IN MEMORIAM FLORIAN CAJORl ^^/cE^^ <:?^'>^ Digitized by the Internet Arciiive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/colawarithmeticOOcolarich SCHOOL ARITHMETIC ADVANCED BOOK BY JOHN M. COLAW, A.M. ASSOCIATE EDITOR OF THE AMERICAN MATHEMATICAL MONTHLT, MONTEREY, VA. AND J. K. ELLWOOD, A.M. PRINCIPAL OF THE COLFAX PUBLIC SCHOOL, PITTSBURG, PA., AUTHOR OF TABLE BOOK AND TEST PROBLEMS IN ELEMENTARY MATHEMATICS. t RICHM02TD, VA. B. F. JOHNSON PUBLISHING CO. 1900 Copyright, 1900, by J. M. COLAW AND J. K. ELL WOOD CAJORl /. •..:• • PREFACE. To meet the needs of progre8si\Le teachers^in the best pub- lic schools has been the first aim of the authors in writing the present work. In the. effort to carry out this purpose great care has been taken to make the book modern yet con- servative. It is believed that an examination will show the book to be sound in theory, modern in method, strong in in- ductive work, clear and accurate in all statements, correct in its teaching of business practice, and copious and varied in its supply of practical work and problems. Special attention is called to the following features : 1. Inductive Method. The inductive method is applied throughout the major part of the book. New topics are in- troduced by carefully prepared questions and suggestions designed to develop in the pupiFs mind a correct understand- ing of the principles to be taught, and to give clear insight into arithmetical relations and processes. 2. Oral and Written Work are given with every appro- priate subject, not only to injpress the princii)les thereof, but also to provide ample practice in making solutions and abundant material for mental discipline. To lead the pupil to think in general terms, as well as to familiarize him with the use of symbols and lay a foundation for the study of alge- bra, numerical exercises are followed by others involving the same processes with letters. 3. Rules and Definitions. Few rules are given, and these usually in the form of directions. At the same time there has been no hesitancy in giving a few rules where it was thought they would assist the pupil in gaining a correct 4 PREFACE. and intelligent understanding of the processes, or help him to neat and orderly methods, but in no place is dependence on rules encouraged. The definitions are brief but accurate ; unnecessary ones are omitted. 4. Form of Solutions. The solutions given are intended as far as possible to suggest the reasons for the various steps and to render unnecessary the lengthy, cumbrous, and tedious explanations and analyses often given in arithmetical text- books. 5. Solutions of Problems, with Examples for Prac- tice. This chapter affords the pupil not only examples of neatness and orderly procedure in making solutions, but also a basis for the recognition of similar problems and an aid in devising definite processes for their solution. 6. Practice Work and Problems. The book is believed to be unsuri^assed in the abundance, variety, and excellent character of its practice work and problems, yet the teacher may find it advisable at times to supply additional work along special lines. 7. Arrangement of Topics. The topical plan of treat- ment has in the main been adhered to as giving the best adaptation to a wider range of schools ; but where it could be advantageously done, much valuable information relating to subsequent topics has been introduced in the examples of the earlier sections. In the Primary book the order of topics is not preserved, but the subject matter is presented in the order that has been found to be most helpful, most stimulat- ing, and most practical, the arrangement being determined by the order of the child's mental development rather than by the logic of the subject itself. In this book the usual arrangement of topics has been departed from in a few instances. We note the following : (a) Factoring is clearly taught, but the G. C. D., which is now rarely met with in practical life and which is difficult for beginners to understand, is presented where it can be PREFACE. 5 given in review at a time when the pupil is better able to grasp the logical reasoning it presents. (b) U. S. Money, easily understood by young pupils, is used as an introduction to decimal fractions. (c) Decimal Fractions are treated before common frac- tions, because now more generally employed, and because the formal treatment in connection with integers is more easily understood, the notation on the right of the decimal point being as easily understood as that on the left if the decimal fractions are compared with integers instead of with common fractions. In the Primary book, however, the common fraction is first introduced, because the first fractions with which very young children become acquainted are the half, quarter, etc., and because these are more simple and within the range of the child's visualizing power. In a formal treatment, however, designed for older pupils, the reasons are reversed. (d) Ratio and (Simple) Proportion. Ratio is given in connection with Relation of Numbers on account of their intimate connection. Simple Proportion, furnishing as it does a good basis of arithmetical reasoning, is given earlier than usual as an aid in improving the reasoning ability of the pupil. The treatment is simple and not burdened with unnecessary matter. Compound proportion being little used is given in the Appendix for later review. (e) Practical Mensuration is treated in connection with square and cubic measures, thus enriching the treatment, giving greater variety, and adding interest to the work by showing the practical uses to which these measures are put. (f ) The Metric System is given directly after Compound Numbers for purposes of comparison as well as to provide a variety of problems in the supplementary exercises intended for advanced classes. (g) Subjects of minor importance and such as are suited to 6 PREFACE. later reviews are placed in the Appendix, while subjects of no practical value in a modern arithmetic are omitted. 8. Chapter on the Equation. The importance of in- troducing this simple and easy chapter cannot well be over- estimated. Arithmetical methods are adhered to, and by- simple inductive exercises and comparative solutions the con- ceptions of the use of letters as the. general representatives of positive numbers, and of the equation to express their relations, are developed as far as the purpose of the chapter demands. By the aid of this simple chapter, equations which have a meaning to the pupil can be substituted for the dead for- mulas sometimes used in percentage problems and interest problems, and a much clearer understanding can be had of such subjects as the greatest common divisor, the square and cube roots, etc. The subject is further developed in the brief but attractive chapter on ^^ Introduction to Algebra"' given in the Appendix. 9. Division of Problems. Experience has shown that the same problems should not be solved by the same class from term to term. It is wiser to add fresh fuel than to be constantly stirring the old coals. The pupil's interest must be kept alive. With tliis in view, we have divided the prob- lems into two parts ; the first to be used when the class first goes over a subject, the second when it reviews that subject. For a similar reason the reviews have also been so divided. 10. Character of Problems. In the preparation of problems tlie actual business practice of to-day has been kept in mind. In addition to their practical and matliematical value, many of the problems furnish mucli useful and scien- tific information that is reliable and strictly up-to-date. The aim has been to make them as practical and «^5e/V«» 3,000 yeaTs' --: The European:, a S:?' ''^""^ "^« «'!• cL'tr^ donng the twelfth century "««'^J^tem from the ArabI NOTATION AND NUMERATION. 17 10. In this system of notation ten figures, or digits, are used to express numbers, viz.: 01234567 89 zero one two three four five six seven eight nine The first figure, zero, is also called naught and cipher, and signifies no7ie. It is used with other figures to express num- bers larger than 9. Figures are not numbers ; they are characters or symbols used to express or represent numbers. ONES, TENS, AND HUNDREDS. 11. In counting up to nine we count by ones. After 9 we say ten, but we have no single figure to express this number. Hence to express ten ones, or 1 ten, we combine zero with 1, and write the number thus : 10. 12. Principle. — When a figure stands alone, it expresses ones. When two figures stand side hy side, the one on the right expresses ones, the other tens. Thus, in 2\ the 4 represents 4 ones, and the 2 represents 2 tens. 3 tens, or 20, is called tiventy. 3 '' '' 30, '' '' thirty. 4 .. .. 40, " " forty. 5 " " 50, " " fifty. 6 '' '' 60, '' " sixty. 7 '' '' 70, '' '' seventy. 8 '' '' 80, '' '' eighty. 9 '' " 90, '' " ninety. 13. In counting more than nine we count by tens, as above, or by tens and ones, as follows : 1 ten and 1 one, or 11, is called eleven. 1 ten " 2 ones, or 12, '' '^ twelve. 1 ten '' 3 ones, or 13, " " thirteen. 1 ten " 4 ones, or 14, " '' fourteen. 1 \Qn ^' 5 ones, or 15, " " fifteen. 2 18 SCHOOL ARITHMETIC. 1 ten and 6 ones, or 16, is called sixteen. 1 ten " 7 ones, or 17, '' (< seventeen. 1 ten '' 8 ones, or 18, '' (( eighteen. 1 ten '' 9 ones, or 19, '' it nineteen. 2 tens, or 20, '' a twenty. 2 tens '' 1 one, or 21, '' a twenty-one. 3 tens '' 2 ones, or 32, " (( thirty-two. 4 tens '' 3 ones, or 43, '' i( forty-three. 8 tens '' 7 ones, or 87, " <{ eighty-seven 9 tens '' 9 ones, or 99, " a ninety-nine. 14. What does each figure in the following express p 1. 15 5. 38 9. 6L 13. 84 17. 92 2. 19 6. 43 10. 68 14. 88 18. 97 3. 21 7. 47 11. 73 15. 79 19. 90 4. 26 8. 54 12. 59 16. 81 20. 99 16. When we count one more than 99, we have nine tens and 10 ones, or 10 tens, which is called one hundred. To ex- press this the figure 1 is written at the left of two ciphers, thus : 100. 16. Peinciple. — When three figures are written side hy side, the one at the left expresses hundreds. Thus, in 324 the 4 represents 4 ones, the 2 represents 2 tens, and the 3 represents 3 hundreds. 17. A number expressed by three figures is read without the word and. Thus, 324 is read three hundred twenty-four. 18. Read the following : 17 24 48 27 32 39 41 46 29 50 53 66 65 77 84 90 72 124 231 132 213 346 427 536 175 381 450 619 680 700 798 800 870 660 739 808 987 711 444 330 602 101 507 600 789 Write the third vertical column in words. NOTATION AND NUMERATION. 19 19. Express the following by figures : 1. Fifty. 14. One hundred fifty. 2. Forty-two. 15. Two hundred fifty-four. 3. Sixty-nine. 16. Three hundred seven. 4. Seventy-six. 17. Six hundred forty-five. 5. Thirty-seven. 18. Eight hundred sixteen. 6. Ninety-five. 19. Four hundred fifty-six. 7. Eighty-eight. 20. Five hundred twenty. 8. Sixty-seven. 21. Five hundred nine. 9. Twenty-four. 22. Seven hundred eighteen.. 10. Ninety-nine. 23. Nine hundred three. 11. Thirty-eight. 24. Six hundred sixty-six. 12. Seventy-three 26. Eight hundred eighty. 13. Eighty-seven. 26. Three hundred three. 27. Nine hundreds, three tens, seven ones. 28. Seven hundreds, eight ones, nine tens. 29. Three ones, seven tens, four hundreds. 30. Write from dictation the numbers in Art. 18. Queries. — 1. Wlien 1 stands alone, how many ones does it express ? How many when it stands at the left of a cipher ? At the left of two ciphers ? 2. In the number 11, which 1 expresses the greater value ? How many times as great ? In the number 111 ? 20. Principle. — A 7iy figure represents ten times the value it would represent in the next place to the right, or one tenth of the value it would represent m the next place to the left. 21. The system of counting by tens is called the Decimal System. In the Arabic notation ten of any place or order make one of the next higher order, hence the system is a decimal one. This system derives its greatest importance from the use of zero, which renders possible the giving of place value to the figures. 22. The ones of a number are called units of the first order, or simply imits; the tens are called units of the second 20 SCHOOL ARITHMETIC. order; the hundreds, are called iinits of the third order, and so on. 23. For convenience in writing and reading numbers the figures are divided into groups of three figures each, begin- ning at the riglit. Each group is called ii period, and con- tains ones, tens, and hundreds of that period. The right- hand group is called the period of units ; the second group, the period of thousands ; the third group, the period of mil- lions, and so on. 24. The system of notation is shown in the following TABLE : CO Cfl CO 1 CO C to to . ^ t3 r;^ to ^ r^ CO o^ /— V Orders. -v o 15§ .„ CO O) ^ CO ed-m illion IS ed-th ousai mds i2 CO -^ 1 ts ^X5;2 ±! "S f^ rS £ O ^^1 |§§ ^ Oi-G g CO to W^H WrHpq WHS Khh ffiHO 2 5 6 1 9 7 8 40 5 3 2 ^"stiT ^TtiT ^T ^sT' 1st Period. Period. Period. Period. Period. Trillions. Billions. Millions. Thousands. Units. The number in the table is read two hundred five trillion, six hundred nineteen billion, seventy million, eight hundred forty thousand, five hundred thirty-two. 1. The left-hand period may have only one or two figures ; the others must contain three figures. 2. In writing large numbers, the periods may be separated by com- mas or written slightly apart, as an aid in reading them. 3. Each period is read as if it stood alone, the name of the period being added except in the case of the units' period. 4. The names of periods above trillions are, in order, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, etc. Notation and numeration. 21 25. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. lines. EXERCISES IN NUMERATION. Read the following numbers : 2,375, 5,732, 8,360, 4,07G, 8,007, 52374, 48106, 90245, 13004, 86730, 71,248,359,024, 42,095,384,217, 14,703,692,581, 60,482,604,826, 90,007,080,500, 1,928,374,650,437,689, 1,234,567,891,098,765, 327195, 245680, 503276, 100538, 736451, 370,689,421, 124,986,073, 581,470,369, 726,934,851, 900,604,003, 6,382,541, 2 906 738. 7,328,911, 1 540 006. 5,374,802, 8 000 325. 2,704,190, 6 030 008. 1,398,453, 5 111 290. 4,210,893,657. 7,563,980,124. 2,581,470,369. 8,276,354,090. 3,415,628,072. 7,129,384,756,140,632,687. 3,470,153,647,005,296,304. Write in words the numbers in the 1st, 10th, and 12th EXERCISES IN NOTATION. 26. The periods are written in regular order, beginning with the highest. If any order of units of a period is lack- ing, a cipher must be put in its place ; and if any entire period is lacking, its three places must be filled with ciphers. 27. Write the following in figures : 1. Three thousand, eight hundred forty-five. 2. Eighty-three thousand, seven hundred forty. 3. Seventy-two thousand, three hundred five. 4. Thirty-seven thousand, five hundred twenty-one. 5. Ninety thousand, ninety. Four thousand, six. 6. Fifty thousand, five. Seventeen thousand, one. 7. Ten thousand. Six thousand, one hundred nine. 8. Three hundred forty-two thousand, six hundred sev- enty-five. 9. One hundred six thousand, four. 10. Three hundred thousand, three hundred. 11. Eight hundred six thousand, seventy-five. 22 SCHOOL ARITHMETIC. 12. One hundred seventeen thousand, four hundred two. 13. Eight million, eight thousand, eight. 14. Fifteen million, fifteen thousand, thirty. 15. Five hundred two million, three hundred four. 16. Two million, six thousand, ninety-eight. 17. Six hundred twelve thousand, four hundred sixty-two. 18. Five hundred sixty thousand, four hundred fifty-six. 19. Ten million, one thousand, one hundred. 20. Sixty-five million, one hundred eight thousand. 21. Nine hundred eighty-seven million, four. 22. Four hundred thousand, fifty-six. 23. One hundred ten thousand, ninety. 24. Nine hundred million, three hundred eighty-one. 25. Six million, six hundred thousand, sixty. 26. Fifteen million, nine thousand, seventeen. 27. Four thousand, three hundred six. 28. Fifty-four thousand, three hundred fifteen. 29. Five hundred seven million, sixty-thousand, two. 30. Nine billion, four hundred million, eighty. 31. Write from dictation all the numbers in Art. 25. UNITED STATES MONEY. 28. The money of the United States has a decimal system ; dollars are written as integers, cents and wills as decimal parts of a dollar. Instead of writing the word dollars after a number, we write the '^ dollar mark'' ($) before the number. Thus, 10 dollars is written $10. 29. A point, called the Decimal Point, is placed at the left of the cents, but at the right of the dollar mark. Thus, 25 cents is written $.35. Five cents is written $.05. 1. In writing both dollars and cents, the decimal point is placed between them. Thus, 6 dollars, 15 cents, is written $6. 15. 2. The first two places at the right of the decimal point are read as cents, and the third as mills. Thus, $4,205 is read 4 dollars, 20 cents, 5 mills. NOTATION AND NUMERATION. 23 30. Read the following : 1. $2.75, $12.82, $375.40, $.87, |-^30.05, $100. 2. $16.08, $43.00, $501.09, $6.90, $.54, $9.86. 3. $107.09, $36.80, $275,005, $690.13, $7.56, $5,245. 4. $260,306, $150.05, $309,401, $3060.40, $58,039. til. Write ill figures : 1. Eight dolhirs, ten cents. Twelve dollars. Eight cents. Nine dollars, six cents. 2. Fourteen cents. Three dollars, sixty cents. Fifteen dollars, four cents. Five cents. 3. Sixty dollars, six cents. Two hundred ninety dollars, seventy-two cents. Six dollars. 4. Five hundred one dollars, eight cents. Nine hundred dollars, thirty cents. 5. Nine thousand, seven hundred four dollars, forty, cents. Five dollars, nine cents. 6. Seven thousand four dollars, fifty-one cents. Sixty- five cents. Nine dollars. 32. Read and name the abstract numbers : 1. $275. 6.7016. 9. $.05 13. 24 birds. 2. $3.24. 6. $1. 10. 200. 14. 12 feet. 3. 500. 7. 3 boys. 11. 6 cows. 15. 3250. 4. 5 cents. 8. 3. 12. 1 man. 16. 3 pounds. THE ROMAN NOTATION. 33. This system employs seven capital letters to express numbers, viz.: I V X L C D M 1 5 10 50 100 500 1000 Other numbers are expressed by combining these letters in accordance with the following 34. Prin^ciple. — 1. Repeating a letter repeats its value. Thus, III represents three ; XX, twenty. 24 SCHOOL ARITHMETIC. 2.. When a letter precedes one of greater value, the differ- ence of their values is the number represented. Thus, IV represents the difference between ^ye and one, or four. 3. Wien a letter foUoivs one of greater value, the sum of their values is the number represented. Thus, VI represents the sura of Jive and one, or six, A dash over a letter increases its value a thousand-fold. Thus, V represents five thousand. These principles are illustrated in the following table : I... ...1. II .. ...2. III... ...3. IV... ..A. V... ...5. VI... ...6. VII... ...7. VIII... ...8. IX... ...9. X... ..10. XI.. XII.. XIII.. XIV.. XV.. XVI.. XVII. . XVIIL. XIX.. XX.. 11. 12. 13. 14. ,15. 16. 17. 18. 19. 20. XXI XXIV XXV XXIX XXX XL. L, LX. LXX. LXXX. .21. ,24. 25. ,29. ,30. ,40. 50. 60. 70. 80. xc c. cc ccc CD D DC M Mx\I X ....90. ...100. ...200. ...300. ...400. ...500. ...600. ..1000. ..2000. .10000. This method of notation is called the Roman, because the Romans invented and used it. It was in common use prior to the introduction of the Hindu system, but its use is now very limited. Let the pupil ascertain by observation for what purposes it is now used. 36. Express by the Arabic notation : 1. XXVII, XVIII, XLIX, XLVII, LVI, LIX. 2. LXII, LXIX, LXXV, LXXXIX, XCII, XCVIL 3. CVIII, CXIV, CCIV, CCCLXV, DVII, DCVII. 4. DXLVI, DXCI, DCXI, DCXLIX, MDCCCC. 5. MCDXCII, MDCCLXXV, MDCCCXCIIL 36. Express by the Roman notation : 1. 18, 27, 36, 45, 54, 63, 72, 81, 94, 97, 101. 2. 110, 203, 470, 509, 747, 931, 1009, 1865, 1789. 3. 1607, 1682, 1498, 1620, 1783, 1812, 1901, 2222. ADDITION. 37. Ill one box I liave 4 biisliels of corn, in another 2 bushels, in a third 3 busliels. If I put all into one box, how much corn will be in that box ? Here we have three numbers to be added (addends), and the result — 9 bushels — is the how m?(ch, the quantity, which we wished to find. It is the sum of the parts or addends. 38. The process of finding the sum of two or more like numbers is called Addition. 1. What is the sum of 5 feet, 3 feet, and 7 feet ? 2. What is the sum of 6 quarts and 4 pounds ? 3. Are 6 quarts and 4 pounds like numbers ? 4. Is the sum of two or more numbers the same in what- ever order they are added ? Find out by trial. 39. liike Numbers are those that have the same unit. Thus, 5 yards and 3 yards are like numbers ; but 5 yards and 3 pecks are not like numbers. 1. The sum of 3 feet and G feet is 9 what ? 2. Is the sum always like the numbers added ? 40. Principle. — Onli/ like mnnhers can he added. Find the sum of : 1. 3 and 5. 4. 2, 3, and 6. 7. G boys and 2 boys. 2. 6 and 2. 5. 3, 5, and 4. 8. 3 birds and G birds. 3. 4 and 7. 6. 4, 2, and 7. 9. a cents and a cents. 41. The Sign of Addition (+) is called ^jZ^,"?, and is placed between the numbers to be added. Thus, 5 + 3 is read 5 plus 3, and means that 3 is to be mlded to 5. 42. The Sign of Equality ( = ) is read equals, or is equal to. Thus, 5 + 3 = 8 is read five plus three equals eight. SCHOOL ARITHMETIC. 10. 11. 12. 13. 14. 15. 43. A statement that two numbers or expressions of num- ber are equal is called an Equation. Thus, 5 + 3 = 8 is an equation, 44. Find the sum of the following, and read the complete expressions : 16. 2 + 8 = ( ). 31. 1.7. 2 + 9 = ( ). 32. 18. 3 + 3 == ( ). 33. 19. 3 + 4 = ( ). 34. 20. 3 + 5 = ( ). 35. 21. 3 + 6 = ( ). 36. 22. 3 + 7 = ( ). 37. 23. 3 + 8 = ( ). 3&. 24. 3 + 9 = ( ). 39. 25. 4 4- 4 = ( ). 40. 26. 4 + 5 = ( ). 41. 27. 4 + G == ( ). 42. 28. 4 + 7 = ( ). 43. 29. 4 + 8 = ( ). 44. 2 4- 7 = ( ). 30. 4 + 9 = ( ). 45. All the combinations of numbers from 1 to 9, taken two and two, a given in these 45 equations. The pupil should practice adding the nui bers until able to state the sums at a glance. 45. In blackboard exercises various devices may be used to advantage. Place these diagrams on the blackboard, and indicate with a pointer the numbers to be added. 3 7 "^ 4 1 + 1 = ( ). l + 2==( ). l + 3.= ( ). l + 4 = ( ). 1 + 5 - ( ). l + 6=( ). 7. 1 + 7 = ( ). 8. 1 + 8 =: ( ). 9. 1 + 9 == ( ). + 2=( ). + 3-( ). + 4=(). + 5 - ( ). + G = {). + 5 r= + C = + 7 = + 8=. 5 + 9 = G + G = G + 7 = 6 + 8 = G + 9 = 7 + 7 = 7 + 8 = 7 + 9 = 8 + 8 = 8 + 9 = 9 + 9 = 5 3 8 7 4 G 2 4 9 1. Combine the numbers written at the center and points ADDITION. 27 of the star into a convenient number of exercises; as 4 + 7, 5 + 8 4- 6, etc. 2. Add the number in tlie center of the square to each of the other numbers. 3. Using tlie diagram at the right, begin at a certain number and add in oitlier direction to 50, 100, etc. These exercises should be continued until quick and correct answers can be given. ORAL EXERCISES. 46. 1. Edna lias G bool^s and Ida has 7. IIow many have they both ? 2. Mario paid 8 cents for paper, 5 cents for ink, and 7 cents for pencils. How much did she pay for all ? 3. James is 9 years old, and Tom is 6 years older. How old is Tom ? 4. Six chickens are on the fence, 8 are in the barn, and 5 are picking grass. How many are there in all ? 5. 1 paid $4 for a hat, 19 for a coat, and had $7 left. How much had I at first ? 6. A boy walked 3 miles to the station, then rode 10 miles in the cars, and then went 8 miles in a coach. How far did he travel ? 7. A farmer killed 3 pigs and 2 sheep, and had 9 pigs and 4 sheep left. How many of both had he at first ? 8. How many days are 6 days, 5 days, and a week ? 9. How many letters in the names of all the days of the week ? 10. What is the sum of the numbers represented by the ten digits ? 11. Harry ran twice around a house 3 rods long and 2 rods wide. How far did he run ? 12. Mary has 5 cents, Kate has 4 cents, and Jane has 7 cents more than Kate. IIow much have Jane and Mary to- gether ? 28 SCHOOL ARITHMETIC. 13. A lias $3, B has $5, and C has as much as both. How much have they all ? 14. One man has a dollars and another has 2a dollars. How much have both ? 15. Ella has 2^ cents. May has 3^ cents, and Tillie has as many as both. How many cents have they all ? 16. What is the sum of a, 2«, ia, and 5« ? 47. In the following exercise add 3 to each number in column A, then to each in column 10 4- 3, 20 + 3, etc.; then, 31 + 3, E 24 64 54 14 74 84 34 94 - 44 When the class can announce these results at sight, the exercise may be varied by substituting one of the other digits for 3. Using all the digits gives 729 examples. 48. 1. Add by 2's from 2 to 50. From 1 to 49. 2. Add by 3'*s from 3 to 30. From 1 to 31. 3. Add by 3's from 2 to 32. From 52 to 70. 4. Add by 4's from 4 to 40. From 1 to 45. 5. Add by 4's from 2 to 34. From 3 to 51. 6. Add by 5's from 1 to 41. From 2 to 52. 7. Add by 5's from 3 to 43. From 4 to 64. 8. Add by 6's from 6 to 72. From 1 to 49. 9. Add by 6's from 2 to 50. From 3 to 45. 10. Add by 6's from 4 to 46. From 5 to 53. 11. Add by 7's from 7 to 84. From 1 to 50. A B C D 10 31 52 83 20 41 92 53 60 81 12 73 80 91 42 23 30 21 62 13 90 51 32 63 50 71 82 43 40 11 72 33 70 61 *22 93 t B, and so on. '\ 'hus, :1 -f- 3, etc. F G H I 95 46 67 78 35 16 77 88 45 36 27 98 75 66 57 38 55 86 97 48 15 76 47 28 25 96 17 68 65 26 87 58 85 56 37 18 ADDITION. 2^ 12. Add by 7's from 2 to 58. From 3 to 66. 13. Add by 7's from 4 to 60. From 5 to 75. 14. Add by 7's from 6 to 83. From to 49. 15. Add by 8's from 8 to 96. From 1 to 97. 16. Add by 8's from 2 to 82. From 3 to 83. 17. Add by 8's from 4 to 100. From 5 to 93. 18. Add by 8's from 6 to 94. From 7 to 167. 19. Add by 9's from 9 to 108. From 1 to 109. 20. Add by 9's from 2 to 92. From 3 to 102. 21. Add 9 eight times to 4. To 5. To 6. To 7. WRITTEN EXERCISES. 49. Copy and add each column from bottom to top, then from top to bottom ; also add each line, (as a), from left to right, then from right to left. In adding, do not say 7 and 2 are 9, and 4 are 13, etc., but give results onl> -. T] bus, 7, 9,13, 19, etc. 1 2 3 4 5 6 7 8 9 10 11 a 5 3 6 i 5 9 8 7 4 9 3 b 3 6 5 4 7 6 4 6 8 4 8 c 6 4 rv 5 6 4 5 9 7 6 • 7 d 4 7 3 9 4 7 9 6 5 6 e 2 8 4 3 9 8 7 8 3 7 9 f 7 5 9 8 8 5 3 4 9 8 5 12 13 14 15 16 17 18 19 20 21 22 693889754 7 6 h 5 8 4 7 3 5 3 1 8 9 7 i 8 4 5 6 9 7 8 4 7 6 5 J 7 6 8 3 6 8 4 9 5 3 8 k 3 7 7 4 5 6 5 2 9 8 4 1 2 5 6 5 8 3 9 8 7 5 9 m 9 9 9 9 6 9 6 6 6 9 6 n 7 3. 7 5 7 5 7 7 8 2 8 4 5 8 7 5 6 8 3 7 7 30 SCHOOL ARITHMETIC. 23 24 25 26 27 28 29 30 31 32 33 p 2 6 4 5 6 7 8 9 8 7 6 q 4 5 6 7 8 9 7 6 5 4 3 r 6 7 8 9 < 6 5 4 3 2 9 s 8 9 7 6 5 4 9 7 6 8 7 t 3 4 5 7 6 5 7 5 4 5 8 u 5 8 9 8 4 8 6 7 7 9 6 V 9 6 5 4 3 7 9 6 9 6 9 w 7 9 6 9 8 9 8 8 8 7 7 X 4 5 7 6 9 6 5 9 6 8 6 y 8 7 8 7 6 4 7 7 5 3 9 z 6 8 9 8 7 3 4 6 7 9 8 ORAL EXERCISES. 50. 1. IS'ell paid 20 cents for a slate and 30 cents for a book. How much did she pay for both ? 2. There are 30 days in April and 31 in May. How many days in both months ? 3. In an orchard there are 40 apple trees, 25 peach trees, and 12 pear trees. How many trees are there in the or- chard ? 4. William is 12 years old, and his father is 27 years older. How old is his father ? 5. How many days are there in May, June, and July ? 6. If Edgar has $25, and Allen has 120, and I have $15 more than they both have, how much have I ? 7. An army marched 25 miles one day and 4 miles far- ther the next day. How far did it march in both days ? 8. A drover sold 25 sheep to one man and 42 to another, and had 23 left. How many had he at first ? 9. A field is 40 rods long and 16 rods wide. What is the distance around the field ? 10. A newsboy sold 40 morning papers, 25 evening papers, and had 12 left on his hands. How many did he buy ? ADDITION. 31 11. A dealer sold 25 bushels of coal to one man, 35 to another, and 32 to another. How many bushels did he sell to all ? 12. After paying $33 for a cow and $51 for a wagon, I had left a ten-dollar bill, a five-dollar bill, and a silver dollar. How much money had I at first ? 13. Forty-one years ago Jane's grandpa was 36 years of age. How old will he be if he lives 5 years longer? 14. A lady bought 3 webs of cloth, the first containing 32 yards, the second 41 yards, and the third 36 yards. How many 3^ards did she buy ? 15. On Monday Ada's teacher gave her 20 words to spell, on Tuesday 25, on Wednesday 25, on Thursday 30, and on Friday 18. How many words were given in the week ? 16. Willie has a marbles, Harry has 4«, and Rob has as many as both. How many have they all ? 17. A man had a pigs and bought b more. How many did he then have ? (« + b). 18. Sarah has a cents, Alice has b cents, and Ellen has as many as Sarah. How many have they all ? 19. What is the sum of «, da, da, and 2b ? 20. If one stone weighs ]) pounds and another weighs q pounds, what is the weight of both together ? WRITTEN EXERCISES. 51. 1. What is the sum of 7597, 1368, and 643 ? 7597 May the numbers be added without placing them in vertical 1368 columns? Which way is the more convenient? 