174.33? 8. On London for $573.30? 9. What is the rate when a draft on Loudon for £150 costs $732? 10. What is the rate when a draft on Paris for 1200 francs costs $232 50? 11. When a draft on Vienna for 800 florins costs $334? 12. When a draft on Helsingfors for 640 marks costs $121.60? 322. EQUATION OF PAYMENTS. The Equation of Payments is the process of finding a time at which several sums due at different times, and not bearing interest, can be paid, without loss to debtor or cred- itor, and without the transfer of money for interest. If the several sums bear interest, all could be paid at any time by chscharging the principal and accrued interest, and no one would lose. 296 NEW ADVANCED ARITHMETIC. The principle upon which the operations are based is very simple. If money is paid before it is due, interest should be allowed upon it ; if not paid until after it is due, it should bear interest from maturity until date of payment. Illuslrative Problem. I owe SoOO due in 4 months; S600 due in seven months; and Si, 000 due in nine months: what is the average term of credit? These sums do not bear interest until after maturity. If any one of them should be paid before its maturity the debtor would lose the use of the mone}' for the remaining time. If it should not be paid until after maturit}^, tlie creditor would lose interest. Assume money to be worth 6% interest, and the several debts to be paid to-day. On the first the debtor would lose the interest for four months, which is 810. On the second he would lose interest for seven months, or S21 ; and on the third, §45. Consequently, if the debts were paid in full to-day, the debtor would lose S76. He may pay them together T>-ithout loss by keeping them long enough after to-day to earn S76. The interest on §2,100 for one day at 6% is 35 cents. §76 H- .35 == 217+. They should be paid 218 days from to-day. This method of finding the time of paj^ment is called The Interest Method. The time sought is called the Equated Time. It may be found also by THE PRODUCT METHOD. The interest on S500 for 4 months equals the interest on $2,000 for 1 month. The interest on 8000 for 7 months equals the interest on 84,200 for 1 month. The interest on 81,000 for 9 months equals the interest on 89,000 for 1 month. The interest on the several amounts equals the interest on §15,200 for 1 month. But the interest on 815,200 for 1 month equals the interest on 82,100 for as many months as §15,200 is tinios §2,100. §15,200 is 7j\ APPLICATIONS OF PERCENTAGE. 297 times $2,100; hence, the equated or average tune is 7 months aud 8 days. PROBLEMS. 1. Find the average time for the payment of $325 due in 30 days; $650 due in 40 days; $500 due in 60 days; and $825 due in 90 days ; all without interest. 2. Find the equated time for the payment of $275 due in 3 months; $580 due in 5 months; $1,020 due in 7 months; $1,260 due in 10 months; no interest being charged on any account. 3. Find the equated time for the payment of the balance of a debt of $1,250 which was to run one year, without interest, one half of it having been paid at the end of 3 months ; one fourth of the remainder at the end of 6 months ; aud one fourth of that remainder at the end of 9 months. 4. When could the following non-interest-bearing debts be paid at one time without loss ? $840 due May 1; $650 due June 15; $900 due July 8; $1,275 due August 1. Note. Assume May 1 as the date of payment. 5. On March 1, 1888, A owed B $2,500 due December 15, without interest. On June 12 he paid $500, and September 10, $1,000. When should he pay $400 to entitle him to keep the remainder until Dec. 15, 1889? 6. What was the equated time for paying the following non-iuterest-bearing bills ? _ March 1, 1889, a bill of $300 for 60 days. April 15, 1889, " " $400 for 30 days. June 10, 1889, " " $583.50 for 4 months. July 1, 1889, " " $962.80 for 3 months. 7. Jan. 1, 1890, A bought a bill of goods amounting to $2,560, on 90 days' time, without interest. On January 16, 298 NEW ADVANCED ARITHMETIC, he paid $850 ; on February 1, $725. If he settled the bill at maturity, liow much should the balance be discounted, money being worth 6 % ? 8. A man owes a bill of $1,200 due in 8 months, without interest. How much must he pay at the end of 4 months, to extend the balance 2 months? 9. Find the average time of payment for the following bills, which do not bear interest. January 1, $400 on 3 months. February 15, $600 on 30 days. May 10, $560 on 4 months June 12, $800 on 60 days. June 20, $250 on 20 days 10. A merchant, on the first cf March, bought goods 1x) the amount of $1,000. He agreed to pay $250 cash, $250 on the 3d of May; $250, July 4; and $250, Sei)tember 15, all without interest. He prefers to pay the whole at one time ; when should it be ? 11. William Jones of Council Bluffs buys goods of Mar- shall Field as follows : 1. May 1, bill of $.300 on 3 mo. credit. 2. May 15, " $800 " 4 mo. '•' 3. June 1, " $500 " 6 mo. " 4. June 9, " $900 for cash. Marshall Field agrees to take Mr. Jones's note for the whole amount, for 30 days, with interest. When should the note be dated? XoTK 1. First find the equated time of payment. Assume for this purpose the earliest date on which a payment falls due. (1)3 mo. credit from May 1. Due .\u 3 " * loC _ 7pt. : 12pt.) 4 : 7 r ^- r-^l\ ^- 4 A. : 9 A. ; 5. 7 pt. : 4 A. : 6. 18 : 24 30 : 9 125 : 600 2.5 : 75 § : 2 n : -10 S : I ) $3 : $8 ) 18 : 24 ) ^ : f y * 4 ft. : 7 ft. I 30 : 9 f 7i : 82 ) 7. 15 : 2h > 329. 1. A Compound Proportion is a proportion in which there is a compound ratio. 2. The following problem involves a compound pro- portion : If a man, working 8 hours a day, build 60 feet of fence in 2 days, how many feet of fence can a man build in 6 days, working 10 hours a day? Note. What is assumed aboui the two fences ? About the men ? Analysis. If the days were of the same length, tlie proportion would read, 2 days : 6 days : : 60 feet : x feet. Since the 2 days are j% as long as the 6 days, the woric done in 2 days will not be to the work done in 6 days as 2 is to 6, but will be jg of that ratio, which is found by multiplying 2 . 6 by 8 : 10. The form : " ' ^ i . GO . r 8 . 10 ) Because there are two elements involved in determining the time that each man worked (the number of days and their length), the relation of the times is expressed by a compound ratio. RULE. IPo solve a probletn invotring a conrpound if^oi>orfion, take for the third term the antecedent of the ratio of mhich the required term is the eonsequentm With one pair of the remaining terms for the first ratio, stMe the proportion aeeording to the con€litions of the probtenu Proceed with the feniaining 2>airs in the same way PROPORTION. 313 until all of the conditions are stated. Find the product of the third term and all of the second terms, and divide it by the product of the first terms, PROBLEMS. 1. If 4 men, in 5 days of 8 hours each, can dig a ditch 120 yards long, 3 feet wide, and 4 feet deep, in how many days of 10 hours each can 12 men dig a ditch 300 yards long, 4 feet wide, and 4^ feet deep? Statement : 120 yd. : 30U yd.] 3 ft. : 4 ft. 4 ft. : 4^ ft. I : : 5 days : x 12 men : 4 men 10 nr. : 8 hr. Why is 12 put first in the fourth ratio? Why 10 in the last? What cancellation can be employed? Why? 2. If S180 be paid for the work of 5 men for 24 days, what should be paid for the work of 17 men for 36 days? 3. If 15 men, in 16 days of 9 hours each, can do a piece of work, how many men will be needed to do the same piece of work in 8 days of 6 hours each? 4. If 15 men, in 16 days of 9 hours each, can do a certain piece of work, in how many days of 6 hours each can 45 men do the same work ? 5. If 45 men can do a piece of work in 8 days of 6 hours each, how many hours a day must 15 men work to do the same work in 16 days? 6. If 50 tons of coal are required to run 4 engines 15 hours a day for 6 days, how many tons will be required to run 7 engines 18 hours a day for 11 days, with 3 times as heavy a load ? 7. If it cost $50 to make a walk 8 feet wide and 60 feet long, what will it cost to build a walk 7| feet wide and 72 feet long ? 514 NEW ADVAXCED ARITHMETIC. 8. If it cost $50 to make a walk 8 feet wide and 60 feet loug, what is the width of a walk that is 72 feet long, and costing $57.50? 9. If a walk that is 72 feet long and 7§ feet wide cost $57.50, how long a walk that is 8 feet wide can be built for $50? 10. If 83 horses eat 933 bushels 3 pecks of oats in 30 days, how many bushels will 125 horses eal: in 45 days? 11. If 44,640 bricks, 4 inches by 8 inches, will pave a court-yard, how many tiles 8 inches wide and 15 inches loug will pave it? Note. Solve these problems by straight-line analysis. Additional problems may be formed from those given, as illustrated in 7 and 8. Remark. No problem can be solved by proportion that cannot be more easily solved by straight-line analysis. 330. PARTNERSHIP. 1. A and B went into business together, A investing $5,000, and B $7,000. They agreed to share gains and losses in proportion to their investments. The net gain was $2400. What was each one'o share? (a) Solve by analysis. What was the whole investment? What part of it did A invest? What part of it did B invest? What was A's share? B's? (6) Solve by proportion. Explain these proportions. (1) $12000 : 85000 : : $2400 : A's share. (2) $12000 : $7000 : : $2400 : B's share. 2. The action of A and B is called "the formation of a partnership." The amount invested is called the Capital. A and B are called Partners. The agreement into which they enter is called the Conditions of Partnership. 3. A Partnership is an association of persons for the prosecution of business on joint account. PROPORTION. 315 PROBLEMS, 1. A. B, and C formed a two-year partnership, agreeing to share gains and losses in proportion to their investments. A put in $5,000 ; B, $6,000 ; C, $7,000. Their net gain was $8,000. Find the share of each by analysis and by proportion. 2. If their net loss had been $2,1 GO, what would have been the loss of each? 3. A, B, and C formed a partnership, C being a silent partner. A invested $6,000; B, $8,500; C, $10,000. By the conditions of partnership, A was to receive a salary of $1,000, and B, $700. The net profits were to be divided in proportion to investments. At the end of the first year the profits, exclusive of all expenses but salaries, were $4,150. What was the share of each? 4. A, B, and C formed a partnership for 3 years. They were to draw equal amounts as salaries, and were to share the net profits equitably. A invested $5,000 at the begin- ning, added $3,000 to it at the end of the first year, and $2,000 more at the end of tlie second. B mvested $6,500 at the beginning, withdrew $2,000 at the end of the first year, and $1,500 at the end of the second. C invested $5,000 at the beginning, and did not change it. At the end of the time their net profits were $4,850. What was the share of each? 5. The investments of three partners are in the ratio of 3, 4, and 5. If they gain $3,600, what is the share of each? 6. A, B, and C owned a mill valued at $18,000. A owned J of it : B, I of it ; and C, the rest. It was insured for f of its value. If it should be destroyed by fire what would each partner lose? 7. A, B, C, and D constructed a street railroad costing $135,000. A furnished J of the capital; B, ^ of it; rnd C and D each furnished ^ of the remainder. The company / 316 NEW ADVANCED ARITHMETIC. sold to E t's of the road for S9,000 ; what part of this amount should each receive ? What part of the stock would each of the original partners own after the sale? 8. A, B, and C tuok a contract for excavating a railroad cut. A furnished 60 men for 25 days; B, 50 men for 48 days; C, 75 men for 56 days. They received ^20,250 for the work. What was the share of each? 9. A, B, and C engaged in business for one year, agreeing to share the profits in proportion to their investments. On January 1, A put in $3,000 ; B, $3,500; and C, 82,500. On March 1, A increased his share $500, B diminished his $500, and C increased his $250. On July 1, A withdrew $1,000, B put in $800, and C increased his $1,000. On October 1, A put in $600, B withdrew $400, and C withdrew $750. They gained $3,000. What was the share of each? 10. A, B, and C engaged in business for 2 years, with a capital of $16,000. A furnished a and B -^ of the capital. C conducted the business for one half the net profits. The gross earnings were $4,800. The expenses were 12i%. What was A's share? B's? 11. What would have been A's share, if at the end of the first year he had transferred to B one thu'd of his interest? INVOLUTION. 317 SECTION X. 331. INVOLUTION. 1. Multiply each of the following numbers by itself: 7, 9, 13, 24, 48, 69. The result in each of these cases is a Square. 2. The Square of a number is the product arising from multiplying that number by itself. The result is also called the Second Power of the number. 3. Learn the squares of all whole numbers from 1 to 25. 4. The expression 8- indicates that 8 is to be used twice as a factor. 8''^ = 8 X 8 = 64. The 2 as here used is called aai Exponent. 5. An Exponent is an expression placed at the right of and above a number to indicate how many times it is to be used as a factor. 6. 12^ = ? 18^=? 1252=? (|)2=r? 21-=? .0152=? 7. 3x3x3=? This result is the cube, or thii'd power, of 3. What is the cube of 4 ? of 5 ? of 8 ? of 10 ? 8. The Cube of a number is the product arising from using that number three times as a factor. 9. Make a definition for the Fourth Power of a number; for the Fifth Power ; for the Eighth Power. 10. 53 = ? 83 =? (3)3=? 73=? 2*=? 3^ = ? a)* = ? (§)' = ? -05" = ? 11. 25'-=? 16^ = ? (^1)3 = ? 32*=? 2.5*=? .02^=? 12. Learn the cubes of all integers from 1 to 10, 13. Recite the following rapidly: 19^, W, \2\ b\ U\ 7\ 222; 43^ 18S 2P, 103, g3^ 232, 132, 83, 172, 15^ 93, 20S 14^, 25*. 318 NEW ADVAXCED ARITHMETIC 333. EVOLUTION. 1. Name one of the two equal factors whose product is 4, 9, 25, 49, 81, 169. 2. Name one of the three equal factors whose product is 8, 27, 64, 125, 343, 729. 