643 Are the tens all in the same column? In which column are ggQg the ones? The hundreds? What is the sura of the ones? Where is the 8 written? What is done with the 1 ten? What is the sum of the tens? What is done with theO? With the 2 hundreds? Explain bow the 9 thousands is obtained. 372 = 300 + 458 = 400 + 765 = 700 -f 70 + 50 + 60 + 2 8 5 1595 = 1400 + or 1500 + 180 + 90 + 15 5 32 • SCHOOL ARITHMETIC. 2. Find the sum of 372, 458, and 765. We here see that the sum of the ones is 15 ; the sum of the tens 18, or 180 ones ; the sum of the hundreds 14, or 1400 ones. Since 15 = 1 ten and 5 ones, we have 18 + 1, or 19 tens, or 190 ones. And since 190 = 1 hundred and 90 ones (9 tens), we have 15 hundreds, or 1500 ones. Hence, 1500 + 90 + 5 = 1595, the sum. Direction. — Arrange the numbers so that units of the same order shall stand in the saine colufnn. Begin at the right and add each column separately. If the sum is less than 10, write it tinder the column added. If it is 10 or more, place only the right-hand figure under that column, and add the remainder with the next column. To insure accuracy, add each column from top downwards as well as from hottoin upwards. Write from dictation and add : 3 4 5 6 7 8 476 418 374 293 416 567 358 736 627 431 907 891 924 375 258 827 692 234 265 632 196 546 375 432 When long columns are to be added, the following method will be found practical. Explain the process. 327 841 576 439 765 192 30 31 28 3140 ADDI riON. 33 Add in two ininu tes : 9 10 11 12 13 14 oG43 1234 9876 9 568 6677 7328 5678 5432 87 7329 896 3576 9123 2347 654 8695 54 8794 4567 678 3210 87 7 4887 8923 4706 123 9764 88 6452 4567 934 45 9 945 7968 9876 87 6 846 8776 5679 5432 5878 5000 6375 9438 9785 7777 6660 9999 8888 7897 Find the value of the following : 15. 728 + 2076 + 7583 + 7127 + 928 + 8267. 16. 83267 + 48 + 2498 + 6735 + 9476 + 10783. 17. 9847 + 3084 + 86540 + 4306 + 89 + 9008. 18. 63075 + 350 + 4703 + 3715 + 937 + 15008. 19. 16160 + 6048 + 260 + 4826 + 50598 + 8763. 20. 357 + 913 + 579 + 135 + 791 + 638 + 794 + 867. 21. 5894 + 876 + 93 + 4672 + 8975 4- 8456 + 9784. 22. 9768 + 87 + 69 + 764 + 893 4- 5768 + 88 + 8769. 23. Add 8568, 3864, 6846, 5976, 7249, 4839. 24. Add 3925, 6868, 4857, 8394, 8426, 9397. 25. Add 3987, 4876, 9254, 7983, 8427, 2935. 26. Add 5763, 3854, 87, 609, 975, 4508, 7358. 27. Add 4978, 9834, 734, 5627, 8, 3764, 47, 835. 28. Add 6984, 8592, 5807, 74, 96, 8958, 64, 789. 29. Add 5926, 7859, 4768, 8729, 384, 769, 8943. 30. Add 8592, 5678, 6854, 4673, 7968, 6843, 4396, 78. 31. Add 476, 5814, 6820, 21, 657. 32. Add 5215, 4863, 9211, 20781. 33. Add 48'i2, 9712, 48, 63, 14. 34. Add 97, 86741, 2248, 6127, 487, 57120. 35. Add 282, 4789, 63175. 3 34 SCHOOL ARITHMETIC. RAPID ADDITION. In rapid addition various methods are employed by ac- countants, one of the most commonly used being what may be called the '^grouping method/' which is here briefly illustrated. *(a) (b) (c) W (e) (f) (g) (h) 5-| 91 r 6'] 9" J8 7 4 7 2 iJ iJ 3_ (9" 3 6 5 3- 8 (7 8~ (^-- 1 9 2 7" 3 8 6-1 (4" 1 ]o 3J (3- 6_ 9_ (5J 6 4 2 2 3 7 8 6 9 6- 4_ ^1 j^- Sj 18 5" 7_ is-] 6_ 4 2 1 4 4 6 38 "37 Yo 51 51 8 5 8 In adding column (a), we may group two or more numbers whose sum is 10, and add thus : 10, 18, 28, 38. In (b) the sum may be 10 or less, and we say, 9, 19, 27, 37. In (c) we may say, 8, 17, 26, 33, 40. Or, making the sum greater than 10, we may add thus : 14, 26, 40. In (d) we may proceed thus : 12, 25, 39, 51. In (e) our count may be, 14, 27, 43, 51 ; or, beginning at the top, 17, 32, 45, 51. * For the first practice in grouping, the teacher should provide num- bers that readily fall into groups whose sum is 10, as in (a). These exercises should be followed by others where the Sum is 10 or less, as in (b), and these in turn by more difficult ones where the sum of the group is from 10 to 19. As a preparation for this work, the pupil should be given abundant practice in announcing at sight the sum of any two numbers between 10 and 20 ; as 11 + 11, 11 + 12, 12 + 13, etc. There are 81 of these com- binations. Exercises in rapid addition should be regular, not spasmodic. Sur- prising results may be obtained in a single term by devoting ^I'e minutes each day to intelligent practice. ADDITION. 35 36. Try to group the mimbers in (f). Which do you find the better place to begin — top or bottom ? 37. Group the numbers in (g), and add. 38. Group the numbers in (h) in as many ways as you can. AVhicli way is best ? DOUBLE-COLUMN ADDING. Many persons add two columns at once, some three columns. The process is as follows : 5731 Beginning at the bottom we add tens and ones, saying (or 3287 thinking), 6-4, 10-7, 16-3 (that is, 15-13), 19-2 (or 18-1*2), 27-9, 31-0 (or 30-10). The sura is 31 tens and no ones. When we get 10 or more ones, we "carry" the 1 ten over to the tens cohimn. Thus, when our sums are 15 tens-13 ones, we say 16-3. In this way we avoid having to remember 7664 more than one figure in tlie right-hand cohimn. ^^77^ Taking tlie second two columns we count thus : 7-6, 14-5, ^^^ 19-3, 27-9, 31-1, 36-8. The sum is 36 thousands and 8 3^8 hundreds. 37110 ^^ before, whenever we have a sum of 10 or more in the right-hand column, the 1 ten is added with the other column. Thus, instead of 13-15 we have 14-5 ; instead of 18-13 we have 19-3. Add each of the following in two minutes : 39. 40. 41. 12345 (> 78 1234567890 78517 6 89 7486 3 925 987654 5 6 78 56395467 57687436 2357898765 3 417 3245 83796894 4 2 13456843 129 5 102 3 46878513 87 6 5987 6 54 89628790 89454789 3989739437 563954 6 7 65987665 876 6 874989 2 3 062134 5 4668937 7898765346 9573987 8 787 9 9876 6 3 74942105 7788699 2 3 784 5 498 8387659872 86947687 9957 6 747 9876543218 68598769 678987 6 5 87 6 5466789 99679578 8629 4856 6943 8g SCHOOL ARITHMETIC. 41 Add 382 thousand, 4 hundred 53 ; 514 thousand, 6 hundred 85 ; 684 thousand, 3 hundred 25 ; 298 thousand, 5 hundred 76 ; 176 thousand, 7 hundred 92. 43. Add 24 million, 356 thousand, 8 hundred 13 ; 92 million, 75 thousand, 3 hundred 46 ; 7 million, 310 thousand, 1 hundred 6 ; 30 million, 30 thousand, 3 hundred 30 ; 8 million, 8 thousand, 8 hundred 8. 44. Add 376 million, 724 thousand, 9 hundred 86; 4 million, 315 thousand, 8 ; 591 million, 304 thousand, 81 ; 79 million, 58 thousand, 627 ; 83 million, 726 ; 819 million, 10 thousand, 50. 45. Add six million, two thousand, five hundred forty- one ; eight million, seven hundred thirty-eight thousand, four hundred three; one million, seven thousand, nine; thirty million, eighty-nine thousand, fourteen; five hundred nine thousand, eight hundred thirty. 46. Find the sum of two million, nine thousand, foilr hundred seventy-six ; seven million, forty thousand, sixteen ; three million, twenty-four ; nine hundred three thousand, ten ; six million, six thousand, six hundred six. 47. A railroad train ran 376 miles on Monday, 298 miles on Tuesday, 437 miles on Wednesday, 326 miles on Thursday, 265 miles on Friday, 368 miles on Saturday, and 20 miles on Sunday. How many miles did it run in the week ? 48. In one year a man pays $375 for rent, $537 more than that for other expenses, and has $513 left out of his salary. What is his salary ? 49. A farmer sold 289 bushels of corn to one man, 397 to another, 568 to another, 197 to another, and then had 685 bushels left. How many bushels had he at first ? 60. A man owns five lots. The first cost $325, the second $275, the third $450, the fourth $580, and the fifth $240 more than the other four. How much did they all cost? 51. The area of New York is 49,170 square miles; of ADDITION. 37 Pennsylvanhi, 45,215; of New Jersey, 7,815; of Delaware, 2,050. What is the entire area of these four States ? 62. A mail traveled 328 miles a day for three days, and 276 miles a day for the next three days. How far did he travel in the six days ? 53. The first year a man worked in an iron mill he re- ceived $450. If his salary was increased $125 a year, for four years, how much did he earn in the five years ? 54. In a stock-yard there are as many cows as calves, and as many sheep as hogs. If there are G8 calves and 97 hogs, how many animals are in the yard ? 55. The school-yard fence has 289 pickets on the front, and an equal number on the back. On each end there are 257. How many pickets are there on the whole fence ? 56. One side of a square farm is 2854 feet long. What is the distance around the farm ? 57. A book-case has six shelves. The first two contain 28 books each, the second two 34 each, and the last two 39 each. How many books are there on the six shelves ? 58. A merchant sold goods to the amount of $485 on Monday, and during the remainder of the week he increased his sales $98 each day. What was the total value of his * week's sales ? 59. A girl has a dollars, her brother has h dollars more than she has, and their father has h dollars more than both. How much have all three ? ^ 60. A train makes the round trip between Philadelphia and Pittsburg every two days. If the distance between these cities is 354 miles, how many miles does the train run in 4 days ? 61. Mr. A bought a lot for S875, and sold it so as to gain $750. At what price did he sell ? 62. A man owning a large tract of land divided all of it 38 SCHOOL ARITHMETIC. except 160 acres amoDg his 4 sons and 3 daughters. To the eldest son he gave 320 acres, and to each of the others he gave 180 acres. To each of his daughters he gave 240 acres. How many acres were in the tract ? 63. The fiumber of lobsters caught in 1889 was as follows; Maine, 12,552,866; Massachusetts, 2,624,218; Connecticut, 687,994; Rhode Island, 538,315; New Hampshire, 176,733; New York, 206,875 ; New Jersey, 74,866 ; Delaware, 3,750. How many were caught ? 64. The peanut crop of Virginia in 1890 was 1,171,624 bushels ; of West Virginia, 39 bushels ; of Georgia, 624,528 bushels ; of Pennsylvania, 22 bushels ; of New York, 106 bushels ; of Illinois, 481 bushels ; of Tennessee, 523,088 bushels ; of North Carolina, 421,138 bushels. What was their combined production ? 65. In the year ending January 1, 1895, Kentucky pro- duced 183,618,425 pounds of tobacco ; Virginia, 35,593,984 pounds ; Ohio, 32,468,938 pounds ; Massachusetts, 3,449,655 pounds ; Maryland, 7,010,380 pounds ; and West Virginia, 2,634,585 pounds. How many pounds did all six States produce ? 66. The bank clearings June 11, 1900, in Charleston, S. C, were as follows : People's National Bank, 132,125 ; Bank of Charleston, $30,219 ; South Carolina Loan and Trust Co., $21,041 ; First National Bank, $25,968. What was the total clearinofs of these banks ? SUBTRACTION. 62. 1. If Kate has $5 and spends $3, how much has she left? 2. How many apples are left when 4 apples are taken from 6 apples ? 3. Seven marbles are how many more than 5 marbles ? 4. What number of cents added to 5 cents will make 9 cents ? 5. How many dollars are left when $5 are taken from $8 ? 53. The number that is left when one number is taken from another is called the Remainder or Difference. 54. The process of finding the remainder or the difference between two like numbers is called Subtraction. 1. Can two yards be taken from 3 pounds ? Why not ? 2. What is the remainder when $2 is taken from $5 ? How does $5 compare with the sum of $2 and this re- mainder ? 3. Subtract one number from another and see what the remainder added to the smaller number equals. 55. Pkinciples. — 1. JVumbers can be subtracted from like numbers only. 2: The larger number is equal to the sum of the remainder and the smaller number. 56. The larger number, or the one from which another is subtracted, is called the Minuend. 57. The smaller number, or the one subtracted^ is C9;lled the Subtrahend, 40 SCHOOL ARITHMETIC. 58. The Sign of Subtraction ( — ) is called 7ninus, and is placed before the number to besnbtracted. Thus, 5 — 3 is read 5 minus 3, and means that 3 is to be subtracted from 5. 59. Practice should be exercises until pupils can gi 1. 1 + ( ; 2. 2. 2 - 1 =^ ( ). 3. 1 -f ( ; ) = 3. 4. 3 - 2 =z ( >• 5. 3 - 1 = ( )• 6. 1 + ( ) := 4. 7. 4- 3 =1 ( ). 8. 4 - 2 = ( ). 9. 4 - 1 z=z ( ). 10. 1 + ( ; = 5. 11. 5-4 = ( ). 12. 2 + ( , =. 5. 13. 5 - 3 = ( )■ 14. 5 - 2 — ( )• 15. 6 - 5 ■=. ( )• 16. G - ( ) = 2. 17. 6 - 3 = ( )• 18. (3 - ( ) =: 5. 19. 7 - G - ( )• 20. 7 - ( ) — 2. 21. 7 - ( ) ■= 3. 22. 7 - 3 =: ( )• 23. 7 - 2 = ( )■ 24 8 - 7 z=z ( )• 25. 8 - ( ) = 2. 26. 8 - ( ) zz: S. 27. 8 - 4 = { ). 28. { )- 3 = 5. continued upon the following ve quick and correct answers. 29. 8 - ( ) =: 6. 30. 9 - 8 =( ). 31. 9 - ( ) = 2. 32. 9 - 6 = { ). 33. 9 - ( ) = 4. 34 ( )- 4 =. 5. 35. 9 - 3 = i ). 36. 9 - ( ) = 7. • 37. 10 - 9 = ( ). 38. 10 - ( ) = 2. 39. 10 - 7 = 40. 10-6 =r 41. 10 - ( ) = 42. 10 - 4 = 43. 10 - 3 rr 44 11 - 9 nr 45. 11 - 8 = 46. 11 - ( ) = 47. 11 - ( ) :^ 48. 12 - 9 = 49. 12 - 8 = 50. 12 - 7 = 61. 12 - ( ) = 52. 13 - 9 r= 53. 13 - 8 = 54 13 - 7 == 55. 13 - ( ) =r 7. 56. 13 - 5 ={ ). 6. SUBTRACTION, 41 67. 13 - ( ) == 9. 65. 15 - ( ) = 8. 58. 14 - 9 = ( ). 66. 15 - G = ( ). 59. 14 - 8 = ( ). 67. 10 - 9 =( ). 60. 14 - ( ) = 7. eS. 16 - { )=z 8. . 61. 14 - G =( ). 69. IG - 7 =( ). 62. 14 - ( ) == 9. 70. 17 - 9 = ( ). 63. 15 - 9 ={ ). 71. 17 - ( ) = 9. 64. 15 - 8 ={ ). 72. 18 - 9 = ( ). 73. Subtract by 2's from 30. From 31 to 1. 74. Subtract by 3's from 3B. From 34 to 1. 75. Subtract by 4's from 40. From 35 to 1. 60. In tl)e following exercises name only sums and differ- ences. Thus, in the first example, say, 8, 12, 7, 13, 6, 1. Find the value of : 1. 8 + 4-5 + 6-7-5. 7. 8 + 9-4-7 + 12-9 2. 7-3 + 8-5-4 + G. 8. 17-8 + G-7 + 9-8 3. 4 + 5-G + 8-7 + 9. 9. 7 + 9-8 + 5-7 + 14 4. 9-7 + G + 5-8-3. 10. G + 8-9 + 6-4 + 17 5. 3 + 8-6 + 9-G-5. • 11. 5 + 11-7-5 + 9-8 6. 7 + 6 + 2-6 + 7-9. 12. 9-7 + 8 + 6-7 + 11 61. In the following mime only remainders, thus : 46, 43 38, 36, etc. 1. 50 -4 -3 -5-2-4-3-5-6-4- 3 -2-5 = ( ) 2. 50 -5-4-2- 6 -4 -7-5-3-5-2- 3 -4:=( ) 3. 50 - 3 - 5 - 5 - 3 - 2 - 3 - 5 - 5 - 6 - 4 - 7 - 2 r= ( ) 4. 50 -6-3-4-5-4 -6-7-4- 3-5 -2-l = ( ) 5. 50 -7-4-5-6- 3 -4-2-3-5-4-3 -3=.( ) 6. 50-3-2-8-5-4-6-l-9-3-4-3-2i:=( ) 62. When several numbers are included in a Parenthesis ( ), they are to be treated as a single number. Thus, (5 + 3) — (8 — G) means that the difference between 8 and 6 is to be taken from the sum of 5 and 3. The parenthesis as a sign of aggregation was first used by Girard in 1629. 7. 4 + 5 + 6- (8 - 3). 8. 3 + 7 + (6- ■ 4) - 5. 9. 5 + 8 -(9 + ■7-6). 10. (16- 9) +8 - (12 - 5). 11. (16- 9) + 8 + (12- 5). 42 SCHOOL ARITHMETIC. 63. Find the value of : 1. 8 + 5 - (3 + 6). 2. 7 + 9 + (G - 4). 3. 15 - (2 + 7) - 4. 4. (17-9)-(13-7). . 5. (16 - 7) + 6 - 8. 6. 15-6 + 5-7. 12. (18 - 4) - (13 -9 + 8). 13. Willie bought 12 marbles and gave 5 of them to his brother. How many did he keep ? 14. Mary earns 112 a month and spends 17. How much does she save a month ? 15. Tillie has 9 cents and Lottie has 4. How many more cents has Tillie than Lottie ? 16. Howard wants to buy a book for 17 cents, but has only 9 cents. How many cents must he get before he can buy the book ? 17. Tv/o pieces of cloth contain 16 yards. If there are 7 yards in one piece, how many arg in the other ? 18. In a class of 14 boys and girls there are 5 boys. How many girls are in the class ? 19. Henry paid $19 for a cart and sold it for $9. How much did he lose ? 20. If a farmer avIio had 19 sheep sold 11, how many had he left ? 21. Mrs. E had two ten-dollar bills, and spent $4 for shoes and $8 for a dress. How much money had she left ? 22. A farmer sold 7 pigs and 2 more died. If he had 11 left, how many had he at first ? 23. Carl is 11 years old and his sister is 7 years younger. How old is his sister ? 24. Sarah is 6 years old and her brother is 14. What is the difference in their ages ? 25. Alice has 12 plants, of which 7 are geraniums and the remainder pinks. How many pinks has she ? SUBTRACTION. 43 26. A hunter shot 15 rabbits and squirrels. If he shot 5 rabbits, how many squirrels did he shoot ? 27. A man has $19 in gold, silver, and paper. If he has 4 silver dollars and 5 paper dollars, how much has he in gold ? 28. A boy has 18 rows of potatoes to hoe. In the fore- noon he hoed rows, in the afternoon 7 rows, and finished the next day. How many rows did he hoe the next day ? 29. A has 19 acres of land and B has 9 acres. How many acres must B purchase from A so that B may have 10 acres more than A ? 30. Harry had 17, Carrie had *9, and Ella had $8 less than both together. How much had they all ? 31. A lady who had 18 chickens sold 5 to one man and 8 to another, and gave a pair to her sister. How many had she left ? 32. A farmer having 7 horses sold them, and bought 9 from one man and G from another. . He afterwards sold 8, and 1 died. How many had he then ? 33. James had oa cents and spent a cents. How many cents had he left ? 34. AYarren having bb marbles lost 2b of them. How many had he left ? 35. Hattie is a years old and Martha is b years old. What is the difference in their ages ? (a — b). 36. Walter has a dollars, George has 2a dollars, and Albert has b dollars less than both. How many dollars has Albert ? 37. A lady who had 7a hens sold a hens to one man and 4a hens to another. How many had she left ? 38. Nettie weighs a pounds, Ida weighs b pounds less, and Mabel weighs a pounds more than both. What is MabeFs weight ? 39. Find the value of a + 3a -\- a — 4:a + b. 40. AVhat is the value of 6a + (3« — a) — 4:a — 2b ? 41. A man having 16 sheep sold 9, and after buying some more had 13. How many did he buy ? 44 SCHOOL ARITHMETIC. 42. Tom was 5 years old 10 years ago. How old was he 5 years ago ? 43. Jerry was a years old J years ago. How old was he a years ago ? 44. David will be 18 years old 7 years hence. How old was he 7 years ago ? WRITTEN EXERCISES. 64. 1. What is the difference between 978 and 435 ? 978 Is it necessary to write the smaller number under the larger ? 435 Is it convenient ? w .„ Under what is the 5 ones written? Where is the 3 tens placed? The 4 hundreds? 5 ones from 8 ones leaves ones; 3 tens from 7 tens leaves tens; 4 hundreds from 9 hundreds leaves hundreds. The required difference is 543, Query. — Is the sum of 543 and 435 equal to 978? Does that prove the work to be correct ? Find the differences, and prove : 2. 3. 4. 6. 84327 76594 69748 $86.75 53124 43251 27413 14.32 6. 7. 8. 8374967H 9997785( ) 66887795 51427343 05424351 L 34465672 9. 10. 9668987957845 8379568978694 4357534825532 1234567876543 11. 8978564837695874968977896708 3526153726460530733845421602 Find the valne of : 12. 9889 - 6345 + 234. 17. 7968 - 6725 + 3009. 13. 7658 - 324 - 5123. 18. 2436 + 7432 - 6143. 14. 9768 - 43 - 6402. 19. 3254 + 435 - 634. 15. 6995 - 4070 - 825. 20. 8776 - 4345 - 2231. 16, 5683 - 5461 + 1012. 21, 9000 + 678 - 7075. SUBTRACTION. 45 22. Mr. C bought a lot for $5375 and sold it for $6798. How much did he gain ? 23. A farmer who raised 876 bushels of corn kept 243 bushels for his own use and sold the remainder. How many bushels did he sell ? 24. A man was born in 1873 and married in 1898. At what age did he marry ? 25. A contractor agreed to build a house for $5875. If his expenses were $3725, how much did he make ? 26. In a school of 2879 pupils there are 1324 boys. How many girls are in the school ? 27. A man gave a wagon and $135 for a horse worth $189. How much was he allowed for his wagon ? 28. A and B were 198 miles apart. They traveled toward each other one day, A going 42 miles and B 35 miles. How far apart were they then ? 29. Twelve years ago Ella^s grandpa was 59 years of age. How old will he be if he lives till 1909 ? 30. Tom ran around a barn 45 feet long and 23 feet wide, and Pat ran 287 feet. How much farther did Pat run than Tom ran ? ORAL EXERCISES. 65. 1. Subtract by 5's from 49 to 4. From 47 to 2. 2. Subtract by 6's from 42 to 6. From 41 to 5. 3. Take 7 from 58 eight times. From 60. 4. Take 8 from 49 six times. From 51. 5. Subtract by 9's from 54 to 0. From 64 to 1. 6. Count by 7's from 2 to 65, and back from 65 to 2. 7. How many are 4 tens less 3 tens ? 40 — 30 ? 8. How many are 7 tens less 4 tens ? 70 — 40 ? 9. How many are 8 tens less 3 tens ? 80 — 30 ? 10. How many are 9 tens less 6 tens ? 90 — 60 ? 11. If you buy a dozen eggs for 17 cents, how much change should you get out of a quarter ? 46 SCHOOL ARITHMETIC. 12. I paid $27 for a chair and a clock. If the cliair cost $18, what did the clock cost ? 13. From a cask containing 25 gallons, IG gallons were drawn. How many gallons remained ? 14. An agent purchased books for $17 and sold them for $32. How much did he gain ? 15. A farmer sold a cow for $41, which was $13 more than the cost. Find the cost. ; ,. 16. James earns $35 a month and spends '$17. How much does he save monthly ? ..^ 17. A man having 49 sheep sold 19 and killed 11. How many had he left ? 18. A farmer having 21a sheep sold 13a and killed 2a. How many had he left ? 19. When the minuend and the remainder are given, how can the subtrahend be found ? Why ? 20. If the minuend is 7a, and the remainder 2a, what is the subtrahend ? 21. When the remainder and subtrahend are given, how is the minuend found ? Why ? 22. If the remainder is h and the subtrahend 5b, what is the minuend ? 23. Mr. B had a twenty-dollar bill, a ten-dollar bill, and a five-dollar bill. He bought a hat for $6 and a coat for $13. How much money had he left ? WRITTEN EXERCISES. 66. 1. Subtract 375 from 632. 632 Since we cannot take 5 ones from 2 ones, we add 1 of the 3 375 tens, or 10 ones, to the 2 ones, making 12 ones. Then 5 ones from ^T^ 12 ones leaves 7 ones. Having taken 1 ten from the 3 tens, but 2 tens remain, from which we cannot take 7 tens. Hence, we take 1 of the 6 hundreds, or 10 tens, and add it to the 2 tens, making 12 tens. Then 7 tens from 12 tens leaves 5 tens. SUBTRACTION. 4'^ From the G hundreds we have already taken 1 hundred, leaving 5 hundreds, from which wc now take 3 hundreds. Queries. — Is the sum of the subtrahend and the remainder equal to the minuend ? What does that prove ? If we say 5 and *t, 12 ; 7 and 1 and 5, 13 ; 3 and 1 and 2, 6, we subtract by the common method^ of *' making change." Can you explain the process ? 2. From 543 take 29G. 543 =^ 5.00 + 40 + 3 = 500 + 30 + 13 = 400 + 130 + 13 296 = 200 + 90 + 6 = 200 + 90 + 6 = 200 4- 90 + 6 247 = 200 + 40 + 7 Subtract and prove : 3. 4. 6. 6. 7. 8. 9. 211 311 773 521 626 889 342 199 279 539 234 481 691 146 10. 11. 12. 13. 14. 15. 16. 817 824 609 508 712 793 690 677 459 368 146 299 679 664 17. 18. 19. 20. 21. 22. 23. 820 924 873 856 964 903 735 698 554 297 344 299 315 285 24. 25. 26. 27. 28. 29. 30. 355 871 664 124 110 485 672 179 73 493 97 20 142 210 31. 32. 33. 34. 35. 36. 37. 527 956 415 133 384 694 955 329 492 116 42 115 295 61 38. 39. 40. 41. 42. 43. 44. 469 783 865 614 468 821 613 252 694 572 156 79 209 108 48 SCHOOL ARITHMETIC. Find the value of : 45. 73251 - 23G79. 46. 52175 - 37896. 47. 04037 - 45069. 48. 30524 60. 51. 52. 756328 231562 936061 53. 238013 54. 832415 8765. 49. 54321 - 12345. 55. Subtract 236 from 500. 500 = 400 + 100 + = 400 + 90 + 10 236 = 200 + 30 + 6 = 200 + 30+6 264 = 467439. 87968. 847076. 199815. 243749. 200 60+4 We cannot take G ones from ones, neitlier can we add one of the tens, for there are no tens. Hence, we take one of the hundreds and change it to 10 tens, then take one of the tens and change it to 10 ones, as illustrated in tiie operation. Then 6 ones from 10 ones leaves 4 ones ; 3 tens from 9 tens leaves 6 tens ; 2 hundreds from 4 hundreds leaves 2 hundreds. Hence the re- mainder is 264. Subtract as indicated and prove : 56. 7250 - 2894. 68. 1962057 - 873698. 57. 5460 - 3291. 69. 212.3456 - 1745798. 58. 8300 - 5867. 70. 7300892 - 2006975. 59. 6500 - 5678. 71. 5437001 - 1008024. 60. 7000 - 3269. 72. 8200345 - 6832456. 61. 8000 - 4753. 73. 1000001 - 101002. 62. 6007 - 2308. 74 6347543 - 945876. 63. 12003 - 7036. 75. 3702674 - 2803879. 64 37200 - 17501. 76. 9008007 - 900809. 65. 17500 - 2351. 77. 7685342 - 6776387. 66. 91306 - 4007. 78. 4433225 - 3344668. 67. 10000 - 6789. 79. 8076539 - 8067845. Find the value of : 80. 6792051 - (139678 + 2005128) - 1076053. 81. 8076004 - 139678 + 2005128 - 1076053. 82. 1000000 - 54321 - (326000 + 240807). SUBTRACTION. 49 83. 7236408 - (6487543 ~ 5328796) - 4009007. 84. 9001 + (7200 - 6835) - (8003 - 3009) - 4372. 85. 59007 - (32046 4- 3423) - (4008 + 13009 - 8729). 86. 43700 + 56300 - (3725 + 6275) - (90000 - 1234). 87. Add one million to the difference between four thou- sand seven and seven thousand four. 88. Subtract the difference between five thousand eighty- one and three thousand ninety-seven from the difference between four thousand nine and nine thousand four. 89. A man bought a farm for $17325 and sold it for $20000. How much did he gain ? 90. How long is it since the discovery of America by Columbus in 1492 ? 91. A man owes $2150, but has only $975. How much must he borrow to pay the debt ? 92. Mont Blanc is 15572 feet high, and is 3572 feet higher than Pike's Peak. What is the height of the latter ? 93. Independence was declared in 1776. How long was that after the discovery of America ? 94. The area of England is 50922 square miles, and that of Pennsylvania is 45215 square miles. How much larger is England than Pennsylvania ? 95. Mr. B having $32700 gave his son $10320, and his daughter $8367. I^ow much had he left ? 96. In 1890 the population of Pittsburg was 238617, and that of Philadelphia was 1046964. Which had the larger population, and how much ? 97. A train left Atlanta with 273 passengers. At the first station 24 got off and 9 got on ; at the next 18 got off and 12 got on; at the third 17 left and 23 got on ; at the fourth 69 got off. How many yet remained on the train ? 98. A, B, and C bought a store for $19325. A paid $6105, and B paid $753 more than A. How much did pay ? 99. A man had $10000 in bank. He bought a house for 4 50 SCHOOL ARITHMETIC. $5G70, paid $1125 for repairs, 137 for insurance, and then sold the property for 18500, putting the money in bank. What sum did he then have in bank ? 100. Mr. B bought two horses, paying $120 for one and $145 for the other. He kept them six weeks, the feed for each costing him $13, and then sold one for $170 and the other for $210. How much did he gain ? 101. The remainder is 1786, the subtrahend 24G7. AVhat is the minuend ? 102. The remainder is p, the subtrahend q. What is the minuend ? 103. The difference between two numbers is 796, and the larger number is 1275. What is the smaller number ? 104. The difference between two numbers is a, and the larger number is h. AVhat is tlie smaller number ? 105. From what must 189 be taken to leave 981 ? 106. From what must a be taken to leave h ? 107. If the moon is 240000 miles from the earth, and the sun 92 million miles, how much farther is it to the sun than to the moon ? 108. At an election there were 21635 votes cast for A, B, and C. A got 9675, B got 327, and the remainder. How many people voted for C ? 109. The signs + and — were used by Widman in an arithmetic j^nblished at Leipzig in 1489, and the symbol = by Recorde in an algebra published in 1557. How many years elapsed from the time + and — were first used until the = was used by Recorde ? 110. The greatest depth of water yet measured is 29400 feet, and the greatest height to which a balloon has ascended is 37000 feet. By how many feet does the greatest height reached exceed the greatest depth measured ? 111. The area of Virginia is 42450 square miles, and that of West Virginia is 24780. How many more square miles has Virginia than West Virginia ? SUBTRACTION. 51 112. The number of bushels of corn raised in the U. S. in 1889 was 1,924,185,000, and the number of bushels of wheat was 675,149,000. How many more bushels of corn were raised than of wheat ? 113. In 1889 the total production of maple sugar in the U. S. was 32,952,927 pounds. In 1879 the production was 36,576,061 pounds. What was the decrease in 10 years ? 114. The total oyster product of the U. S. in 1898 was 25,349,668 bushels. Maryland produced 10,282,752 bushels and Virginia 6,572,493 bushels. How many bushels did all the other states produce ? Adding and subtracting equal numbers. — If we add 10 to each side of the equation 12 + 8 - 9 = 25 - 16 + 2, we have 12 + 8 - 9 + 10 = 25 - 16 + 2 + 10. Or, subtracting 6 from each of the equals in the same equation, we have 12 + 8 - 9 - 6 = 25 - 16 + 2 - 6. 1. Does adding equal numbers to equals destroy equality ? 2. Does subtracting equal numbers from equals affect equality ? 67. PRiisrciPLES. — 1, If equal numbers are added to equals, the sums are equal. 2. If equal numbers are subtracted from equals, the re- mainders are equal. MULTIPLICATION. 68. 1. Suppose you have a roll of $5 bills and wish to know how many one-dollars you have. How can you find out ? 2. Laying the bills out one by one, you may count by 5's, thus : 5, 10, 15, 20, 25, 30. In doing this you think merely of the sum of the addends, and not of the number oi $5 bills. 3. Or you may first count the bills — 1, 2, 3, 4, 5, 6. Six bills, each %o. How many times is 15 repeated to make your $30. Did you tliink of this six in the former process ? Observe that this process introdnces the idea of times — an idea not present in addition. 4. Your money is measured in two ways, by 15 and by 11. When the unit of measure is $5, how many times is it re- peated ? How many when the unit is II? 5. It should be remarked that this higher process is founded upon addition. AVe learn that six 5's are 30 (ones) by first finding the sum, and then noticing hem many times the 5 is repeated to make that sum. 6. Six times 2 quarts are how many '^ one-qnarts^' ? How many '^ six-quarts ^^ ? 7. A man measuring his corn filled a 2-peck measure 8 times ; how much corn had he ? 8. In the last example what is the unit of measure ? How many times was it taken ? What quantity did he measure ? Did he wish to find how many ^^ two-pecks"^ he had, or how many pechs 9 9. Ella bought 10 yards of cloth at $2 a yard. How much did it cost her ? MtJLTlPLiCAtioiJ. 53 How many times is $2 repeated or taken to make $20 ? What is here used as a unit of measure ? Is |2 itself a meas- ured quantity ? By what unit is it measured ? The cost is 10 x $2 — ten units of $2 each— but by the process of mul- tiplication we change this to $20, that is, to 20 units of $1 each. 09. The process of taking one number as many times as there are units in another is called Multiplication. 70. The number multiplied by another — the number taken so many times — is called the Multiplicand. It is regarded as a unit of measure. 71. The number that tells how many times the multipli- cand is taken is called the Multiplier. It is pure number. 72. The result obtained by multiplying is called the Product. It is regarded as measured quantity. 73. The multiplicand and multiplier are the Factors of the product. 74. The product of two numbers is the same whichever is taken as the multiplier. Of course it is absurd to say that 3 times 15 is the same as $5 times 3 ; but the meaning is that 3xl5 = 3x5x$l=:5x3x$lrrr5x|3. • 75. The Sign of Multiplication ( x ) is read times when it follows tlie multiplier, and multiplied by when it precedes the multiplier. Thus, 3 X $5 is read 3 times $5 ; $5 x 3 is read $5 multiplied by 3. This symbol was first used by Oughtred in 1631. 1. When we multiply $5 by 3, is the multiplicand concrete or abstract ? Is the multiplier concrete ? Is the product ? 2. Can we multiply U by 13 ? 4 by $3 ? $5 by 2 feet ? By what can we multiply 15 ? u SCHOOL ARITHMETIC. 76. Prin"Ciples. — 1. The multiplier is ahmys an abstract number. 2. The product is always like the multiplicand. MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 11 20 30 40 50 60 70 80 90 100 110 120 132 22 33 44 55 66 77 88 99 110 121 12 24 36 48 60 72 84 96 108 120 132 144 This table may be used in two ways. Find out the two ways. 77. The following exercises should be practiced daily until the pupil is able to announce at sight the product of any two numbers from 2 to 12. Announce products at sight : 1. 5 X 7 4x9 2 X 11 7x2 11 X 4 2. 9 X 5 3x6 3x9 6 X 10 7x8 3. 5 X 11 7 X 10 3 X 12 7x4 2x9 4. 4 X 4 6x3 8x6 10 X 5 9x8 5. 5 X 3 4x5 4 X 12 7x6 11 X 11 6. 7 X 9 5x8 6x5 8x7 4x6 7. 5 X 6 3x8 10 X 2 5x9 11 X 8 MULTIPLICATION. 55 8. X 4 3 X 7 6 X 12 8 X 10 9x7 9. 5 X 5 12 X 3 8 X 8 7 X 5 9x9 10. 6 X 6 7 X 12 10 X 11 12 X 5 10 X 4 11. 9 X 4 10 X 3 8 X 4 4 X 11 3 X 11 12. 4 X 10 12 X 11 10 X 7 7 X 3 11 X 6 13. 9 X 2 8 X 3 10 X 6 6 X 8 12 X 12 14. 4 X 8 10 X 10 8 X 8 5 X 10 8 X 11 15. 7 X 7 12 X 9 4 X 7 8 X 12 10 X 9 16. 9 X 12 11 X 3 12 X 8 9 X 10 12 X 10 17. 6 X 11 2 X 12 11 X 9 12 X 4 6x9 18. 5 X 12 11 X 10 10 X 8 9 X 11 6x7 19. 8 X 5 12 X 7 7 X 11 5 X 4 12 x 2 20. 5 X 11 11 X 2 12 X G 10 X 12 11 X 12 For further practice in rapid work place these diagrams on the ^ blackboard, and then, . using the numbers at the centers as multi- pliers, indicate the 7 multiplicands with the pointer. ORAL EXERCISES. A V 78. 1. At $4 a day, how much can I earn in 6 days ? 2. There are 12 things in a dozen. IIow many things are in 8 dozen ? 3. How much will 9 yards of calico cost at 6 cents a yard ? 4. How far does a man walk in 10 hours at the rate of 4 miles an hour ? 5. At $6 each, how much must he paid for 12 sheep ? 6. Ben picked 9 quarts of cherries, and Fred picked 8 times as many. IIow many quarts did Fred pick ? 7. If my hens lay 7 eggs each day, how many will they lay in 9 days ? 8. There are 8 quarts in a peck. How many quarts in 7 pecks ? In a bushel ? 56 SCHOOL ARITHMETIC. 4 9. Frank went to the post office 8 days, and each time got no letters. How many letters did he get in the 8 days ? Then 8x0 = what ? 10. A man bought a dozen slates at 10 cents each. How much change did he get out of a two-dollar bill ? 11. Find the cost of 8 yards of velvet at $5 a yard. (a). Since one yard costs $5, 8 yards cost 8 times $5, or $40. (b). At $1 a yard 8 yards would cost $8 ; hence, at $5 a yard, the cost is 5 times $8, or $40. Note. — The pupil should compare these solutions and make himself thoroughly familiar with the process in each. 12. James bought 15 papers at 3 cents each and sold them for 5 cents each. How much did he make ? 13. Dick earns $23 a week, and Charles earns $10 a week. In 6 weeks Dick earns how much more than Charles ? 14. Two men start from the same place, one going west at the rate of 7 miles an hour, the other going east at the rate of 4 miles an hour. How far apart are they in 4 hours ? 16. John has twice as many apples as Mark. How many have both, if Mark has 2 times 8 apples ? 16. A and B start from Boston and travel in the same direction, A going 25 miles an hour, and B 11 miles. How far are they apart in 3 hours ? 17. A has $4, B has 3 times as much, and C has twice as much as B. How much have they all ? 18. How much will 5 pencils cost at h cents each ? 19. If a man walks a miles an hour, how far will he walk in h hours ? 20. A man having 12a dollars bought 5 sheep at 2« dollars each. How much had he left ? 21. If a man spends h dollars a day for c days, how much does he spend ? 22. A man having a dollars paid p dollars each for q cows. How much had he left ? 23. Harry has jt? marbles, Ira Jias q marbles, and Sam has r times as many as both together. How many has Sam ? MULTIPLICATION. 57 24. When hats are worth $5 each, how much are 3 hats worth ? How do you know ? 26. Why do 3 hats cost 3 times $5 ? 26. Complete the following : When hats are worth a dollars each, 7 hats are worth dollars, because 7 hats are worth as mucli as . 79. Announce products at sight : G. 2. 2. 3. X 2. X 3. X 11. X 12. 9. 10. 11. 12. 13. 14. 15. 16. 12. 4. 3. 2. 3. 2. 3. 17. 18. 19. 20. 21. 22. 23. 24. 11. 12. 20. 3. 2. 3. 11. 2. WRITTEN EXERCISES. 80. 1. Multiply 365 by 4. 365 = 300 4 = 60 + 1460 = 1200 + 240 + 20 or 1200 + 260 + or 1400 +60+0 (b) 365 365 365 365 1460 (a) (b) 4 times 5 ones are 20 ones, or 2 tens and no ones. We write the in ones' place in the product, and add the 2 tens to 4 times 6 tens, making 26 tens, or 2 hundreds and 6 tons. We write the 6 in tens' place in the product, and add the 2 hundreds to 4 times 3 hundreds, making 14 hundreds, or 1 thousand and 4 hundreds. How is the same result obtained in (b) ? Why do we write the multi, plier under the multiplicand ? Find by trial if you can multiply, begia ning at the left. Multiply the following : 2. 253 by 3. 8. 697 by 8. 14. 287 by 9. 3. 426 by 3. 9. 853 by 9. 15. 913 by 8. 4. 736 by 4. 10. 437 by 4. 16. 368 by 7. 5. 594 by 6. 11. 978 by 5. 17. 627 by 6. 6. 8.67 by 5. 12. 634 by 6. * 18. 598 by 5. 7. 539 by 7. . 13. 549 by 7. 19. 817 by 9. 58 SCHOOL ARITHMETIC. 20. 62, 417, 685 by 4. 26. 98, 765, 432 by 6. 21. 85, 236, 417 by 6. 27. 24, 681, 357 by 9. 22. 47, 395, 173 by 5. 28. 36, 925, 814 by 8. 23. 18, 236, 456 by 7. 29. 62, 847, 418 by 7. 24. 54, 923, 687 by 9. 30. 56, 847, 389 by 5. 25. 37, 281, 459 by 8. 31. 12, 345, 679 by 9. 81. In expressions like 4 + 2x3, the operation indicated by the sign x must be first performed. Thus, 4+3x3 means 4 + 6, not 6 x 3. 18 - 8 x 2 means 18 - 16, not 10 X 2 ; but (18 - 8) x 2 means 10 x 2. (4 + 2) x 3 = 18 ; but 4 + 2 X 3 = 10. Of course 4 + (2 x 3) =r 10, but when two or more num- bers are connected by the sign x the parentheses are superfluous. Find the value of : 20 - 9 X 2 + 3 X 5 - 7. 25 + 4 X 3 - 6 x 6 + 9. 12 X 4 - 3 X 7 + 10 X 5. 12 X (4 - 3) X 7 + 10 X 5. 36 + 175 X 4 - 89 X 6 - 200. 1000 - 97 X 8 + 1 - 3 X 75. 82. To multiply by lO, 100, lOOO, etc. 1. When a number is multiplied by 10, what is annexed to it ? Then what is the shortest method of finding 10 times any number ? 2. AVhen a number is multiplied by 100, what is annexed to it? Then what is the shortest method of finding 100 times any number ? 3. When the multiplier is 10, 100, 1000, etc., how many ciphers must be annexed to the multiplicand ? Find out by trial. 83. Principle. — A number is multiplied hy 10, 100, 1000, etc., hy annexing m many ciphers to the multiplicand as there are ciphers in the multiplier. 1. 6 + 4x2. 7. 2. 7 + 3x6. 8. 3. 2 + 8x3. 9. 4. 8-3x2. 10. 5. 16 - 5 X 3. 11. 6. 12 + 4 X 7. 12. MULTIPLICATION. 59 WRITTEN EXERCISES. 84. Copy, and complete the equations : 1. 25.xl0=:( ). 37xlOO = ( 2. 50xlO=( ). 175xl00=( 3. 100 X 10 == ( ). 387 X 100 = ( 4. 41G X 10 = ( ). nOO X 100 = ( 5. 708 X 10=( ). 914 X 100=( 6. Multiply 743 by 5000. 743000 5000 is 5 times 1000. ITence we first annex 3 ciphers, 5 which multiplies 743 by 1000, aiul then multiply by 5. 59xl000 = ( ). 324 X 1000 = ( ). 872 X 1000 =( ). 903 X 1000 = ( ). 1000 X 1000 = ( ). T^ouKi we luuiuiniy nrst oy 3715000 ^ ^ ^ ana tnen oy luuu r Find the products of : 7. 578 X 30. 12. 924 x 700. 17. 8125 by 6000. 8. G39 X 20. 13. 538 x 400. 18. 5346 by 8000. 9. 825 X 40. 14. 190 x GOO. 19. 4325 by 9000. 10. 347 X 50. 15. 467 x 800. 20. 6913 by 7000. 11. 718 X GO. 16. 28G x 900. 21. 9000 by 6000. 85. To fiiKl the product when the right-hand figure of the multiplier is significant ; that is, is not 0. 1. Multiply 327 by 245. 327 327 245 = 200 + 40 + 5 245 1635 =z 5 X 327 1635 13080 = 40 X 327 1308 65400 = 200 X 327 654 80115 = 245 X 327 80115 The multiplier 245 = 200 + 40 + 5. Hence we multiply first by 5, then by 40, then by 200, and then add the partial products. Or, since 40 is 4 tens, we may multiply by 4 tens instead of 40. The product is 1308 tens, hence the 8 must be written under the tens, and the ones' place left vacant. Since 200 is 2 hundreds, we may multiply by 2 hundreds instead of 200. The product is 654 hundreds, hence the right- hand figure must be written under hundreds, leaving vacant two places to the right. 60 SCHOOL ARlTttMETiC. DiR^CTto^.-^ Write the multiplier under the multiplicand, ones under ones, tens under tens, etc. Multiply the multipli- cand first hy the ones of the mMltiplier, then by the tens, and so on, placing the right-hand figure of each product directly under the figure of the multiplier used to obtain it ; then add the several products thus obtained. To insure accuracy, review the work carefully, or multiply the multiplier by the multiplicand, and compare results. WRITTEN EXERCISES Find the product of : 2. 375 X 24. 11. 943 X 235. 20. 8076 X 607. 3. 628 X 35. 12. 627 X 364. 21. 4109 X 385. 4. 296 X 16. 13. 548 X 621. 22. 7340 X 468. 5. 418 X 29. 14. 703 X 513. 23. 5897 X 903. 6. 537 X 32. 15. 864 X 207. 24. 8645 X 795. 7. 704 X 58. 16. 597 X 648. 25. 9857 X 606. 8. 650 X 43. 17. 486 X 753. 26. 7469 X 729. 9. 486 X 74. 18. 398 X 462. 27. 7208 X 905. 10. 825 X 86. 19. 987 X 789. 28. 8900 X 547. Multiply as indicated : 29. 37 X 425 X 8000. 35. 456 X 723 X 800. 30. 72 X 618 X 6000. 36. 925 X 860 X 400. 31. 43 X 703 X 5000. 37. 378 X 409 X 700. 32. 40 X 900 X 7823. 38. 846 X 520 X 290. 33. 83 X 750 X 8600. 39. 750 X 693 X 804. 34. 69 X 583 k 4009. 40. 623 X 954 X 287. 41. 421896 X 215. 48. 516057 X 579. 42. 373842 X 327. 49. 439104 X 806. 43. 654083 X 456. 50. 182564 X 975. 44. 937584 X 274. 51. 869375 X 658. 45. 160835 X 621. 52. 82047 X 4306. 46. 395827 X 768. 53. 47638 X 5219. 47. 284936 X 492. 54. 68049 X 3168. MULTIPLICATION. Ql 65. 37583 x 6574. 69. 18254 x 9651. 66. 38065 X 4018. 60. 75864 x 3972. 67. 92736 x 8539. 61. 68357 x 4098. 68. 72069 X 3964. 62. 37528 x 6573. Find the value of : 63. (703 + 815) X (5403 - 765) - 860 x 1354. 64. 876 X 234 - 98 + 56 x 1009 - 12895 x 8. 65. (345721 - 18693) x 8700 - 63594 x 27. 66. (5389 - 76 X 59) x 86 - (51839 + 11 x 1987). 67. (5678 - 79) x 84 x 68 - 52389 - 8760 x. 937. 68. 62007 X 503 - 8460 x 790 - 37625 x 89. 69. 4587 + 7854 x 79000 - 78 x (673005 + 378900). 70. (99 - 88) X 180 4- 10000 - 77 x 66 - 179 x 38. 71. I paid $2678 for 426 bushels of clover seed, which I sold at S7 a bushel. How much did 1 gain ? 72. John drives the cows home to be milked every morn- ing and evening. If the pasture is 80 rods from home, how far does John travel in 4 weeks ? 73. Two trains left Chicago at the same time, one going west at the rate of 36 miles an hour, the other east at the rate of 28 miles an hour. How far apart were they in 17 hours ? 74. A has $95, B has three times as much, and C has twice as much as both. How much more than $1000 have they all? 76. There are 5280 feet in a mile. If steel rails weigh 36 pounds to the foot, what is the weight of the rails in -two miles of double-track railway ? 76. If a cow is worth 7 sheep, and a horse is worth 6 cows, how many sheep are worth as much as 19 horses and 28 cows ? 77. A boy who lives 167 rods from the schoolhouse goes to school regularly. If he goes home for dinner every day, how far does he walk in a week ? 78. A drover bought 286 sheep for $1430. After feeding 62 SCHOOL AKITHMETIC. them a week at an expense of $175, he sold 97 of them at $8 each, and 10 of them died. He sold the remainder for $900. How much did he gain on all ? 79. A lady bought 15 yards of calico at 6 cents a yard, 3 yards of velvet at $1.49 a yard, and 27 yards of muslin at . 16. Find | of 24. Of 40. J of 32.* Of 56. Of 80. 17. What is ^ of 27 ? | of 27 ? -J of 45 ? f of 54 ? To find one of the equal parts of a number requires division. Thus, to find i of 8 we must divide 8 by 2. To find i of 12 we must divide 12 by 3 ; that is, find one of the 3 equal parts of 12. 102. The first 13 of the following examples involve the process mentioned in (B). The quotient in each is concrete; it corresponds to the multiplicand. 1. Mr. A divided $10 equally between two boys. How much did each boy get ? Since two boys get $10, each boy got \ of $10, or $5. $10 -r- 2 = $5. 2. I paid $1 5 for 3 chairs. How much was that apiece ? 3. If 4 bags contain 28 bushels, how many bushels are in each bag ? 4. John earned $30 in 6 weeks. How much was that per week ? 5. If 45 people live in 9 houses, what is the average num- ber in a house ? 6. How much was calico a yard when 8 yards cost 88 cents ? 7. A merchant paid $96 for 8 stoves. How much did he pay for each stove ? 8. A drover has 84 cattle in 7 stables. How many has he in each stablo ? 9. There are 10 rows of trees in an orchard containing 120 trees. How many trees in a row ? In 2 rows ? 10. For 9 clocks a jeweler paid $108. How much did he pay for one clock ? For 2 ? For 3 ? 11. If 5 hats cost $15, how much will 8 hats cost ? 12. How much will 10 sheep cost, if 7 sheep cost $28 ? 13. A lady paid 25« dollars for 5 yards of fine cloth. How much did she pay for one yard ? DIVISION. 71 14. When wheat is h cents a bushel, liow much must bo paid for a busliels ? 15. I divided GZ> dollars equally among 3 boys, and then gave one of the boys a dollars more. How much had that boy then? 103. In the following exercises let the pupil determine which use of division eacii problem illustrates ; that is, whether one of the equal parts or the number of them is to be found. 1. At $10 each, how many carts can be bought for $80 ? 2. If 8 coats cost $80, how much does one cost ? 3. My milk bill for one week was 9G cents. If I paid 8 cents a quart, how many quarts did I get ? 4. A lady gave 48 apples to 13 boys. How many apples did each boy get ? 6. If 8 apples cost 24 cents, what is the cost of a. dozen apples ? 6. Seven hunters shot 84 rabbits. What was each cue's share of the game ? 7. If 5 tons of coal cost $30, how many tons can be bought for $54 ? 8. Mr. A paid 110 cents for cloth at 11 cents a yard, and Mr. B paid 9G cents for cloth at 12 cents a yard. How many yards did both buy ? 9. One jeweler paid $'«'2 for watches, and another paid $132 for some of the same kind. If the first got 6 watches, how many did the second get ? 10. How many pencils at 3 cents each can be purchased for a quarter ? How many cents will be left ? 11. At 4 cents a yard, how many yards of cloth can be bought for half a dollar ? 12 yards cost 48 cents. What part of a yard will the 2 cents that remain buy ? 12. Why is not 8J pencils the answer to the 10th problem ? Y2 SCHOOL ARITHMETIC. 13. In how many days can a man earn $50, if he receive $6 a day ? 14. How many sheep, at $7 a head, can be purchased by a man who lacks $5 of having 150 ? 15. What is the cost of one table, if 4 tables cost $34 ? 16. In how many days can a man earn 54a dollars, if he receives 6« dollars a day ? 17. In 8 weeks a boy earned 12h dollars. How much did he earn in a week ? In 2 weeks ? WRITTEN EXERCISES. 104. 1. Divide 8255 byG. (a) 6)8255 13751 (b) 6)8255(1375 quotient 6 22 18 45 42 35 30 5 For convenience we write the divisor at the left of the dividend, with a line between them, and the quotient either as in (a) or in (b). In 8 there is one 6, with a remainder of 2. Since the 8 is thousands, the quotient 1 and the remainder 2 are thousands. 2 thousands = 20 hundreds, and 20 hundreds and 2 hundreds — 22 hundreds. In 22 there are three 6's, with a remainder of 4. Since the 22 is hun- dreds, the quotient 3 and the remainder 4 are hundreds. 4 hundreds = 40 tens, and 40 tens + 5 tens = 45 tens. In 45 there are seven 6's, with a remainder of 3. Since the 45 is tens, the quotient 7 and the remainder 3 are tens. 8 tens = 30 ones, and 30 ones + 5 ones = 35 ones. In 35 there are five 6's, with a remainder of 5, which may be written as in (a) or left to stand as in (b). Pboof. — 1375 X 6 + 5 = 8255, the dividend, Hence the work js correct. DIVISION. 73 The process as sliown in (a) is < sailed Short Division; as shown in (b) it is called Long Division. Wherein do the two processes differ ? Find the quotients by short division : 2. 1237 -T- 3. 18. 4405 -^ 5. 34. 268735 H- 3. 3. 1934 ^ 4. 19. 3264 -h 6. 35. 281076 -^ 6. 4. 2180 ^ 5. 20. 1896 -T- 8. 36. 340125 H- 5. 6. 3265 -^ 4. 21. 6188 -=-4. 37. 532024 -^ 4. 6. 5375 -^ 5. 22. 1809 ^ 9. 38. 659134 -^ 7. 7. 6894 -^ 6. 23. 2808 ^ 9. 39. 386937 -^ 9. 8. 4506 -=- 3. 24. 3629 H- 7. 40. 726859 -^ 8. 9. 6025 -f- 5. 25. 8736 -f- 6. 41. 400002 ^ 6. 10. 6132 -^ 4. 26. 6320 ^ 5. 42. 590023 -^ 7. 11. 4306 -V- 7. 27. 9144-^8. 43. 646398 -^ 9. 12. 4955 ^ 6. 28. 5706 -^ 6. 44. 200000 -f- 3. 13. 5005 -^ 5. 29. 3800 -H 5. 45. 486018 --6. 14. 1216 -^ 8. 30. 4008 -7- 8. 46. $598524 - -$6. 15. 2032 -^ 4. 31. 7398 -- 9. 47. $567014 - -$7. 16. 7294 -- 7. 32. 4543 -^ 7. 48. $630927 - -$9. 17. 3735 -^ 3. 33. 8888 -T- 6. 49. 1803449 - -$8. 60. 9306^ -11 53. 10186 ^ 11. 51. 3432 -f -12. 54. 10188 -^ 12. 52. 5964 -f -12 55. *69 576 -^ $12. RULE FOR LONG DIVISION. 105. Find hotu many times the divisor is contained in the number represented by the fewest left-hand figures of the dividend that ivill contain it. Multiply the divisor by the quotient thus obtained, write the product under the left-hand figures used, and subtract. To the remainder annex the next figure of the dividend, and then proceed as before, until all the figures have been annexed. If any partial dividend is less than the divisor, place a cipher in the quotient, annex the next figure of the dividend, and proceed as before. u SCHOOL ARITHMETIC. Proof. — Multiply the divisor by the quotient, and add the remainder, if any, to the product. If the result is equal to the dividend the work is correct. Queries. — 1. Why must each remainder be less than the divisor ? 2. When the product of the divisor and the quotient figure is greater than the partial dividend from which it is to be subtracted, what must be done ? 1. Divide 1728 by 24. 24)1728(72 168 48 48 ones. 48 ones remainder ? As 24 is greater than 17, it is necessary to take the number represented by three figures of the dividend for the first partial dividend. 172 tens -s- 24 = 7 tens, and a remainder. 24 x 7 tens = 168 tens. The remainder is 4 tens, and the new dividend is 4 tens and 8 ones, or 48 ^ 24 = 2 ones. 24 x 2 ones = 48 ones. Is there any Find the quotients 2. 4536 -T- 21. 3. 9175 ^ 25. 4. 7998 -^ 31. • 5. 7052 -^ 41. 6. 21879 -^- 51. 7. 11792 -V- 22. 8. 15136^32. 9. 26250-^42. 10. 42224 -r- 52. 11. 37015 -^ 55. 12. 11914^23. 13. 14058-^33. 14. 14964 -^ 43. 41. 376859328 ^ 48. 42. 384710564 -i- 58. 43. 238311937 -^ 29. 44. 287135862 h- 39. 45. 429436049 ^ 49. 46. '}f28395002 -^ 59. by long division 15. 13409-^53. 16. 17400 -^ 24. 17. 21556 -V- 34. 18. 22836 -^ 44. 19. 50058^54. 20. 30660-^35. 21. 32715-^45. 22. 61620 -^ 26. 23. 22212-^36. 24. 24472-^46. 25. 32088-^56. 26. 88020 -^ 27. 27. 16206 -^ 37. 28. 35344 h- 47. 29. 21888 -^ 57. 30. 10388-^28. 31. 18696 -^ 38. 32. 52542-^63. 33. 10725-^75. 34. 26487^81. 35. 33943-^91. 36. 43368 -=- 74. 37. 66864-^84. 38. 52776 ^ 72. 39. 25568 ~ 68. 40. 64108^94. 47. 169135679 48. 207407256 49. 181481349 50. 219752926 51. 057662754 52. 197393584 137. 168. 147. 178. 201. 304. DIVISION. 75 63. 3049G3cS22 -^ 400. 64. 108825952 ^ 4528. 54. 304123450 -f- 510. 65. 230884080 -^ 5684. 55. 442680498 -^ 611. 66. 597126784 -^ 6788. 56. 584093472 ^ 712. 67. 941108532 -^ 7638. 57. 263090640 -^ 819. 68. 265283625 ^ 8725. 58. 383748326 -^ 923. 69. 397046588 ^ 9337. 59. 677510968 -4- 937. 70. 308196056 -^ 3962. 60. 018875970 -f- 745. 71. 535673956 -^ 8009. 61. 353628594 ~ 1023. 72. 810891081 -^ 9009. 62. 512763462 ^ 2186. 73. 103031370 ~ 8346. 63. 498933150 -- 3275. 74. 462017992 -^ 4678. 106. 1. A drover bought some cattle for $17616. If the average price of each was $48, how many did lie buy ? 2. The salary of a Congressman is $5000 a year. How much is that a day ? 3. A owes B $5200. If he pays him $650 a year, in how many years will the debt be canceled ? 4. The circumference of the earth is about 8 million rods, and tliere are 320 rods in a mile. How many miles is it around the earth ? 6. A grocer bought 368 barrels of flour for $2208, and sold them for $2944. How much did he gain per barrel ? 6. TheW. Y. R. R. is 268 miles long, and cost $5,660,728. What was the average cost per mile ? 7. The salary of the President of the United States is $50000 a year. How much is that per day in leap year ? 8. The product of two numbers is 1,259,375. One of the numbers is 97 less than 500. What is the other ? 9. If a man receives $1600 a year for his labor, and spends $832, in how many years can he save enough to buy a farm of 132 acres, at $24 an acre ? 10. By selling a farm of 240 acres for $12720, I gained $1200. How much did I pay per acre for the farm ? 11. How often can 436 be subtracted from 34444 ? 76 SCHOOL ARITHMETIC. 12. The dividend is 9689, the quotient 134, and the re- mainder 41. What is the divisor ? 13. A double-track street railway is 5 miles long. How many rails does it contain, if each rail is 24 feet long, there being 5280 feet in a mile ? 14. A train of fifteen cars contained 279300 pounds of flour in barrels. How many barrels were in each car, a barrel of flour weighing 196 pounds ? 15. Two men leave Memphis, Tenn., to travel around the earth, one going east at the rate of 154 miles a day, the other going west at the rate of 144 miles a day. In how many days will they meet, if the distance around is 16390 miles ? 16. A railroad train makes 2 round trips daily between New York and Philadelphia. How far apart are these cities if the train runs 131400 miles in a common year ? To divide by lO, lOO, 1000, etc. 107. 1. 80 H- 10 = what ? 750 ^ 10 = what ? 3200 -^ 10 = what ? How do the quotients compare with the divi- dends ? 2. 700 -^ 100 = what ? 2400 -^ 100 = what ? Compare quotients with dividends, and tell how the latter have been changed. 3. 75 -^ 10 = what ? What is the remainder ? 4. 3825 -^ 100 = what ? -What is the remainder ? 5. 43875 -i- 1000 = what ? What is the remainder ? Where are these remainders seen in the dividends ? Can the quotients be seen in the dividends ? Where ? 108. Pkinciple. — A number may be divided by 10, 100, 1000, etc., by cutting off from the right of the dividend as many figures as there are ciphers in the divisor. The part cut off is the remainder, and the rest of the dividend is the quotient. Thus, 4100 -^ 100 = 41 ; 4125 -=- 100 = 41, with a remainder 25. The quotient may be written 41fo\- DIVISION. YY Divide the following : 1. 380 by 10. 2. 275 by 10. 3. 420 by 10. 4. 600 by 10. 6. 775 by 10. 6. 905 by 10. 7. 8. 9. 10. 11. 12. 500 by 100. 320 by 100. 875 by 100. 7200 by 100. 2450 by 100. 4315 by 100. 13. 14. 15. 16. 17. 18. 6G000 by 1000. 9300 by 1000. 2460 by 1000. 8725 by 1000. 35000 by 1000. 40009 by 1000. 19. Divide 31275 by 500. 5 1 00 )312 I 75 Cutting off two figures from the right of the dividend divides it by 100, the quotient being 312, with the remainder 75. Since 500 is 5 timts 100, the quotient is 5 times as large as it should be. Hence we divide it by 5, getting a quotient of 62 and a remainder of 3, which is hundreds. 2 hundreds 4- 75 (the first remainder) = 275, the entire re- mainder. Hence the quotient is 62§^XB. Find the quotients : 26. 78960 27. 62845 80. 90. 28. 59320^300. 29. 32856 -J- 500. 30. 47623 -v- 600. 31. 89974-^800. 32. 102030 -^ 900. 33. 510075 H- 700. 34. 246783 -^ 800. 35. 987654 -^ 600. 36. 100000-4-800. 37. 808080 -h 700. 20. 2765 -^ 20. 21. 4275 -r 30. 22. 5180 -^ 40. 23. 3625 -4- 50. 24. 7338 -^ 60. 25. 6774 -4- 70. 38. Divide 1728 by 12, by dividing first by 2, and then the quotient by 6. 1728 4- 2 = 864. 864 4- 6 = 144. 39. Divide 2625 by the factors of 15. 40. Divide 4536 by 21 ; also by 3 and 7, and compare re- sults. 109. In expressions like 18 + 24 4- 6, the operation indi- cated by the sign -^ must be first performed. Thus, 18 + 24-5-6 means 18 + 4, not 42 -^ 6. 120 - 80 -h 2 means 120-40. 78 SOHOOL ARITHMETIC. Find the value of 1. 43 + 18 ^ 3. 2. 19 - 36 -^ 4. 3. 27 ^ 9 -1- 8. 4. 35 - 21 -^ 7. 5. 22 + 9 X 3 - 49 -T- 7. 6. 68 - 35 -^ 5 + 7 X 10. 7. 54 - 57 -^ 19 - 3 X 17. 8. 36 4- 7 X 9 - 63 -f- 9. GENERAL PRINCIPLES OF DIVISION. no. The value of the quotient depends upon the relative values of the dividend and divisor. Hence, if either divi- dend or divisor is changed, the quotient will be changed. If both are changed equally (as to ratio), the quotient will not be changed, as may be seen in equations (e) and (f) below. The following equations illustrate all the changes : Given Equation, 24 -^ 6 = 4. (a) Multiplying the divi- dend by 2 multiplies the quotient by 2. (b) Dividing the dividend by 2 divides the quotient by 2. (c) Multiplying the divisor by 2 divides the quotient by 2. (d) Dividing the divisor by 2 multiplies the quotient by 2. (e) Multiplying both divi- dend and divisor by 2 does not change the quotient. (f) Dividing both dividend and divisor by 2 does not change the quotient. Changing (a) 48 ^ 6 = 8. dividend. (b) 12 ^ 6=2. Changing (c) 24 -^ 12 = 2. divisor. (d) 24-- 3 = 8. Changing (e) 48 -^ 12 = 4. both equally. (f) 12 -T- 3 = 4. DIVISION. 79 From these examples are deduced the following general 111. Principles. — 1. Multiplying the dividend multiplies the quotient, and dividing the dividend divides the quotient, 2. Multiplying the divisor divides the quotient, and divid- ing the divisor multiplies the quotient. 3. Multiplying or dividing both dividend and divisor hy the same number does 7iot change the quotient. Queries. — 1. If a number equal to the divisor should be added to the dividend, what change would occur in the quotient ? 2. Subtracting twice the divisor from the dividend would have what effect on the quotient ? 3. Would adding the same number to both dividend and divisor in- crease or diminish the quotient? 4. If the same number were subtracted from dividend and divisor, would the quotient be increased or diminished ? 5. Does subtracting any number from the divisor increase or diminish the quotient ? REVIEW WORK. ORAL EXERCISES. 112, 1. Whafc number is represented by 45 ? 2. What number is represented by 8 ones of the first period and 7 tens of the second period ? 3. In 42 tens liow many ones ? 4. How many ones in 3 hundreds, 6 tens, and 5 ones ? 5. If 9 hats cost 127, what will 5 hats cost ? Since 9 hats cost $27, 1 hat costs -^ of $27, or $3; since 1 hat costs $3, 5 hats will cost 5 times $3, or $15. Query. — Why will 5 hats cost 5 times $3 ? 6. If 6 sheep cost $30, how much will 11 sheep cost ? 7. A man bought 10 books for $40, and sold 7 of them at the same rate. How much did he receive for them ? 8. How much will 12 yards of cloth cost, if 5 yards cost 55 cents ? 9. How much will 13 pounds of meat cost, if 9 pounds cost 72 ceuts ? 10. A sold 5 pigs and B sold 3, each getting the same price per head. How much did each get if both got $64 ? 11. I sold a calf for $19, which was $7 more than it cost me. How much did I pay for it ? 