3. Name one of the four equal factors whose product is 16, 81, 256, 625. 4. Each of the preceding results is a Roo.t. 5. A Root of a number is one of the equal factors whose product is the number. 6. The Square Root of a number is one of the two equal factors whose product is the number. 7. Define the Cube Root of a number; the Fourth Root; the Sixth Root. 8. What is the square root of 49? of 81 ? of 144? of 324? of 441? of 625.^ 9. Evolution is the process of finding the root of a number. 10 ^4 = 2. ^25 = 5. \/iU = 12. The sign placed before these numbers is called the Radical Sign. It indicates that a root of the number is to be found. 11. ^8 = 2. ^64 = 4. ^81-3. ^32 = •-?• The expression placed above the sign is called the Index. The radical sign, when used alone before a number, indicates that its square root is to be extracted. 12 The index is read as its corresponding ordinal number. ^8 is read, " the third root of 8," or " the cube root of 8." Read ^fe; -^32; Vh VW- 13. ^324, a/^, \/T2], V^, \/r2T, VI^ST, V-0016. 14. Instead of the radical sign, a fractional exponent may be employed. 9= = \/9. 27^ = -^27^ 64^ = ^64^ EVOLUTION. 319 15. To understand how numbers may be separated into equal factors, let us study their composition. 16. Since the square of 1 is 1, and of 9 is 81, it is clear that the square of a one-place number cannot be more than a two-place number. The square of 10, the smallest two-place number, is 100, a three-place number. The square of 99, the largest two-place number, is 9,801, a four- place number. 17. Similarly it may be shown that the square of any in- tegral number contains twice as many places as the number, or one less than twice as many. Conversely, if any integral number be separated into periods of two places each, begin- ning with units, the number of periods thus formed will be the same as the number of places in the square root. The left-hand period may have but one place. 18. What is the square of.l ? of .9 ? of .01 ? of .09 ? The square of any decimal fraction will contain how many times as many decimal places as the fraction? Why? 19. To find the number of places in the square root of a decimal fraction : a. If the fraction has an odd number of places, annex one zero. Why? h. Beginning with tenths' order, separate the expression at the right of the decimal point mto periods of two places each. 20. Study the form of the square of 64. X 4 6 + 6 64 64 6 X X 6 + 4x4 4 6 X ^ + ■2 X 4 X 6 + 4 x 4 It is clear that the square of any two-place number must take this form: The square of the tens + twice the product of the tens and units + the square of the units. Remem- 320 NEW ADVANCED ARITHMETIC. bering this fact, the square root of a four-place number may be found as follows : Illustrative Problem. 21. V'44:89 = ? Since 4489 is a four-place number, its square root must be a two-place number. Siuce the square of tens is hundreds, it must be fouud in 44 hundreds. The largest square in 44 hundreds is 36 hundreds. Its square root is 6 tens. Sub- tracting 36 hundreds, the remainder is 8 hundreds. The entire remainder is 889. If the original number is a square, 889 is the sum of twice the tens of the root by the units, and the square of the units. The tens' term is 6. Twice the tens is 12 tens. Tl:e product of the 12 tens and the units is tens ; hence, it must be in 88 tens. 7 is probably the units' term of the root. 7 X 12 tens is 84 tens. 88 tens — 84 tens =: 4 tens. 4 tens + 9 = 49. 49 is the square of 7; hence, the square root of 4489 is 67. Note. If the number should be a pure or mixed decimal, it may be considered an integer, and the result mny be corrected. FORM. 4489 I 67 36 1^ 1^88 84 49 Note. Explain the following form, showing that it is briefer than the {oTjuer. 4489 |"67 36 127 I 889 889 22. How can this plan be extended to larger numbers? Illustrative Problem. V288369 =-- ? EVOLUTION. 321 By examining the number it is found that its root is a three-place number. We may first deal with 2883. FORM. 2883 [-53 25 103 1^83 309 53 may now be regarded as the tens* teiJQ of the root. To the remainder, 74, the remainder 69 may be annexed, and the work continued as before. FORM. 288369 I 537 25 103 |~^^ 309 1067 I 7469 333. RULE FOR THE EXTRACTION OF THE SQUARE ROOT OF A NUMBER. 1. Beginning at the decimal point, group the figures into periods of ttvo orders each. 2. Find the largest square in the left-hand period and place its root at the right, as the first term of the root. 3. Subtract the square from the left-hand period, and to the remainder annex the next period. 4. Double the root already found, and using it as a trial divisor find how many times it is contained in the new dividend exclusive of its right-hand term. Place the quotient as the second terin of the root, and also annex it to the trial divisor. Multiply the complete divisor by the second term of the root, subtract the product from the partial dividend, and proceed as before. 322 NEW ADVANCED ARITHMETIC. 5. If the trial dividend tvill not contain the trial dit'isorf annex a sero to the root and to the trial divisor, annex a new period to the trial dividend, and proceed as before. 6. If there is a remainder after the last operation, to continue the tvork reduce the remainder to hundredths, ten-thousandths, etc., continuing the tvork as before. 7. To extract the square root of a common fraction where the terms are squares, extract tJie square root of each term. If only the denominator is a square, extract the approximate root of the numerator and divide it by the root of the denominator. If the denominator is not a square, change the fraction to a decimal and extract its approximate root. 334. 1. Since the number of units in the area of a square is the square of the number of units in one side, the square root of the number of units of area is the num- ber of units in a side ; hence, if the area of a square be given, a side may be f Dund by applying the preceding rule. The process of applying the rule may also be illustrated by diagrams. 2. What is one side of a square whose area is 1225 square feet? First, confining our attention to the left-hand period, 12, it appears that this square contains at least 900 square feet. Each side of such a square is 30 feet. E F M A 30 ft. J 900 sq. ft. i c H EVOLUTION. 323 There yet remain 325 square feet with which to make additions to this figure. They must be made so as to i-etain the form of a square ; hence, the length and width must be equally increased. If a rectangle a foot wide were to be added to the sides A B and B C, 2 X 30 square feet (twice the tens) would be needed. The remainder is large enough to make these additions 5 feet wide, and leave 25 square feet with which to fill out the space F M G B. Note. Use the figure to illustrate the extraction of the square root of 541696. PROBLEMS. Find the square root of each of the foiiowing numbers: 1. 4489. 8. 143641. 15. 40640625. 2. 7921. 9 214369. 16. 9036036. 3. 9216. 10. 450241. 17. 23261329. 4. 15625. 11. 466489. 18. .6889. 5. 42436. 12. 519841. 19. .355216. 6. 82369. 13. 567009. 20. 76.3876. 7. 93025. 14. 622521. 21. 49 64* Find the root to hundredths. 22. if. 26. 28. 30. 3.6. 23. \n- 27. 6. 31. 4.9. 24. l%%- 28. 2. 32. 8.1. 25. m- 29. .7. 33. 12.1. See Art. 333, 7. 34. A|. 38 §• 42. 121. 35. fs. 39 7 43. 81. 36. I. 40. H- 44. ^. 37. it\. 41. n- 45. 161. 335. 1. Find the length of one side of a square field containing 17 A. 89 sq. rd. 324 NEW ADVANCED ARITHMETIC. 2. The entire surface of a cubical block contains 223^5 square feet. "Wliat is the length of one edge? 3. A square contains 900 square inches. What are the width and length of an equivalent rectangle whose width is to its length as 1 to 4 ? 4. Supply a mean proportional in each of the following proportions : a. I2ix':ix:4:8. c. 4.8:x:tx. 432 6. 192: a;:: a;: 27. KoTE. A mean proportional is a number that is the second and third term of a proportion. 5. How many rods of fence will enclose a square field containing 10 acres? 60 A body of 7,921 soldiers is arranged so that there are as many in rank as there are in file. How many are there in each ? 7. A rectangular surface whose width is | ot its length contains 1^470 square feet Find its width and length 336 THE RIGHT TRIANGLE 1. A triangle, one of whose angles is a right angle, is called a right triangle. The side opposite the right angle is the hypote- nuse. The other sides are the base and alti- tude The sides form- ing the right angle are called the arms. 2. Draw a right tri- angle, ABC, on a sheet of pasteboard. EVOLUTION. 325 Draw the squares on the three sides and subdivide as shown in this figure, by extending the sides of the largest square through the smaller squares, and drawing a line at right angles to the longer extension. With a sharp knife cut out the five pieces and place them in the positions 1', 2', 3', etc. We thus see that the square of the hypotenuse is equal to the sum of the squares of the arms. PROBLEMS. , " Base. Altitude. Hypotenuse. 1. 8 inches. ? 10 inches. 2. 20 feet. 15 feet. ? 3. 224 yards. ? 260 yards. 4. ? 272 mUes. 353 miles. 5. 192 rods. 144 rods. ? 6. The top of a ladder that is 30 feet long rests against n telegraph pole 24 feet from the ground ; how far is the foot of the ladder from the foot of the pole ? >^ 7. A and B start from the same point at the same time. ^ A travels north and B east, the former traveling at the rate of four miles an hour and the latter three. How many feet apart are they in 15 minutes? 8. A rope is attached to the top of a 96-foot pole. It touches the ground 28 feet from the foot of the pole. What is its length ? 9. What is the diagonal of a rectangle whose dimensions are 6 yards and 8 yards? Note. Tlie diagonal of a rectangle is the straight line joining opposite vertices. 10. A 13-foot ladder rests with its top against a window sill 12 feet from the ground. How many feet from the wall to the foot of the ladder? 11. Against the top of a pole 15 feet high are braced in opposite [directions a 17-foot ladder and a 25-foot ladder. How far apart are the feet of the ladders. Make diagram. 22A 326 NEW ADVANCED ARITHMETIC. 12. A 25-foot ladder stands erect against the wall of a building. The foot is pulled out until the top is lowered one foot. How far from foot of ladder to wall ? 13. If the foot of the ladder is pulled out 8 feet farther, how much more is the top lowered? 14. What is the area of a rectangle whose length and diagonal are respectively 15 and 17 rods? 15. What is the length of the longest wire that can be stretched straight in a room 20' X 15' X 12'? 16. What is the diagonal of an inch cube? 17. What is the shortest line that can be traced on the surface of a cube, joining the extremities of a diagonal? 18. What is the diameter of a circle that will just enclose four silver dollars arranged in the form of a square ? 19. How much is saved by going diagonally across a sec- tion instead of along the boundary? 20. How many acres in a square field whose diagonal exceeds its side by 16.568 rods? Note. The valiie.s of V2 aud V'3 should be memorizeiL 21. A rectangular field three times as long as wide con- tains 87 acres. What is its length? 22. A triangular field, A B C, is 60 rods long and 20 rods wide. At what distance from B may a fence be built divid- ing the field into two equal areas? B EVOLUTION. 327 Queries. A C is what part of A B ■? X O is what part of X B' What is the area of rectangle O X B T 1 What is the area of tlie square O X K S ? What is the length of O X ? of X B ^ 23* At what point shall a triangular board, 12 feet long and 12 inches wide, be sawn into two equivalent pieces? 24. At what point, if the 12-foot board be a trapezoid, 15 inches wide at one end, 3 inches at the other ? 25. Show that 128 stakes a foot apart can be driven on a ten-foot square. 26. The sides of an equilateral triangle are six feet each. What is the altitude ? 27. What is the area of an equilateral triangle, each side of which is 4 feet ? 28. Draw the three altitudes of an equilateral triangle from the vertices to the opposite sides. They meet at a common point. Join this point with the vertices forming three equal triangles. Show that the altitude of each is i the altitude of the equilateral triangle. 29. What is the diameter of the circle that will just enclose three silver dollars arranged in the form of a triangle? 30. What is the diameter of a circle whose area is 314.16 square feet? 31. With what length of rope shall a horse be tethered to graze over one fourth of an acre? 32. A road two rods wide about a square field contains one acre. What is the area of the field ? 33. A road two rods wide about a circular field contains one acre. AVhat is the area of the circle? 34. About a circular field 80 rods in diameter is a road of uniform width containing six acres. Width of the road? 35. How much must the diameter of a 36-inch grindstone be reduced, to reduce the weight one fourth? (No allowance for the opening.) * A riglit triangle. Use figure, page 326. A C is ^^ of A B. Assume X dividing point. X O is j^j of X B. Divide X B into 12 parts. Area of \'Rn'2 Of XBTO? Of each of the 12 squares? Length of X ? 328 NEW ADVANCED ARITHMETIC. 337. CUBE ROOTc 1. What is 5 X 5 X 5 ? 9x9x9? 23 X 23 X 23 ? 86 X 86 X 86 ? 2 X § X § ? i X i X i ? .8 X .8 X .8 ? .04 X .04 X .04? The product arising from each of these indicated multiplications is a Cube. 2. The cube of a number is the product arising from using that number three times as a factor. 3. What is the cube of 79 ? 93? 207? 300? |? -j'^? U? 1.6? .24? 10^=? f5=? (})' = ? .06^ = ? 4. What is one of the three equal factors whose product is 8? 27? 125? 729? ^y^? ^? .216? .000512? 3a? 5. The Cube Root of a number is one of the three equal factors whose product is the number. 6. 4^8='} 'V^27=? \^Wi='^ \/:729=? 7. Define Exponent, Radical Sign, Index, Fractional Exponent. 338. EXTRACTION OF THE CUBE ROOT OF NUMBERS. 1. The method of extracting the cube root of any number above 1000 will be ascertained by studying the form of the cube of a two-place numbero 2. 