12. A man bought 6 barrels of flour for $30, and gave half of them for potatoes at $3 a barrel. How many barrels of potatoes did he get ? 13. Which is cheaper, and how much per dozen — eggs at 25 cents a dozen, or at 3 cents each ? 14. How long will it take A to earn $99, if he earns $18 in 2 weeks ? REVIEW WORK. 81 15. How many days can three men live on the provisions that 5 men require for 9 days ? 16. If 7 men can dig a ditcli in 9 days, how long would it take 3 men ? 17. If a load of hay lasts 8 cows a week, how long would it last 14 cows ? 18. Twelve times 7 are how many times 4 ? 19. If 3 apples are worth 1 lemon, and 2 lemons are worth 13 pears, how many pears are worth 1(S apples ? 20. When rice is 6 cents a pound, how many pounds should I receive in exchange for 9 dozen eggs at a cent apiece ? 21. In 85 days how many weeks ? 22. How many days in 8 weeks and 5 days ? 23. Jack bought a dollar's wortii of apples at the rate of 2 for 5 cents. How many did he get ? 24. May bought 2 dozen eggs at 25 cents a dozen, and sold them at the rate of 3 for a dime. How much did she gain ? 25. Owen bought 9 oranges for 7 cents each and 11 lemons for 5 cents each ; he gave in exchange 9 pounds of butter at 15 cents a pound. How much was due him ? 26. I gave half a dozen dozen pencils worth 5 cents each for 6 knives. What was each knife worth ? 27. How many letters are required to write $2.41 in words ? 28. At $2.88 a dozen, what is the value of 10 hoes ? 29. If a boys earn ba cents per day, how much do b boys earn in one day ? 30. If q cows eat a tons of hay in a month, how much will p cows eat ? 31. John was h years of age a years ago. How old will he be in « + J years ? 32. At %a each, what will be the cost of c rabbits ? 33. A has %b, B has %c more than A, and C has as much as the difference between A^s and B's money. How much have they together ? SCHOOL ARITHMETIC. WRITTEN EXERCISES. 113. 1. Miss B teaches 9 months in the year at a salary of $1350. How long does it take her to earn 1900 ? 2. A and B bought a farm of 80 acres for $7360. If A paid $3128, how many acres did he pay for ? 3. The President of the U. S. receives $50000 a year. If his salary were increased $5 a year, how much would he receive a day ? 4. How long a string will it take to reach around a barn 42 feet long and 36 feet wide ? 5. Mr. A bought a piano for $450, paying one-half in cash, and the remainder at the rate of $15 a month. If he made the purchase January 1, 1899, when did he make the last payment ? 6. Rome was founded 753 years before the birth of Christ. How long was that before Columbus discovered America ? 7. There are 369600 feet in 70 miles. How many feet in 5 miles and a half ? 8. What number besides 269 will exactly divide 36853 ? 9. In 100 years the population of the U. S. increased from 3,929,214 to 62,622,250. What was the average increase per year ? 10. If I spend a quarter a day for books, a dollar a day for rent, and $35 a month for groceries, how much do I save in a leap year if my salary is $2000 ? 11. Find the sum of the five largest numbers that can be expressed by the figures 9, 8, 0, 4, and 2. 12. The minuend is 7019, the remainder 3107. The sub- trahend is how many times the sum of 3, 2, and 7 ? 13. The divisor is 437, the. quotient 86, and the remainder 50. What is the dividend ? 14. Two men had $7583 divided between them. The dif- ference between their shares was $223. How much did each man get ? REVIEW WORK. 83 16. How many times can 461 be subtracted from 57820, and what is tlie remainder ? 16. Marvin read from chapter LXXVII to chapter XCIX. How many chapters did lie read ? 17. How many quarts of oats will two horses eat in 30 days, if each horse eats 4 quarts 3 times a day ? 18. The skull has 8 bones, the face 14, the ear 3, the trunk 53, the shoulders 4, an arm 3, the wrists 16, the hands 38, the legs 8, the ankles 14, and the feet 36. Allowing 33 teeth, how many bones are in the body ? 19. If a lot of hay lasts 18 horses 27 months, how long would it last 27 horses ? 20. The product is 5832 and the multiplier is 324. What is the multiplicand ? 21. If a horse travels 8 miles an hour and a locomotive 40 miles an hour, how much sooner can a man go 120 miles by traveling on the cars than by going on horseback ? 22. Tlie product of three numbers is 13824. Two of the numbers are 18 and 32. What is the third ? 23. Find a number to which if 369 be added the sum will be 1001 less than 9090. 24. A man bought 68 horses at $84 each ; 11 of them died. At what price must he sell the others to gain $444 ? 26. Mr. H bought 160 acres of land at $75 an acre. After spending $1200 dollars for improvements, he sold it at a gain of $2000. At what price per acre did he sell ? 26. A newsboy bought papers at 3 cents each, and sold them at 5 cents each, thereby gaining 90 cents. How many papers did he sell ? 27. A mile is 5280 feet. How many steps of 2 feet each will a boy take in walking 5 miles ? 28. How many years does it take to make the difference between saving $2 a month and $6 a month amount to a sav- ing of $100 ? 84 SCHOOL ARITHMETIC. 29. I traded 120 head of cattle at 164 a head for 1 60 acres of land. What price per acre did I pay ? 30. If 8 horse shoes weigh 16 pounds, how many horses can be shod with shoes that weigh 152 pounds ? 31. Horace rode 31680 yards on his bicycle, the wheel of which was 12 feet in circumference. How many turns did the wheel make ? 32. What number multiplied by twice 37 will produce 2664 ? 33. A wagon weighing 1000 pounds contains 6 barrels of flour and 7 of pork, and is drawn by two horses. A barrel of pork weighs 200 pounds and a barrel of flour 196 pounds. How many pounds does each horse draw ? 34. The distance from Pittsburg to Philadelphia is 354 miles. If a railroad conductor makes a round trip every two days, how many miles does lie travel in 4 weeks ? 35. At $15 per uniform, how many companies of 85 soldiers each can be uniformed for $10200 ? 36. The smaller of two numbers is 3782, and their differ- ence is 1218. What is tlie larger number ? 37. Mr. E paid $125 an acre for 80 acres of coal land. He sold the coal for $64000, and the land at $75 an acre. How much did he gain ? 38. How many weeks would a man take to walk 1344 miles, if he walks 4 miles an hour, 7 hours a day, and 6 days a week ? 39. The sum of 18 equal numbers is 96346 less than a million. Find one of the numbers. 40. What is the value of 175 + 92 x 105 ? 41. If a train runs 28 miles an hour, in how many hours can it run to a place 420 miles distant and return ? 42. A miller owing $500 gave in part payment 250 bushels of wheat at $1.50 a bushel, and paid the remainder with flour at $5 a barrel. How many barrels were required ? 43. A drover bought 45 liorses at $85 each, and sold them so as to gain $720. How much a head did he receive for them ? 44. An extension table is 12 feet long when its four boards are in, and 7 feet long when they are out. How wide is each board ? 45. Henry was 30 years old when Edward was born. Edward was 14 years old in 1876. How old will Henry be in 1910 ? 46. If one hen lays 180 eggs in a year, liow many dozen eggs slionld 2 dozen hens lay in 2 years Y 47. From New York to Havana is 1260 miles, from Havana to Aspinwall is 1046 miles, from there to Panama is 60 miles, and from Panama to San Francisco is 3616 miles. What is the distance between San Francisco and Havana ? 48. A party of 64, eight more than half of whom were ladies, took a boat ride at an expense of $3 each. If all expenses were paid by the gentlemen, how much did each pay ? 49. I paid $7500 for two lots, one of them costing $1000 more than the other. What did I pay for each ? 50. A and B together have $500, and A has $100 more than B. How much has each ? 51. The area of Texas is 265780 square miles, and that of Pennsylvania is 45215 square miles. Into how many States of the size of Pennsylvania could Texas be divided, and how many square miles would be left over ? 52. At $4 a ton, what is the value of a carload of coal weighing 17920 pounds, counting 2240 pounds to a ton ? 53. There are 640 acres in a square mile. How many acres in Rhode Island, whose area is 1250 square miles ? 54. Each front wheel of a carriage is 10 feet in circum- ference, and each hind wheel 12 feet. The front wheels will make how many more turns than the hind wheels in going 5 miles, there being 5280 feet in a mile ? B6 SCHOOL ARITHMETIC. 55. If steel rails weigh 72 pounds to the yard, and SOOO pounds are a ton, how many tons of rails will be required to lay 2 miles of railroad, half of which is to have double track ? SUPPLEMENTARY EXERCISES. (FOR ADVANCED CLASSES.) 1 14. 1. Prove that if we multiply by 4, and divide the product by 100, we obtain the result of dividing by 25. 2. Find the value of 720 + 964 x 8 - 154 x x 6. 3. Prove that $17 x 11 = $11 x 17. 4. By what number must we divide a given number to obtain the same result as when the given number is multi- plied by 2 and divided by 70 ? 5. The remainder is 723. What is the minuend if it is twice as great as the subtrahend ? 6. The minuend is « + a, and the remainder is equal to the subtrahend. Find the remainder. 7. What number multiplied by 100 and divided by 4 gives the same result as is obtained by multiplying LS by 25 ? 8. The sum of two numbers is 60, and their difference is 24. What are the numbers ? 9. The sum of two numbers is a, and their difference is d. What are the numbers ? 10. What is the qnotient when $12 is divided by $4? When $« is divided by U ? 11. A man living at the rate of $3500 a year for 6 years finds that he is exceeding his income, and reduces his ex- penditures to $2500 a year. At the end of 4 years he finds that he is just out of debt. What is his income ? FACTORS AND MULTIPLES. 115. 1. What two numbers multiplied together will make 6 ? Then what are the factors of G ? 2. Is each factor of 6 an exact divisor of G ? 116. The integers which multiplied together will produce a number are called the Factors of that number. Thus, 5 and 6, or 2, 3, and 5 are the factors of 30. The factors of a number are exact divisors of it, 1. What are the exact divisors of 8 ? Of 7 ? Of 10 ? Of 13 ? Of 18 ? Of 19 ? Of 23 ? 117. A number whose only exact divisors are itself and 1 is called a Prime Number. A number that has other exact divisors is called a Composite Nimiber. Thus, 3, 5, 11, 17, etc., are prime numbers, and 4, 9, 12, 20, etc., are composite numbers. 1. Make a list of all the prime numbers between and 100. 2. Make a list of all the numbers from 1 to 100 that are exactly divisible by 2. 3. Make a list of all the numbers between and 144 that are not exactly divisible by 2. 118. A number that is exactly divisible by 2 is called an Even Number. All other numbers are called Odd Numbers. 119. The exact divisors, or factors, of a number must be found by inspection or by trial. The following facts are very helpful in finding factors : 88 SCHOOL ARITHMETIC* Any number is exactly divisible 1. By 2, when the right-hand figure is 0, 2, 4, 6, or S. 2. By d> when the sum of the numbers represented by its digits is divisible by 3. ^ 3. By 4^ when the number represented by the two right- hand digits is divisible by 4. 4. By 5^ when the right-hand figure is or 5. 5. By 6, when it is divisible by 2 and 3. 6. By 8, when the number represented by the three right- hand digits is divisible by 8. 7. By 9, when the sum of the numbers represented by its digits is divisible by 9. Find some divisors of the following by inspection : 1. 324. 6. 8406. 11. 9072. 16. 84,306. 2. ] 75. 7. 7300. 12. 8100. 17. 52,146. 3. 2G0. 8. 2904. 13. 3285. 18. 93,528. 4. 513. 9. 5344. 14. 7824. 19. 60,000. 6. 4Q0. 10. 4563. 15. 5259. 20. 78,327. FACTORING. 120. 1. What prime numbers multiplied together will produce 6 ? 10 ? 14 ? 22 ? 12 ? 2. What prime numbers will exactly divide 18, or what are the prime factors of 18 ? 121. Prime numbers used as factors are called Prime Factors. Thus, 3 and 7 are the prime factors of 21. 1. Can 11 and 18 both be divided by the same number ? Can 12 and 25 ? 2. Which of the numbers in the preceding example are prime numbers ? 122. Two numbers that have no common factor except FACTORINGS. 89 tinity are said to be Prime to each other, though one or both of them may be composite. 1. Since 3 is a factor of G, must it be a factor of two 6*s, or 12 ? Of three 6's, or 18 ? Of any number of G's ? 2. Since 13 is a factor of 36, are all the factors of 12 also factors of 36 ? Find by trial. 3. Can all the numbers of which 12 is a factor be exactly divided by the factors of 12 ? Investigate. 4. Any exact divisor of a factor is always a factor of what ? 123. Principle. — A71 exact divisor of a factor of a num- ber is a factor of the mimber itself. An exact divisor may be a fraction, but in '* factoring '' only integral divisoi's or factors are considered. 124. The process of finding the factors of a number is called Factoring. WRITTEN EXERCISES. 125. 1. What are the prime factors of 360 ? 2 360 Since the prime number 2 is a divisor of 360, it is o TfiO ^^^ ^^ ^^^ factors, and 180 is another. Since 2 is an exact divisor of 180, it is a factor of 360 (Art. 123), 2 90 as is 90 also. Since 2 is an exact divisor of 90, it is 3 45 a factor of 360. Likewise 3 and 5 being exact „ ~rz divisors of 45 and 15 are also factors of 360. Hence 2, 2, 2, 3, 3, and 5 are the prime factors of 5. 360. Note, — The number of times any factor occurs in a product may be indicated by an exponent. Thus, 2^. S'', 5 are the prime factors of 300. The small figures written above and to the right of the factors 2 and 3 are exponents. Find the prime factors of : 2. 60. 8. 480. 14. 2956. 20. 2310. 3. 108. 9. 672. 15. 4620. 21. 7644. 4. 144. 10. 1056. 16. 9170. 22. 64,384. 5. 180. 11. 1872. 17. 5432. 23. 20,000. 6. 315. 12. 2310. 18. 2002. 24. 242,424. 7. 308. 13. 3204. 19. 6006. 25. 714,510. LEAST COMMON MULTIPLE. 126. The product of two or more integers is called a Multiple of those numbers. It follows that any number is a multiple of another when it is exactly divisible by that number. Every number is a multiple of its factors. 1. Does 10 exactly contain both 2 and 5 ? 2. What number will contain 7 and 3 without a re- mainder ? 3. Kame a multiple that is common to 5 and 11. To 4 and 6. To 2, 3, and 4. 127. A multiple that is common to two or more numbers is called a Common Multiple. 1. What is the least number that will exactly contain 3 and 5? 4 and 6 ? 2, 3, and 4 ? 2. Is 24 a common multiple of 3 and 4 ? Is it their smallest common multiple ? What is their least common multiple ? 128. The least number that is exactly divisible by each of two or more numbers is called their Least Common Multiple, ivritten L. C. M. 1. What are the prime factors of G ? Of 10 ? What is their L. CM.? 2. What are the prime factors of 30 ? How do they com- pare with those of 6 and 10 ? 3. What factor is common to 6 and 10 ? Does it occur twice in the factors of 30 ? 4. Since 30 contains both 6 and 10, must it contain all their prime factors ? LEAST COMMON MULTIPLE. 01 5. The factors of are 2 and 3, and those of 14 are 3 and 7. Which of these factors must be multiplied together to produce the L. C. M. of 6 and 14 ? 129. Principle. — The least common multiple of two or more numbers contains all the prime factors of those number s, and no others. If a factor is common to two or more numbers, it is contained in the L. C. M. only the greatest number of times it enters into any one of the numbers — not as often as it occurs in all of them. WRITTEN EXERCISES. 130. 1. Find the least common multiple of 25, 30, and 42. 25 = 5 X 5. The least common multiple must 30 = 2 X 3 X 5. contain all the prime factors of 25, 30, 42 = 2 X 3 X 7. ajMl 42, that is, 2, 3, 5, and 7. Each 2x3x5x5x 7 = 1050. of these must be contained as often as it occurs in any one set of factoi-s. The only factor that occurs twice in one number is 5. Hence the factors of the L. C. M. are 2, 3, 5, 5, 7. Their product is 1050, the L. C M. The following method, which is in common use, is based upon the same principle : Since 2 is an exact divisor of some of the num- bers, it is a factor of the L. C. M. Since 3 is an exact divisor of some of the quotients, it is a factor of the L. C. M. (Art. 129). We find in the same manner that 5 is also a factor of the L. C. M. The last quotients which are prime to each other are also factoi-s of the L. C. M. Hence the L. C. M. is 3 x 3 x 5 x 5 x 7, or 1050. Find the L. C. M. of the following : 2. 3. 4. 5. 6. 1. When one of the numbers is a factor of another, it may be dis- regarded, as its multiple contains the same factors. 2. When several numbers have no common factor, their product is the L. C. M. 2 25, 30, 42. 3 25. 15, 21. 5 25, 0, 7. 5, 1, 12, 24, 30. 7. 4, 5, 9, 8, 12, 6. 18, 97 32. 8. 7, 3, 4, 5, 6. 22, 33, 55. 9. 6, 7, 8, 10, 14, 16. 28, 30, 60. 10. 4, 6, 8, 16, 24, 48. 36, 50, 70. 11. 3, 5, 7 11, 13, 17. CANCELLATION. 131. Cancellation is a process of shortening the work in problems that involve multiplication and division. It is based on two principles. 1. What is the product of 6 x 5 ? Of 3 x 5 ? How do the products compare ? 2. What is the product of 8 x 2 ? Of 4 x 2 ? How do the products compare ? How could you get the second product from the first ? 3. Does dividing one factor by any number divide the product by the same number ? Find by trial. 132. Principles. — 1. Dividing any factor of a series of factors hy any number divides the product by the same number. 2. Dividing both dividend and divisor by the same number does not change the qnotient. (Art. 110.) WRITTEN EXERCISES. 133. 1. Divide the product of 3, 21,, and 25 by tlie prod- uct of 3, 7, and 10. 3 5 The division is indicated by writing the dividend 3 X ^Z X ^^ above and the divisor below the line. Dividing both by 3 X J X ^0 3 cancels that common factor. Dividing both by 7 2 cancels 7 in the divisor and 21 in the dividend, leaving the qnotient 3 in the latter. Dividing both by 5 cancels 10 and 25, leaving the quotient 2 in the divisor and the quotient 5 in the dividend. The product of the remaining factors of the dividend is 15. Hence the quotient is 15 -5- 2, or 7|. CANCELLATION. 93 Find the quotients of the following : ^ 4 X 5 X 6 X 10 45 X 8 X 11 X 73 3. 2 > : 3 X 5 X 8 * 8 X 14 X 9 X 12 6 X 15 X 7 X 4 12 X 8 X 9 X 30 4 X 80 X 6 X 9 * 25 X 32 X 18 X 7 16 X 15 X 28 X 9 13 X 14 X 15 X IG 24 X 18 X 15 X 33' ()3 X 13 X 93 X 23 39 X ^1 X 21 X 69* 121 X 54 X 28 X 35 44 X 219 X 30 84 X 65 X 55 X 49 56 X 63 X 70 X 22' 132 X 52 X 68 X 45 10. 6. . 11. 7 X 8 X 9 X 10 77 X 65 X 51 X 20 12. How often is 12 x 13 x 50 contained in 65 x 10 x 84 X 3 ? 13. How many pounds of butter at 28 cents a pound must be paid for 25 yards of cloth at 56 cents a yard ? 14. How many barrels of apples, each containing 3 bushels, worth 70 cents a bushel, are worth as much as 20 boxes of crackers, containing 15 pounds each, if 2 pounds are worth 28 cents ? 15. A miller sold 20 barrels of flour, 196 pounds each, at 3 cents a pound, and received his pay in wheat at 84 cents a bushel. If there were 2 bushels in a bag, how many bags did he get ? 16. If 36 men, working 8 hours a day, can do a piece of work in 57 days, how long would it take 27 men, working 9 hours a day ? 17. The factors of the dividend are 10, 14, 9, 25, and 32 ; the factors of the divisor are 5, 16, 7, and 25. What is the quotient ? UNITED STATES MONEY. (An Introduction to Decimal Fractions.) 134. United States money has a decimal currency. It is written as dollars and decimal parts of a dollar, called dimes, cents, and mills. 1. A dime is what part of a dollar ? How is it written ? ($.1.) How may this be read? (One tenth of a dollar.) Then how would you write 2 dimes, or 3 tenths of a dollar ? 2. How is 3 tenths of a dollar written ? 5 tenths ? 7 tenths ? 9 tenths ? 8 dimes ? 15 dimes ? 25 tenths ? 3. Write 3 dollars and 5 dimes. 7 dollars and 9 tenths of a dollar. 20 dollars and 3 tenths of a dollar. 5 and one- tenth dollars. 4. What is always written in the first place to the right of dollars ? (Tenths of a dollar.) What separates the dollars from the tenths of a dollar ? 5. How is $.01 read ? A cent is what part of a dollar ? Then how would you write 2 cents, or 2 hundredths of a dollar ? 6. Write 3 cents, or 3 hundredths of a dollar. 5 hun- dredths. 7 hundredths. 8 cants. 9 hundredths. 7. Write 2 dollars and 6 cents. 5 dollars and 8 hundredths of a dollar. 9 dollars and 9 hundredths. Why put a cipher between dollars and hundredths ? 8. What is always written in the second place to the right of dollars ? (Hundredths of a dollar.) 9. Since there are ten cents in a dime, what is the difference between %.\ and $.10 ? Are they read in the same way .^ UNITED STATES MONEY. 95 10. How many hundredths of a dollar in a tenth of a dollar ? In 2 tenths ? In 2 tenths and 5 hundredths ? 11. $.25 may be read 25 cents ; or 2 dimes and 5 cents ; or 2 tenths and 5 hundredths, or 25 hundredths of a dollar. 12. A cent is what part of a dime ? Then a huiidredtli of a dollar is what part of a tenth of a dollar ? One tenth is equal to how many hundredths ? 13. In $.22, which 2 has the greater value ? Its value is how many times the value of the other ? 14. Write 24 hundredths of a dollar, 3 tenths and 5 hundredths. How many tenths of a dollar can be written with one figure ? How many hundredths ? With two figures ? 15. How is S.OOl read ? Since there are 1000 mills in a dollar, what part of a dollar is 1 mill ? Then how is 2 mills, or 2 thousafidths of a dollar, written ? 16. Write 3 mills, or 3 thousandths of a dollar. 5 thou- sandths. 9 thousandths. 7 mills. 5 dollars and 5 thou- sandths of a dollar. 17. What is always written in the third place to the right of dollars ? (Thousandths of a dollar.) 18. How many mills in a cent ? Then how many thou- sandths of a dollar in a hundredth of a dollar ? In 10 hun- dredths, or a tenth ? One dollar equals how many tenths of a dollar? How many hundredths ? How many thousandths ? 19. $.375 maybe read 375 mills; or 37 cents, 5 mills ; or 3 dimes, 7 cents, 5 mills ; or 3 tenths, 7 hundredths, 5 thou- sandths of a dollar, or 375 thousandths of a dollai'. 20. Since 5 mills equal half a cent, $.375 may be read 37 and one-half cents, and written $.37 J. 135. Copy and complete the following : 1. 7 dimes, or 7 tenths of a dollar = $.7, or $.70. 2. 3 dimes, or '' " " = ( ). 3. 5 cents, or " '' " = ( ). 4. 9 cents, or — '' '' ** = ( ). 96 SCHOOL ARITHMETIC. 136. The processes of adding and subtracting in U. S. money are the same as in simple numbers, or integers. Dolhirs should be written under dollars, cents under cents, mills under mills. The decimal points should be in a vertical line. Perform the operations indicated : 1. 172.65 + $18.23. 8. $G + $.6 + $.08 + $.008. 2. $39.47 + $26.82. 9. $271.83 - $187.93. 3. $53.90 + $18.25. 10. $1,000 - $100.75. 4. $91.03 + $4,775. 11. $86.37 - $24.80. 5. $.325 + $10,584. 12. $9,875 - $5,312. 6. $.875 + $.75 + $.093. 13. $5,003 - $2,008. 7. $3.40 -f $.205 + $80. 14. $100 - $.975. 15. $73,806 + $16,194 - $89.98. 16. $1 + $.1 - $.01 + $10. 17. To 3 tenths of a dollar add 7 hundredths of a dollar. 18. From 7 tenths of a dollar subtract 35 hundredths of a dollar. 19. What is the difference between 4 hundredths of a dollar and 9 tenths of a dollar ? 20. From the sum of 8 tenths and 5 thousandths of a dollar subtract 9 hundredths of a dollar. 21. A farmer bought two cows, giving $29.50 for one and $36.75 for the other. He gave in payment a wagon worth $42.25, and the rest in cash. How much money did he give ? 22. One month a man worked 24 days at $2.75 a day. His expenses were $41.70. How much did he save ? Dollars and decimal parts of a dollar are multiplied and divided just as integers are. Care must be taken to point off the proper number of places for cents and mills, 23. From $12,375 x 25 subtract 12 times $20.50. J37. United States money has a decimal scale; that is, UNITED STATES MONEY. 97 1 of any order or denomination is equal to 10 of the next lower order. Thus, $1 = 10 dimes ; 1 dime = 10 cents ; 1 cent = 10 mills. 1. How many cents in 5 dimes ? How many mills ? 2. How many dimes in 4 dollars ? How many cents ? How many mills ? 138. Carefully examine the following : $ dimes cents mills (a) 8. = 80. = 800. = 8000. (b) 3.25 = 32.5 = 325. = 3250. Query. — 1. In (a), how have we changed dollars to lower denomina- tions ? By annexing what ? 2. In (b), how have the changes been made ? By moving what ? In which direction ? Principle. — Any denomination is changed to a lower hy annexing one or more ciphers, or by moving the decimal point one or more places to the right. Change to mills, or thousandths of a dollar : 1. 15. 6. 370 dimes. 11. 12.05. 2. 3 dimes. 7. 435 cents. 12. $6,005. 3. 9 cents. 8. $7.65. 13. $.34. 4. 21 dimes. 9. $1.5. 14. $2,125. 5. 75 cents. 10. 7 dollars. 16. $.875. 139. Carefully examine the following : mills cents dimes $ (a) 2000 =200 =20 =2 (b) 3125. = 312.5 = 31.25 = 3.125. Query. — 1. In (a), how have we changed mills to higher denomina- tions ? By cutting off what ? 2. In (b), how have the changes been made ? By moving what ? In which direction ? Prikciple.i^^w^/ denomination is changed to a higher by cutting off one or more ciphers, or by moving the decimal point one or more places to the left. 7 98 SCHOOL ARITHMETIC. Change to dollars, or to dollars and decimal parts of a dollar : 1. 7000 mills. 7. 865 cents. 2. 500 cents. 8. 750 mills. 3. 80 dimes. 9. 1000 dimes. 4. 600 dimes. 10. 8625 mills. 5. 9000 cents. 11. 1250 cents. 6. 35 dimes. 12. 12375 miHs. 140. It was learned in multiplication of integers that an- nexing one cipher to a number multiplies it by 10 ; annexing two ciphers, multiplies it by 100 ; and so on. The same is true in IT. S. money. Thus, $2.00 X 10 =• $20.00 ; $2.00 x 100 = $200.00. 141. Moving the decimal point one place to the right multiplies expressions of U. S. money by 10 ; moving it two places, multiplies by 100 ; and so on. Thus, $3.25 X 10= $32.50 ; $3.25 x 100 = $825. 142. Moving the decimal point one place to the left divides expressions of U. S. money by 10 ; moving it two places, divides by 100 ; and so on. Thus, $125.50 -- 10 = $12.55 ; $125.50 -?- 100 = $1,255. Find the value of the following : (Omitting the dollar mark does not affect the operation.) 1. $52 X 10. 8. 13.245 x 10. 15. $300 x 100. 2. $7.25 X 10. 9. $.625 x 100. 16. $60.50 x 100. 3. $1.50 X 100. 10. $.004 X 1000. 17. $1,627 x 1000. 4. $.75 X 100. 11. $330.00 -^ $10. 18. $700.00 -^ $100. 5. $4.07 X 10. 12. 73.5 -f- 10. ' 19. 300.00 -^ 100. 6. $6 X 100. 13. 47.3 -^ 100. 20. 82.50 -4- 10. 7. 8.25 X 100. 14. 219 -^ 100. 21. 513.7 ~ 100. FRACTIONS. 143. 1. In measuring milk Kate uses a can that holds half a gallon. She fills it twice. How many lialf-gallons has she ? How many gallons ? 2. A merchant measures a piece of silk with a ruler one third of a yard long, and finds the piece to contain two thirds of a yard. How many times did he apply the measuring unit ? (a). Is the piece of silk a yard in length ? (b). What unit of measure did he use ? (c). How many times did he take the unit ? 3. A grocer selling molasses filled a jar three times. If the jar held one fourth of a gallon, how much molasses did he sell ? (a). What measuring unit did he use ? (b). What number tells how many times he took or repeated the unit ? (c). Is the unit of measure one of the equal parts of a gallon ? (d). What expresses the quantity of molasses sold ? 4. A farmer used a half -bush el to measure his wheat, filling it five times. (a). What was the measuring unit ? (b). How many such units in a bushel ? (c). How many half-Mishels had he ? 144. A Fraction is a number whose unit of measure is one of the equal parts of a certain whole or quantity. Thus, three fourths of a yard (3 fourth-yards) is a fraction, its unit of measure being one fourth-yard — one of the four equal parts of a yard. 100 SCHOOL ARITHMETIC. ^ Five half-bushels (5 halves of a bushel) is a fraction ; its unit of measure is one half-bushel — one of the two equal parts of a bushel. 1. An integer (Art. 6) is a number whose unit of measure is an entire quantity — not one of the equal parts of a larger quantity. 2. The fraction f yard may be regarded as 3 fourth-yards or as three fourths of a yard. The unit of measure is one of the four equal parts of tiyard, and this unit is repeated 3 times in measuring the quantity, which compared with a yard is three fourths as great. 145. The general method of expressing fractions is by two numbers, written one above the other, with a line be- tween them ; as, |. But a special class of fractions, called Decimal Fractions, is expressed in a notation peculiar to themselves. DECIMAL. FRACTIONS. 146. 1. When anything is divided into ten equal parts, what is one part called ? 2. When each of these ten parts is divided into ten equal parts, how many parts are there ? What is one part called ? 3. When each of these 100 parts is divided into ten equal parts, how many parts are there ? 147. AVhen anything is divided into tenths, limidredtJis, thousandths, etc., the parts are called Decimal Parts; that is, tenth-parts, the word decimal being derived from decern, the Latin word for ten. 148. A Decimal Fraction is a number whose unit of measure is one of the decimal or tenth parts of a certain quantity. Thus, 9 tenths (unit 1 tenth), 25 hundredths (unit 1 hundredth), 13 thousandths (unit 1 thousandth), etc., are decimal fractions. Decimal fractions are often called simply decimals. 5^. FRACTIONS. ,"%H 149. The decimal fractions one tenths one hundredth, one thousandth, etc., are obtained by dividing a quantity or whole into 10 equal parts {tenths), and each of these into 10 equal parts {hundredths), and each of these again into 10 equal parts {thousandths) ; hence, 1 (whole) = 10 tenths, 1 tenth = 10 hundredths, 1 hundredth = 10 thousandths. 10 thousandths = 1 hundredth, 10 hundredths = 1 tenth, 10 tenths == 1 (whole). 150. It is seen that tenths, hundredths, thousandths, etc., taken in order, decrease in value from left to right by the scale of tens, just as integers do. That is, 1 in any place or order is equal to 10 in the next place to the right ; and 10 in any place is equal to 1 in the next place to the left. 151. Since the notation of decimals follows the same law as that of integers, an integer and a decimal fraction may be written as one expression, as in U. S. money. The first place to the right of ones is tenths ; the second, hundredths; the third, thousandths, etc. 152. A point (.), called the Decimal Point, is placed before tenths to locate ones. Thus, three tenths is written .3 153. An integer and a decimal written together as one number is called a Mixed Number. In writing mixed numbers the decimal point is placed between the in- tegral part and the fractional part, thus : 6.9: 2.03. When there is no integral part, what may be written in ones' place ? ■jte:; SCHOOL ARITHMETIC. 154. The relation of decimals to integers is clearly shown by the following diagram : ] NTEGERS Decimals 00 00 fl f> .„ e8 2 w -4-3 f3 1 Etc. Millions £ i 9 = 3 i ^ 1 i , o 00 1 Q M O o w to -t-3 B Thousandths Ten-thousand Hundred-thoi Millionths Etc. ( /) (e )(d )W 1 «f) J«) 1 WO i)(, (. n It will be noticed that : 1. Orders at equal distances to the left and right of ones^ place have corresponding names ; the names at the right having the fractional ending ths. 2. The increase and decrease according to the decimal scale go right along, without regard to the decimal point. 3. For every jy«^r^ of the unit expressed by any order on the right, there is a corresponding multiple of the unit ex- pressed by the corresponding order on the left. 4. All orders, both higher and lower, are derived from ones. Tens denotes tens of ones, and tenths denotes tenths of one. READING DECIMALS. 156. In reading integers, we give the numbers, but omit the name ; thus, 25 is read tweyity-five. The name omitted is ones, which is the name of the right-hand order of the integer. FRACTIONS. 103 150. In reading decimals, we give both number and name. Thus, .25 is read, not twenty-five^ but twenty-five hundredths. Tlie name given is hundredths, which is the name of the rifrht-hand order of the decimal. With the exception of giving the name, decimals are read pre- cisely as integers are read. Rule. — Read the decimal as an integral number, and give it the name of the right-hand order. Read the following decimals : 1. .3. 6. .87. 11. .23742. 16. .003761. 2. .9. 7. .087. 12. .00013. 17. .009042. 3. .15. 8. .235. 13. .00304. 18. .0007103. 4. .35. 9. .101. 14. .01238. 19. .00000001. 6. .07. 10. .3005. 15. .000013. 20. .327604385, 1. In reading mixed numbers, read the integral part first, then the decimal part, connecting them with the word and. Thus, 205.03 is read, two hundred five and three hundredths. 2. It is sometimes necessary to make a pause before giving the name of the decimal. Thus, in .300, read three hundred thousandths ; and in .00003, read three hundred-thousandths. 3. Expressions like .12^ and .33^^ are read twelve and one-half hun- dredths, and thirty-three and one-third hundredths, respectively. The former may be written .125. 157. Since per cent means hundredths, in reading deci- mals we may say per cent instead of hundredths. The symbol ^ means either per cent or hundredths. Thus, .25 = 25 per cent = 25^. .12i = 12^ per cent = Vit\%. .50 = 50 per cent = 50^. .05 = 5 per cent = 5^, Read the following mixed numbers : 1. 2.25. 6. 800.2035. 11. 136.00004. 2. 50.07. 7. 70.005. 12. 4000.004. 3. 13.033. 8. 30.078. 13. 3050.0507. 4. 310.09. 9. 3826.7. 14. 17005.017. 5. 7,394. 10. 4002.006, 15, 12345.12345, 104 SCHOOL ARITHMETIC. Read both ways : 16. .15. 19. .01. 22. .37i. 25. 10^. 17. .27. 20. .09. 23. .80. 26. 35^. 18. .40. 21. .75. 24. .62^. 27. 33^^. Remark, — Decimals may be read in different ways. Thus, .125 may be read 125 thousandths; or 1 tenth, 2 hundredths, 5 thou.sandths; or 12 hundredths, 5 thousandths. In practice, however, it is desirable to follow the method indicated in the Rule. WRITING DECIMALS. 158. 1. Express decimally thirty -four hundred -thou- sandths. Hundred-thousandths is the fifth decimal order, hence 5 places are needed to express the decimal. But 34, written as an integer, occupies only 2 places, leaving 3 places on the left to be filled with cipher's. Hence the decimal is written .00034. l^ Pupils should become thoroughly familiar with the names of the decimal orders at least to millionths. Since the name or denomination of the decimal is indi- cated by the position of the right-hand figure with respect to ones' place, we have the following Rule. — The deiiomination of the decimal determines the number of places necessary to express it; therefore, Write the decimal as an integral iiumher, and prefix ciphers, when necessary, to supply the required number of places, placing the decimal point directly before tenths. Express decimally : 2. Three tenths. One tenth. Nine tenths. Six tenths. 3. Twelve hundredths. Four hundredths. Fifty-five hun- dredths. Ten hundredths. 15^. 9 per cent. 12|^. 4. Six thousandths. Fifteen thousandths. Two hundred three thousandths. Twenty-five ten-thousandths. Seven ten-thousandths. Four hundred fifty-one ten-thousandths. 6. Eight hundred-thousandths. 725 hundred-thousandths. FRACTIONS. 105 5 millionths. 75 hundred-millionths. 351 thousandths. 15 ten-millionths. 12 hundred-thousandths. 6. Four, and four hundredths. Three hundred-thou- sandths. One, and five thousandths. Six millionths. Five hundred thousandtlis. 7. Thirteen hundredths. 47005 billionths. Six hundred, and seven thousandths. Six hundred seven thousandths. 8. Sixty-nine, and 903 thousandths. Forty-nine, and 000 ten-thousandths. One millionth. Write in words : 9. .7; .05; .01; .016; .203 ; .25 ; .324; 8^. 10. .001; .0015; .0125; .2405; .00025; .00123. 11. 2.7; 5.04; 6.008; 4.0010; 12.02301; 202.0202. 12. How many tenths can be expressed by one figure ? How is 10 tenths written ? 13. How many hundredths can be expressed by one figure ? By two ? How is 100 hundredths written ? How is 100 per cent Avritten ? 14. Since 1 tenth equals 10 hundredths, 100 thousandths, 1000 ten-thousandths, etc., it is plain that .1 = .10 = .100 = .1000 = .10000, etc. Hence, 159. Principle. — Annexing ciphers to a decimal reduces it to a lower denomination without changing its value. Query. — Does omitting ciphers from the right of a decimal change its value ? 1. Change .5, .03, .027, and .4850 to thousandths. .5 = . 500 In the first two decimals we annex ciphers enough .03 =: .030 to make the 3 places required to express thou- .027 = .027 sandths. The third needs no changing. Why ? The .4850 = .485 last is changed by omitting the cipher at the right. This process is called reducing to a common name or de- nomination (or denominator). 2. Change .8, .25, .030, .4600, and .07 to thousandths. 3. Reduce .75, .013, .020, and .0146 to ten- thousandtlis. 106 SCHOOL ARITHMETIC. 4. Reduce .09, .0240, .3275, .1, .00010 to millionths. 5. Change .30, 5, .400, .8000, and 1.7 to tenths. 6. Change .0032, .2, .470, and .835000 to ten-thousandths. 7. Reduce 5 ones to tenths, 3 ones to hundredths, 10 ones to tenths, and 2 ones toper cent. 8. How does .1 compare in value with .01 ? How in form ? Then what is the effect of prefixing a cipher to .1? 9. How does .01 compare in vahie with .001 ? Then how is .01 affected hj prefixing a cipher ? Prefix ciphers to other decimals, and compare values. 160. Pbinciple. — Prefixing a decimal cipher to a decimal divides the value of the decimal hy ten. Queries. — 1. How does prefixing a cipher affect the place of each figure in the decimal ? 2. Does a figure in that place express as much value as it did before being moved ? 3. What part of its former value does it express ? Then by what has the decimal been divided ? ADDITION AND SUBTRACTION. 161. In addition and subtraction of decimals the opera- tions are the same as the like operations in integral numbers. 1. What is the sum of .613, .0176, .2, and .601 ? • 613 By arranging the decimal points in a vertical line, .0176, we make units of the same order stand in the same • 2 vertical column. The numbers are added precisely as •oQl in integers, and the decimal point is placed before 1.4316 tenths. (Is 14 tenths a fraction? How is it written?) 2. From .3 subtract .1235. (^) (b) By arranging the decimal points in a verti- .6 = .oOOO cal line, we cause units of the same order to •I^^^ ^^ .2235 stand in the same column. We subtract as .1765 .1765 in integers or U. S. money. Queries. — Why may .3 be written as in (b) ? (See Art. 159.) Is it necessary to annex ciphers to the minuend ? When the rejuainder is q, mixed puraber, where is the decimal point placecl ? FRACTIONS. 107 Find the value of the following : 3. .17 + .002 + .2509. 11. .75 - .25. 4. .005 4- .301 + .29. 12. .5 - .005. 6. 19.909 + 100.01 + 199. 13. 100.01-25.001. 6. .375 + .048 + 255.0. 14. 10 - .0678. 7. 4.372 + .4293 + 3.87. 15. .10 -.06814. 8. 5.0008 + 124 4- .010. 16. 1000 - .1000. 9. 86.45 + .001 + .05. 17. .6504 - .067. 10. 2.3 + .004 + .2 + .88. 18. .1 - .0053. 19. 94.61 + .00421 + .0003 + .0044 + 10. 20. 84.56 + 9.245 + .8703 + 8.009 + 7.7. 21. 1 million — one millionth. 22. 10. — 10 ten-thousandths. 23. 94 thousandths — 253 ten-millionths. 24. 25 thousandths — 25 ten-thousandths. 25. 1 — 1 thousandth + 1 tenth + 100 hundredths. 26. The minuend is the sum of .3 and .003 ; the subtra- hend is .02875. What is the remainder ? 27. From what number must .0105 be subtracted to leave the remainder 1.807 ? 28. The larger of two numbers is 3822.078 ; their differ- ence is 1934.124. What is the less number ? 29. Find the least decimal which added to 1.4142 — .0022 will make the result an integer. 30. A owes 11,000 to B, and $1,347.55 to C. He has in cash $1,955.75. If he jmys C in full, how much will he lack of having enough to pay B ? 31. Mr. Slaven bought an organ for $85.50 on a credit of three months. He concluded to pay cash, and was allowed a discount of $1.27. How much had he left out of a $100 bill ? 32. Bishop Brothers sold goods amounting to $190.50 on Monday, $250 on Tuesday, $117.25 on Wednesday, $57 on Thursday, $135.75 on Friday, and $427.37 on Saturday, What were the total sales for the week ? 108 SCHOOL ARITHMETIC. 33. Find the sum of 345 millionths, forty and 40 millionths, seven and 7 thousandths, thirty-eight and 87 ten-thou- sandths. 34. In a corncrib that will hold 572.5 bushels of corn there are 329.375 bushels. How many bushels will be re- quired to fill it ? 35. One side of a square field is 42.375 rods long. If 12.5 rods of the fence around it are blown down, how many rods will remain standing ? 36. A tank that will hold 1050.75 gallons contains 396.7 gallons. If 135.5 gallons be added, how much will still be needed to fill the tank ? 37. A man bought a farm for $1,750 and a lot for $975. 75. For what amount must he sell both to gain $289.50 ? MULTIPLICATION AND DIVISION. 162. The processes of multiplication and division of deci- mals are the same as the like processes in integers, the locat- ing of the decimal point being the only thing that needs special attention. (a) (b) (c) (d) .1 1. .25 25 1. In (a), the figure 1 expresses 1 tenth, in (b) it ex- presses 1 one. What has been the effect of moving the deci- mal point one place to the right ? 2. In (d), the 2 expresses 2 ones, in (c) 2 tenths. The 5 in (d) expresses 5 tenths, in (c) 5 hundredths. What has been the effect of moving the decimal point one place to the left ? 163. Principle. — Each removal of the decimal point one place to the right multiplies the decimal hy 10 ; each removal one place to the left divides the decimal hy 10. Thus, by moving the point one place to the right, .825 becomes 3.25 ; that is, 3 tenths have become 3 ones, the 2 hundredths have become 2 tenths, and the 5 thousandths have become 5 hundredths. Since the FRACTIONS. 109 value of each figure has been multiplied by 10, the value of the entire decimal has been multiplied by 10. IJ^" Have the pupil illustrate the second part of the principle, which is the converse of the first. 164. To multiply or divide a decimal by lO, lOO, lOOO, etc. Rules. — 1. To multiply a decimal hy 10, 100, 1000, etc., move the decimal point as many places to the right as there are ciphers in the multiplier, annexing ciphers when necessary. 2. To divide a decimal hy 10, 100, 1000, etc., move the deci- mal point as many places to theleft as there are ciphers in the divisor, prefixing ciphers when necessary. 1. Multiply .275 by 100 ; also by 10000. .275 X 100 = 27.5. .275 x 10000 = 2750. 2. Divide .275 and 62.5 each by 100. .275 -^ 100 = .00275. 62.5 H h 100 = .625. Find the value of : 3. 3.25 X 10. 8. 37.5 ~ 10. 13. .37685 X 1000. 4. 69.3 X 10. 9. 6.25 -^ 10. 14. 52.16 X 1000. 5. .75 X 100. 10. .314 ^ 10. 15. 7.013 -7- 100. 6. .486 X 1000. 11. .209 H- 100. 16. 3875 -^ 1000. 7. 1.625 X 100. 12. 632 -^• 100. 17. 41.065 -^ 100. 165. To multiply or divide a decimal by .1, .Ol, .001, etc. Rules. — 1. To multiply hy .1, .01, .001, etc., move the deci- mal point as many places to the left as there are decimal places in the multiplier. 2. To divide hy .1, .01, .001, etc., move the decimal point as many places to the right as there are decimal places in the divisor. 1. Multiply 1.093 by .1 ; also by .01. 1.093 X .1 = 1.093 -J- 10 = .1093. 1.093 X .01 = 1.093 -- 100 = .01093. To multiply a number by .1 is to take one tenth of it ; that is, to 110 SCHOOL ARITHMETIC. multiply by .1 is to divide by 10 ; to multiply by .01 is to divide by 100, etc. By comparing the products with the multiplicands, we find that the decimal point has been moved to the left as many places as there are ciphers in the multiplier, (See Art. 164.) 2. Divide 32.5 by .1 ; also by .01. 32.5 ~- .1 = 32.5 X 10 = 325. 32.5 -^ .01 = 32.5 X 100 = 3250. Since there are 10 tenths in 1, one tenth is contained in any number 10 times as often as one is contained in it. But dividing a number by 1 does not alter its value. Hence to divide a number by .1 is to multiply it by 10 ; to divide by .01 is to multiply by 100, etc. Multiply : Divide : 3. .258 by 10. 14. 37.5 by 100. 4. 7.07 by 100. 15. 436 by 1000. 5. 3.916 by 1000. 16. .900 by 100. 6. .846 by 10000. 17. 24.57 by 1000. 7. 7.5 by .1. 18. 5 by 1000. 8. 83.7 by .01. 19. .99 by .1. 9. 3.25 by .01. 20. .0075 by .01. 10. .3004 by .001. 21. .0003 by .001. 11. 179.5 by .001. 22. 4444 by .0001. 12. 3.428 by .0001. 23. 18 by .01. 13. .5 by .0001. 24. 100 by .1000. 25. Which is the greater, .5 x 100, or .5 -^ .01 ? 26. How much greater is .75 x 1000 than .25 -^ .001 ? 166. To niviltiply or divide in tlecimals — universal case. Principles. — 1. Tlie product of tioo decimals contains as many decimal places as there are decimal places in hoth factors. 2. The quotient of two decimals contains as many decimal places as the number of decimal places in the dividend exceeds the number in the divisor. The number of decimal places in the dividend can be increased as you please, by principle in Art. 159. FRACTIONS. Ill 1. Multiply .036 by .27. 036 The multiplier .27 = 27x .01. "We therefore multiply ^27 first by 27, then the resulting product by .01. 36 thousandths oKo X 27 = 972 thousandths, or .972. Multiplying this product wo by .01 moves the decimal point two places to the left (Art. 155). Hence the required product is .00972. It has as many 00972 decimal places as both factors have. If we multiply as in integers, we get the product 972, to which we pre- fix two ciphers to make the required five places. 2. Divide .00972 by .27. .27).00972(.036 The dividend being the product of divisor and 81 quotient must contain us many decimal places as ]62 both of them. Since the dividend contains 5 decimal 162 places and the divisor 2, the quotient must contain 5 — 2, or 3 decimal places. Dividing as in integers, we get the quotient 36, to which we prefix a cipher to make the required three places. 167. EuLES. — 1. In the muUipUcation of decimals multiply as in integers, and from the right of the product point off as many decimal places as there are in hoth factors, prefixing ciphers, if necessary, to make the required number of decimal places. 2. In the division of decimals, divide as in integers (annex- ing ciphers, if necessary, to the dividend), and point off from the right of the quotient as many decimal places as those of the dividend exceed those of the divisor. If the quotient does not contain a sufficient number of decimal places, ciphers must be prefixed to make the required number. (a) Find the product of : 1. .28 X 4.8. 9. 10000 X .0001. 2. .6 X .7. 10. 7.5 X .0005. 3. .35 X .16. 11. 1000000 X .000001. 4. 10 X .1. 12. .001 X 10000. 5. .134 X 25. 13. .1 X .1. 6. 216 X .24. 14. .5 X .5. 7. .478 X .152. 15. .5 X .05. 8. .0017 X .09. 16. .05 X .005. 112 SCHOOL ARITHMETIC. (a) Find the product of : 17. .01 X .001. 18. 150 X .1. 19. $1 X .1. 20. 7 X 1.1. 21. 2.5 X 2.5. 22. $100 X .06. 23. .017 X 3.7. 24. 101 X 1.01. 25. 1.03 X 1.09. 26. 5.005 X .005. (b) Find the quotient of 1. .00125 -T-.5. 2. .0075-^1.5. 3. 1 -- .1. 4. .01 ^ 100. 5. 16.84 -f- .02. 6. .00884 -^ .34. . 7. .0355 ^ .71. 8. 16.025 -^ .045. 9. 10000 -^ .0001. 10. .000375 -^ .0005. 11. 1000000 -f- .000001. 12. .000001 -^ 1000000. 13. 1150 ^ $ .06 14. II -^ $ .05. 15. 159.750^ .00375. 16. 14400-^.32. 17. 14400 ^ 3.2. 18. 200 -^ .002. 19. .735 -J- 500. 20. 78.13^5. 21. 78.39 -f- 3. 22. 125 ~ 25000. 27. .008 X 800. 28. 5 tenths x 50 hun- dredths. 29. .01 X .1 X 1. 30. .05 X 5 X .50. 31. 72.5 X 10. 32. .1225 X .1. 33. 25.6 X .20. 34. .054 X 100. 35. 125 X 1.05. 23. 12 -=- .0012. 24. 5.4768-^22.82. 25. .025-4- 250. 26. .0567 -^ 43. 27. 1 -i- 3.1416. 28. ten -=- .01. 29. 1 millionth -f-. 01. 30. 300 hundredths ~ 15 tenths. 31. 3.1416 -^ .31416. 32. .25 -^ .0025. 33. 9 ones -^ 40 tenths. 34. 25 tenths -^ 25 hun- dredths. 35. 25 hundredths ~- .025. 36. 27.45 -r 1.5. 37. 250 ~- .025. 38. 2750 -4- .25. 39. 3.609 -4- .9. 40. 27.63 -^ .003. 41. 4.914-4- 70. 42. .026 -^- .000013. FRACTIONS. . 113 108. 1. Multijily 4.Ge5 by 700, and divide the product by 300. 4.G5 X 100 = 4r,5. Then 405 x 7 = 3255. 3255 -^ 100 = 32.55 ; and 32.55 -f- 3 = 10.85. Find tlie vahie of : 2. 52(3.53 X 50. 7. 030 -^ 500. 3. 245.0 X 400. 8. .844^ 400. 4. .804 X 900. 9. 307.2 -i- 1200. 6. .7854 X 700. 10. 2607.5 -^ 8300. 6. 150 X 25^. 11. 150 -^ 25^. 12. What is the value of .05 x .07 + .28 ^ .5 ? 13. If I give 3 i:>igs for 17.50, how many must I give for $37.50? 14. A man paid $17.25 for 300 pounds of sugar. What did it cost per pound ? 15. How many eggs in a crate containing 24.5 dozen ? 16. If 8 pounds of coffee cost $1.74, what will 5 pounds cost ? 17. At 2|^ each, how much will 3.25 dozen lemons cost ? 18. A man paid $15 for rice, at the rate of 4 pounds for a quarter. How many pounds did he get ? 19. A has $1.40 and B has 2.5 times as much. How much must B give A so that each may have the same amount ? BILL.S AND ACCOUNTS. 169. Prof. Samuel Andrews bought of J. R. Weldiii & Co. the following : 6 dozen lead pencils at 1.30 a dozen, 2 gross pens at $.85, 5 reams note paper at 11.50, and 20 arithmetics at $.75. In a few days he received the following hill : Columbia, S. C, May 1, 1899. Mr. Samuel Andrews, Bought of J. R. Weldin & Co. To 6 dozen Lead Pencils @ $.30 1 80 '' 2 gross Pens '' .85 1 70 '' 5 reams Note Paper '' 1.50 7 50 " 20 Arithmetics '' .75 15 00 26 00 When this bill was paid, the following was written on it as a receipt : *^ Received payment, J. R. Weldin & Co. (The " G " is the initial of Mr. Greene, who receipted the bill.) 1. Mrs. R. D. White ordered the following from Davis & Russell, New Orleans, La. : 18 yd. Scotch Gingham @ 21^. 36i yd. Calico @ 6^. 12i yd. India Silk @ 45^. 25 yd. Cashmere @ $1.25. Make out her bill, and receipt it. BILLS AND ACCOUNTS.. 115 170. The following is a specimen of a receipted bill, with a discount, and credits : Jackson, Miss., Oct. 1, 1899. Mr. T. B. DeArmit, To Gordon, Hay & Co., Dr. 1899. Jan. 13 May 21 Aug. 9 To 50 Grammars... $.40 '* 24 Arithmetics. . .60 '* 42 Histories.... 1.00 Less 10^ 20 14 42 00 40 00 76 7 40 64 Cr, By Cash $25.00 " '^ 25.00 July 25 Sept. 8 68 50 76 00 18 76 Received payment, Gordon, Hay & Co. By WiLSOK. 171. A Debt is the amount which one person owes another. A Debtor is a person or firm that owes a debt. 172. A Credit is the amount paid on a debt. A Cred- itor is a person or firm to whom a debt is due. In the transaction mentioned in Art. 170, who is the debtor ? Who is the creditor ? Name the credits. 173. An Account is a record of the debts and credits be- tween two parties — a debtor and a creditor. 174. A Bill is a creditor's written statement of the quan- tity and price of each item in liis account with a debtor, together with the discount and credits, if any, and the net amount due. Bills are commonly called invoices. 116 SCHOOL ARITHMETIC. 175. A Statement is a written summary of an account between two parties, rendered at stated intervals, usually monthly. 1 76. Make out and receipt the following bills. Supply dates and names where needed. 1. Mr. R. P. Lougeay bought of McAllister & Co. 25 pounds of coffee at 28 cents a pound, 75 pounds of sugar at 5^ cents a pound, and 20 pounds of prunes at 12 cents a pound. 2. Mrs. M. B. Kifer bought of Campbell & Smith 10 yards of silk at 11.50 a yard, 36 yards of muslin at 7 cents a yard, 15 yards of flannel at $.75 a yard, jind 2 pairs of shoes at $3.25 a pair. 3. Mrs. A. C. McLean bought of Kauffman Bros. 3 table- cloths @ $3.50, 1 piano cover @ $4.75, 4 pairs of lace cur- tains @ $5.25, 2 doz. towels @ $3.60 a dozen, and 12 yards cashmere @ $1.25 a yard. 4. Miss Xiinnie Mackrell bought of W. M. Laird 2 pairs ladies' shoes @ $2.75 a pair, 6 pairs overshoes @ $.75 a pair, 1 pair Oxford ties @ $1.25, 3 pairs misses' shoes @ $2.15, and 1 pair gum boots @ $3.25. 5. Mr. S. M. Brinton bought of Hopper Bros. & Co. 2 doz. silver knives @ $36 a dozen, 4 doz. silver teaspoons @ $16 a dozen, 2 doz. silver tablespoons @ $10.25 a set, and 1 silver spoon bolder for $9. 6. Mr. Wm. Hasley bought of Fred Gray 12.5 tons of coal @ $3.25, 40 bushels of apples @ $.75, 200 lb. grapes @ 3 cents a pound, and 25 bushels of potatoes @ $.85. 7. On May 25, J. M. Logan bought of W. II. Keech 5 bedsteads @ $14, 1 bookcase for $35, and 18 chairs @ $15 a dozen. On July 3, he bought 3 hammocks @ $2.25, and a leather couch for $45. On June 15, he paid $50 in cash, and on the 10th $37.50 more. REVIEW WORK. ORAL EXERCISES. 177. 1. How Tiijiny tenths in 80 hundredths ? 2. How many hundredths in 7 tenths and 15 liundredtlis ? 3. I paid 3 tenths of a dollar for 3 cakes of soap. At the same rate, how much would I pay for a dozen cakes ? 4. A lady spent .1 of her money for a hat, and A for a shawl, and the remainder for a dress which cost l?15. How much had she at first ? 5. At $.09 each, how many slates can be bought for $3.00 ? 6. If 80 is divided into 10 ecjual parts, what is one jiart called? Three parts ? Nine parts ? How many are 7 tenths (.7) of 80? 7. If 18 is .3 of some number, what is the number ? 8. Thirty-five is .5 of what number ? 9. Of wliat nnmber is 9 three tenths ? 10. The sum of .2 and .05 is .5 of what number ? 11. A has $1.50, and .3 of his money is .1 of B's money. How much has B ? 12. B and C together have $40. If .3 of B's money equals .9 of C's, how much has each ? 13. How often must .3 be added to itself to make 3 ? 14. How many times must .7 be subtracted from 3.5 to leave a remainder of 1.4 ? 15. How many hundredths can be taken from 25 tenths ? WRITTEN EXERCISES. 178. 1. If seven sheep are worth $31.50, how many sheep can be bought for $184.50 ? 118 SCHOOL ARITHMETIC. 2. A man divided his farm of 227.5 acres into 14 equal fields. How many acres in 5 of the fields ? 3. At $2,625 a yard, how many yards of cloth can be bought for $55,125 ? 4. If a person's taxes are 5.8 mills on $1, how much will they be on $2500 ? 5. Find the cost of 237.25 bushels of oats at .42 of a dollar a bushel. 6. At $.08 each, how many copy books can be bought for $24? 7. Gold weighs 19.36 times as much as an equal bulk of water, and a cubic foot of water weighs 62.5 pounds. How many cubic feet of gold weigh a ton, or 2,000 pounds ? 8. One pound of dry oak' wood when burnt yields .023 of a pound of ashes. How many pounds must be burnt to pro= duce 46 pounds of ashes ? 9. Every day a newsboy buys 70 papers at 30 cents a dozen, and sells them at 5 cents each. How much money does he make in 6 days if 40 papers remain unsold ? 10. If a boy saves 6 dimes a week, in how many days can he save enough to buy a suit worth $5.40 ? 11. How often can .013 be subtracted from 26 ? 12. Find the cost of 8 bushels 3 pecks of turnips at $.125 a peck. 13. The divisor is 27.125, the quotient 7.32, and the re- mainder 18.0825. What is the dividend ? 14. Divide 3 ten-millionths by 10 millionths, and multiply the quotient by 30. 15. At $1.50 a thousand, what will 1,750 envelopes cost ? 16. The distance around a circle is about 3.1416 times the distance across it through the center. If the distance around a circular pond is 50 feet, what is the distance across it? 17. Two men start from the same place at the same time and travel in the same direction, one going 3.28 miles an REVIEW WORK. 119 hour, the other 4.07 miles an hour. How far apart will they be in 9 hours ? 18. The circumference of the wheel of a bicycle is 11.28 feet. How many times will it turn in going 2.5 miles, there being 5280 feet in a mile ? 19. Find the cost of 8375 feet of lumber, when lumber is worth $18 a thousand feet. 20. In a city of 240,000 inhabitants .125 of tlie population are school children. If each teacher has 50 pupils, how many teachers are in that city ? 21. Add 155 ones, 155 tenths, 155 hundredths, 155 thousandths. 22. The divisor 5.125 is 5 times the quotient; what is the dividend ? 23. The product of three factors is 78. GG ; two of the factors are respectively 6.9 and 7.125; what is the third factor ? 24. Find the least decimal fraction which added to the sum of 87.43 and 1G9.578 will make the sum an integer. 25. A man paid .15 of his money for rent, .02 for wood, .18 for clothing, and had $812.50 left. How much had he at first? 26. Find the product of the two smallest decimals that can be expressed by the figures 0, 0, 9, and 3. 27. Gunpowder is composed of .76 nitre, .14 charcoal, and .10 sulphur. How much of each is required to make 2000 pounds of powder ? 28. At $0.34 a bushel, how many barrels of apples can be had for $13.60, allowing 2.5 bushels to the barrel ? 29. How many pounds of butter could be made from 46 cows during the month of June, each cow averaging 2.5 gallons of milk daily, and each gallon making .5 of a pound of butter ? 30. If 4 cords of wood are worth as much as 13.4 bushels of rye, how much rye can be obtained for 15 cords of wood ? 120 SCHOOL ARITHMETIC. 31. If a Mexican dollar is worth 10.85, how many Mexican dollars equal the value of |G80 in U, S. money ? 32. If the land that produces a bale of cotton yields 30 bushels of cotton seed, what is the value @ $.30 per bushel of the cotton seed produced by the land that yields 21 bales of cotton ? 33. In one manufacturing establishment the average weekly wages paid to 2()2 operators was $12.85 ; in another, to 355 operators, $13.84; and in a third, to 128 operators, $15.11. Find the average weekly wages in all three establishments. 34. If a railroad train runs 350 miles in 19.5 hours, but makes three stops of 20 minutes each, and ten stops of (5 minutes each, what is the average rate per hour while run- ning ? 36. A franc is 19.3 cents. Find the cost in United States money of goods bought in Paris amounting to 1,000 francs. 36. A cubic foot of water weighs 1000 ounces. IIow many pounds does a cubic foot of gold weigh, gold being 19.4 times as heavy as water ? 37. If oysters yield 1.25 gallons to the bushel, how many bushels in the shell must I buy so that when opened they will fill a 10-gallon can ? 38. In the year 1897, the total ordinary expenditures of the United States government were $365,774,159, which was $5.02 to each person. What was the population in that year, to the nearest 1000 ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 179. 