46^ = 46 X 46 X 46 = (4 ^ + 6) (4 ^ + 6) (4 ^ + 6). 1 + 2 « (4 0== 4^+6 4^+6 4 < X 6 + 62 + At X Q (4 0^+2 X 4 f X 6 + 6- 4< + 6 (4 0« X X (4 0^ (4 0' X 6 + 2 X 6 + 1 X 4 « X 6- + 6^ X 4 « X 62 C4 0' + 3 X (4 0*^ X 6 + 3 X 4 « X 6- + 6^ EVOLUTION. 329 3. Stating the above result in words, we have the following : The cube of a t-wo-place number consists of (1) the cube of the tens, (2) plus three times the square of the tens by the units, (3) plus three times the tens by the square of the units, (4) plus the cube of the units. Note. Verify the above statement aud fix it iu the memory. 4. Illustrative Ex-ample., The cube of 35 = SO** or 27000' + 3 X 30- X 5, or 13500 + 3 X 30 X 5", or 2250, + 5% or 125. Give the several parts of the cube of 24 ; of 32 ; of 41 ; of 66; of 87; of 93. 5. The cube of 1 is 1. The cube of 9 is 729. Therefore, the cube of a one-place number is not more than a three- place number. 6. The cube of 10 is 1000. The cube of 99 is 970299. Therefore, the cube of a two-place number cannot be less- than a four-place, nor more than a six-place, number. 7. Similarly, it may be shown that the cube of any in- tegral number contains three times as many orders as the number, or three times as many less one or two. 8. The number of orders in the root of an integer may be ascertained by beginning at the decimal point, and, so far as possible, grouping the figures into periods of three orders each. The left-hand period may contain only one or two figures. 339. Illustrative Example. ^^103823 =? Since this is a six-place number, the cube root of the largest cube in it is a two-place number. The root, there- fore, consists of some number of tens plus some number of units. I first withdraw from the number the largest cube in 103 thousands. This is 64 thousands. Its cube root is 4 tens. 103823 — 64000 = 39823, We have seen that the second part of the cube is "three times the square of the tens by 330 NEW ADVANCED ARITHMETIC. the units." Three times the square of the tens is 48 hun- dreds. If this were multiplied by the units of the root, the product would be hundreds ; hence, would not be of a lower denomination than 398 hundreds. The 398 hundreds, then, may be used as a trial dividend with which to find the units' term of the root. The trial dividend seems large enough to indicate that the units' term is 8, but it is to be remembered that there are two more parts to the cube. Trying 7 as the units' term, "three times the square of the tens by the units" is 336 hundreds. "Three times the tens by the square of the units" is 588 tens. "The cube of the units" is 343. The sum of these numbers is 39823; hence, 103823 is a cube, and its root is 47. 3 X (4 fi) = 48 hundreds 103823 1 47 _64 330 = 7 X 48 hundreds. 622 588 = 3 X 4 i X 72 343 343 = 73 Since "three times the square of the tens" is to be mul- tiplied by the units, and since " three times the tens by the square of the units " may be found by multiplying three times the tens by the units, and that product by the units., and since " the cube of the units " equals the square of the Mnits by the units, it is clear that the units' term is a factor ©f each of these parts of the cube. The three parts may be reduced to one by arranging them as follows: (-^ f^ + 3fn + w'^) n. Here 3 t'^ is the trial divisor, and (3 i^ + 3 tu + u'-) is the complete divisor. EVOLUTION. 331 FORM. 103823 {_47 G4 3 X /2 = 4800 3 < u = 840 u- = 49 5689 39823 Note. 3 f u is one order lower tlian 3 /-. Why 1 u- is one order lower than 3i u. Why i Why are the two zeros placed at the right oi 48 ? 39823 PROB LEMS. 6. 7. 8. 9. 10. 1. ^^79507 = ? \ 6585U3 = '^ 804357 = ? 2. '^ 157464 = ? V 3. -^314432 = ? 'V^884736 = -^970299 = ? 4. -v/357911 = ? y 5. V 551368 = ? ■V'912673 = V Note. Pupils should work on similar problems until they can be solved easily at the rate of one a minute. 340. DECIMAL FRACTIONS. 1. The cube of .1 is .001. The cube of .9 is .729. The cube of .01 is .000001. The cube of .09 is .000729. The cube of tenths is thousandths ; of hundredths is millionths ; of thousandths is billionths, etc. 2. To find the number of orders in the cube root of a decimal fraction, begin at the decimal point and group the figures into periods of three orders each. If the right-hand jteriod is not full, annex zeros. (Why?) PROBLEMS. Proceed as with whole numbers. 1. -^.032768 = ? 3. '^. 000195112 = ? 2. -v^.07950T = ? 4. v^. 000000493039 = ? 332 NEW ADVANCED ARITHMETIC. 341. COMMON FRACTIONS. 1. How is the cube of a common fraction found? How, then, may the cube root of a common fraction be found? V27 — • V512 — • *'T"5 74g4 — . Make a rule for the extraction of the cube root of a com- mon fraction. 2. If the denominator is not a cube, change the common fraction to a decimal fraction, and proceed as in Art. 340. 342. '^I0l847563 = ? OPERATION. 4800 720 36 101847563 64 3 X42 = 3X4x6 = 6X6 = 37847 5556 33336 3 X 46-^ = 3 X 40 X 7 = 7X7 = 634S00 9660 49 4511563 644509 4511563 467 First consider the first two periods only, and proceed as in Art. 339. Having found the first two terms of the root, consider them as the tens' term, and proceed as before. 343. RULE FOR FINDING THE CUBE ROOT OF A NUMBER. 1. Point off the number into periods of three jjJaces ■each, beginning at the decitnnl point and counting to the left for integers and to the right for decimals. 2. Find the largest cube in the left-hand period and jtlace its root at the right. Subtract the cube from the left-hand period and annex the second period to the ■vemainder. EVOLUTION. 333 3. Find three times the sqtiare of the first term of the root, annex two meros, and place it at the left as a trial difisor. Compare it tvith the second dividend and place the Quotient as the second term of the root. 4. Find three times the product of the first and second terms of the root, anncjc one zero, and write it under the trial dirisor. Sqtiare the second term of the root and write the result tinder tlie preceding prodiict» Find the sum of these three results and jniiltiply it by the second term of the root. Subtract the product thus found from the partial dividend and to the remainder annex the next period. 5. Find three times the square of the root already found, annex tujo zeros, and tvrite it at the left as a trial divisor. Find the third term of the root and complete the divisor as before. Notes. (1) If the number is not a cube, the work may be continued to any extent by reducing the last remainder to thousandths, millionths, etc., by annexing periods of three zeros each. (2) If the denominator of a common fraction is not a cube, both terms may be multiplied by some number that will make the denominator a cube. Why is it more important to have the denominator a cube than the numerator ? (3) If the partial dividend at any time will not contain the trial divisor, write a zero in the root, annex two zeros to the preceding divisor as a new divisor, and proceed as before. Show the reason for annexing these zeros by going through the work with zero as a term of the root. (4) The rules for squaring numbers from 25 to 100 will be found con- venient in forming the trial divisor for the third term of the root. (5) Mixed numbers should usually be changed to improper fractions or to mixed decimals. 344. The following rules will be found convenient : RULES FOR SQUARING NUMBERS FROM 25 TO 100. 1. To square mimbers frotn 25 to 50, (a) Subtract 25 from the number. (b) Subtract the remainder from 25. (c) Square the last result, and (d) Add it to the first result considered as hundreds. 334 NEW ADVANCED ARITHMETIC. Illustrative Example. 42- = ? (a) 17. (h) 8. (c) 64. (d) 1764. 2. To SQ-uarc numbers from 50 to 75. (a) Subtract 50 from the number, (b) Add the result to 25. (c) Square the first result, and (d) Add it to the second result considered as hundreds. Illustrative Example. 69^ = ? (a) 19. (&) 44. (c) 361. (c?) 4761. • 3. To square numbers from 73 to lOO. (a) Subtract the number from lOO. (b) Subtract the first result from the number. (c) Square the first result, and (d) Add it to the second result considered as hundreds. Illustrative Example. 88" = ? (a) 12. (6) 76. (c) 144. {d) 7744. Note. Inquisitive students -n-ill desire to find the reasons for these rules. The first is an application of this formula: (a — b)- — a- — Inh -\- ^A a here represents 50, and b the difference between 50 and the given number. The second is an application oi [a + h]" = a- + 2ab -\- b-; the letters being used as in the first. The third employs {a — b)-; a representing 100, and b the difference between 100 and the given number. 345. PROBLEMS. 1. -^86004573 = ? What is the remainder? 2. v^.0685 = ? Carry root to thousandths. 3. ^2 = ? Carry root to hundredths. 4_ /y^:5 = ? 6. 15.625^ = ? 5. '^^3698.400375"= ? 7. \/Ui = ? EVOLUTION. 335 8. ^2Uji = 'i 16. -^1 = ? 9. '^^U = ? 17. ^/^ = ? 10. ^27189441343 = ? 18. \^^7 = ? 11. 34328125^ = ? 19. '^QA = ? 12. 194104539» = ? 20. V^0l25 = ? 13. 223648543^ = ? 21. '^^ = ? 14. 736314327' = ? 22. 9^ = ? 15. V1003003001 = ? 23. Vf = ? Note. Teachers should dictate many problems uutil pupils can work rapidly and accurately. 24. A cubical block contains 96 feet of lumber. How many inches long is each edge ? 25. A cubical cistern holding 3,600 gallons is how deep? 26. Find the dimensions of a cubical bin whose capacity is 2,000 bushels. 346. 1. The method of extracting the cube root of a number may be illustrated bv tlie use of blocks. 2. V50653 = ? A cubical figure contains 50,6.53 cubic inches. What is the length of one edge ? 3. Tiie number of inches in the edge is a two-place number. The largest cube in .50,000 is 27,000. 27,000 inch-cubes will foim a figure each of whose edges is 30 inches. This 27,000 is " the cube of the tens," the first part of the expansion shown in Art. 338, 2. 23,653 inch-cubes remain with which to enlarge the figure. The size must be increased in such a way as to retain the form of a cube. Since the length, breadth, and thickness are equ;\l, the additions must be made to three adjacent faces, and must l)e equal. If a layer of cubes were placed upon one face, 30 X 30 = 900 cubes would be required. Since three such additions are to be made, 2,700 cubes would be needed to make the additions one inch thick on each face. We thus find the illustration of " three times the square of the tens as a trial divisor." It is called a " trial divisor " be- cause we wish to ascertain how many such additions may be made to each face with the remaining blocks, and yet leave enough to fill out the figura 336 NEW ADVANCED ARITHMETIC. The iudications are that 7 such layers may be added to each face. This would use up 3 X 30- X 7 olocks. The expression is " three times the square of the teus by the uuits," or the second part of the expausion siiown iu Art 338, 2. 3 X 30- X 7 n= 18,900. 23,653 — 18,900 = 4,753. the number of inch- cubes remaining. The figure is now 37 inches wide, 37 inches long, and 37 inches high, but is not a cube, since additions are still to be made along three edges and on one corner. Each of these additions to the edges must be 7 inches by 7 inches and 30 inches high. These will require 3 X 30 X T''^ = 4,410 blocks. This expression is " three times the teus by the square of tiie units," or the third part of the expausion showu iu Art. 333, 2. 4,753 — 4,410 = 343, the remaining blocks. An unfilled corner, 7 inches by 7 inches by 7 inches, remains. Since 343 blocks are needed to make the figure a cube, it is clear that 50,653 is the cube of 37. The 343 is " the cube of the uuits," or the last part of the expansion shown in Art 338, 2. Fig. 1. A Cube. FiG. 2. Additions for Cube. Ililllilllilil Fig. 3. Additions made to Cube. Via. 5. Small Cube for Comer. EVOLUTION. 337 EXPLANATION OF THE FIGURES. Fig. 1 represents the cube formed from 27,000 blocks. Fig. 2 represents the three additions made to the faces of Fig. 1. Fig. 3 represents the figure resulting from the three additions to the faces of Fig. 1 . Fig. 4 represents the three additions made to the edges, and Fig. 5 the final addition made to the corner to complete the cube. Apply this method of illustration to Problems 1-6, Art. 339. 347. MISCELLANEOUS PROBLEMS. 1. "What is the area of a piece of ground arranged in the form of a triangle, the length of one side being 124: rods, and the distance to the vertex of the opposite angle being 86 rods? 2. How many feet of lumber are there in a two-inch plank in the form of an isosceles triangle, 18 feet long and 14 inches wide at the base? 3. Find the area in acres of the irregular field A B C D E. Distance : 'AD = 4.0 rd. EG=4:6 rd. AB =4:U rd DF=S7 rd. BII=SS^ rd DC = 52'rd. 4. A board in the form of a rhombus is 19 inches on each side and 10 inches high. Draw the figure. "What is its area? Compare it with a square that is 19 inches on each side. Note. A Ehombus is an oblique-angled equilateral parallelogram. 338 NEW ADVANCED ARITHMETIC. 5. A field in the form of a trapezoid contains 23^ acres. One of its parallel sides is 95 rods and the other 65 rods long. What is its width? XoTE 1. A Trapezoid is a quadrilateral only two of whose sides are parallel. Its area is equal to the prod- _B uct of its height aud half the sum of its parallel sides. Note 2. Drawing G E parallel to B C makes G B C E a parallelogram. AGE = EE1); hence A B C D is equivalent to GBCE, and EC = \(^AB+ CD). 6. What is the area of a circular pond whose diameter is 15 rods? 7. What is the circumference of a circus ring whose area is 872f square yards? 8. What is the area of a field in the form of a trapezoid, the parallel sides being 42 rods and 88 rods respectively, and the distance between these sides being 36 rd. 3 yd. ? 9. A cubical box contains 79,507 cubic inches. What is the length of each edge? 10. A cubical tank has a capacity of 7 60 J gallons. Find the length of one side. (Approximate.) 11. A cubical cistern is 9 feet deep. What will it cost to construct it at 75 cents a barrel (31^ gallons) ? 12. A corn-bin whose width equals its height is 4 times as long as it is high. If it will hold 3,200 bushels of shelled corn, what is its length? 