1. Can 75 tenths be written as a decimal fraction ? Why not ? Can it be written as a mixed number ? 2. Divide one by seven, carrying the quotient to 12 decimal places, and carefully note the result. 3. Investigate the result of dividing one by 3, 11, 13, and 17, carrying the division as far as may be necessary. REVIEW Work. 121 4. After spending .015 of liis mouey, and 2 tenths of the remainder, B luid $15.70 left. How much did he spend ? 5. Find the cost, at *S 75 per thoiisund, of the rails for 040 panels of 14- rail fence. 6. The nniliipliciand is .005, and tlie product is 1. hy what must the multi[)lier he divided to give a quotient ecjual to the product ? 7. I gave .44 of my money for a fjirm, and .75 of the re- mainder for a store. If the farm cost ^250 more than the store, how much did I pay for the store ? 8. Cork weighs 15 pounds per cubic foot, and its weight is .24 of the weiglit of water. Find tiie weight of 10 cubic feet of oak, if the weight of oak is .934 of the weight of water. 9. The distance of the moon from the earth is 50.07 times tlie earth's radius. If this radius is 30()2..S24 miles, find the distance to the moon. 10. In 1800 the native population of the United States was 85.23^ of tlie whole. Tlie foreign born was what decimal part of the native ? 11. The population of Italy is 29,090,785. The total in- debtedness of the country is $2,324,826,329. Find tlic rate of debt for each person. 12. The population of New Orleans in 1807 was 280,000. The assessed valuation of taxable property was $140,054,475. Supposing the whole population to be divided into families of five, compute the average wealth of each family. COMMON FRACTIONS. 180. 1. When anything is divided into ten equal parts, what is one part called ? What name is given to tenth-parts of anything ?^ What kind of fraction is one expressing deci- mal parts of any whole or quantity ? Then what kind of fraction is one-tenth (.1) ? 2. When anything is divided into 100 equal parts, what is one part called ? 25 parts ? What kind of fraction is 6 hun- dredths ? Fifty thousandths ? How are they written ? 3. Is one fourth yard a decimal fraction ? $4 ? 5 half- hushels 9 Why not ? 181. When a measuring unit is a decimal part of a certain whole or quantity, any number of these units (parts) ex- pressed in decimal notation is called a decimal fraction ; but when the Unit of measure is one of any number of equal parts, one or more of such parts, expressed by two numbers one above the other with a line between them, is called a Common Fraction. Thus, 5 tenths (.5) and 25 hundredths (.25) are decimal fractions, while 3 fourths (f) and 10 thirds (^3^) are common fractions. Notes. — 1. The decimal point was first used about the beginning of the 17th century, but 100 years elapsed before decimal fractions were exten- sively employed. 2. During the introduction of this new class of fractions, the "old style" fractions were given the name "vulgar "or "common" frac- tions to distinguish them from the others, which were written in a new and special way by the aid of the decimal point. 3. Strictly speaking, a number of decimal parts, to be called a deci- mal fraction, must be written in the form peculiar to decimals, since it is only in that form that the practices exemplified in decimals are applicable. FRACTIONS. 123 182. The Unit in fractions is v^part of one whole, and is called a Fractional Unit; that is, a fourth is the unit of measure in three fourths. 183. The Denominator (nanier), the number written below the line, shows the number of equal parts into which a certain whole or quantity is divided in order to obtaisi the fractional unit. Thus, in f bushels, the denominator 2 shows that the one bushel is divided into two equal parts and the unit is a half-bushel. How many- fractional units are used to express the value of the quantity ? 184. The Numerator {nnmberer), the number written ahove the line, shows how many of the fractional units are used. Queries. — In the fraction $}, which is the denominator ? What does it show ? What does the 3 show ? Of what does %l express the value ? Note. — A fraction whose numerator is less than the denominator is called a proper fraction ; otherwise it is usually called an improper frac- tion. 185. A Mixed Number is the sum of an integer and a fraction, and is expressed by writing the fraction imme- diately after the whole number. Thus, $5 + If = $5|, is a mixed number. Notes. — 1. The integer and fraction must be like numbers, 2. Five silver quarters are equivalent to $li ; but in this case the mixed number is 7iot the sura of an integer and a fraction. The quan- tity is ?k fraction ($|) made up of 5. units of measure (quarters). In prac- tice, however, no attention is paid to this distinction. 3. The definition of a fraction in Art. 181 includes only fractions whose 2 5 denominators are integral, hence excludes such expressions as — - and — -, which are properly expressions of unexecuted division. (See Art. 95.) However, in algebra the definition of a fraction is extended so as to em- brace any expression in the fractional form. 186. An Integer or a Mixed Numler may be expressed in the form of a fraction, and treated as a fraction. Thus, |4 = 1^ = 4 one dollars ; $3^ = $| = 7 half-dollars ; 1 = i = l=ietc. 124 SCHOOL ARITHMETIC. 187. The number of equal parts into which anytliing la divided gives the parts their name. Thus, any quantity divided into 2 equal parts = 2 lialves, or J; any quantity divided into 3 equal parts = 3 thirds, or ^ ; any quantity divided into 4 equal parts = 4 fourths, or J; any quantity divided into 5 equal parts = 5 fifths, or § ; etc. Therefore, to read fractions. Direction. — State the numher of fractional units (units of measure) and give tlteyn the name indicated by the denomi- nator. (a) Read, and write in words : 1. h 1. !• 4. U> M. If 7. u, :.'„'6. H- 2. h i> *• 5. ih ih itV- 8. M. ,%%, .42. 3. i ■1. A- 6. -?\«> ih' /A- 9. ?' 1' ««• (b) Write in figures : 1. Three fourths. 7. Fifty-two hundred tlis. 2. Four fifths. 8. Sixty two-hundredths. 3. Ten elevenths. 9. Nine eightieths. 4. Eight twenty-firsts. 10. Eighty ninetieths. 5. Twelve thirty-thirds, 11. Forty three-thousandths. 6. Two hundredths. 12. Two and one-half thirds. CHANGE OF FORM. 188. To change fractions to larger denominators. 1. How many fourths in a gallon ? In ^ of a gallon ? Then ^ is equal to how many fourths ? 2. How many eighths in an apple ? lu ^ of an apple ? Then ^ is equal to how many eighths ? 3. Since ^^ = |, and I — g, what has been done to the numerator and denominator of tlie fraction | without chang- ing its value ? How has it been done ? 4. Change ^ to sixths. To twelfths. To 16ths. To 20ths. FRACTIONS. 125 5. Change § to sixtlis. To niiitlis. To 12tlis. To ISths. 6. Ill 1 liow many 24ths ? Then how many 24ths are there in ^ ? In ^ ? In f ? In j\ ? 7. Name three fractions each equal to J. 8. What may be done to the numerator and denominator of a fraction without changing its value ? 9. Express I with nunieratorand denominator 4 times as large, and explain why the value of the fraction is not changed. Multiplying numerator and denominator by 4, we have |f . Since the fraclional units are ^ as large, while the number of them taken is 4 times as great, jg is equivalent to f . Hence the following 189. Principle. — Multiplying numerator and denomina- tor hy the same number does not change the value of the frac- tion. WRITTEN EXERCISES. 190. 1. Change | to twelfths. j2 _i_ 3 ::::: 4 Multiplying numerator and denominator by the same 2 j^ o number does not change the value. Since the denomina- — = — tor is to be 12, both numerator and denominator must 3x4 12 be multiplied by 12 -^ 3, or 4 ; therefore, f = h- Analysis. — In |, or one- there are 12 twelfths, hence in \ there are J of 12 twelfths, or 4 twelfths, and in f there are 2 times 4 twelfths, or ]*2. Direction. — Multiply numerator and denominator hy the quotient arising from the division of the required denomina- tor hy the given denominator. Change ^' h 3' i' ''^^^^ i ^^ twelfths. 3. ^, i, |, and. -^^ to twenty-fourths. 4. 1, f, -^Q, and ^f to fortieths. 5- 3, f, J, ^, and ^ to 36ths. 6. h h /o. ih and A to lOOths. 7. A, j\. 1% ih and ^ to 144ths. 8- f h t\, fj, and U to 385ths. 126 SCHOOL ARITHMETIC. 191. To change fractions to smaller denominators. 1. How many twelfths in 1 ? How many halves ? Then 12 twelfths = how many halves ? 2. How many halves in 6 twelfths ? 3. How many thirds in 1 ? Then how many thirds in 12 twelfths ? In 4 twelfths ? In eight 12ths ? 4. Since A == i, and -^ = i, what has been done to the numerator and denominator of the fractions -j% and -^^ with- out changing the values ? How has it been done ? 5. Change j\ to fourths, -f^, \\, \%. 6. How many fourths in f ? Is f = f ? What change has been made in the denominator ? What corresponding ghange in the numerator ? Dividing numerator and denominator by 3, we have \. Since the fractional units are twice as large, while the number of them is ^ as great, f is equivalent to f . Hence the following 192. Principle. — Dividing denominator and numerator of a fraction hy the same number does not change the vahie of the fraction. WRITTEN EXERCISES. 193. 1. Change ^} to twelfths. Dividing th^ given denominator 24 -h 12 =: 2 by the required denominator we get 16 -i- 2 8 2, which must be used as divisor 24 _^ 2 12 ^^ ^^^^ numerator and denomi- nator. Queries. — 1. In this process, how has the size of the fractional unit been changed ? 2. How correspondingly has the number of fractional units been changed ? 3. Which gives more definite idea of the value of the quantity, || yd. or | yd, ? Change to smaller denominators : 4. if;f*;a;«;4f;«;«. Queries. — (a) When should you express a fraction with smaller denomi- FRACTIONS. 127 nator ? (b) Can ,^2 l>e changed to smaller denominator without a change of value ? Can J ? Can ^ ? Are 2 and 3 prime to each other ? Why ? 194. A fraction is expressed with its stnallest denominator when its numerator and denominator are prime to each other. WRITTEN EXERCISES. 195. 1. Change }f to its smallest denominator. 30 -^ 3 = 10 45 -^ o = 10 Divide numerator and denominator Therefore, by 3, then by 5. Since no number will f ^ — il* exactly divide both numerator and de- 10 -T- 5 = 2 nominator of the fraction f, the frac- 15 -4- 5 = 3 tion is changed, or reduced, to its small- Therefore, «*^ denominator. IS = I- Direction. — Divide numerator and denominator by com- mon factors successively until they are prime to each other. Change to smallest denominators : 14. HI. m. m- 15. iM. m, m- 16- ttf. f«. m- 17. m, m> m- 18. m, Mf. m- 19- fh> 1%. m- 196. To change integers or mixed numbers to frac- tional form, and the reverse. 1. How many quarter-dollars in one dollar ? 2. How many fourths in a dollar ? In 2 dollars ? In 2^ dollars ? In 2f apples ? Illustrate with objects. 3. How many apples have you when you have eight quar- ters, or fourths ? Ten fourths ? 17 fourths ? 4. How many eigliths in 1 ? In 2 ? In 3 ? In 5 ? In 2f ? In 3f ? 5. How many l^s in eight eighths ? In 16 eighths ? In Y ? In ^ ? In 1/ ? In ^ ? 2. if. if, H- 8. T%, m, iVt- 3. n, ft, !%• 9. IM, Hi IM- 4. If, li 41- 10. ili, iM, iJi- 5. ft, if, -ji 11. Mi, ^, m- 6. «, 15, It- 12. m, tVV iH- 7. M, li M. 13. Ul. iff, ili- 128 SCHOOL ARITHMETIC. 6. Twenty-five half-dollars are how mauy dollars ? 7. How many quarters will pay for a pig that costs $2^ ? 8. How many fractional units, thirds, are there in 3^ ? In 5| ? 9. How many ones are there in ^jt p j^ s^_ ? in ^ ? In 4j9 ? In V ? I" V ? WRITTEN EXERCISES. 197. 1. Change lof to fourths. ^ - 4 P In 1 there are 4 fourths, and in 15 there are 15 ^ ~ ^ times 4 fourths, or 60 fourths; 60 fourths + 3 6J) 4- .a — fit fourths = 63 fourths. Why do we add 3 fourths t -^ t — i ^^ QQ fourths ? Cliange to fractional form : 2. 1|. 9. 9f. 16. 12|i 23. 115 jV 3. 4f. 10. 3^%. 17. 18^. 24. aOSeV 4. Gf. 11. 514,. 18. 25^V 26. 365/^. 5. 71. 12. 8. 19. 29^f. 26. oOOff. 6. 5f. 13. 7^. 20. 37^V 27. 710|^. 7. 8|. 14. 6tV 21. 72^. 28. 802.^ 8. 7^. 15. 10. 22. 90|f 29. 613.25. 30. Change ^^ to a mixed number. OQ . q _ 02 Since there are 3 thirds in one, in 29 thirds there ~ ^ are as many ones as there are 3's in 29, or 9| ; therefore, ^3^ _ gj. Change to integers or mixed numbers : 31. i, f, V- 35. ^^ HK 32. ^, V, V- 36. If?, Y/. 33. ¥. ¥, \'- 37. W, W-. 34. 5^, 5ji, ig4. 38. YJ5. -7^.. Complete the following equations 43. 38| = ( ). 46. 24:j\ = ( 44. 1^3 ^ ( ). 47. i^si = ( 45. 10-[^ ^ ( ). 48. 365i = ( 39. i|ji 2j^. 40. -L|p, ^ifii. 41. 5^1-8, 1||1. 42. i^s, ''If''. )• )• )• 49. %' = ( ). 50. lOOlrV = ( ). 51. 4;|^ = { )• FRACTIONS. 129 198. To change decimal fractions to common frac- tions. 1. (a) Express .7 as a common fraction ; (b) change .125 to a common fraction. / \ 7 _ 7 ^® write the figures of the decimal ^ / T^' for the numerator, and 1 with as [o) .1^0 = xffij = ru — ¥• many ciphers after it as there are figures after the decimal point for the denominator. When desirable, we change to smallest denominator, as in (6). Change to common fractions with smallest denominators : 2. .25. 9. .39. 3. .35. 10. .43. 4. .85. 11. .38. 6. .75. 12. .50. 6. .24. 13. .95. 7. .72. 14. .05. 8. .60. 15. .03. 16. .375. 23. .0125. 17. .425. 24. .3750. 18. .500. 25. .0875. 19. .625. 26. .1872. 20. .205. 27. .4020. 21. .875. 28. .0075. 22. .945. 29. .15625. 199. To change common fractions to decimal frac- tions. 1. How many tenths in 4 twentieths ? In ^ ? In ^ ? JJ = how many lOths ? 2. How many tenths in ^ ? In f ? In f ? f = how many lOths ? 3. How many hundredths in ^^^ ? In ;jy_ p Jn .^^ 9 .^g^ = how many 100th s ? 4. How many hundredths in ^^ ? In 2V ? I^ ^ ? ^^ A ? ^ = how many lOOths ? 5. What have you been changing in these examples ? Since lOths and lOOths are decimal divisions, to what have you been changing the common fractions ? 200. To change a common fraction to a decimal fraction is to change it to larger or smaller denominators, and to express it in the notation peculiar to decimals. Thus, i = ,5, = .5 ; i = ,\^ = .75 ; -^^^ = jhj = .02. 130 SCHOOL ARITHMETIC. 1. Change f to a decimal fraction, that is, to a fraction whose denominator is 10 or 100 or 1000, etc. 1 000 -^8 19^ '^^^ required denominator must be a ' multiple of 8. The least decimal denom- 3 X 125 _ 375 _ ^^^ inator that is a multiple of 8 is 1000. 8 X 125 1000 ' * Hence | must be changed to lOOOths. (Art. 189.) The required denominator can be found only by inspection or by trial. Since it must be divided by the given denominator, and the given nu- 8)3.000 ^"erator multiplied by the quotient, we may, for convenience, — — — combine the processes and obtain the required decimal at .6 lb Qj^gg^ jjy dividing the numerator by the denominator, annex- ing ciphers, and pointing off decimal places as in division of decimals. (b) 200)26.00 2. Change -^^ to a decimal fraction. In many cases, reducing a fraction (^) to a smaller denominator changes ^^ = ^1^= .13 it to a decimal fraction, as in (a). It may also be solved as in (b). •^^* Rule. — Annex ciphers to the numerator and divide hy the denominator, pointing off decimal places as in division of decifnals. Note. — When the division will not terminate, the remainder may be expressed as a common fraction, or the sign + may be placed after the decimal figures to show that the division is not complete. Thus, i = .33i, or .33 +. Change to decimal fractions : 3. }. 11. i\ 4. i. 12. 5V 5. |. 13. if, 6. i. 14. U 7. |. 15. H 8. f 16. A 9. J. 17. U' 10. ^\. 18. V, 19. tV- 27. If 20. f. 28. If. 21. f 29. U- 22. |. 30. A- 23. j%. 31. il 24. A. 32. li. 25. a. 33. ^. 26. il 34. Tij. FRACTIONS. 131 201. To change dissimilar to similar fractions. 1. Have I and | the same denominators ? Have f and | ? Have f and 5 ? Have f, J, and J ? 202. Fractions that have the same denominators are said to have a common denominator^ and are called Similar Fractions. They have a common unit of measure. 203. Fractions that do not have the same denominators are called Dissimilar Fractions. Their units of measure are not the same. Tell which are similar fractions : 1. h h h h h h 3. I j\, f, i, t5^, 4. 2. h h h f h I 4. tV, A, A, A, iV A- 5. How many sixths in 1 ? In ^ ? In ^ ? Then what common denominator may ^ and ^ have ? 6. How many eighths in 1 ? In ^ ? In | ? Then what common denominator may | and f have ? 7. When fractions have a common denominator, what are they called ? Change to similar fractions : 8. i and i. 17. 1 and f. 9. i and h 18. f and f . 10. i and i. 19. t and f il. i and i. 20. t and f. 12. i and i. 21. h h and i. 13. i and f. 22. h h and |. 14. t and f. 23. h f, and f. 16. 1 and f; 24. f, i, and J^. 16. 1 and |. 25. f, I and /o- 204. A common denominator of two or more fractions is a Common Multiple of their denominators. 132 SCHOOL ARITHMETIC. WRITTEN EXERCISES. 205. 1. Change f, f, and f to similar fractions. 2 X 20 40 Since the product of the given denominators TT" ^^7^7 is a common multiple of each, 3x4x5, or 60, is a common denominator. Hence the fractions 3 X 15 __ 45 must be changed to GOths, which is done by 4 X 15 60 multiplying numerator and denominator of each 4 X 12 48 ^^ ^^® quotient that arises from dividing the ^= — required denominator by its given denoraina- 5 X 12 60 tor. Rules. — 1. Multiply numerator and denominator of each fraction hy the number of times its denominator is contained in a common multiple of all the given denominators. Or, 2. Multiply numerator and denominator of each fraction hy the product of the denominators of all the other fractions. Change to similar fractions : 2. h i, f. 3. i, I, A. 4. J, i, J. 5. h h f. 6- h i A- 7. f, .1, ^. 8. f, i U- 9. tV -8. H- Ifi 11 7 8 E> 7 •LO. "25^ 27' ^¥? W^y "6"0- 206. Fractions may have more than one common denomi- nator. The smallest one they can have is called their Least Common Denominator. The smaller the denominator, the greater the unit of measure. 207. The least common multiple of the denominators of several fractions is their least common denominator (L. C. D.). 10. h h A. 11. h h 4. 12. A. h 1. 13. \h h f. 14. i Ih .9. 15. i^> A' A- 16. ^h .5, h U' 17. ii' ¥Trj A- FRACTIONS. 133 WRITTEN EXERCISES. 208. 1. Change f, f, and ^ to similar fractious having their least common denominator. 3 X 18 _ 54 We find 72 to be the L. C. M. of the donoini- ^ ^ jg 1^2 nators, and the least common denon)inator K Q AK ^^ ^^*^ fractions, whicli must, therefore, be = — changed to 72nds. This is done by multiply- 8x9 75& iiig numerator and denominator of each 7 X 8 _ 56 fraction by tlie number of times its denomi- q ^ ft^ nator is contained in the L. C. M. B^^ Explain why fractions having their least common denominator have their greatest common fractional unit. Change to similar fractions with their L. C D. : 2. h i, i- 8. i A, I. 14. I «, i4, 18. 3. h I I 9. H' -5. i 18- l h A. A- 4. h tV. a- 10- i i' h f 18- M. -8' H, iS- 6. ^, S, i. IX. ?, ^\, A, A. 17. +, ^, A, T^. 6. i, I, j\. 12. A. A. ih il 18. A, 4, ,'j- 7. I, i, T^. 13. f, f, I, -rV- 19. i I, I- 20. Change f to a number whose fractional unit is ^. 21. How many fractional units are there in f if changed to 16ths ? 22. A fraction whose fractional unit is an eighth has the numerator 6. What is the value of the fraction in fourths ? 23. What fraction has sixteen fractional units, each of which is ^^ ? ADDITION. 209. 1. Ella has If, and Jane has $f . How much have they both ? 2. Mary paid || for a knife and $| for a book. How much did she pay for both ? 3. George bought f of an acre from one man, and f of an acre from another. How much land did he buy ? 134: SCHOOL ARITHMETIC. 4. "When fractions have different denominators, what must be done to them before they can be added ? Why ? 6. Can :|^ of a dollar and -J of a bushel be added ? Why not? 210. Principle. — Fractions can he added only when they express like quantities, and have a common denominator. WRITTEN EXERCISES. 211. 1. Find the sum of |, f, and \\. ^ * ^ Changing the fractions to similar ones hay- 's 4 b ing their least common denominator, we have i^— ff 24, 85, and 22 fortieths, and the sum of these 2. What is the sum of 3^, 7^, and 12f ? ^ 1^ Changing the fractions to similar ones •1) • iV having their least common denominator, we 13f = 12|f find their sum to be f|, or IH; the sum of yoj^i the integers is 22 ; and the sum of both is 23|i , Supply the words that are wanting in the following Rule. — Change the given fractions to , add their , and write the sum over the . When there are integers or mixed numbers, add and separately, and then the two sums. Add the following: 3. h h i, i- 12- i\' H. ih -35 4. I, f, tV. «• 8. h .7, A, A- 6- f A. i, T%- 8. I, i, J^, «. 9- A. \h ih A- 10. i A, ih A- U- h i A. i- 13. n, 4|, 2t. 14. 61, 5i, 3. 15. 8i Q-A, 2.5. 16. 3i 8.7, 121. 17. 131 , 17tV ^0,5^. 18. 221. . 181, 7t\. 19. 30f , 20^, 40.3. ^0, 5t, 2A, 8Jf. FRACTIONS. 135 21. One week Jasper earned |;3|, Homer, $4.7, David $12|, and James, $0^^. How much did all earn ? 22. Mary weighs 75^ pounds, Edna 12f pounds more than Mary, and Charles as much as both. How much does Charles weigh ? 23. The product of two numbers is the sum of 258^ and 173|, and one of the numbers is 24. Find the other. 24. Find the distance around a rectangular field whose length is 281f rods, and whose width is 190f rods. 25. A certain minuend would be iJ, if it were ^ less. What would it be if increased by ^ ? 26. A locomotive runs 354f miles every other day. How far does it run in a week, not counting Sunday ? 27. What is the minuend when the remainder is 17^, and the subtrahend 10^^ ? 28. The number of fractional units in the greater of two fractions is 15, the number in the less is G. The denomi- nator of the greater fraction is 27, and that of the other is 18. Find the sum of the fractions. SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 212. 1. Mr. A has 5 fields, the smallest of which contains 6f acres, and the largest 12.375 acres. Each of the others contains 9^ acres. How much land has Mr. A ? 2. John has $3j'^, Harry has $li more than John, Ben has 13.05 more than both, and their father has $10.25 more than the three boys together have. How much have all four ? 3. A man earned $16.66f in January. Each month there- after he earned $16,125 more than he earned the preceding month. How much did he earn in a year ? 4. A lady spent ^ of her money for shoes, f for a dress, and f for a hat. If she spent $37.30 for all, how much had she left ? 5. Find the value of 3 x (6f + 5.2) - 7.5 ^ (.5 + 1). 136 SCHOOL ARITHMETIC. 6. 'A can do | of a piece of work in a clay, B | of it, C J of it, and D ^ of it. In what time can they do the work, all working together ? 7. I sold to different customers 13f gallons, 11.5 gallons, 15^ gallons, 3^ gallons, and 11.75 gallons of oil. How many gallons did I sell to all ? 8. Find the sum of all the proper fractions that can be formed having one figure each in the numerator and denom- inator. 9. Ten is added to a certain mixed decimal. The point is then moved one place to the left and 10 is added. The sum is equal to 4.5 times the original number. Find the original number. SUBTRACTION. 213. 1. Ella has $f, and Jane has $f. How much more has Ella than Jane ? 2. Mary paid $^ for a knife and $f for a book. How much did the book cost more than the knife ? 3. A man who had | of an acre of land sold f of an acre. How much had he left ? 4. When fractions have different denominators, what must be done to them before they can be subtracted ? Why ? 5. Can f of a dollar be subtracted from f of a foot ? A¥hy not ? 214. Principle. — Fractions can he subtracted only tvhen they express like quantities, and have a common denomi- nator. WRITTEN EXERCISES. 215. 1. Subtract f from |. 1 = 11 3 2Jl Changing to similar fractions, we have 32 „ thirty-sixths, and 27 thirty-sixths, whose ■^ difference is 3-5. FRACTIONS. 137 2. From 6| take 4|. (a) (b) The fractions must be made similar, as in (a). g3 _. go :::^ 533. ^^ caniiot be taken from 9^4, hence 1, or ||, is taken 42 _ 41.6. =3 4'R ^*'0"^ ^ **"*^ added to /<, making \\, as in (b). ^ ~ 2^^ "" ^^ Then i2 ^''""i i5 leaves i^. tt"^ ^ from 5 leaves 1. l^J Hence the remainder is UJ, 1^ In practice, the numbers under (b) should not be written; the work should be done mentally. Supply the blanks in the following Rule. — Change the fractions to ; fi7id the difference between the , and write it over the . Wheii there are mixed numbers or integers, subtract frac- tions and integers separately. Query. — May integers or mixed numbers be changed to fractional form and subtracted according to the first part of the rule ? Find the value of : 3. f- f. 10. l+i-h 17. G.7 - ^. 4. A ■ -A- 11. .9 + 3-f 18. 7f - 0|. 8. -,V - -sV 12. f - A + 4- 19. 21.7 - 16jV 6. A - -i- 13. A-i--i- 20. 14i - 11. 7. H- -A- 14. H + 1 -1- 21. 12J - 3ft. 8. 5- ■n- 15. "i - n- 22. 2i + 3^ + 5^. 9-M- -i-v 16. 13j^ - 4.26. 23. 4i + 6f - 3.7. 24. Ella is lOf years old, and Nellie is 19| years old. How much oUler is Nellie than Ella ? 25. A. pole 14| feet long was broken into two pieces, one of which was 5| feet long. How long was the other ? 26. A train ran from Lynchburg to Roanoke in 1| hours; it reached Roanoke at 9 a.m. At what time did it start ? 27. Carl bought a book for $2yV, a knife for $|, a hat for $1|, and a coat for $2^. If he had a ten-dollar bill at first, how much had he left ? 28. The sum of two numbers is 21f, and one of them is 8f . What is the other ? 29. What fraction added to ^ + | + f will make 2^^ ? 138 SCHOOL ARITHMETIC. 30. The minuend is J|, and tlie remainder is |. What is the subtrahend ? 31. Two fractions have the common denominator 12. One has 8 fractional units, the other 11. How much greater is one than the other ? 32. Subtract ^ from the greatest possible fractional unit. 33. If 4 is added to numerator and denominator of f, how much is the value of the fraction increased or diminished ? How much if the fraction is | ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 216. 1. If numerator and denominator of | are each diminished by 2, will the value of the fraction be increased or diminished, and how much ? 2. What number is that, f of which exceeds ^ of it by llf ? 3. A owns f of a store, and B the remainder. If f of the store is worth $575 more than .5 of it, what is the value of B's share. 4. What number is that, to which if its f and its .25 are added, the sum wiirbe 170 ? 5. The difference between the subtrahend and minuend is 5|-. If the subtrahend is 8^^^ less than 9y\, what is the min- uend ? 6. AVhat fraction is as much larger than f, as f is less than I ? 7. The highest score in an inning was f of the total, and the next highest was ^^-g^ of the total less. The scores in these two innings differed by 5 runs. What was the total score ? MULTIPLICATION. 217. 1. James has $^, and Henry has twice as much. How much has Henry ? Then 2 times -^ = { ). 2. Rose has ^ of a pie, and Orville has three times as much, How much has he ? Then 3 x I = { ), FRACTIONS. 139 3. May worked f of a week, and Arthur worked 4 times as long. How long did he work ? 4. How much is 5 times 3 twentieths ? 6. 5 X j^ = J^. Have ^ and ^f the same denominators ? How do their numerators compare ? How do the fractions compare in value ? Which has the greater number of frac- tional units ? 218. Pkinciples. — 1. Multiplying the numerator of a fraction hy any number multiplies the value of the fraction hy that number . For, since the numerator tells how many fractional units are taken, multiplying it multiplies the number of fractional units, each of which has the same size or value as before. Conversely, 2. Dividing the numerator of a fraction hy any number divides the value of the fraction by that number. For, since the numerator tells how many parts are taken, dividing it divides the number of parts, each of which is of the same size as before. WRITTEN EXERCISES. 219. 1. Multiply ^ by 9. q 1.— 27— 17 ^ times 3 twentieths = 27 twentieths, just » X 2-0 - tl5- - -LjV j^g 9 tjjjjgg 3 tops = 27 tops. 2. Multiply 3i by 7. 3|- When the multiplier is an integer, it is often more _7_ convenient to multiply the integer and fraction 21 =7x3 separately, and then add the products. Or we may 1|^ = 7 X :^ proceed as in the first example, thus : 7 x 3^ = m = 1 x3i 7x^.^ = ^.^ = 221. Multiply the following : 3. I by 12. 8. ^^ by 15. 13. 5f by 10. 4. I by 13. 9. \l by 11. 14. llf by 14. 5. 1^ by 8. 10. t\ by 21. 15. 13| by 5. 6. y«3 by 11. 11. II by 38. 16. 15| by 7. 7. f^ by 16. 12. 2J by 8. 17. 23f by 12. 140 SCHOOL ARITHMETIC. 18. Multiply eacli of the following by its own denomi- nator, and note the results : |, f, f, -g, f, ^, j, -^g. 19. How does it affect a fraction to multiply it by its de- nominator ? Find by trial. 20. Multiply eacli of the following mixed numbers by the denominator of the fractional part, and compare each mul- tiplicand with its product : 2^, 1:^, 4^, 3|, 6f. 21. How does it affect a mixed number to multiply it by the denominator of the fractional part ? Note. — As regards their form, it is usual to make a distinction be- tween simple and complex fractions. They are called simple when numerator and denominator are integers ; otherwise complex. 22. Multiply the complex fraction — by 4. 4 X I ^ V\^ 3 5 5 5* 23. Change -^ to a simple fraction. 2 ^ Q 6 2 1 A fraction is made integral by multi- j o = "T^ ^^ T^ ~ fi^ plying it by its denominator. Hence we multiply both numerator and denomi- nator by the denominator of the fraction §; that is, by the denominator which is in the numerator of the complex fraction. Change to simple fractions ; -!■ 28. i 8* "t 29. 15* 26. |-. 30. 5* «.i. 31. H 4' 32. 33. 34. 35. 5| 7' ±0. 1 1 20' 17| 22' 20^ .21 * FRACTIONS. Ul 36. Multiply 12 hyl To multiply 12 by | is to find | of 12, or 2 12 -^ 3 = 4 times ^ of 12. 2x4=8 ^ of 12 = 12 -h 3, or 4 ; and J of 12 = 2 or times 4, or 8. 12 x J = V = : 8 Or, since f x 12 = 12 J, or i/, or 8. X f, we have 12 times Find the vali Lie of: 37. 15 X |. 44. tV X 23. 61. 7f X 30. 38. 16 X f. 46. -^ X 17. 62. of X 42. 39. 12 X f. 46. 2i X 16. 63. 9.8 X 14. 40. 14 X f 47. 18 X 3i. 64. 18f X 12. 41. 18 X f. 48. 21 X 6f. 66. '^ X 16. 42. 24 X f. 49. 24 X 6f. 66. 25 X y. 43. 27 X 4. 60. 25 X ^. 67. 42 X 4. 68. Multiply 1 by 4. 3 X 4 _ V- _ 12 Multiplying the numerator of a fraction by ! anv number multinlies the fraction bv that 5 5 35 number. (See Art. 218.) Hence we multiply 3 by ^ (as in example 36), which gives \\ Writing this product over the denominator, this is readily changed to the simple fraction ^f . (See example 23.) Multiply the following : 59. 4 by h 63. jV by f 67. 6 by f. 60. I by i. 64. f^ by f. 68. 12 by |. 61. I by i 65. 2i by H. 69. 15^ by I 62. I by f . 66. 3f by 2^^. 70. f by 2^. (U^* In the product of ? x f , or in the product of any two or more fractions, it will be seen that the numerator of the product is the prod- uct of the numerators of the factors, and the denominator the product of the denominators of the factors. Hence the following convenient method may be used : 'RvLE.^Change all integers and mixed numbers to frac- 142 SCHOOL ARITHMETIC. tional form ; then write the product of the numerators over the product of the denominators. Notes. — 1. The work may often be very much shortened by cancella- tion. 2, The sign x after a fraction is sometimes read "of." Thus, | x $3 may be read "four fifths of three dollars." In expressions like ^ of ^ the sign x may be substituted for "of." Find the value of : 71. f xf 76. f x|. 81. S^a^ X 34. 72. 4xf 77. u xif. 82. lOj^ X 33. 73. f xf 78. if xU- 83. 21f X 6|. 74. tV X ^^. 79. H XtV 84. 13t X IVj-. 75. A X A. 80. .9 xA. 85. 324 X ^. 86. if X :f X Y. 91. f X 4^ X i|. 87. if X : U xf. 92. 3i X 2^ X ^. 88. 41 X n xi 93. f X 3.1 X 5i 89. 1 X |x 3^. 94. 90 X 3i X If 90. f X 1 X n- 95. if X f X 4^. 96. Find the weight of 2f bushels of oats, allowing 32 pounds to a bushel. 97. If a man can walk 3 A miles in an hour, how far can he walk in 7f hours ? 98. A and B paid 1160 for a horse. If A paid /„ of the cost, how much money did B pay ? 99. The multiplier is -J, and the multiplicand is -^. AVhat is the product ? 100. Mr. E owned f of a mill which w^as valued at $12000. He sold B y\ of his share. What was the value of the part retained ? 101. A farmer bought 12f acres of land at $37^ an acre, and sold it at $52^ an acre. How much did he gain ? 102. A man who had f of an acre of land sold | of his share at the rate of $300 an acre. How much did he get for it ? FRACTIONS. 143 103. What part of a gallon is f of f of a gallon ? 104. If a ton of hay is worth |15f, what is the value of 7.5 tons ? 105. When cloth is worth $1^ a yard, how much must he paid for | of a yard ? 106. A has f as much money as B, who has § of $80. How much money has A ? 107. Find the cost of 12^ pounds of butter at 18f cents a pound. $ . 18f In examples like this it is often better to multiply as in the 13i^ solution given. Thus, 2iQ 12 X 18 cents = 216 cents. 9 I X 18 cents = 9 cents. 9 12 X f of a cent = 9 cents. |. i X i of a cent = | of a cent. The sum of all is 234| cents, or $2.34|. $2.34f 108. What is the value of 7 barrels of sugar, each con- taining 344^ pounds, at |.04f a pound ? 109. What is the price of a 1000-mile book of tickets at If cents a mile ? 110. If a cubic foot of water weighs 62| pounds, and iron is 7^Q times as heavy as water, what is the weight of a cubic foot of iron ? 111. How many pounds of ship-biscuit will be required for a cruise of 30 days, if the daily allowance is j\ of a pound to each of the 250 men in the crew ? 112. If the freight rate is If^ cents a ton for each mile, what will it cost to ship 5^ tons of produce 100 miles ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 220. 1. A ship sailed 3 days, each day sailing l^j times as far as on the preceding day. If she traveled 55 miles the first day, how far did she travel the last two days ? 2. How much must be paid for the rent of a store for 2f years at $62^ a month ? . 144 SCHOOL ARITHMETIC. 3. If a man plants J of an acre of corn in a day, how many acres will 7 men plant in 3.5 days ? 4. I bought 51 pounds of sugar at 6| cents a pound, and | as much coffee at 4 times the price paid for the sugar. What did I pay for the coffee ? 5. Divide $4.09 between two boys so that one will receive 40 cents more than twice what the other receives. 6. If I breathe 17 times a minute, and take in at each breath f of a quart of air, how many quarts of air do I need in ^ hour ? 7. If each soldier walks | of a mile in ^ of an hour, what is the combined distance marched by a regiment of 800 soldiers in the same time ? 8. The factors of the multiplicand are 2, 3, 4, 5, and f ; those of the product are f, and the prime factors of 144. What is the multiplier ? 9. A clock loses lj\ minutes in a day. If it is correct at noon on the 4th of 'July, what time will the clock indicate at noon on the 14th of July ? 10. A clock loses J of a second every 5 minutes. How much will it lose in 5 days ^nd 12 hours ? 11. Prove that the sum of two fractions, the numerator of each of which is 1, is equal to the siwi of the denominators divided by their product. Divisioisr. 221. 1. A man having 4 fifths of an acre divided it into 2 equal lots. How much land was in each lot ? Then *-3 = ( )■ 2. A lady divided 3 fourths of a pie equally among 3 boys. How much did each boy get ? Then ^ ~ d = ( ). 3. Mrs. A gave | of a cake to 4 girls. How much did each get ? Then « ^ 4 = ( ). 4. -| -V- 4 = -f. Are the fractional units of this dividend FRACTIONS. 145 and quotient equal in size ? Are they equal in number ? The number in the quotient is what part of the number in the dividend ? Then, how is a fraction divided by an inte- ger ? (See Art. 218.) WRITTEN EXERCISES. 222. 1. Divide j* ^7 3. Dividing the numerator of a fraction by any number 2V "^ ^ ~ ^TF divides the value of the fraction by that number. (Art. 218.) 2. Divide f by 6. Dividing the numerator by 6, a.s in the pre- „ . r, f_ 3__ jL ceding example, we get a complex fraction as ** 8 a quotient, which is changed to a simple frac- tion, as in Art. 219. Find the quotients of : II - 15. 17. «^ - 35. U - 18. 18. 4 - 12. U - 14. 19. t - 9. A-f-4. 20. ill -24. if - 5. 21. If § - 28. il-9. 22. 11^^32. i - 11. 23. ei - 36. 24. Find the quotient of 127f divided by 9. The dividend may be changed to fractional form 9)127f a,nd divided as in the preceding examples. But it JT is often more convenient to proceed as follows : 14— = 14^T 9 is contained in 127 fourteen times, with a re- mainder of 1^, and 1^ -i-9 = ^-. Hence the quotient is UjV Divide the following: 25. 32i by 4. 31. 134^ by 12. 37. ^£ by 5. 26. 48| by 3. 32. 176f by 5. 38. 2f by 19. 27. 65i by 5. 33. 248^ by 6. 39. 7| by 18. 28. 73| by 6. 34. 325^ by 3. 40. if by 5. 29. 92| by 7. 35. 540| by 7. 41. 170f by 20. 30. 93f by 9. 36. 809^^ by 8. 42. 2001- by 10. 10 3. 1 -^ 2. 10. 4. H-5. 11. 5. if -4. 12. 6. li-7. 13. 7. f i -^ 9. 14. 8. IS -14. 15. 9. U - 13. 16. 146 SCHOOL ARITHMETIC. 43. A man earned $13594^ in 9 years. What was his aver- age yearly income ? 44. How often is f contained in 9 ? 9 r= ^ Changing 9 to fourths, we have 36 fourths -t- 3 sfi _i_ 3 _ 22 fourths. 36 fourths contains 3 fourths 12 times, just as $36 contains $3 twelve times. 45. How often is -fj contained in -| ? ■9 = f^ Changing dividend and divisor to similar fractions, ^ = ^ we proceed as in example 44. Any example in divi- 88 _i_ 1 8 -_ ^ sion of fractions may be solved in this manner. Find the quotients of : 46. i^i. 52. a -^ |. 58. 0-i--^. 47. i -i- i. 53. 5 -^ f . 59. 3f - 3i. 48. t -^ f . 54. 8 - f 60. 4f -^ oi 49. A -i- i. 55. 2i -=- i. 61. 9i - IJ. 50. 1 - f 56. 7i -^ |. 62. 7| H- Jf 51. T^ ^ f 57. 3f - |. 63. ^\ -^ 2i. 64. How many times is f contained in | ? Since i is contained in 1 eight ^ "^ E^ ^^ " times, I is contained in 1 one-third of 1 -7- f = 1^ (divisor inverted) 8 times, or f times ; and in i of 1, |^-f-^=:4 X 1= |-|, or 2y\ o^' o> it is contained f of | times, or 'li times. The inverted divisor shows how many times the divisor is contained in 1 ; it is called the reciprocal of the divisor. Rule. — Multiply the inverted divisor hy the dividend. Mixed numbers and integers must be expressed in fractional form. Cancellation should be used whenever possible. Find the value of : 65. rV - -tV 72. Jt ^ M. 79. 51} -^ 7f. 66. A - -A. 73. ?| -^ |}. 80. A - 12. 67. A- -^■ 74. 7 - A- 81. Jf H- 20. 68. J} - -i|. 75. 8 - A. 82. 1^ -J- 60. 69. H - ■r{h 76. 3f -- |. 83. 15| -r 7. 70. M - -if. 77. 8| - i 84. 181^ -V- 2.2. 71.11- -»• 78. Uf ^ A. 85. 7.5-^ 22 J. - FRACTIONS. 147 86. Find the value of -^. I This expression is equivalent to 2^ -f- 3, and may be treated accord- ingly. (Note 3, Art. 185.) 87. What is the quotient of ^8^ - f ? ■y\ -4- 1^ = f This process is the converse of that employed in multiplication of fractions, second rule. The dividend -,*5 is the product of two factors, one of which is §. Since the numerator 8 is the product of two factors, one of which is 2, the other factor is 8 -j- 2, or 4, which is the numerator of the required quotient. Since the denominator 15 is the product of two factors (or denominators), one of which is 3, the other factor is 15 -i- 3, or 5, which is the denominator of the required quotient. Hence the quotient is ^. ^° This is the most convenient method when the numerator and de- nominator of the dividend are respectively multiples of the numerator and denominator of the divisor. Divide, using this method : 88. A by f . 92. U by A- »6. ^t\ by 2|. 89. It by |. 93. M by A- 97. 7^ by 1^. 90. if by A. 94. Hbyf 98. U^yU- 91. if by f . 95. 3A by |. 99. «§ by if 100. At $2f a day, how long will it take to earn $37^ ? 101. If Thomas can walk 3^ miles au hour, in what time can he walk 20 miles ? 102. A man gave |.5G for sheep, paying $5J a head. How many did he buy ? 103. If a bankrupt's property is worth $3100, and his debts amount to $7000, how many cents on a dollar can he pay ? 104. A man paid $55 for 3f tons of hay. What was the price a ton ? 105. A horse traveled 24J miles in 4 hours. At what rate an hour was that ? 106. How many times must f be added to itself to make 7^? 107. How many times can f be subtracted from 7 ? 148 SCHOOL ARITHMETIC. 108. A man having 10 acres of land sold each of several men f of an acre, and had 2^ acres left. To how many men did he sell ? 109. The distance around a barn is 213|^ feet. How wide is the barn, if it is 60} feet long ? 110. I bought 25 bushels of i^otatoes at $f a bushel, and sold them for $18|. How much did I gain on one bushel ? 111. What must 3| be multiplied by to make 11^ ? 112. A man bought 396 pounds of oats, at $f a bushel. Allowing 32 pounds to a bushel, what did he pay for them ? 113. A man paid $1359f for 21f acres of land. How much was that an acre ? 114. The divisor is .8, the dividend -f. Find the quotient. 116. When eggs are worth 18J cents a dozen, how much must be paid for 5 dozen and 6 eggs ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 223. 1. A boy bought lemons at the rate of 8 for 7 cents, and sold them at the rate of 7 for 8 cents. If he made $1.35 in 7 days, how many did he sell each day ? 2. A merchant purchased a cargo of flour for $2173^, and sold it for If of its cost, thereby losing $^ on a barrel. How many barrels did he purchase ? 3. The product of a number multijolied by 3 is how many times the product of the same number multiplied by f ? 4. A owns 2^ times as much land as B, C owns 1^ times as much as both A and B, and D owns 3f times as much as B and C. If B has 36^ acres, how much has J) ? 5. A boy bought |^ of a bushel of nuts, and sold -| of them for what he paid for all, and the remainder at cost. If he gained $1| by the transaction, how much money had he invested ? 6. Change jf-^ to a decimal, ^nd divide by 5000. 7. The owner of ^j- of a mine sold -^q of his share for $40,500. What should he who owns f of the mine get for I of his share ? FRACTIONS. 149 8. A can mow a field in 3 days, B in 4 days, C in 5 days, and D in 6 days. If A can earn $20 a week, how much can B, C, and D together earn in the same time ? 9. If 8f tons of hay are worth 36 sheep, and 11 sheep are worth 2 cows, and 9 cows are worth 1200, how many dollars is hay worth a ton ? 10.i + L + i:-^^^_ = ( ). 5 1 + 1.5 11. (a) .00001 -^ 10000 = ( ) ; (b) 1000 -^ .0001 = ( ) ; (c) .001 -7- .000001 = ( ) ; (d) 400 -^ 10000 = ( ). 12. Find value of 2f -^ f x 24^\ x 0. 13. The sum of two numbers is 1000 ; the difference is 648. Divide twice the larger by J. 14. A owns f of a farm, and B the remainder, f of the difference between their shares is $10500. What is the value of the farm ? 15. The minuend is 8 times ^ x .6 ; the remainder is i of .7 -^ -J. What is the subtrahend ? 16. A can dig 16|- rows of potatoes in a day, and B can pick 33 J rows in a day. If A has 93.5 rows dug when B begins, how many rows must B pick before he overtakes A ? ILLUSTKATIVE SOLUTIONS, With Problems for Practice. o« 524. The solution of all problems in the applications of arithmetic requires Analysis of some kind. A type of aritJi- metical analysis much employed involves the process of reasoning from a given number to one, and then passing from 07ie to the required numher. 225. As many formal solutions begin with a '' since ^' and end with a "therefore/' it is convenient to represent these two words by signs. By general usage the sign (*.•) is read "since/" and the sign (.-.) is read ''therefore." 226. 1. In a town ^^ of the people are sick and 512 are well. How many are sick ? §^ of the people — -j^- of the people = ^| of them. •.• ^1 of the people = 512 persons, t/s of the people = 16 persons, and /s of the people = 48 persons. 2. If electricity passes through 7200 miles of wire in | of a second, what is its rate a second ? 3. If a horse trots | of a mile in 2^ minutes, in wliat time can he trot a mile ? 4. One half of a post is in the water, ^ in mud, and 4 ft. 6 in. in the air. Find length of the post. 5. After selling .32 of his slieep to one man and .88 of them to another, a drover had 312 remaining. How many had he at first ? ILLUSTRATIVE SOLUTIONS. 151 6. A boy gave J of his marbles to A, | of the Remainder to B, and thea had 15 marbles left. At first he had how many ? After giving ^ of his marbles to A he had ^ of them left. To B he gave § of this |, or ^, and had ^ — i, or ^ of them left. . •. i of his marbles = 15, and f of them = 90. 7. A man invested J of his money in land, J of the re- mainder in cattle, and had $1000 left. How much money had he at first ? 8. In a certain school f of the. pupils belong to the third grade, | of the remaining pupils belong to the second grade, and the remainder, which is 20, belong to the first grade. How many pupils are there in the school ? ' 9. A man gives ^ of his property to one son, J of it to another,^ of it to the third son, and the remainder, $550, to his daughter. What is the value of the whole property ? 10. A person loses ^q^ of his fortune and then ^^ of the re- mainder, and then | of what he tlien had, and finds that he has $3600 left. How much had he at first ? 11. Find the average weight of three men weighing, respectively, 130 lb., 145 lb., and 175 lb. They together weigh 130 lb. + 145 lb. + 175 lb. = 450 lb. .-. the average weight of the three men is ^ of 450 lb., or 150 lb. 12. If 130 pupils attend school on Monday, 126 on Tues- day, 122 on Wednesday, 125 on Thursday, and 122 on Friday, what is the average attendance for the week ? 13. If the temperature as indicated by a thermometer at eight different times on a certain day was, respectively, 37°, 36°, 36°, 38°, 40°, 38°, 36°, and 35°, what was the average temperature for the day ? 14. The pe?^ capita indebtedness of France is $116, of Prus- sia, $37, Great Britain and Ireland, $88 ; Russia, $31 ; Spain, $74, and the U. S. $15. What is the average per capita indebtedness of these six countries ? 152 SCHOOL ARITHMETIC. 15. If the heart beats 140 times a minute during the first 3 years of life, 120 times a minute for the next 3 years, 100 times a minute for the next 6 years, 90 times a minute for the next 10 years, and 75 times a minute for the next 28 years, what is the average number of beats a minute in a life of 50 years ? 16. The sum of two numbers is 84, and their difference is 14. Find the numbers. The greater + the less = 84. The greater — the less = 14. .*. 2 X the less = 70, and the less — 35. 17. The sum of two numbers is 603, and their difference is 273. Find the numbers. 18. The difference between A's money and B's is $16.50, and they together have $166.50. How much has each ? 19. George has J doz. eggs more than Kate, and both have 42 eggs. How many has each ? 20. The sum of two numbers exceeds their difference by 198. What is the smaller number ? 21. A crew can row 60 miles down stream in 3 hours, but requires 4 hours to row back. What is the rate of the cur- rent ? Rate in still water + rate of current = 20 miles an hour. Rate in still water — rate of current = 15 miles an hour. .'. 2 X rate of current = 5 miles an hour, and the current's rate = 2| miles an hour. 22. A steamboat goes 72 miles down stream in 6 hours, but is 8 hours returning. What is the rate of the stream ? 23. Going down stream A can row 11 miles in 2 hours. Going up stream he can row 1^ miles in a half hour. Find the rate of the current. illustrative: solutions. 163 1i4. If a man can row GJ miles an hour down stream and 4J miles an hour up stream, how far can' he row in an hour in still water ? 25. A garrison of 1200 men had provisions to last 90 days, but 30 days later 300 more men arrived. How long did the provisions last after the increase in the number of men ? They would have lasted 1200 men 60 days, or 1 man 1200 x 60 days. .-. they lasted 1500 men l??!^-^, or 48 days. 1500 26. A garrison at Manila, consisting of 2500 men, had pro- visions for 30 days, but 1500 men were withdrawn. How long did the provisions last the remainder ? 27. A can do a piece of work in 3 days, and B can do it in 5 days. How long will it take both together ? In 1 day A does ^ of the work. In 1 day B does i of the work. In 1 day A and B do i + i, or ^g of it. To do if, or the whole work, will require \^ -j- ^\- or 1| days. 28. A can do a piece of work in 5, B in 6, and C in 8 days. How long would it take them to do it together ? 29. A and B can build a house in 3 months. B alone can complete it in 8 months. In what time can A build the house ? 30. If A can lay a certain wall in 4| days, and B in 5^ days, how long will it take both together ? 31. Two pipes lead into a tank. One can fill it in 50 min- utes, the other in 40 minutes. There is a discharging pipe which can empty the tank in 25 min. In what time will the tank be filled if all three pipes are in operation ? The L. C. M. of 50, 40, and 25 is 200. Hence one pipe fills the tank 4 times in 200 min., the other 5 times, and both 9 times. The third pipe empties the tank 8 times in 200 min. .'. the tank is filled 9 — 8, or 1 time, in 200 min., or 3 hr. 20 min. 154 . SCHOOL AEITHMETIC. 82. One pipe will fill a cistern in 3 hours ; a waste-pipe will empty it in 2 Hours. If the cistern is full and both pipes are opened, in what time will the cistern be emptied ? 33. An empty cistern has two pipes. One fills it in 40 minutes, the other empties it in 60 minutes. If both are opened, in what time will the cistern be filled ? 34* If 8 horses eat 48 bushels of corn in 24 days, in how many days will 4 horses eat 38 bushels ? 8 horses eat 48 bu. in 24 days. 1 horse eats 48 bu. in 8 x 24 days. 1 horse eats 1 bu. in — ^^— i- days. 48 . •. 4 horses eat 1 bu. in — ^ days, 48 X 4 and 4 horses eat 38 bu. in ^ ^ 24 x 38 ^ ^ ^g . 48 X 4 ^ ^ 35. If 8 men in 7 days can reap a field of 40 acres, how many acres will be cut by 24 men in 28 days ? 36. If 3 men earn $150 in 20 days, how many men will earn $157.50 in 9 days, at the same rate ? 37. If 5 yards of cloth f yd. wide cost $3.12^, how much will 15 yards of that cloth 1 yd. wide cost; at the same rate ? 38. If a 5-cent loaf weighs 1.5 pounds when wheat is 50 cents a bushel, what should it weigh when wheat is $.75 a bushel? 39. When a certain number is multiplied by 9, the product divided by 12, the qxiatient increased by 96, and the sum divided by 3, the quotient is 36. What is the number ? 36 X 3 =108 ; 108-96 = 12; 12 X 12 = 144 ; 144 -f- 9 = 16. 40. If a certain number is diminished by 76, the remainder multiplied by 2, the product increased by 148, and the sum divided by 12, the quotient is 21^. Find the number. ILLUSTRATIVE SOLUTIONS. 155 41. At what time between 1 and 2 o^clock are the honr and minute hands of a clock together ? First Solution. In 1 hour the minute hand moves over 60 minute-spaces, and the hour hand over 5, Hence the former gains 55 minute-spaces in that time. To gain 1 space requires A" of an hour. At 1 o'clock the hands are 5 spaces apart. Hence to gain tliis 5 spaces will require 5 x ^- hr. = tV hr., or 5 A" min. Hence the time is 5/f min. past 1 o'clock. Second Solution. The minute hand gains 11 hour-spaces in going 12, that is, in 1 hour. Hence to gain 1 space requires -,V of an hour. At 1 o'clock the hands are one hour-space apart, which the minute hand will gain in I'r hr., or 5^^ min. Hence it will overtake the hour hand at 5/,- min. past 1 o'clock. 42. At what time between 4 and 5 o'clock do the hour and minute hands of a cIocIj: coincide ? 43. At what time between 10 and 11 o'clock do the hour and minute hands of a watch coincide ? 44. At what time between 1 and 2 o'clock are the hands of a clock exactly opposite each other ? 45. A and 13 start from the same point and travel in the same direction. A goes 7 miles an hour and B 3 miles an hour. If B has a start of 5 hours, when will he be overtaken by A? RELATION OF NUMBERS. 227. The Relation of Numbers is their comparative value. Thus, comparing 2 with 4, we say 2 is ^ of 4. 228. To find a number when part of it is given. 1. 82 is J of my money. How much have I ? 2. Nettie spent $5, which was ^ of her money. How much money had she ? 3. Harry lost 3 marbles, which was ^ of all he had. How many had he ? 4. Two is J of what number ? ^ ? i ? 5. Five is J of what number ? i? 6. Eight is f of what number ? ^ of the number = 8. .-J" *^ ^^ = Jof 8, or 4. . I ^^ '' '' = 3 X 4, or 12. 7. Nine is f of what number ? 8. Ten is | of what number ? Find the number of which Since 8 is f of some number, 1 third of that number is | of 8, or 4 ; and since 4 is 1 third of the number, 3 thirds, or the number, equals 3 times 4, or 12. Hence 8 is f of 12. 9. 12 is f 17. 72 is f |. 10. 15 is f . 18. 65 is If. 11. 24 is |. 19. 120 is f . 12. 29 is I. 20. 217 is |. 13. 32 is j\. 21. 210 is f 14. 39 is A. 22. 225 is f f 15. 40 is If 23. 414 is If. 16. 52isf 24. 1000 is .5. RELATION OF NUMBERS. 157 25. 2 is f of what number ? f of the number = 2. .i- - - =iof2, or|. p. .. " = 4 X I, or |. Since 2 is J of some number, 1 fourth of that number is i of 2, or f ; and since ^ is i of the number, 4 fourths, or the num- ber, equals 4 times f, or f. Hence 2 is J of 2J. 26. 9 is 4 of what number ? 27. 16 is f of what number ? 28. A. lady spent $20, which was | of her money, much had she at first ? 29. I is f of what number ? How ^ of the number = ^. '.\ '' " '' = |of f, or 4. ■. ^ - - .. ,, 7 X i or 3^. Since | is f of some number, 1 seventh of that number is i of §, or i ; and since ^ is | of the number, }, or the number, equals 7 times ^, or V, or 3Jf. Hence § is ? of 3^ 30. ^ is 4 of what number ? 31. A hat cost 11^, which was f of the cost of a vest. How much did the vest cost ? 32. William walked 27 miles in one day, which was J of the distance Joseph walked. How far did Joseph walk ? 33. A lot cost $450, which was ^ of the cost of a house. Find the cost of the house. 34. Three fifths of the distance from Pittsburg to Phila- delphia is 213 miles. How far apart are the two cities ? 35. A bought a horse for $126, which was ^ of what he sold him for. How much did he gain ? 36. In fj of a mile are 216 rods. How many rods in a mile ? 37. $18 is f of what B paid for a cow. His horse cost twice as much as the cow. How much did he pay for the horse ? 158 SCHOOL ARITHMETIC. ALIQUOT PARTS. 229. The Aliquot Parts of a number are the parts that will exactly divide it. Thus, 2 and 5 are aliquot parts of 10. The relation of 50 to 100 is J. Hence 50 = ^ of 100. 334 to 100 isi i( m = i of 100. 25 to 100 isi. i( 25 = i of 100. 20 to 100 hi. a 20 = 1 of 100. 16| to 100 hi. <( m = I of 100. 142 to 100 hi. i ( 14f = 1 of 100. 12J to 100 isi. te 12^ = i of 100. Hi to 100 hi. te 1H = ^ of 100. 10 to 100 isiV- i( 10 = 3^^ of 100. 8J to 100 isiV- {< 8* = ^j of 100. 6i to 100 iST>,. (( H = iV of 100. The numbers in the first column are aliquot parts of 100. From them may be found the following other parts of 1 ?0 : 40 = I of 100. 37^ = f of 100. 83^ = | of 100. 60 = I of 100. 62 J = 1 of 100. 41| = ^\ of 100. 80 = I of 100. 87^ = i of 100. 5.8^ = j\ of 100. 75 = f of 100. 66f = I of 100. 31^ = j% of 100. 230. To multiply by the aliquot parts of lOO. 1. Multiply 2856 by 25. 4)285600 Since 25 is i of 100, we multiply by 100— which is done WVA()f) by annexing two ciphers — and take ^ of the product. Multiply in a similar manner : 2. 3576 by 33^. 3. 2748 by 50. 4. 1728 by 12f 6. 3270 by 16f. 10. At $.62 J a bushel, what must be paid for 1648 bushels of wheat ? 6. 4368 by 8^. 7. 5138 by 14|. , 8. 2946 by 66|. 9. 35768 by 37^. RELATION OP NUMBERS. 159 11. Find the cost of 75 dozen shovels at $.87^ each ? 12. When potatoes are worth 66f^ a bushel, how much must be paid for 240 barrels, each containing 3^ bushels ? 231. To divide by the aliquot parts of lOO. 1. Divide 1257 by 33^. 12.57 Since 33^ is i of 100, we divide by 100— which is done by 3 pointing off two decimal places — and multiply the quotient Divide in a similar manner : 2. 2576 by 16f. 7. 3344 by 11^. 3. 2718 by 25. 8. 76512 by 6^. 4. 3592 by 12f 9. 37584 by 37.5. 6. 5340 by 14|. 10. 21360 by 66^. 6. 4825 by 8^ 11. 576900 by 62 J. 12. If pears are worth $.33^ a peck, how many can be bought for $12.50? 13. When butter is selling at $.16| a pound, how much can be purchased for $1.50 ? 14. At $^ a yard, how many yards of cloth can be bought for If ? 15. James can walk 33^ miles in a day. How far can he walk in six weeks ? 16. A farmer has 2400 bushels of corn to husk. In a day he can husk 75 bushels. In how many days can he husk the entire crop ? 17. In an orchard ^ of the trees bear apples, ^ peaches, I pears, and the rest, 5 of them, cherries. How many trees in the orchard ? 18. A father willed f of his estate to one son, f of i-he re- mainder to another, and the rest to his wife. If one son re- ceived $900 more than the other, how much did the widow receive ? RATIO. 232. To find the relation of one number to another. 1. 2 feet is what part of 4 feet ? Of 6 feet ? 2. How does $5 compare with $10 ? With 120 ? 3. What is the relation of 3 to 6 ? 4 to 8 ? 5 to 15 ? 4. Howdi3es$10compare Avith$2 ? 12 with 3 ? 16 with 8 ? 12 with 4 ? 6. What is the relation of 3 to 8, or what part of 8 is 3 ? Since 1 is ^ of 8, 3 is 3 times i of 8, or | of 8, Hence | is the rela- tion of 3 to 8. 6. What is the relation of 4 to 9 ? Of 8 to 12 ? Of 6 to 14 ? 233. The relation of one number to another of the same kind is Ratio. Note. — There is no ratio between $3 and 6 hats, nor can the ratio between ^ feet and 8 yards be determined in this form ; but if we reduce the 8 yards to feet, the ratio is found to be ^^ or \. 234. The Sign of ratio is ( : ). Thus, the ratio of 3 to 8 is written 3 : 8. This was first used as a sign of division by Leibnitz. The ratio of 3 to 8 may be expressed in three ways— 3 : 8, 3 -j- 8, and |. 235. The two numbers compared are together called a Couplet. 236. The first is called the Antecedent, the second the Consequent. Thus, in the ratio 3 : 8, 3 is the antecedent, and 8 the consequent. Notes. — 1. The ratio of one number to another is found by dividing the antecedent by the consequent. 