13. A building is 36 feet wide. If the attic is 9 feet high, what is the length of the rafters, allowing for a projec- tion of 18 inches? 14. If the area of an equilateral triangle is 12 square feet, what is the area of a similar triangle each of whose sides is twice as long? Four tunes as long? 7^ times as long? EVOLUTION. 339 Note. Similar plane figures are those having the same shupe ; e. g., a square is similar to a square, a circle to a circle. Two similar surfaces are to each other as the squares of like lines. 15. A circle whose diameter is 4 feet is what part of a circle whose radius is 6 feet? 8 feet? 9 feet? \Q\ feet? 16. A circle whose diameter is 6 feet is how many times a circle whose diameter is 6 inches? 9 inches? li feet? 4 feet? 17. A cube 2 feet high is how many times one that is 1 foot high ? Note. Similar solids are those having the same shape ; e. g., a cube is similar to a cube, a sphere to a sphere. Similar solids are to each other as the cubes of like lines. 18. A sphere whose diameter is 1 inch is what part of a sphere whose diameter is 2 inches? 4 inches? 7 inches? 1 foot? 19. How many 2-inch spheres contain as much volume as a 4-inch sphere ? an 8-inch sphere ? a foot sphere ? 20. Compare the volumes of earth and moon, the diameter of the former being about 8,000 miles, and of the latter about 4,000 miles. Compare their surfaces. 21 o Compare the volumes of sun and earth, the diameter of the former being about 880,000 miles. Compare their surfaces. 22. If a cannon-ball weighs 36 pounds, what will one weigh whose diameter is 3 times as great? 23. Height of cylinder 6 feet ; diameter of base 21 feet. Find convex surface. Find entire surface. Find volume. 24. The volume of a cylinder is 72 cubic feet. The diameter is 4 feet. What is the convex surface? Entu'e surface? 25. The volume of a cylinder is 196.35 cubic feet. Its height is 10 feet. What is the area of the base of a similar cylinder whose volume is 27 times as great? 340 NEW ADVANCED ARITHMETIC. Note. Tlie surface of a sphere is 3.1416 tiroes the square of its diameter. 26. What is the surface of a sphere whose diameter is 4 inches? 2^ feet? 1 yard? 27. What is the area of the earth's surface, counting the diameter 7,912 miles? 28. The surface of a sphere that is 1 inch in diameter is what part of the surface of a sphere whose diameter is 2 inches? 4 inches? 6 J inches? 29. What is the diameter of a sphere whose surface is 31,416 square miles? 201.0624 square feet? 337 square inches ? 30. Compare the surfaces of two spheres whose diameters are as 3 to 7. As 2 to 5. As 2^ to 3^. 31. The volume of a sphere is 3.1416 times J of the cube of the diameter. What is the volume of a sphere whose diameter is 6 inches? 32. What is the approximate interior diameter of a sphere that will hold a gallon? 33. What is the weight of a sphere of gold 2 inches in diameter ? 34. What is the weight of a sphere of stone whose diame- ter is 30 inches, the stone weighing three times as much as water? 35. Steel weighs about 7.84 times as much as water. What is the weight of a hollow steel sphere whose interior diameter is 12 inches, the shell being ^ of an inch thick? 36. The State in which you live is what part of the sur- face of the earth ? 37. What is the diameter of a circle whose area is equal to the area of tlie State in which you live ? 38. What is the surface of a sphere whose diameter is one foot? EVOLUTION. 341 39. If the above sphere represents the world, for how many square miles does a square inch of its sui'f ace stand ? What, then, is the scale? 40. What is the diameter of a circle that will represent the area of the State in which you live, on the above globe? 41. What is the weight of a cast-iron street-roller, 8 feet long and 5 feet in diameter, the shell being 2 inches thick, cast-iron being 7.2 times as heavy as water? 348. THE CONE. 1. A Cone is a solid formed by the revolution of a right triangle upon its base or perpendicular as an axis. 2. A cone is one third of a cylinder having the same base and altitude (height). 3. The convex surface of a cone is equal to one half the product of the circumference of the base and the slant height. Why ? PROBLEMS. 1. Height, 20 inches; radius, 15 inches. Find convex surface and volume. 2. Slant height, 10 inches; height, 8 inches. Find volume and entire surface. 3. Cii'cumference of base, 75.3984 feet; slant height, 20 feet. Find volume and entire surface. 4. The radius of a cone is 7 inches and its altitude 15 inches. What is the convex surface of a similar cone whose radius is 10 inches? What is its volume? 5. What is the weight of a steel paper-weight in the form of a cone, the diameter of the base being 3 inches and the height 4 inches, steel being 7.83 times as heavy as water ? 23A 342 A'iiTr ADVANCED APdTHMETIC. 3i9. 1. The Frustum of a cone is that portion of the cone included between the base and a plane passing through the cone parallel to the base. 2. The convex surface of a frustum of a cone is a Trapezoid. How may its area be found ? 3. The frustum of a cone is equivalent to three cones having the altitude of the frustum, and whose bases are the upper base of the frustum, the lower base, and a m.ean pro- portional between them. 4. To find the volume of a frustmn of a cone, find the sum of til e upi>er base, the lotrer base, and the square root of their product, and mnJtiply the result by one third of the altitude {distance betiveen the bases). PROBLEM. Find convex surface of a frustum of a cone, the radius of whose upper base is 10 inches, of the lower base 15 inches, and whose altitude is 21 inches. Find the volume. 350. PRISMS AND PYRAMIDS. 1. A Right Prism is a solid two of whose faces are equal and parallel polygons, and the rest of whose faces are rectan- gles. 2. The volume of a right prism is found in tlie snmo way as the volume of a right parallelopiped. 3. A pyramid is a solid lliat is bounded by a poly- gon called the base, and three or more triangles that meet at a point. EVOLUTION. 343 4. The volume and convex surface of a pyramid are found in the same way as the volume and convex surface of the cone. 5. Define a Frustum of a Pyramid. Art. 349, 1. Tell how to find the volume and convex surface of a frus- tum of a pyramid. Art. 349, 2, 3. Describe the above figures. Find a frustum of a pyramid. PROBLEMS. 1. What is the volume of a prism whose base is 11 square feet and height 10 feet? Of a pyramid of the same dimen- sions? 2. What is the volume of a prism whose altitude is 6 feet and whose base is a triangle each of whose sides is 2 feet? of a pyramid of the same dimensions ? 3. What are the volume and convex surface of a pyramid whose altitude is b\ feet, and whose base is a square the diagonal of which is 8 feet? 4. Find the convex surface and volume of a frustum of a square pj-ramid, a side of the lower base being 3^ feet, of the upper base 2^ feet, and whose height is 4^ feet. 5. Find the volume of a pyramid whose base is an equi- lateral triangle, each side measuring 6 feet, and whose alti- tude is 24 feet. 6. Compare the volume of this pyramid with that of a pyramid of the same height, but whose base is 6 feet square. 344 NEW ADVANCED ARITHMETIC. 351. GENERAL REVIEWS. I. 1. Define Multiplicatiou, Denominator, Decimal Fraction, Interest, Sphere. 2. What common fractions can be changed to pure deci- mals? Why? 3. 8243 ^ ? ^^80964. 73807 = ? 1.18 3.64 4. Smiplify ;j5-2 X ^^- 5. Derive a rule for dividing by a fraction. 6. Find the interest on $824.60, at 7%, from June 12, 1887, to Sept. 5, 18'JO. 7. How many gallons of water will a cylindrical cistern hold whose diameter is '1\ feet and depth 9i feet? 8. Change § of a square mile to integers of lower denom- inations. 9. What is the cost of the lumber and posts to fence the N. W. \ of the S. \ of the N. E. ^ of a section with a four- board fence, posts 8 feet apart and costing 23 cents each, fencing at $18.50 per M.? 10. What is the cost of a 40-day draft for $580, exchange being \% premium, and interest at 6% ? 11. What is the difference between seven hundred and two thousandths and seven hundred two thousandths ? II. 1. Divide 3,744 into three parts in the proportion of h h h 2. Define Ratio, Proportion, Commission, 2sotation, Division. GENERAL REVIEWS. 345 3. How many square feet of sheet-iron \ of an inch thick can be made from a cylindrical shaft 20 feet lono- and 4 inches in diameter? 4. If 52 men can dig a canal 355 feet long, GO feet wide, and 8 feet deep in 15 days, how long will a canal be that is 45 feet wide and 10 feet deep which 45 men can dig in 25 days ? 5. What is the selling price of an article bought at a dis- count of 33^%, and sold at an advance of 12i%, yielding a gain of $8? 6. How many gallons of liquid will a hollow sphere hold whose inner diameter is 22 inches? 7. A 45-foot ladder placed between two poles reaches one of them 24 feet from the ground, and the other 28 feet. How far apart are they? 8. What is the difference of time between two places whose difference of longitude is 46° 18' 46"? 9. If -/j of an acre of land cost S33|, what will 36| acres cost? 10. A, B, C, and D together own a tract of land 2 miles square. A owns |- as much as B ; B § as much as C ; C | as much as D. How many acres has each? III. 1. Define Antecedent, Consequent, Mean Proportional, Complex Fraction, Compound Interest. 2. What is the average time for the payment of $600 due in 3 months, $480 due in 5 months, $390 due in 8 months, and $850 due in 14 months? 3. Find the 1. c. m. of 174, 485, 14,065. 4. Give the rule for " pointing" the quotient in division of decimals, and give the explanation of it. 5. Change 18 mi. 124 rd. 4 yd. to feet, and use three forms of analysis in the reductions. 346 NEW ADVANCED ARITHMETIC. 6. A public square is surrounded by a walk 2 rods wide. The area of the walk is an acre. What is the area of the square? Make a figure. 7. For what amount shall a 90-day note be made that the proceeds shall be $358.60, interest at 7% ? (Bank Discount.) 8. State the three general problems of percentage. 9. -^j; is what per cent of ^5? Give an analysis. 10. A, B, and C form a partnership, and make a gross gain of 816,440. A invests 85,000 for 12 months ; B, 89,000 for 16 months; C, 87,100 for 6 months. The total expenses were 84,110, which they agreed to share equally. What was each partner's share of the net gain? IV. 1. Define Subtraction, Minuend, Subtrahend, Remainder, Partition. 2. Divide 83,600 by 37^, and give the analysis. Multiply 564 by 83 J, and give the analysis. 3. Give tests of divisibility by 3, 4, 8, 9, 11. 4. If a school-room is 15 feet high, how many square feet of floor must it have to furnish 60 persons 300 cubic feet of air? If the length is to the breadth as 4 to 3, what will each be? 5. A cubic bin with a square bottom holds 164,025 cubic inches. Depth is to width as 9 to 5. What is the depth? The width ? 6. 40 men agree to do a piece of work in 50 days, but after working 9 hours a day for 30 days only half the work is completed. How many additional men must be employed to finish the work on time by putting in 10 hours a day? 7. What is the present worth of a non-interest-bearing debt of 8728.40, due in 3 years, 7 mouths, and 19 days, money being worth 7 % ? 8. V860473. 02986 = ? GENERAL REVIEWS. 347 9. Find the value of the foUowhig lumber at §21 a thousand : 4 6x8 sills, 16 feet long. 26 2 X 8 joists, 18 " " 30 2 X 4 studs, 22 " " 18 2 X 6 rafters, 20 " " 10. Find the premiums on the following policies of insurance : 62,100, at l\%. 82,800, at 2\% . 63,150, at |%. 1. "When do you conclude that a number is prime? Why? 2. Give the demonstration of the test of divisibility by 9. 3. The proceeds of a note for 8265.50, discounted on June 12, 1891, at 7 %, were 8263. When was the note due? 4. The proceeds of a 90-day note for 8480 were 8470.08. What was the rate of discount? (With grace.) 5. Bought goods for 8729. What must they be marked that the merchant may fall 10%, lose 10% on bad debts, and still gain 10% ? 6. A circular piece of laud 16 feet in diameter is to be divided into 3 equal parts, the inner part being a circle, and the second and third parts being circular strips. What is the diameter of the inner circle ? What is the width of each of the circular strips? Note. What is the area of the whole circle ? What is the diameter of the inner circle ? 7. Give two ways of changing a common fraction to a decimal. Change 1%-^ to a decimal, and explain each stop. 8. A can do a piece of work in 2| days ; B, in 3* days ; and C, in 4i days. How long would it take them to com- plete the job working together? If ZQ> is paid for the whole work, what is the share of each? 348 ^'EW ADVANCED ARITHMETIC. 9. A note of $1,200, dated April 1, 1886, and bearing in- terest at 8% , bad the following indorsements : Sept. 12, 1886, $130. March 20, 1887, $240. Aug. 24, 1888, $325. What was due April 1, 1890? 10. When it is noon in Boston, what time is it at San Francisco ? 11. At what rate must 4% bonds be purchased to yield o^7o on the investment? 12. Write the tables of long, square, and cubic measures ; also of Troy, Apothecaries', and Avoirdupois weights. VI. 1. A man travelled at the rate of 3 mi. 165 rd. 4 yd. 2 ft. an hour. How far did he go in 36 hours? 2. How many revolutions will a wheel, whose diameter is 4i feet, make in rolling 3 miles? 3. A hollow brass sphere, whose diameter is 4 inches, weighs I as much as a solid sphere of the same size and material. How thick is the shell? 4. Express the ratio of a pound Troy to a pound Avoirdu- pois ; of an ounce; of a cubic foot to a bushel; of a quart liquid measure to a quart dry measure. 5. Solve the following by compound proportion : If 15 men in 12 days of 10 hours each can dig a ditch 180 rods long, 6 feet wide, and 4 feet deep, how many hours a day must 10 men work to dig a ditch 200 rods long, 8 feet wide, and 2 feet deep in 10 days? 6. Add : 5 A. 120 sq. rd. 21 sq. yd. 6 sq. ft. 12 " 96 " 22 " 83 " 17 " 74 " 7. Find the interest on $1,580, at 7i%, from Dec. 18, 1889, to May 1, 1891. 18 (,( u 7 25 u (( 4 28 (( a 8 GENERAL REVIEWS. 349 8. What is the cost of a UO-day draft ou London for £850, exchange being $4.86, and interest 5% ? (With grace.) 9. What is the value of a pile of wood 360 feet long, 12 feet wide, and 6 feet high, at §3.20 a cord? VII. 1. Put the following items into the form of a receipted bill: R. D. Smith bought of Cole Bros., Newark, N. J., on June 1, 1892, 16 yards silk, @ $1.85. June 12, 56 yards cotton cloth, @ 9 cents. June 15, 8 yards broadcloth, @ S2.25. June 20, 24 yards carpet, @ 96 cents. July 1, 31 yards matting, @ 40 cents. July 10, 5 sets curtains, @ $3.85. 2. Find the g. c. d. of 1127 and 6581. 3. A man bought a house and lot for $5,088. 3 of the cost of the house was | of the cost of the lot. What was the cost of each? 4. At $4.75 a cord, what is the cost of the following piles of cord wood : a. 18 feet long, 6 feet high. b. 23 " " b\ " " c. 17 " " 7 " " 5. Change 2 rd. 4 yd. 2 ft. to the decimal of a mile. 6. What must be the rate of taxation in a town to yield a net return of $16,660, if the real estate is assessed at $531,000, the personal property at $200,182.80, 7 % of the tax being uncollectible, and the collector's commission being 2%? 7. In what time will $469.50 yield $36.80, at 7% ? 8. What is the volume of a sphere whose diameter is 7^ inches ? 9. If a block of stone 18 inches long, 4 inches wide, and 2 inches thick, weighs 12 lb. 15 oz., what is the weight of a 350 NEW ADVANCED ARITHMETIC. block of the same material 2J feet long, 2 feet wide, and 9 iuclies thick? 10. What is the volume of the frustum of a cone the diameters of whose bases are respectively 28 inches and 16 inches, and whose height is 30 inches? VIII. 1. Anal^'ze each of the following: a. 2 is what part of 5^? b. f is what part of \\? c. .015^ is what part of .63? 2. Add/y, 2of 51, §^|, f X31. 3. Give the rrle for "pointing ' the product in Multipli- cation of Decimals, and explain it. 4. Change 61,368 seconds to integers of higher denomina- tions, and give the analysis for two reductions. 5. f of an inch is what part of a rod? Analyze. 2i 6. "What number multiplied by jy '"'i^l gi'^'e ~ foi' ^ product? 7. Change y\ of a mile to integers of lower denominations. 8. "What will it cost at 24 cents a square yard to plaster a hall AQ^ feet wide, 82 feet long, and 24: feet high, no allow- ance being made for openings? 9. Bought 42 shares of stock at lOoi^. Received a 4% dividend, and sold the stock at 103 J-. The gain was what per cent of the investment? 10. "What principal will amount to Si, 690 from Jan. 12, 1890, to June 17, 1892, at 7% interest? 11. A railroad train moves a mile in 65 seconds. "What is its rate per hour? IX. 1. What is the area of an equilateral triangle each of whose sides is 42 rods? GENERAL REVIEWS. 351 2. Find the number of acres in the following tracts of land : a. N. i of S. W. I of a section. b. S. iof X. W. i- of S. E. 1. c. S. W. \ of S. E. \ of N. AV. ^. Make a figure showing each tract. 3. A is 26 rd. 4 yd. north of C. B is 13 rd. S^ yd. east of C. AVhat is the distance from A to B? 193 3i I3i 4. From "•* take the sum of § X — and | X — ?, and divide the result by 2|f . 5. How many -iO-gallon barrels will a cylindrical cistern hold, the diameter of which is 9i feet, and whose depth is 10 feet? 6. Which is the better investment, 5% bonds at 98, or 4% bonds at 95? Give the rate per cent of interest on each investment. 7. What is the per cent of gain if ^ of an article is sold for ^^ of its cost? 8. Explain the rule for " pointing " a number to ascertain the number of terms in its cube root. 9. Bought 15 railroad bonds at H% discount, brokerage 1|%. For what must a 90-day note be drawn, interest at 8 7c , to obtain the amount of the purchase at a bank ? (With grace.) 10. Find the cube root of 3 to within .001. X. 1. Define a ratio. Define each term. What is the differ- ence between a ratio and a fraction? Define a proportion, and each term. By what principle is any term of a propor- tion found when three are given? » 2. A can do a piece of work in 6 days; B, in 7 ; and C, in 8. In what time can they do it working together? 352 NEW ADVANCED ARITHMETIC. 3. What is the capacity of a hollow sphere whose outside diameter is 15 inches, and whose walls are ^ of an inch thick? 4. A man bought four articles for $896.45. For the first he gave 821 more than for the second; for the second, $62.50 more than for the third; for the third, $81.75 more than for the fourth? What did each cost? 5. Sold 160 acres of land at $87.50, commission 2|-%. Directed the agent to iuv^est the proceeds in 5% bonds at 98, reserving his commission at 2%, and returning the surplus of less than $100. How many bonds did he purchase, and how much did he return ? 6. V|=.? \/i = ? 7. The interest on two sums of money for 4 j^ears and 8 months at 6% was $256. § of the first sum equalled the second. What was each? 8. A steamer can sail 10 miles an hour with the current, and 5 miles an hour against it. AVhat is the rapidity of the current? How long a trip up stream and down can it make in 6 hours? 9. Find the volume of a cone whose altitude is 15 inches, and radius of base 3h inches. 10. A commission merchant sold goods at 2% brokerage. He invested the proceeds at 2%, reserving his commission. His commissions amounted to $149. What was the amount of the first sale? XI. 1. Give the laws of the Roman Notation. 2. Explain the philosophy of " pointing " for the extrac- tion of the cube root. 3. How many rolls of paper are needed to cover the walls and ceiling of a room 16 feet by 18 feet, and 11 feet high, deductions being made for three windows 3 ft. 2 in, by GENERAL REVIEWS. 353 7 ft. 4 in., 2 doors 3 ft. by 8 ft. 2 in., and a 10-inch base board ? 4. Find the compound interest on 8483.93 for 4 yr. 5 mo. 13 d., at 6%, compounded semi-annually. 5. A room is 15 feet by 18 feet, and 10 feet high. What is the length of a line extending from an upper corner diago- nally through the room to an opposite lower corner? 6. What is the diameter of a circle containing 20 acres of land? What is the area of a strip 18 feet wide, lying next to the circumference and reaching around the field on the outside ? 8. On a certain farm the barn cost f as much as the house, and the house ^ as much as the land. The tenant raised 3,600 bushels of corn and 3,000 bushels of oats. The landlord received % of the corn and \ of the oats for rent Corn sells for 44 cents, and oats for 3H cents. The land- lord's income was ^\l\% of his investment. Find cost of barn, house, and farm. 9. How many bushels of corn in the ear will a crib hold that is 46 feet long, 8 feet wide, and 10 feet high, counting the bushel at | of true capacity ? XII. 1. What is the face of a 60-day note the proceeds of which are $2,654.38 when discounted at a bank at 7% ? (With grace.) 2. A and B can do a piece of work in 24 days. A can do I as much as B. In how many days can each do it alone? 3. A rectangular field contains 12^ A. Its width is f of its length ; what is the distance around it ? 4. A, B, and C were partners in business. A's capital was § of B's, and B's was f of C's. A's capital was in 8 months; B's, 9 months; C's, 10 months. Their net gains were §2,674; what was the share of each? 354 NEW ADVANCED ARITHMETIC. 5. An agent sold a house and lot for his principal. After reserving his commission of 2% for selling and 2% for buy- ing, he invested the remainder in corn at 51 cents a bushel. His total commissions were $400 ; how many bushels of corn did he buy? 6. How many gallons of water will a hollow sphere hold whose interior diameter is 3^ ft. ? 7. A piece of land in the form of a trapezoid is 120 rd. between its parallel sides, one of which is 45 rd. long, and the other 60 rd. long. What is the land worth at §62.50 an acre? 8. A merchant sold a customer 7 pieces of cloth, each containing 50 yd. He made a reduction of 20% from the retail price, and a further reduction of 5% for cash. The retail price was 40% above cost. He received $532. What was the retail price per yard? 9. A certain note for $600 is dated June 1, 1890. It is due two years from date, and bears 6% interest. What should be paid for it Sept. 1, 1890, in order that the invest- ment shall yield 10% per annum? xni. 1. Define Division. Define a Common Fraction. Effect of multiplying numerator and denominator by the same number? Explain fully. 2. Give and fully explain the rule for the multiplication of Decimal Fractions. 3. What sum of money put at interest for 3 years 7 months 18 days, at 6%, will amount to $648.36? 4. A tank can be filled by keeping one pipe open 4 hours, or by keeping a second pipe open for 5 hours. The tank has a pipe by means of which it can be emptied in 2^ hours. In what time will the tank be filled if the three pipes be left open? GENERAL REVIEWS. 355 5. Change ^j of a mile to integers of lower denominations. 6. A farmer has a -iO-aere field in the form of a square. He has it planted in corn, the rows being 3 feet G inches apart. The first row is 3 feet from the line. How far does he walk in plowing it once, taking a row at a time? 7. A, B, and C form a partnership for one year. A put in 83,000 for the first six months, when he withdrew $1,000. B put in 83,000 at first, and when A withdrew he made the deficit in their joint capital good. C put in 85,000 for the year. Their net gain was 82,750. What was each one's share ? 8. I spent 25% of my money, 33 J % of the remainder, and 8^-% of the remainder. I then had 8550. How much did I have at first? 9. A man was paying rent at the rate of 815 a mouth. He borrowed an amount from a Building Association which enabled him to build a house as good. He made a monthly payment of $18 for six years, when his house was paid for. How much more than his rent did the house cost him, count- ing interest on his money at 6% ? 10. A house is 30' by 40'. The cistern connected with its roof is cylindrical in shape, 9 feet deep, and has an average diameter of 10 feet. At the end of a rain the cistern was found to be half full. How many inches of rain had fallen? 11. A man left $10,000, to be invested for his three sons, aged 12, 15, and 18, at 4% compound interest. He directed that the money should be so divided that the children would receive equal amounts when 21 years of age. How much was set aside for each one? 12. AVhat is the rate of interest received on an investment in A\ bonds purchased at 106? 13. At what rate must 3^% bonds be bought to yield 4% on the investment? 2. 9 + x= 16 3. a^-3 = 12 4. 12 — a; = 2 356 NEW ADVANCED ARITHMETIC. 352. ALGEBRAIC QUESTIONS. Find what number x stands for, if 1. X + 7 = 10 5. Ix = 21 9. 3x + ox = 32 6. 3.« + 5 = 29 10. 9.C — 2.?; = 35 7. 7 + 2x = 17 11. 7x + 3x — 5x = 30 8. 17 — ■2x = 13 12. 5x—2x-\- 11-17 13. If the base of this rectangle is b units, the altitude a units, ex- press the sum of the base and altitude. 14. Express the difference of base and altitude. 15. Express the length of the perimeter (entire boundary) o 16. Express the area. 17. Express what part the altitude is of the base. 18. Express what part the base is of the perimeter. 353. THE LITERAL NOTATION. 1. Numbers are often represented by letters, as in the questions above. Any letter may stand for any number, but in any particular problem a letter must stand for the same number throughout. 2. A letter, or a combination of letters, standing for a number is called an algebraic expression. 3. The numbers expressed by algebraic expressions may be sums, differences, products, quotients, powers, or roots. A power is the product of equal factors ; the degree of the power is determined by the number of equal factors, and is indicated by a small figure or letter (written above and to the right), called the exponent or index. ALGEBRAIC QUESTIONS. 357 Thus, 23 =2X2X2 = 8. Eight is the third power of two. A root of a iinniher is oue of the equal factors whose product is the given number. 2 is the 3d root of 8 ; 5 is the 2d root of 25. Roots are indicated by the radical sign, V. Tlie index of the root is a numeral written above tlie radical sign. -^16 = 2. The fourth root of 16 is 2. Y r/^ — d. The third root of d third power is d 2d powers and 3d powers are called squares and cubes. 2d roots and 3d roots are called square roots and cube rootSr The index of square roots is usually not written. \/9 = 3. The square root of 9 is 3. 4. In reading algebraic expressions involving more than one letter, it is often best to name the kind of number be- fore reading it ; thus, (1) a + 3 6, the sum of a and 3 J, or a plus 3 h. (2) 6 a — 3 c, the difference of 6 a and 3 c, or 6 a minus 3 c. Note. Such expressions for sums and differences are called binomials. The expressions connected by tlie signs + or — are called terms. 7 a and 5 c- are the terms of the binomial 7 a — 5 c. An expression of three terms is a trinomial, as 3 a + 2 c — 5 .r. Any expression of two or more terras may be called a polynomial, although the name is usually applied to ex- pressions of more than three terms. A monomial contains one term only. Polynomials are sometimes called compound expressions. Compare com- pound numbers with polynomials. (3) a X (6 — ■), or a (b — c), the product of a and the binomial b minus c, or a times the binomial b minus c. (4) (3x — a) -f- c, the quotient of the binomial 3 x minus a divided by c. ■VT . 3 .r — a Note. A quotient written is called a fraction. (5) a — (.r — ?/)3, a minus the cube of the binomial x mi- nus y. (6) a ■— {x — y^), a minus the binomial x minus y cubed.. (7) Va^ —2 c, the cube root of a squared minus 2 c 24A 358 NEW ADVAXCED ARITHMETIC. (8) ('V^a)'^ — 2 c, the square of the cube root of a minus 2 c (9) ^a^ —2 c, the cube root of the binomial a squared minus 2 c. ,^^. V {3(t — b) (2x + y), the fraction, the square root a + X — y of the product of the binomials 3 a minus b and 2 a; plus y divided by the trinomial a plus x minus y. 354. EXERCISES IN LITERAL NOTATION. Write as algebraic expressions : 1. The sum of a squared and the square root of c. 2. Five times the binomial the fourth root of x cubed minus 7 a b. 3. Five times the fourth root of x cubed minus lab. 4. Five times the fourth root of the binomial x cubed minus lab. 5. a plus the fraction b divided by c plus d. 6. a plus the fraction b divided by the binomial c plus d. 7. The fraction a plus b divided by the binomial c plus d. 8. The fraction a plus b divided by c plus d. 9. The square of the fraction a minus c divided by x. 10. The product of 3 times the sum of x and y by the sum of 3 x and y. 355. EVALUATION OF ALGEBRAIC EXPRESSIONS. To evaluate an algebraic expression is to find its numer- ical value ; thus : If a = 1, & = 2, c = 3, re = 10, what is the value of /babe — 2a + hx\^ V 3x — 2c ALGEBRAIC QUESTIONS, 359 FORM. .^s ^5abc — 2a+h x\^ ^^ \ 3x~^^^ ) ^ . / 5-1-2-3-2-1 + 2- lO y __ ^"^ \ 3 ■ 10-2 ■ 3 ) ~ (;)) (2)^ = (G) 8 Description. Substituting in expression (1) the numer ical values of the letters we have exp. (2). Performinc indicated multiplicatious we have exp. (3)*. Performino- ad- ditions and subtractions we have exp. (4). Dividing we have exp. (5). Cubing the quotient, as indicated, we obtain 8. 356. PROBLEMS IN EVALUATION a = 1, & = 2, c = 3, cZ = 10, .T = 0. 1. 3a& + ca;+4d 6. lOc— (6d — 4ac) 2. ^/hc-V d 7. 10c — &d — 4ac 3. (2fZ-&c)2 8. ^4& + 2rZ-a 4. (3 a + &) .r + 2 a c d 9. ^^^4 6 + 2 d — a 5. a.T+ &.'r -f cd 10. (d — by a-{-2b + d' 11. (d^ — b^ 357. EQUATIONS. Problems. 1. A boy has four times as many white mar- bles as brown ones. Of both he has 30. How many of each? 360 NEW ADVANCED ARITHMETIC. Let X = number of brown marbles. Then, since, etc. 4 x = number of -white marbles. And z + 4 X = number of both. But 30 = number of both. Therefore x + 4 z = 30. ox = 30. X = 6, no. of brown marbles. 4x = 24, no. of white marbles. 358. DEFINITIONS. (1) An equation is a statement in mathematical symbols that two expressions stand for the same number. .r + 4 X = 30 in the problem above is an equation, x + Ax and 30 each stand for the total number of marbles. (2) The numbers of an equation are the two equivalent expressions, x + Ax and 30, in the equation above. (3) Equations are used to find the value of unknown numbers represented by a-, y, z, etc. (4) To solve an equation is to find the value of the un- known number involved. 2. In a school of 45 pupils there are 7 more girls than boys. How man}" of each? Let X = number of boys. Then, since, etc., x + 7 = number of girls. And X + X + 7 = number of pupils in school. But 4.5 = number of pupils in school. Therefore x + x + 7 = 4.5. 2 X 4- 7 = 45. 2 X = 38. X = 19, number of boys. X 4- 7 = 26, number of girls. Show that in the above solution we have proceeded in accordance with the following truths : ALGEBRAIC QUESTIOyS. 361 359. Axioms. (1) Things equal to the same thing are equal to each other. (2) If equals be added to equals, the sums are equal. (3) If equals be subtracted from equals, the remainders are equal. (1) If equals be multiplied by equals, the products are equal. (5) If equals be divided by equals, the quotients are equal. 3. Six times John's age exceeds four times his age by 22 years. How old is he? ^'oTE. Let X = the number of years in John's age. If we say let x = John's age, we treat x as a mere quantity of time, not as a number of time-units. 4. 821,000 is divided among three children so that the first receives twice as much as the second, the second twice as much as the third. What is the share of each? XoTE. Let X = no. of dollars in the share of the third. 5. Thomas. Richard, and Heury have 72 marbles. Thomas has twice as many as Richard. Henry has twice as many as both the others. How many has each? 6. How old am I, if three times my age four years ago exceeds twice my present age by 27 years? 7. Equal weights of sugar and flour were bought for 63 cents. The sugar cost 5 cents per pound, the flour 2 cents. How many pounds of each? 8. The perimeter of a rectangular field 80 rods long is 280 rods. What is its width ? 9. The perimeter of a rectangular field, twice as long as wide, is 180 rods. What is its length? 362 NEW ADVANCED ARITHMETIC. V iO. It takes 70 feet of border to enclose a square room. W^hat are its dimensious ? 11. A room 27 feet wide and x feet long requires 99 square yards of matting. What is the value of x? 12. A Sunday-school collection in dimes, nickels, and cents amounted to 200 cents. There were three times as many nickels as dimes and five times as many cents as nickels. How many of each? 13. Grace is 5 years older than May. May is two years older than Ethel. The sum of their ages is 42 years. What is the age of each ? f^ 14. A father is four times as old as his son. Five j^ears ago, he was seven times as old. What is the father's age? Let X — no. of years in son's age. Then (why ?) 4 x = no. of years in father's age. " " X — o = no. of years in son's age 5 years ago. " " 7 (z — .5) = no. of years in father's age 5 years ago. " " 4: X — 5 = no. of years in father's age 5 years ago. Hence (1) 7 (x - 5) = 4 x - 5. (2) 7x-.3o = 4x-5. (3) 7 X = 4 x - 5 + 35. (4) 7x-4x = 35-5. (5) 3x^30. (6) X = 10. (7) 4 X = 40, no. of years in father's age. What was added to each member of (2) ? What was subtracted from each member of (3)? Note. If a term contain ,i " numerical " factor, and one or more literal factors, the numerical factor is called the coefficient of the term. Terms containing the same literal factors are called like terms. In the terms 7 t, 3 a'^ b, 7 and 3 are the coefficients. 3 a x^ and 5 a x- are like terms. 3 d- b and 1 ah- are unlike terms. Why ? I;i describing the solution of the equation in Problem 14 we may say: Performing indicated multiplication in Eq. (1 ) we have Eq. (2). Add- ing 35 toeach member of Eq. (2) we obtain Eq. (3). Subtracting -ix from each member of Eq. (3) we obtain Eq. (4). Collecting like term.s in (4)j we obtain Eq. (5) Dividing each member of Eq. (5) by the coefficient of ALGEBRAIC QUESTIONS. 363 X, 3, we obtain Eq. 6. Multiply iug each member of (6) by 4 we get Eq. (7). (Adding 35 to, and subtracting 4r from, each member of Eq. 2 we obtain Eq. 5.) 15. A man of 35 is 7 times as old as his son. In how many years will he be twice as old? Let X — no. of years hence when the father's age will equal twice the son's age. Then 5 + x = no. of years in son's age at required time ; and 2 (5 + a;) = no of years in father's age at required time. But 35 + a; = no. of years in father's age at required time. Hence (1) 2 (5 + x) = 35 + x. (2) 10 + 2 a; = 35 + x. (3) 2x-x = 35-10. Note. In subtracting 10 and x from each member of Eq. (2), we cause each of these terms to pass to the other member of the equation with change of sign. This transfer of a term to the opposite member with change of sign is called transposition. 16. A has 8 dollars more thafi B. After paying B 12 dollars, A has only ^ as many as Bo How much had each at first? 17 o John has 40 marbles more than Fred. After giving Fred 50, John has only i as many as Fred. 18. A debt of $102 is paid with an equal number of ten- dollar, five-dollar, and two-dollar bills. How many bills were paid in all? 19. In paying 27 cents for an article, I tendered some dimes and received an equal number of cents as change. How many dimes did I tender? 20o Harry and Walter, 62 miles apart, ride towards each other. Harry, starting at 9 a.m., rides 2 miles per hour faster than Walter, who started at 8 A. m. They meet at nooDo What is the rate of each? 21. Take some number, double it, add 20, divide by 2, take away the first number; you have 10 left. Why is this? If you had added 30, instead of 20, how many would you have left? 364 NEW ADVANCED ARITHMETIC. 22. Take some number, multiply by 6, add 30, divide by 3, subtract 4, divide by 2, take away the first number; you have 3 left. Explain. 360. EQUATIONS CONTAINING FRACTIONS. 1, I of what number = 4 ? 2. I of what number =3? 3o f of what number — 6? 4o j\ of what number ■--. 10? 5o ij of a number + J of the same number = 21. What is the number? 6. f of a number — ^ of the same number =10. What is the number? 7o f of a number = 7. What is twice the number? 8o I of a number =11. What is four times the number? X 9. - = 5o Fmd value of x. o By what number must we multiply each member to obtain the value of x ? 10. -— = 36. 5 . By what number must we multijjly each member to obtain the value of 4a;? Will 10 do? What other multipliers? What axiom is involved? x 4x "• 3 + X = ''■ Analysis. To make the first fraction integral we must m.ultiplj it by 3, or some multiple of 3 ; to make tlie second fraction integral we must multiply it by 5, or some multiple of 5; to preserve the equality of the members we must multiply both by the same multiplier. 15 is tlie least common multi])le of 3 and .5. Multiplying the first fraction by 15 (first by 3 to suppress the denominator, and then by 5) we have 5x. Multiply- ALGEBRAIC QUESTIONS. 365 ing the second fraction by 15 (first by 5, and then by 3) we have I2r. Multiplying the second member by 15, we have 255. Our equation now stands : 5x + I2x= 255. Note. This process of transforming a fractional equation into an inte- gral equation is called clearing of fractions. Solve : ,„ 4.x ^x ^^ 3 4 5 12. — + -— = 31. 23. - + - + - = 1. 5 4 13. ^+10 = ^-1. 2*- i^- + i^-^^- = 25„ ^ ^ 25. %x-\- %x~}x^ 82. 3a; 2x _ 3 ^ g ''' 4 3 -'• 26. ^/__^ + ^.34. 4^^ +5 .^^ + 8. 28. 2fa; = 105. 16. ' ' 29. 3lx = 48. z . z z 17, 1 + ^ + - = 39. 30. Ux- 2| .r = 45. 2 3 4 31. 51 a; — 32 a; = 44„ 4^2~5~'"' 32. 5i (.T — 3) — ^ = 0. 18. - + -__ = 22. ,, -, , o, 2a; 32; 33 z 3 a; ^^- T - T0~ "" ^^* 33. X- -^ 2 a; + — = 36. 20 ^ ^ 2/ , '^^ 99 4 a; 7 - a- , '°- T-2 + 5 = 2^' 34. --+-^ = 53. 3a; + 4 2a;-8 _ 10. ^ .r _ 3 ,^ 21- ■ — ^ + — o = ^- 35. -^ + -— ^ >0 2a; 3a;+ 4 3a;— 4 366 NEW ADVANCED ARITHMETIC. 361. DEFINITIONS. 1. If the members of an equation are alike in form, or if they are reducible to the same form, the equation is called an identical equation, or an identity. Thus, 9 = 9, 5 -f 2 — 4 = 3, and 5 a;— 7 = 3 x'— 7 + 2 x are identities. 2. A solution is verified by substituting for x in the given equation the value of x, as found in the solution, and performing all indicated operations in each member. If the equation reduces to an identity, - the solution is correct. Illustration : 3 + 2a; 7a;+6_18a; 5 "^ ~Tl ~^^ 66 + 44 ic + 70 a; + 60 = 198 a; 114a; + 126 = 198a; 126 = 84a; I = X. Verification : f + 6 18 . f 3 + 2 . I 7_ 11 3+3 10^ + 6 n" + :.^— = 6 16| 5^ 11 10 27 10 27 10 6 3 _27 5^2 ""10 27_27 10~10 3. If upon substituting for x its supposed value the equa- tion becomes an identity, the value of x is said to satisfy the e(iuation. ALGEBRAIC QUESTIONS. 367 362. PROBLEMS. Solve and verify : 1. Divide 90 into two such parts that one shall be 3^ times the other. 2. Divide 100 into two such parts that one shall be 2^ times the other. 3. A horse was sold for $80, at a gain of I of the cost. What was the cost? 4. A is 12 years older than B. \ of A's age = ^ of B's. WhaJt is the age of each ? 5. If to John's age there be added its half, its third, and its fourth, the sum is 25 years. What is his age? 6. If to Mary's age there be added its half, its third, and its fifth, the sum is 2^ times her age. What is her age? Query. What is the matter with the foregoing problem ? 7. If to A's age there be added its double, its half, and its third, the sum lacks 7 years of four times his age. What is his age ? 8. A, B, and C received $162 for digging a ditch. A dug 4 rods to B's 3 rods and C's 2 rods. What pay should each receive? 9. Two barrels contain respectively 42 and 50 gallons of oil. After drawing the same amount from each, the first contained § as much as the second. How much was drawn from each? 10. A campaign pole 84 feet high broke at such a point that the top was f of the stump. What was the height of the stump? 11. A campaign pole 100 feet broke at such a point that the top was 6 feet longer than the stump. What was the length of the stump ? 368 NEW ADVANCED ARITHMETIC. 12. The sum of two numbers is s, the difference d. What are the numbers? Let X — the smaller number. Then x + d — the greater number, and -X + X -}- d = s 'i X -}- d ^ s 2x = s — d smaller number. 2 s — d 2d s -\- d , X + d = — 1 — — = — - — ? greater number. ^ Jt £d Note. In solving the preceding problem we have solved every prob- lem in which the sum and difference of two numbers are given to find the numbers ; for s and d are any numbers. By substituting the values of s and d in any particular problem of this type, we avoid a formal solution. Thus : 13. The sum of two numbers is 42, their difference 12. What are the numbers? s + d 42 + 12 54 Greater number = — 5 — = — 9 ~ ^ ~ "^ s — d Smaller number = — - — 14. Sight problems : Sum. Difference. (1) 30 20 (2) 30 e (3) 22 8 (4) 55 5 (5) 44 6 (6) 23 7 42- 12 2 - 30 2 = 15. Sum. ( 7) 88 ( 8) 33 Difference. 12 3 ( 9) 52 12 (10) 26 (11) 13 (12) 51 6 9 19 15. A boat runs 6 miles per hour up-stream, and 16 mileo per hour down-stream. What is the rate of the current? ALGEBRAIC QUESTIONS. -^69 16. John had six marbles more than James. Harry had none. After receiving \ of James's marbles, and f of John's, Harry had as many as John or James. How many had each? 363. POSITIVE AND NEGATIVE QUANTITIES. 1. Quantities are sometimes so related that one tends to neutralize or destroy the other. Thus, a rise in temperature counteracts an equal fall ; debts are opposed to assets ; traveling southward cancels traveling northward; the force of the current destroys an equal force propelling the boat up stream. To these quantities, opposite in kind, the names positive and negative are applied. Either of a pair of oppo- sites may be called the positive quantity ; usually the more familiar of the two is so named. 2. The signs + and — , wheu used as signs of operation, are read "plus" and "minus;" when placed before num- bers to show their character, they are read " positive" and "negative." 3. At -i A. M., Monda}', the thermometer stood at 3° above zero. In the succeeding 12-hour periods it (1) rose 12°, (2) fell 19°, (3) rose 7°, (4) fell 15°, (5) rose 22°, (6) fell 9°. What was the sum, or result- ^-+5 ''' t'- ing temperature, Thursday morning? Explanation. The initial temperature, 3° above zero, is positive. Addhig to it the various positive numbers (rises), we obtain +44°. The sum of the negative , -^ numbers (falls) is — 43^ This sum cancels + 43 of the positive sum, leaving + 1° as the total sum, or final temperature. 364. PROBLEMS. 1. Seven boys pull at a rope ; three pull northward, ex- erting respectively forces of 75, 85, and G3 pounds; four + 3 + 12 — 19 + 7 — 15 + 22 - 9 + 44 - 43 370 NEW ADVANCED ARITHMETIC. • pull southward with forces of 52, 57, 59, and 43 pounds. The rope is moved in what direction and with what force? 2. In the left scale-pan of a balance are two 8-ounce weights, three 4-ounce, and five 2-ounce. In the right pan are two 16-ounce weights and three 2-ounce. What must be added to the left pan to produce equilibrium? 3. A letter-carrier has walked north 7 blocks, east 3 blocks, south 5 blocks, east 2 blocks, north 4 blocks, west 11 blocks, south 8 blocks, east 2 blocks. He is now how far east of his starting-point? How far south? Note. Mark eastings aud northings +, westings and southings — . 4. A surveyor has measured N. 6.22 ch., E. 4.17 ch., N. 2.12 ch., W. 6.96 ch., N. 3.16 ch., E. 11.25 ch., S. 12.58 ch., W. 8.62 ch., N. 2.00 ch. He is now how far north and east of his starting-point? 5. Add 6 a, — 3 a, — 7 a, — 5 a, + 3 o, + 6 a, — 3 a. 6. Add 2aa', — 4aa;, -fSacc, 4-6aa-, — 7aa', -l-3ax. 7. Simplify 3a6 — 6a& + 4a6 — 5a6 — 7a6-|-r2a6. 8. Sunplify hxy — lxy — 22xy — xy-\-2hxy + 2Qxy-' 10 xy. 9. Add 3a-h5 6, 9a-i-36 — 2 c, 6a — 4 c, 12 c— 18 a. FORM. 3a + 5b 9a + 36- 2c 6a — 4c -18 a + 12 c 86 + 6c 10. Add 5a — 3. T, 9x — 5y+2a, Qx — 2y—la, 5 a + 9 ?/. 11. Add 3a6 — 2c-j-c?, 7a& — 6d, 5c— 10a6 — 4d, 10d-3c. ALGEBRAIC QUESTIONS. 371 365. Subtraction. 1. What is the change iu temperature if the thermometer reading changes from + 75° to + 90° ? from + 33° to + 56° ? from — 7° to + 8° ? from — 2° to + 11° ? from — 4° to — 22° ? from + 3° to - 7° ? from + 10° to — 1° ? 2. What must be added to + 3 to make + 10? to — 3 to make +10? to + 10 to make +3? to + 10 to make — 3 ? to — 3 to make — 10 ? to — 7 to make — 2 ? 3. Since the sum of the subtrahend and difference equals the minuend, we may define subtraction as the process of finding from two given numbers called subtrahend and minu- end a third number called difference, which added to the subtrahend produces the minuend. 4. The minuend contains the subtrahend and the differ- ence. If we can destroy the subtrahend in the minuend, only the difference will remain. To destroy, or cancel, a number, we add an equal number with opposite sign. Hence, we add to the minuend the subtrahend with changed sign and obtain the difference. 5. From 5a — 7^ + 4c take 2a -2h — 2c. If we write 5 a — 7 ^ + 4 c — 2 « , we have taken 2 a from the minuend; but we were required to take away, not 2'/, but 2 a dmiinished by 2 5 and 2 c. We have therefore taken away too much, and must add 2 b and 2 c to obtain the true difference. Hence, 5a_7^ + 4c_(2a-2Z'-2c)==5a-7^ + 4c-2« + 25 + 2c. Here we see, as in 4, that to find the difference, we must change the signs of the subtrahend and add to the minuend. 366. PROBLEMS. 1. From lax — 2>hx — 4:cy take ^ax — 2hx ~ cy^ 2. From 5 ??«. + 3 n + 12^ take Sm — 2n — 1 p. 3. From 4 r/ — 3 /> — he take 6 a + 5 c — Ad. 372 NEW ADVANCED ARITHMETIC. 4. a + b + {4:a — 3b)=? 5. 9a'—7b — 6c—(3a + 6b — dc)=? 6. 17 a — 12 « m — 4 c2 — (— 4 a — 14 a m'^ + 2 c^ =? 7 3a + 4^ — 5c — [2a + 3^ — (2« — 6e)] =? 8. 5a — (ox — 4: >/) — [5 a ?/ — 3 a — (2 a? + 3 y)] =? 9. 7 a2 _ (7 ^,2 _ 3 ,.2^ _ 2 c2 — [4 Z/2 4- (2 a^ - 3 c^)] =? 367. Multiplication. 1. If a street-car ride costs m cents, what is the cost of ? ridesi'' 7 rides? 22 rides? a rides? x rides? 2. A dime is tendered in payment for car-fare; b cents are returned as change. What is the cost of the ride ? 3. In paying 4 such fares, how many dimes are tendered? How many cents cliange are returned? In paying in such fares ? Note. In the expression ?n (10 — b) = 10 m — b m, we see that the signs of the product are the same as in the corresponding terms of the multiplicand. 4. A conductor starts on a trip with two dollars, collects 12 such fares, and refunds 3 fares. The number of cents he now has is expressed 200 -I- 12 (10 -b) -S(10- b). The sign before the multiplier shows what is to be done with the product. In receiving 12 fares, he receives 12 x 10 cents and pays out 12 X b, or 12 6, cents. In refunding 3 fares, he pays out 3x10 cents and receives 3 b cents. Therefore he now has 200 + 12 • 10 — 12 ^» - 3 • 10 + 3 b. Note. The last four terms are products. The positive terms, +12 10 and -j- 3 b, are the products of factors with like signs; the negative terms, — 12 6 and — 3 • 10, are tlie products of factors of unlike signs. Would the signs be as they are if other numbers than 12, b, 10, and 3 had been used'' ALGEBRAIC QUESTIONS 373 RULE. Two factors of like sign give a positive product; ttco factors of unlike sign, a negative product. 368. PROBLEMSo 1. A conductor starting with e cents collects a — b fares of m — X cents each. How many cents has he at the en-l of the trip? What do — h and — x signify in this problem r 2. A butcher sold m guaranteed hams at a — h cents each. X hams were returned as spoiled. What were the net receipts ? 3. A rectangle is m + w units long, a + x units wide. What is its area? Explain these diagrams : a + b a ( 25A 4. Construct diagrams for (a + h)"- and (a — 6;-. 374 NEW ADVANCED ARITHMETIC. 5. "VMiat is the area of a square a -{• b units on each siae: 6. What is the volume of a rectangular solid b feet long, a feet wide, and h feet high? How man}- cubic feet in the bottom layer? 7. What is the volume of a cube a + 6 feet on each edge? 8. What is tJie volume of a rectangular solid a + b feet long, m + n feet wide, x + y feet high? 9. What is the area of a rectaugle a -{• b feet long, a — b (eet wide ? 10. What is the area of a rectangle (3 a — x) by (4 a — 3 x) ? 11. (3 a - 4 a: + 7 &) (9 X - 3 a + 2 6) = ? 12. (4 a — 3 a 6) (3 a — 5 6) (a + b) = ? 13. (a — c;)(a — 2x) (a + a-) = ? 14. (la — 6 ax) (Sa + 4iax — ox) = ? 15. da-7 {a + b) + A{a — 2b) = ? 16. 5{x^—xy)+3x(4:X — Dy)—oy{2x _)_ 3y = ? 17. 3 (9ax - 2by) - ba {bx + 4y) + 2y (3 ^/ + lOO = ? 369. Division. 1. 3 6 dollars are paid for b cords of wood, '^liat is the price per cord ? 2. axy trees are planted in a equal rows. How many trees in each row? 3. The area of a rectangle is ax + ay. It is a units long. What is its width? 4. The area of a rectangle is 4 ax — 8 xy. It is 4 a- units long. What is its width? 5. The area of a rectangle is 6 .r^ + 5 a- — 6 squ ire feet. Its length is 3 x — 2 feet, AVTiat is its width ? A rectangle 3 a: — 2 feet long and 1 foot wide contains how many square feet? A rectangle 3 a; — 2 feet long and i x feet wide? How many more square feet in the given rectangle? How manv more " rows" of square feet do they make? ALGEBRAIC QUESTIONS. 375 Zx-2) 6x2 + 5x-G (2x + 3 6 x^ — 4 X yx- d 9x-6 Note. The first term of the quotient is found by dividing the first term of the dividend by the first term of the divisor. The work proceeds as in ordinary long division. 6. Fiud the T\idth of the folloTviug rectangles : 5^/4-3 7^ + 2 28 1/ + 50 y + 12 A X -\- jf \2x" + Uxy + 2f 370. LAW OP SIGNS IN DIVISION. jlemembering that the dividend is the product of the divisor and quotient, and that a positive product is the product of two factors of like sign, write the quotients required. 376 NEW ADVANCED ARITHMETIC. + 12 —ab + 2la'^x 1. ~ = 5. = 9. = = — 7a -Seab'^c + 12 ab — ax + a bx — xy 4 3 4- 12 -3 - 12 4-3 - 12 6. = 10. 3. = 7. = 11. + b -bx + b -be — c + 12 ax 8. = 12. -3 + 6a- —x From the examinatiou of the first four problems we conclude : If dividend and divisor are of like sign, the quotient is positive. If dividend and divisor are of unlike sign the quotient is negative. 371. Perform divisions as indicated : 1. a- 4- 2 a6 4- b' 8. a^ — x" a + b a -\- x' a^ -2ab + b^ a — b 9, a* — X* 2. a--x^ 3. a-'-b^ 10. x^ + x- 12 a — b' X — 3 4. x^ — y^ X — y' 11". x^ - 7a: + 12 a; — 3 5. x^ 4 y^ 12 a^ + Sa-b 4 ^ab' + 6» X + y ' a -{- b €. X* + xh/ 4 y^ x^ + xy + y^' 13. (x^ _ 3.r 4 2) (a- - 3), a-2 _ Sec + 6 7. a* -a-* 14. (^2 _ a- - 12) (x 4- 5) ALGEBRAIC QUESTIONS. 377 a + X ' ' x'^ — l/'^ ^6 (X - yf + ^Xy ^^ (g-^ + y2)2 _ ^2 ^. a;^ + ^f ' X?- ^ xy ^ y"^ 17 i.^-yy{^-^ h) 20 "' - ^' APPENDIX. 379 APPENDIX. 372. GREATEST COMMON DIVISOR. 1- A divisor of a number is one of the integral numbers w^hich being multiplied together -vyill produce that number. 2. Name all of the divisors of each of the following numbers : 4, 6, 8, 12, 15, 24, 36, 39, 40, 48, 56, 64, 72, 96. 3. What number will divide 4 and 6? 9 and 12?' 10 and 15? 6, 9, and 12? 15, 18, and 21? 14, 21, 2S, and 35? 4. What do you call a number that will divide each of two or more numbers? 5. A common divisor of two or more numbers is a number that is a divisor of each of them. 6. Name all of the common divisors of 6 and 12. Which is the greatest? Of 8, 16, and 24, which is the greatest? Of 16, 24, and 32, which is the greatest? What is such a number called ? Define it. 7. The greatest common divisor of two or more numbers is the greatest number that is a divisor of each of them. 8. Examine these groups of. numbers and find of what the greatest common divisor is the product in each case. 9. The greatest common divisor of tvro or more numbera is the product of their common prime factors. Prove the preceding statement. 380 NEW ADVANCED ARITHMETIC. 373. PRINCIPLES. 1. Any number is divisible by each of its prime factors and by the product of any number of them. 2. The product of any of the common prime factors of two or more numbers is a common divisor of the numbers. 3. The product of all of the common prime factors of two or more numbers is theii' greatest common divisor. Find the g. c. d. of 21, 42, and 63. FORM. 21 = 3 X 7 42 = 2 X 3 X 7 63 = 3 X 3 X 7 Explanation. 3 is a prime factor of each of these num- bers. 7 is also a prime factor of each of these numbers. Hence, 3x7 will divide each of them. As they have no other common prime factors, 21 is their g. c. d. Name all of the common divisors of these numbers. Which is the greatest? Of what is it the product? EXAMPLES. Find the g. c. d. of the following: 1. 24, 28, 36. 8. 210, 294, 462. 2. 60, 84, 96. 9. 195, 273, 429, 507. 3. 45, 60, 7.5. 10. 204, 255, 357, 459. 4. 48, 64, 96. 11. 342, 399, 513, 627. 5. 28, 42, 56, 98. 12. 295, 413, 531. 6. 39, 65, 91. 13. 414, 690, 966, 1242. 7. 112, 110. 108. 14. 780, 234, 312, 390. RULE I. For finding the greatest common iUriaor. Separate the ntimhers into their jtritne factors, and find the product of those that are common. APPENDIX. 381 The factoring method may be employed satisfactorily with any numbers, but the process may be shortened when the numbers are large, by devices that render some of the factoring unnecessary. 374. FINDING THE G. C. D. BY AN EXAMINATION OF DIFFERENCE. Illustrative Example. Find the g. c. d. of 2002, 2366, 3367. A divisor of two numbers is also a divisor of their differ- ence ; hence, the g. c. d. of these numbers must also divide the difference between 2002 and 2366. This difference is 364. Its ^^^*'* prime factors are 2, 2, 7, and 13. The g. c. d. of 2366 and 2002 is f^ ^ also the g. c. d. of 364 and 2002. 364 _ 2 X 2 x 7 X 13. (Why?) Hence, we need to compare only these numbers. The prime factors of 364 are 2, 2, 7, 13. By examining 2002, I find that only three of the prime factors of 364 will divide 2002 ; viz., 2, 7, 13. The product of these three factors is consequently the g. c. d. of 364 and 2002, and hence, of 2002 and 2366 also. If these factors are also found in 3367, their product is the g. c. d. of the three numbers. By trial I find that 2 is not a factor of 3367; 7 and 13 are prime factors of 3367; hence, 91 is the g. c. d. of the three numbers. EXAMPLES. (Solve by the above method.) 1. 59449, and 61659. AxALTSis. The difference is 2210. Its prime factors are 2, 5, 13, 17. Onlv 13 and 17 are factors of 59449 ; hence 13 X 17 is the g. c. d. Why need we pay no attention to G1659? 2. 83971 and 79463. 381 NEW ADVANCED ARITHMETICc The difference is 4508. Its prime factors are 2, 2, 7, 7, 23. None of these are factors of 79463, heuce the g. c. d. is 1. 3. 387 and 2754. Multiply 387 by 7. Is the g. c. d. sought a divisor of this product? Why? 2754 — 27U9 r:^ 45. Vill the g. c. d. divide 45? Wh}'? What are the prime factors of 45? Which of these are prime factors of 387 ? What, then, is the g. c. d. of 387 and 2754? How do you know? * 375. RULE II. 1. Find the difference between tico of the mnnbers. Find its prime factors. Determine irhich of them are prime factors of the smaller of the tiro }iit)nhers. Their product is the g. c. d. of the tivo numbers. 2. Compare this product irifh a third number, proceed- ing as before, ami so continue until all of the numbers have been disposed of. 376. FINDING G. C. D. BY DIVISION. Illustrative Example. Find the g. c. d. of 91 and 325. FORM. 91 ) 325 ( 3 273 ~52)91 (1 52 39)52(1 3£ "l3 ) 39 ( 3 39 Explanation. The g. c. d. of these numbers cannot be greater than 91. If 91 will divide 325, it is the g. c. d. of 91 and 325. The quotient is 3, and the remainder 52 ; hence, 91 is not their g. e. d. Since a divisor of a number is a APPENDIX. 383 dhnsor of any of its multiples, the g. c. d. of these numbers must be a divisor of 273. Since a divisor of two numbers is a divisor of their difference, the g. c. d. must divide 52 ; hence, it cannot be greater than 52. Since 52 is a divisor of itself, if it will divide 91, it will divide 273, by Principle 1, and 325, by Principle 2. The quotient is 1, and the remain- der 39 ; hence, 52 is not the g. c. d. sought. Since the g. c. d. of 91 and 325 must divide 52 and 91, it must divide 39, by Principle 3 ; hence, it cannot exceed 39, If 39 will divide 52, it will divide 91, by Principle 2; 273, by Principle 1; and 325, by Principle 2. The quotient is 1, and the remain- der 13 ; hence, 39 is not the g. c. d. sought. Since the g. c. d. must divide 39 and 52, it must divide 13, by Prin- ciple 3. Since 13 will divide itself and 39, it will divide 52, by Principle 2; 91, by Principle 2; 273, by Principle 1 ; and 325, by Principle 2 ; hence, 13 is the g. c. d. of 91 and 325. 377. RULE III. Select two of the niiuihers mid tiivide the greater by the less, and the less by the renittinder, if there is one. Con- tinue the process until there is no retnainder. The last divisor will be the g. c. d. sought. Compare tJiis divisor unth a third nuinber. proceeding in the same manner, and thus continue until all of the numbers are disposed of. Note. Observe that this method discovers numbers tliat are smaller than tlie given numbers, and yet that have the same g. c. d. EXAMPLES. Find the g. c. d. of the following: 1. 340 and 578. 2. 333 and 703. 3. 533, 697, and 779. 4. 1265, 1870, and 8613. 5- 7944, 12247, and 13902< 384 NEW ADVANCED ARITHMETIC. 378. APPLICATIONS OP G. C. D. 1. What is the greatest width that a carpet can be to prevent waste in covering the floors of four rooms that are, respectively, 15, 18, 21, and 24 feet in width? 2. What is the greatest length of floornig that can be used, without cutting, for three halls that are, respectively, 24, 36, and 60 feet in length? 3. What is the length of the longest paving-stones that, without cutting, may be used to build 4 walks, 144 feet, 180 feet, 204 5"eet, and 300 feet long? 4. What is the capacit}^ of the largest box that will be filled an integral number of times in measuring 160 bushels of oats, 304 bushels of wheat, and 400 bushels of rye? 379. LEAST COMMON MULTIPLE. The Method for Large Numbers. In finding the 1. c. m. of large numbers that are not easily factored, the w^ork may be simplified by employing the g. c. d. Observe that the I. c. m. of two or more numbers is the product of their g. c. d. and their uncommon prime factors. Illustrative Example. Find the 1. c m. of 6837, 73o3, 7860. (a) Find the g. c. d. of 6837 and 7353 by the third method. (6) Divide 7353 by it. (c) Multiply 6837 by the quotient. (d) Proceed in a similar manner with the third number and the result thus obtained. Employ the method with smaller numbers until the process is familiar. Form a rule from the solution of the illustrative problem. APPENDIX. 385 PROBLEMS. Find the I. c. m. of the following : 1. 629, 703, 851. 3. 1496, 1768, 2312. 2. 338, 364, 448. 4. 990, 1305, 1188. 5. What is the smallest sum of money that may be ex- pended by using only 3-cent pieces, 5-eent pieces, 10-cent pieces, or 25-cent pieces? 6. What is the shortest distance that will exactly contain an 8-foot measure, a 12-foot measure, a 15-foot measure, or an 18-foot measure? 7. What is the smallest quantity of oats that will fill, an integral number of times, a 5-bushel box, a 9-bushel box, a 15-bushel box, or a 21-bushel box? 8. What is the product of the 1. c. m. of 12, 15, 18, and 24, and their g. c. d. ? 9. Divide the 1. c. m. of 7, 12, 21, 9, 10, and 252, by the g. c. d. of 80, 120, 840, and 960. 10. What is the difference between the 1. c. m. of 10, 45, 75, and 90, and the 1. c. m. of 7, 15, 25, and 35? 11. What is the shortest cord that could be cut into pieces of 9, 12, 15, 18, or 45 inches? 330. Stone and Brick Work. 1. Stone Work is usually measured by the Perch, although in many localities it is estimated by the cubic foot. 2. A Perch of Stone contains 24| cubic feet. It is 1 rod long, li feet wide, and 1 foot thick. 3. In estimating the labor of laying stone and brick the corners are usually counted twice, because of the extra care needed, 4. For 8-inch walls it is customary to count 15 bricks to the foot; for 12-inch walls, 21 bricks are counted for a foot. 386 NKW ADVANCED AIUTllMirriC. 381. Grain. A cubic foot is about .« (jf m bushel. Tlic capacity of a waj(oii-box or a bin may bo found approxiiiiati'ly l)y liii(liu<^ the number of cubic feet wliich it contains, and multiplying tliis result by .«. 382. Ear Corn. A iMishel of ear corn (••Milains aliont 2] eubic; feet. To rnid I lie iinnil)cr (if bushels a bin will contain, lind the num- bci (.f enl)ic leet in the bin and lake I of it. 383. Hay. Hay measurement is only :ippr May 24. " produce, 158.00 Analysis. Let us assume that A. B. S. paid his account in full on May 24. He would then have had a credit of 4 mo. 23 d. on the first, 3 mo. 26 d. on the second, and 2 mo. 12 d. on the third. If interest were paid at 6%, the charge on the first would he S8.78; on the second, $4.35; on the third, S2. 24. Their sum is S15. 37. 'J'he sum of the items is S779.75. If interest were charged, then, on May 24, A. B. S. would pay to Brown Bros. $779.75, and Si 5.37 as interest. If Brown Bros, were to pay A. B. S. on May 24, they would have a credit of 3 mo. 9d. on the first, and 1 mo. 12 d. on the .second. If interest •were paid at 6%, the charge on the first would he S7.43 ; on the second, $2.80. Their sum is S10.23. The sum of the items is 51,008. If Brown Bros, settled with A. B. S., then, on May 24, they would pay him 51,008, and SI 0.23 as interest. But Brown Bros, owe A. B. S. S228.25 more than he owes them. If there were no interest charge, they could settle by pay- ing him that balance. Since interest is to be considered, they would lose S15.37, and A. B. S. would lose 810.23. Their loss would e.xceed his by S5.14. To prevent this lo.ss, they should retain the S228 25 until it would earn them 55.14 at G%. The intere.«t on 5228.25 is about 5 038 per day. It will take 130 days for 5228.25 to gain 55.14 ; hence. Brown Bros, should retain the balance until Oct. 7. 387. ORIGIN OF UNITS. 1. The yard is the standard from 'vvhich nearly all of our units of measure are derived. It was definitely fixed by what is known as the " Pendulum experiment." 2. A pendulum was found which, at London, at the sea level, vibrated once in a second. It was divided into 391 ,393 equal parts, of which 360,000 were about equal to the yard then most commonly used. By law this became the unit of linear measure, under the name of the vard. It was divided / 388 NEW ADVANCED ARITHMETIC. into three equal parts called feet, and each of these was divided into twelve equal parts called inches. 3. Since a square foot is a square, each of whose sides is a linear foot, it is seen that the units of square measure are derived from the same experiment. 4. Since a cubic foot is a cube each of whose edges is a linear foot, the units of cubic measure have the same origin. 5. A gallon is 231 cubic inches. The bushel is similarly derived. The standard of weight is the Troy pound. Its weight is equal to the weight of about 22.8 cubic inches of distilled water. 388. THE METRIC SYSTEM OF WEIGHTS AND MEASURES. 1. On account of the great variety of scales employed in the common system of weights and measures in use in this country, an effort has been made to introduce the French system. 2. The standard unit is the Meter, which was supposed to equal one ten-millionth of a quadrant of the meridian passing through Paris. It is about | of a yard ; but those using this system must learn to think in its units. 3. Decimal parts of the meter are indicated by prefixing to meter, milli, meaning one-thousandth, centi, meaning one- hundredth, and deci, meaning one-tenth. 4. Decimal multiples of the meter are expressed by pre- fixing to meter, deka, meaning ten, hecto, meaning one hundred, kilo, meaning one thousand, and myria, meaning ien thousand. Note. — The first three prefixes are Latin, and the otliers arc Greek The abbreviations for names containing the Latin prefixes are printed in small letters, and those containing the Greek, in capitals. APPENDIX. 3b9 389. Table of Long Measurt. 10 millimeters (ram.) equal 1 cenlimeler (fin.) equal .3937+ in. 10 centimeters •• 1 decimeter (dm.) •• .■5.!)37+ in. 10 decimeters " 1 imier (m.) " 3!j.37+ in. 10 meters " 1 dekametcr (Dm.) " 32.8+ ft. 10 dekanieters •' 1 hektomelcr (llin.) •' 19.927+ rd. 10 hektomelers " I kilometer (l\n\.) " .()21+ mi. 10 kilometers " 1 myriameter (Mm.) " G.213+ mi. Note. — The uuits most commonly used are printed in italic. 1, Long distances are measured in kilometers. This unit is about g of a mile. 2. The symbol denoting the denomination may be placed after the integral part of the expression ; thus 24 m. oG, or after the entire expression, 24. 5G m. 390. Surface Measure. The surface units are squares each of whose sides is a linear unit. It follows that 100 of each order make one of the next higher. TABLE. 100 sf|. mm. equal 1 sq. cm. 100 sq. cm. '' 1 sq. dm. 100 sq. dm. " 1 sq. m. =1.190 sq. yd. 100 sq. m. " 1 sq. Dm. — 119. P. + s(|. yd. 100 sq. Dm. " 1 sq. Hm. = 2.47 + A. 100 sq. Hm. " 1 sq. Km. = 247.114 A. 1. The square meter is used in the measurement of small surfaces as the square yard is used in the ordinary system. When used to measure land it is called a centare (en). 2. The sq. Dm. is called an are when used as a land measure, and the sq. Hm. a hectare when so used. 26A 390 NEW ADVANCED ARITHMETIC. 391. Measures of Volume. The voliune units are cubes, each of whose edges is a linear unit. In the following table 1000 of each deuomi- nation make one of the next higher. TABLE. 1000 cu. mm. equal 1 cu. cm. lUUO cu. cm. " 1 cu. dm. 1000 cu. dm. " 1 cu. m. = 35.31 G cu. ft. 1- The cu. m. is the unit most commonly used. 2. "When the cubic meter is used in measuring wood it is called a stere. One tenth of a stere is a decistere. 10 steres make a deJcaster (Dst.) 3. The stere is a little more than a quarter of a cord. 392. Tables of Liquid and Dry Measure. 1 milliliter (ml.) = 1 cu. cm. 10 ml. — 1 centiliter (cl.) 10 cl. = 1 deciliter (dl.) 10 dl. = 1 liter (1.) =-- 1 cu. dm. 10 1. 1= 1 decaliter (Dl.). 10 Dl. = 1 hectoliter (HI.) 10 HI. = 1 kiloliter (Kl.) = 1 cu. m. 10 Kl. = 1 myrialiter. Dry. Liquid. 1 liter (1.) = .90H qt. = l.()')7 qt. 1 decaliter (Dl.) = 1.13;') pk. =r 2,642 gal. 1 hectoliter (HI.) = 2.837 bu. = 26.417 gal. 1 kiloliter (Kl.) = 28.37 bu. The liter, which is very nearly the liquid quait, and the hectoliter — about 2g bushels — are the units most commoulv used APPENDIX. 391 393. Weights. The unit of weight is the (jram^ which is the weight of one cu. cm. of pure water at the temperature of greatest density. TABLE. 10 milligrams (mg.) = 1 centigram (eg.) 10 eg. — 1 decigram (dg.). 10 dg. = 1 gram (g.) 10 g. =1 decagram (Dg.). 10 Dg. = 1 iiectogram THg,) 10 Ilg. — 1 kilogram (Kg.) — wt, 1 cu, dm. of water. 10 K. = 1 myriagram (Mg.). 10 Mg. = 1 quintal (Q.) 10 Q. =1 tonueau (T.) =■ wt, 1 cu. m. of water. 1 gram (g.). = 15.432 + grains 1 kilogram (Kg.) = 2.204 + Ih. a\% 1 tomieau (T.) = 2204.621 + lb. av. 1. The units most commonly used are the the centigram, the gram, the kilogram, and the tonueau. 2. The kilogram is called the kilo, for brevity. 394. Table of Approximate Equivalents. 1 decimeter = 4 in. nearly. 1 meter = 1^ yd. " 1 decameter = 2 rd. " 1 kilometer = f of 1 mi. " 1 are' = Jg^of an A. " 1 stere = ^ of a cord." 1 liter ' = 1 liq. qt. " 1 hectoliter = 3 bushels " 1 gram = 15i grains " 1 kilo = 2^ 'lb. av. " > 1 tonueau = W^ tons. " 392 NEW ADVANCED ARITHMETIC. 395. Exercises on the Metric Tables. 1. Reduce 2864 m. to mm. ; to dm. ; to Hm. ; to Mm. 2. Reduce 24685 sq. dm. to sq. cm. ; to sq. Dm. ; to sq. Km. 3. Reduce 2.5709853 cu. dm. to cu. cm. ; to cu. m. ; to decisteres; to dekasters. 4. Reduce 47073 1. to c^ ; to Dl. ; to Kl. ; to dl. 5. Reduce 279436 Dg. to g. ; to mg. ; to Mg. ; to Q. 6. How many sq. m. in the floor of a room 12 m. long and 10 m. wide? 7. Find the length of j-our school-room in meters. Find the width of your desk in decimeters. What is the area of the top of your desk in square decimeters? It is what part of a sq. m. ? 8. Find the number of square centimeters in a pane of glass in a window of your school-room. What part of a sq. m. is it? 9. Lay off in your school-yard a square a dekameter on a side. What is this unit used for? How many square meters does it contain? What is its name? Calling its area 120 square yards, what part of an acre is it? 10. Find the volume of your school-room in cubic meters. If it were filled with wood how many steres would it contam? About how many cords ? 11. What would it cost to lath and plaster the ceiling of your school-room at 28 cents a sq. m. ? 12. If your school-room were used for a grain bin, how many hectolitei-s of shelled corn could be put into it? How many bushels ? If it were a tank how many decaliters of water wduld it hold? How many gallons? 13. How many grains are there in a gram? How many grams in an ounce avoirdupois ? A gram is about what part of an ounce? Of a pound? APPENDIX. 393 14. Compare the kilo with the pound ; the touiieau with the ton. 15. Find the cost of 10 kilos of coffee at 3 cents an Hg. 16. A field is 1 kilometer in length and 5 hectometers la width. What is it worth at 82.25 an are? 17. What is the weight of a liter of i)ure water? 18. The atmospheric pressure under ordinary conditions is about 1.5 pounds to the square inch. What is it in tonneaus per square meter? 19. How many bushels in a quintal of wheat? (Wheat 60 lbs. to bu.) 20. What will a liter of mercury (specific gravity, 130 weigh? Find weight also in pounds. 21. What is the cost of a pile of wood 12 m. long, 2 m. v/ide, and 2 m. high, at Si. 25 a stere? 22. A bin is 10 m. long, 3 m. wide, and 3 m. high. It is filled with wheat. What is the value of the vrheat at 210 cents an HI. ? 23. Find the weight in quintals of the above wheat, count- ing it I as heavy as water. 24. Find the capacity in HI. of a cylindrical cistern whose diameter is 3.2 m., and depth 4.1 m. Note. — The hectoliter is about | of a barreL THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AM INITIAL FINE OF 25 CENTS THIS BOOK ON THE DATE DUE_TH ^^^^ OVERDUE. ^P 11 193. OCT 2 1934 JAN 24 1938 FFB 21 19411 M JUL 20 t©45 LU 2i---:0'H-i5;^- ' '^CO 800574 (3Sv UNIVERSITY OF CALIFORNIA LIBRARY