2. The ratio being the quotient of one number divided by another of the same kind is always an abstract number. (Art. 99.) RATIO. 101' Since the antecedent may be regarded a3 a dividend, and the consequefit as a divisor, we have the following 237. Principles. — 1. Multiplying the antecedent or divid- ing the consequent hy any number multiplies the ratio by that number. 2. Dividing the antecedent or multiplying the consequent by any number divides the ratio by that number. 3. Multiplying or dividing antecedent and consequent by the same number does not change the value of the ratio. 238. Since a ratio is the quotient of an antecedent by its consequent, it follows that (a). The antecedent = the consequent x the ratio, (b). The consequent = the antecedent ~ the ratio. Find the ratio of : 1. 12 to 16. 9. 5 to .5. "i 2. 7 to 21. 10. .5 to 5. 3. 8 to 18. 11. 50^ to 100^. 4. $10 to 125. 12. $5 to $2. 5. 30 to 15. 13. 52 to 13. a 20 yd. to 30 yd. 14. 18 to 15. 7. 27 to 9. 15. 12 to 17. 8. 9 to 27. 16. 95 to 19. : 17. What is the ratio of 1 foot to a yard ? 18. What is the relation of $.12|- to %l ? 19. What is the ratio of 2.5 to .25? 20. What is the ratio of f to f ? , .'- •; ■ \i. .;.*f Query. — When the denominators are ; the same, why may they be di's-' •egarded ? 21. What is the ratio of | to | p Suggestion. ^ -^ |^ = | x f = j§, or •i Or, l-^^ = \%-^\l=\h ori What is the ratio of : 22. f to f ? I to i ? I to I ? 23. I to i ? i to t ? 4 to i ? 11 162 SCHOOL ARITHMETIC. What is the fatio of ; 24. 1 to ^ ? 2 to I ? 3 to I ? 25. i to 1 ? i to 2 ? f to 1 ? 26. f to .7? 6 to 1.2? 10 to2J? 27. What number has to 6 the ratio 3 ? To 7 the ratio 5 ? 28. Mention two numbers whose ratio is |. 29. How much is 6 : 2 more than 7:3? 30. Which is greater, f , 2 -=- 3, or 1 : 2 ? 31. Find the value of (2 : 6) x (5 : 2). 32. Find the value of (5 : 3) -h (12 : f ). 33. If the antecedent is 12 and the ratio 3, what is the consequent ? 34. If the consequent is 5 and the ratio 3, what is the antecedent ? 35. What is the effect produced on the ratio 4 : 5 by multi- plying the antecedent by 3 ? Tlie consequent by 3 ? Both by 3? 36. What is the ratio of 4« to 2a ? Of 6^> to 31) ? 37. What is the ratio of 6d to 10Z» ? Of 7a to 21« ? 38. AVhat number has to 5 the ratio 3 ? What number has to a the ratio b ? 39. What is the ratio oi b to a ? Oi p to q ? 40. If the antecedent is Qa and the ratio 3, what is the consequent ? 41. If a is the antecedent and 2 the ratio, what is the consequent ? PROPORTION. 239. 1. What is the ratio of 2 to 4 ? Of 3 to G ? Are these ratios equal to each other ? 2. Since tliey are equal, they may be written thus : 2:4 = 3:0. This may be read in two ways : thus, the ratio of 2 to 4 equals the ratio of 3 to G ; or, 2 is to 4 as 3 is to G. 240. When two ratios are equal, and written as above, they form an Equality of Ratios. 241. An equality of ratios is a Proportion. Thus, 8 : 4 = 12 : 6 is a proportion. It may also be written 8 -T- 4 = 12 -4- 6, or I = V- 242. A proportion being composed of two ratios must use four numbers — two antecedents and two consequents. When any three of these are given, the fourth may be found. Find the wanting number in the following : 1. 2 : 4 = 5 : ( ). 6. 3 : G = 2 : ( ). 2. 1 : 2 = 4 : ( ). 7. 3. 2 : ( ) = 3 : 6. 8. 4. 1 : 3 = ( ) : 8. 9. 5. ( ) : 5 = 8 : 4. 10. ( ) : G = 3 : 9. Queries. — 1. Which numbers are antecedents ? 2. Which are con- sequents ? 3. Is every proportion an equation ? Why ? 243. The sign ( : : ) is sometimes written between the equal ratios instead of the sign of equality ( = ). Thus, 3:6=4:8 may'be written 3 : 6 : : 4 : 8, and is read 3 is to 6 as 4 is to 8. 8 : 4 = : G :( ). 10 :( )-- = 8 :4. 4 : 2 = ^( ): 5. 164 SCHOOL ARITHMETIC. 244. The first and fourth numbers of the proportion are the Extremes, and the second and third the Means. Thus, in the proportion 5 : 10 = 2 : 4, 5 and 4 are the extremes, 10 and 2 the means. 245. The ratio of one number to another is a Simple Katio. 246. A Simple Proportion is an equality of two simple ratios. 247. Principles. — 1. 77ie product of the extretnes is equal to the product of the means. In any proportion, as 2 : 4 — 3 : 6, the ratios may be expressed in fractional form. Thus, 2:4 = 3:6 may be written f = ^. Changing to similar fractions, we have — ^ — - = ^. 24 24 Hence, 2x6 = 3x4. But 2 and 6 are the extremes, and 3 and 4 the means of the given proportion ; therefore, the product of the extremes is equal to the product of the means. 2. The product of the means divided hy either extreme is equal to the other extreme. 3. The product of the extremes divided hy either mean is equal to the other mean. ^^° Have the pupils prove principles 2 and 3. Find the number omitted in each of the following : ( )• 1. 6 : 8 = 9 : ( ). 9. 10 feet : 15 feet : : 1 2. 12 : 10 = ( ) : 5. 10. 1 : 6i : : 8 : ( ). 3. 9 : 15 = 12 : ( ). 11. $14 : 17 : : ( ) : 1. 4. 16 : ( ) = 18 : 27. 12. f : 2 : : 5 : ( ). 5. 50 : 20 = ( ) : 12. 13. ( ) : f : : 13^ : If. 6. ( ) : 15 = 4|^ : 9. 14. 3.5 : ^ : : ( ) : ^. 7. 6.5 : 19.5 = i : { ). 15. i: ( ):: f : i- 8. 7.5 : 2.5 = ( ) : 1.3. 16. 24 : ( ) : : 10 : J^. APPLICATIONS OF SIMPLE PROPORTION. 248. Many problems that are usually solved by analysis can be readily solved by proportion. PROPORTION. 165 1. If 8 hats cost $10, how much will 12 hats cost ? First Statement. Since only like numbers can be compared hats hats $ $ in a ratio, we have for the first couplet 8 R ' 12 ■= K) ' ( ) ^^^^ ^^^ ^^ hats, either of which may be - jj ^ P made the antecedent. For the second coup- __ 24 Ifit we have the cost of 8 hats and the cost 8 of 12 hats. In the first statement we made 8 hats the antecedent, and it is less than the consequent. Therefore, the antecedent in the second couplet must be less than the consequent. It is evident that 8 hats cost less than 12 hats ; hence $16 must be made the antecedent in the second couplet. Solving by Principle 2, we find the cost of 12 hats to be $24. Second Statement. In the second statement we make 12 hats hats hats $ $ the first antecedent, which is greater than -j^2 • 8 = f ) • 16 ^^^ consequent. Hence, the required cost, ^jj P which is greater than $16, must be maile • = 24 the second antecedent. Solving by Prin- *8 ciple 3, we get the same answer as before. It^^ The pupil will note that 8 hats bears the same relation to 12 hats as the cost of 8 hats bears to the cost of twelve hats. Note.— In solving problems of this kind in proportion, we apparently multiply hats by dollars. This is due, however, to the form of work. Since the ratio of 8 hats to 12 hats is equal to the ratio of 8 to 12, we may write 8 : 12 = $16 : ($ ). In practice this substitution is not necessary. 2. If 5 men can build a house in 40 days, in how many days can 8 men build it ? It is evident that 8 men can do the men men^ days ^ days ^^^.^ .^^ j^^^ ^.^^ ^^^^^ 5 ^^^ . ^^^^^ (a) 5 : 8 — ( ) : 40 ^^^^ required number is less than 40, (b) 8:5= 40 : ( ) and the proportions are easily expressed as in the statements. Note. — In example (a) we have a direct proportion. Thus, 8 hats : 12 hats = cost of 8 hats : cost of 12 hats. In example (b) we have an inverse proportion. Thus, 5 men : 8 men = time of 8 men : time of 5 men. Query. — How does an inverse proportion differ from a direct propor- tion ? 166 SCHOOL ARITHMETIC. Rule. — For the first couplet compare the two like numbers. For the second couplet compare the remaining number with the required number. Determine from the conditions of the problem which is the greater. Arrange these two numbers as a couplet^ making the greater or less the antecedent according to the arrangement of the first couplet. Divide the product of the means or extremes hy the sirigle mean or extreme. The quotient will be the required number. The work may frequently be sliortened by cancellation. This rule is often called "The Rule of Three," because three numbers are given to find a fourth. 3. If 12 pounds of tea cost $5, how much must be paid for 24 pounds ? 4. At 8 cents a yard how much will 10 yards of calico cost ? 6. A tree 60 feet high casts a shadow 80 feet long. How long a shadow is cast by a tree 48 feet high ? 6. If 18 horses eat 12 bushels of oats in a day, how many horses would eat 20 bushels in the same time ? 7. If a train runs 9.0 miles in 3 hours, how far will it run in 24 hours ? 8. By working 9 hours a day, B can dig a ditcli in 16 days. In how many days can he dig it by working 10 hours a-day ? 9. If a family uses 6 barrels of flour in 10 months, how many barrels would last a year ? 10. A man earns $1000 in 6 months. In how many months can he earn $2500 ? 11. If f of an acre cost 130, how much must be paid for 6 acres ? 12. A farmer raised 3G bushels of wheat on \^ acres. At this rate, how many bushels can he raise on 10 acres ? 13. In how many days can 12 men build a wall that 5 men can build in 18^ days ? PROPORTION. 167 14. Mr. C paid |5l40 for 7 months' rent. How much did he pay in a year ? 15. How long would it take 3 men to mow a field that 7 men can mow in 3\ days ? 16. Find the cost of 75 sheep at $6.40 each. 17. A paid 115.75 for 3J^ weeks' board. At the same rate, how much did he pay in a year ? 18. If f of a yard of cloth costs $|, how much must be paid for 16 yards ? 19. If 10 tons of hay last 5 horses 8 months, how long would it last 12 horses ? 20. If 20 bushels of wheat produce 6^ barrels of flour, how many bushels will produce 100 barrels ? 21. If 24 yards of carpet cover f of a floor, how many yards will cover the entire floor ? 22. If 9 compositors can set up a 6-page paper in 8 hours, in how many hours can they set up a 20-page paper ? 23. If a cows cost b dollars, how much will 3a cows cost ? 24. When b hats cost c dollars, how much must be paid for d hats ? 25. Complete the equation, a : b = 7a : ( ). SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES), 249. 1. The Washington monument casts a shadow 223 feet 6^ inches long when a post 3 feet high casts a shadow 14.5 inches. Find the height of the monument. 2. Milk is worth 20^ a gallon, but by watering it the value is reduced to 15^ a gallon. Find the ratio of water to milk in the mixture. 3. If 4 is one third of a certain number, what is one half of it ? 4. Sixty men can grade a street in 40 days. After 24 days, one third of the men are discharged. In how many days can the others finish the work ? 5. A piece of work can be done in 5 weeks by 12 men. At 168 SCHOOL ARITHMETIC. ^ the end of 2 weeks ifc is decided to complete the work in 6 days. How many more men must be employed ? 6. If a 4-cent loaf weighs 9 ounces when flour is $6 a bar- rel, how much should a 5-cent loaf weigh when flour is 18 a barrel ? 7. If 300 cats kill 300 rats in 300 minutes, how many cats can kill 100 rats in 300 minutes ? 8. Two men or three boys can plow an acre in | of a day. How long will it take 3 men and 2 boys to plow it ? 9. If 4 horses or 6 cows can be kept 10 days on a ton of hay, how long will it last 2 horses and 12 cows ? 10. If 8 men or 15 boys plow a field in 15 days of 9^ hours, how many boys must assist 16 men to do the work in 5 days of 10 hours each ? 11. Divide $1000 between A and B so that A shall have $3 out of every $5. There are 1000 -r- 5 = 200 fives. . •. A gets 200 X $3 = $600. 1^" A gets $3 for every $2 B gets. The money is divided in the ratio of 3 to 2. 12. Tom and Peter found a watch worth $45, and agreed to divide the value of it in the ratio of f to |. How much was each one's share ? 13. Gunpowder is composed of nitre, charcoal, and sulphur in the proportion of 38, 7, and 5. How many pounds of sul- phur in 180 pounds of powder ? 14. If 20 men can perform a piece of work in 12 days, how many men can do a piece of work 3 times as large in ^ of the time ? 15. A man rides a certain distance at the rate of 6 miles an hour, and walks back at the rate of 3^ miles an hour. If the time of the round trip is 4f hours, what is the distance ? 16. If da sheep cost $24, how much will 5b sheep cost, if a : ^^ : : 2 : 1 ? THE EQUATION. 250. An Equation is a statemeut that two numbers or expressions are equal. Thus, 1 + 1 = 2, 7+3 = 2x5, and 3a; — 2 = 11 are equations. 251. An equation is like a balanced scale-beam — one side representing the thing weighed, and the other side the weiglits. 262. The part of an equation that is written before the sign = is called the first member, and that written after the sign the second member. , Thus, in the equation 7 + 3 = 2x5, 7 + 3 is the first member, and 2 x 5 is the second member. 1. The two members may differ widely in form, but in value they must be the same. 2. In the expression 4 -r- 3 — 2 = ( ), the second member is to be supplied. 253. The numbers that compose the members are called the Terms. The terms of each member are separated from each other by the signs + or — . Thus, in the equation 3+6 x 7— 8-f-4 = 43, the terms of the first member are 3, 6 x 7, and 8 -e- 4. In the expression 5+2x4— ( ) = 10, one term is to be supplied. INDUCTIVE EXERCISES. 254. 1. When will a scale-beam balance ? 2. If two one-pound packages of coffee are placed in one scale, what must be done to restore the balance ? 170 SCHOOL ARITHMETIC. 3. If one of the packages is removed, what must be done to restore tlie balance ? 4. Would the balance be destroyed if the coffee and weight* should change scales ? Why not ? 6. Could anything else of equal weight be substituted for the coffee without destroying the balance ? 6. If 5 pounds are added to one side, what must be done to the other side to keep the balance true ? 7. If one pound is taken from one scale, how can the bal- ance be restored ? 8. What does the expression 7 4- 5 + 3 = 15 mean ? 9. If we add one to the first member, how much must be added to the second to make the sides equal ? 10. If we take 5 from the first member, what must be done to the second to preserve the equality ? 11. Is there any difference in value between 7 + 3 and 15 - 5 ? Then 7 + 3 = 15 - 5. 12. How much is 6 times (7 + 3) ? How much is 6 times (15 — 5) ? Are the products equal ? 13. Then what may be done to both sides of an equation without destroying the equality ? 14. What is"^ the quotient of (7 + 3) -f- 2 ? Of (15 - 5) -^ 2 ? Are the quotients equal ? 16. Then what else may be done to both sides without destroying the equality ? ^ 16. A 10-acre field is .worth $100 an acre, and a 20-acre field is worth $50 an aci'e. In what respects are the two fields equal ? 255. Since one member of an equation is equal in value to the other, whatever is done to one side must be done to the other in order to preserve the equality. 256. Principles. — 1. Expressions which are equal to the same thing or to equal things are equal to each other. 2. If equals are added to or subtracted from equals, the re- sults are equal. THE EQUATION. 171 3. If equals are multiplied or divided by the same number ^ the results are equal. 4. In general, if the same operations are performed upon both members of an equation^ the results are equal. 5. If two expressions are equal, either can be substituted for the other ivherever it occurs. Bt^" These principles are self-evident, and arc called axioms. CHANGE OF FORM. 267. It is frequently necessary to change the form of an equation in order to simplify it. The principal change is transposing terms. 268. Transposiiigr is the process of changing a term from one member of an equation to the other without destroying the equality. 1. In the equation $16 — $5 = $8 + $3, transpose 15 to the second member. Adding $5 to each member (Prin. 2) we have $16 - $5 + $5 = $8 + $3 + |5. But since — $5 + f 5 = 0, we may write the equation $16 = $8 +13 + $5. How does this equation compare with the one given ? It will be noticed that (— $5) has disappeared from the first member, while (+ $5) appears in the second, 2. In the equation 2 + 5 = 4 + 3, transpose 3 to the first member. Subtracting 3 from each member (Prin. 2), we have 2 + 5-3 = 4 + 3-3. But since + 3 — 3 = 0, we may write the equation 3 + 5-3 = 4. How does this equation compare with the one given ? It will be noticed that 3 has disappeared from the second member, and appears in the first, with a dififerent sign before it. 269. Any term may be transposed from one member of an equation to the other by dropping it from the member iu 172 SCHOOL ARITHMETIC. which it stands, and writing it in the other with a different sign. Transpose so that only odd numbers will be in the first member : 1. 7-4 = 8-5. 4. 5x3-2x4 = 14-6. 2. 2 + 9 = 1 + 10. 5. 39 -^ 3 - 4 ^ 2 = 22 -^ 2. 3. 19 + 6 = 30 - 5. 6. 7 + 4 - 8 = 26 - 25 + 2. Transpose so that only like terms will be in each member : 7. 3 times a number — 5 = 2 times the number. 8. 5 times A's money — $10 = 3 times A's money + $20. 9. ^ of my money = $9 — | of my money. 10. I of A's age + 4 years = :^ of A's age + 11 years. 11. $2 + 10 cents = II + $^ + 60 cents. 12. 3 rods + 2 yards = 2 rods -I- 7^ yards. Transpose all terms containing x to the first member, and all others to the second member : 13. 7a; - 5 = 3a; + 15. 14. 5 + 6a; - 9 = 2a; + 8. 16. 5a; -i- 6 - 8 = 4 + 3a;. 16. 9a; - 36 = 24 - 6a;. 17. 8a; - 2y = Uy - 16a;. 18. 3y + 10a; = 10 + lla;. 260. After like terms have been collected into one mem- ber, the equation may often be made still more simple by performing the operations indicated. Thus, the equation, 5 times A's money — 3 times A's monoy = $240 + $60, may be written 2 times A's money : Perform indicated operations : 1. 4 times a number — 3 times the number = 10 + 5. 2. 2 X B's money + 3 x B's money = $1000 - $500. THE EQUATION. 173 3. 7 X 5 - G4 -^ 16 - 11 = 5 X 4. * 4. 6 + 5x0 + 18-2 = 2 + 2x25. d^" This may be called unititig like terms. Notes, — 1. In uniting terms we subtract the 8um of the minus terms from the sum of the plus terms. 2. The sign x may be omitted between factors if one (or more) of the factors is a letter. Supply the wanting term : 5. 32 - 5 X 3 + 45 -r- 9 = ( ). 6. 46 - 18 + 16 + ( ) = 30 + 7 X 10. 7. 75 + 13 X ( ) - 51 = 64 - 2 X 7. 8. l + ix|-( ) = 2-2xi 261. Expressing the conditions of a problem in the form of an equation is called stating the problem. The several steps of the analysis may be expressed in a series of equations, each derived from the one preceding, by a change of form under Principles 1 — 5. 1. If 3 melons are worth 60 cents, how much are 5 melons worth ? 1.) The cost of 3 melons = 60 cents. 2.) .*. the cost of 1 melon = 20 cents. 3.) .'. the cost of 5 melons = 100 cents. Queries. — How is equation 2 derived from 1 ? How is 3 derived from 2 ? 2. 24 is f of what number ? 1.) f of some number = 24. 2.) .-. ^ of the number = 12. 3.) .'. f of the number = 36. Query, — How are equations 2 and 3 derived ? 3. Find f of 40 ? 1,) I of forty = 40. . 2,) .-, i of forty = 5. 3.) .-. i of forty = 15. Query. — How is equation 1 derived ? 174 SCHOOL ARITHMETIC. ^ 4. 35 is how many eighths of 40 ? 40 = 8 eighths of 40. .-. 5 = 1 eighth of 40. .-. 35 = 7 eighths of 40. 262. The following solutions show a few of the many ad- vantages of using the equation in arithmetic. Its utility in solving many of the problems of percentage, etc., appears in later pages of this book. 1. Eight times a number diminished by 46 equals 14 more than 3 times the number. Find the number. First Solution. 8 times the number — 46 = 3 times the number + 14. Transposing 46, 8 X the number — 3 x the number = 14 + 46. Uniting terms, 5 X the number = 60. .'. the number = 12. Second Solution. Let X = the number. .-. 8a; - 46 = 3a: + 14. Transposing, 8a: - 3a; = 14 + 46. Uniting terms, 5a: = 60. .'. a: = 12. 1^*" In the solution of any problem, the number to be found is the one that must be represented by a letter. This letter may be treated just as the number itself would be if known. 2. A coat and a hat cost ^3G. The coat cost 5 times as much as the hat. What was the cost of the hat ? Let X — cost of hat. 5a; = cost of coat. .*. a: 4- 5a; = cost of both. .*. 6a: = $36. .-. x = $6, cost of hat, and 5a; = $30, cost of coat. THE EQUATION. , 175 3. A farmer has 100 hens and chicks. Every hen has 9 chicks. How many of each has he ? Suggestion. — Let x = the number of hens. 4. If three times A^s age plus 12 years equals five times his age less 8 years, what is A's age ? Let X = A's age. 3a; + 12 yr. = 5x - 8 yr. 3a: - 5a; = - 8 yr. - 12 yr. -2a;--20yr., or 2a; = 20 yr. a; = 10 yr. U^" The signs of all the terms can be changed without destroying the equality ; for, by transposition, the members can be interchanged and therefore their signs changed. Queries. — 1. In solving problems, what is the first thing to do ? (State the problem.) 2. What is the second step ? The third ? 3. How do you explain the work after terms have been united ? Let the pupil write a rule. 1^" In solving the following problems great care must be exercised in making the statement. Solve all by using x. 5. A and B have $80, and for each dollar B has, A has $3. How much has each ? 6. $40 is 14 more than ^ of my money. How much have I? 7. 18 is I of what number ? 8. What number added to ^ of itself equals 2x9? 9. I of a number is 5 more than f of the number. What is the number ? 10. A lady bought a dress for $24, and found that she had I of her money left. How much money had she at first ? 11. Mr. E has gold dollars and silver dollars to the amount of $30. He has one half as many silver dollars as gold dol-. lars. How many of each has he ? 12. A got \ of his father's fortune, B got I of it, and C 176 SCHOOL ARITHMETIC. ^ got the remainder. If A got $2000 more than B, how much did get ? 13. If 18 is added to 6 times a certain number, the sum will be 22 less than | of 150. What is the number ? 14. A pole 36 feet long was broken into two unequal pieces, f of the longer piece being equal to | of the shorter. How long was each piece ? 15. Three boys. A, B, and C, have 77 marbles ; B has 10 more than A, and A has 8 more than C. How many has each boy ? 16. Find a number such that if it be added to ^ of itself the sum will be 6Q. , 17. By selling a watch for $36, I gained f of its cost. What was its cost ? 18. f of A^s money diminished by $20 is equal to ^ of his money increased by $5. Find A's money. 19. A and B together sold 728 bushels of wheat, and B sold 3 times as much as A. How much did each sell ? 20. Divide $1000 between C and B, so that C may have | as much as B. 21. What number increased by one half of itself, by one third of itself, and by 18 more, will be doubled ? 22. If with the money I now have, I had 3 times as much, and $25 more, I should have $125. How much money have I? 23. Find a number such that the sum of its half and its third may exceed the sum of its fourth and its fifth by 23. 24. A man left ^. of his estate to his wife, -^ for charity, | to his children, and $1400 to his servants. What was the amount of his estate ? 25. A man gave $100 to his 3 sons ; to the second he gave twice as much as to the first, lacking $8, and to the third he gave 3 times as much as to the first, lacking $15. How much did he give to the first ? THE EQUATION. 177 SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 263. 1. A drover, being asked how many sheep lie had^i said, *^ If to ^ of the number of my flock you add the num- ber 9^, the sum will be 99|." How many sheep had he ? 2. A's money added to ^ of B's money equals $2000. How much money has each, provided A's is to B's as 3 to 4 ? 3. A boy cut off one half of the length of his kite-string. 'He then added 45| ft., and found that the new string was ^ of the original length. What was this original length ? 4. A lady being asked the time of day replied that J of the time past noon equaled f of the time to midnight, minus ^ of an hour. Wluit was the time ? 6. After losing f of my money I earned |5l2, and then spent f of what I had. What I then had left was $36 less than I had at first. How much had I at first ? 6. A thief stole | of A's money and spent f of the amount stolen before he was caught. The remainder, $324.75, was secured. How much money had A at first ? 7. A, B, and C together have $434. f of A's money equals I of B's, and f of B's equals f of C's. How much has each ? 8. If f of A's money is to f of B's as 3 to 4, and together they have $1520, how much has A ? 9. A, B, and C together have $21950. fof A's money, | of B's, and ^ of C's are all equal to each other. How much has each ? 13 REVIEW WORK. ORAL EXERCISES. 264. 1. A wagon cost 150, which was ^ of the cost of a horse. What was the cost of the horse ? 2. If three barrels of flour cost $14.40, what will 2|^ barrels cost ? 3. Mr. A bought f of a mill, and sold f of it to B. What part of the mill did he retain ? 4. Twenty-eight is 7 tenths of how many times 4 ? 6. A had 60 acres of land ; he sold f of it to B, and | of the remainder to C. How many acres had he left ? 6. Eight ninths of 27 are how many times 6 ? 7. A man gave 7 beggars $1.60 apiece, and had $8.90 left. How much had he at first ? 8. If f of a lemon cost f of a cent, how much will 3 lemons cost ? 9. A boy sold a watch for $16, which was $4 less than f of its cost. What did it cost ? 10. A horse cost $180, which was 3 times the cost of a yoke of oxen. What was the cost of an ox ? 11. A has f as much money as B, and both have $50. How much has each ? 12. John and Jane each has ^ as much money as Kate, and all have $35. How much has each ? 13. Two thirds of A's age is | of B^s age, and the sum of their ages is 77 years. What is the age of each ? 14. Ada gave f of her flowers to Eli, and had 12 remaining. How many did she give to Eli ? REVIEW WORK. 179 15. Mr. D gave $45 for shovels at $1.25 each. How many dozen did he get ? 16. A lady bought 2 bushels and 3 pecks of pears at $1.50 a bushel. What did they cost her ? 17. Mrs. A bought a dress for $G.75. If she paid $| a yard, how many yards did she buy ? 18. A farmer has a field containing 16 acres. If he can mow ^ of it in two days, how many acres can he mow in a day ? 19. A can dig a ditch in 2 days, and B can dig it in 3 days. If they work together, how long will it take them ? How much of it can A dig in a day ? How much can B dig in a day? Then how much can both dig ? How many sixths are to be dug ? Then how many days will it take them ? 20. A can build a wall in 2 days, B in 3 days, and C in 4 days. How long would it take all three working together ? 21. A and B can cut a field of wheat in 6 days, and B alone can cut it in 10 days. In what time can A alone cut it ? 22. A farmer bought 9 sheep for $45, and sold them for $14 more than | of what they cost. Did he gain or lose, and how much ? 23. A boy bought apples at the rate of 6 for 5 cents, and sold them at the rate of 5 for 6 cents. How much did he gain on each apple ? How much on 10 apples ? 24. Tom is f as old as his mother, who was married 24 years ago at the age of 25. How old is Tom ? 25. When hay is $13.50 a ton, and coal is $8 a ton, what part of a ton of coal can be bought for -J of a ton of hay ? 26. Jane bought a dozen oranges, of which she ate two, and sold the remainder at 2 cents apiece, thereby gaining 1^ cents on each orange bought. How much did they cost each ? 27. After spending ^ of his money for a cake, and ^ of it for a ball and bat, Henry had $1.40 left. How much had he at first ? 28. A can build a boat in 4 days, B in 5 days, and in IQ 180 SCHOOL ARITHMETIC. days. If all work together 1 day, how long will it take C alone to finish ? 29. A and B hired a rig for $11. A used it one day, and B used it two days. How much should each pay ? 30. A can do a piece of work in half a day, and B can do it in f of a day. How long will it take them if they work together ? 31. Jack can eat a loaf in 1^ days, and Jill can eat it in 2 J days. How long will it last hoth ? 32. What number is that which, being increased by its ^, its ■^, and its -^^ Avill be doubled ? 33. What number will be doubled by adding to it its ^, its ^, and 5 more ? 34. Mr. Kay is 45 years of age, and f of his age is -^ of his wife's age. How old was his wife when she was married 20 years ago ? 35. A and B earned $110. If A earned $10 more than B, how much did each earn ? 36. A man spends -J of his money, and then loses ^ of the remainder ; he then has $400. How much had he at first ? 37. In a school f of the pupils study arithmetic ; ^ of the remainder, algebra ; and the rest, or 12, geometry. How many pupils are there ? 38. A can do a piece of work in 3 days ; B can do the same work in 4 days ; if A earns $2 a day, what does B earn a day? 39. f of my money is 4 times my week's wages ; I have $100. What are my weekly wages ? 40. In traveling 72 miles a man went f of the distance the first day, ^ of the distance the second day, and the remainder the third day. How far did he travel the third day ? 41. Owen is -f as old as his father, and f as old as his mother. If he is 18 years old^ how old are his father and mother ? kEVIEW WORK. 181 42. I of C's money is ^\ of D's, and ^ of D'a is 120. How much money has C ? 43. If Mr. Fox were twice as old as he is, ^ of liis age would be 20 years. What is his age ? 44. It took I of my money to pay a debt ; I then paid |5l2 for a coat, which was f of the money I had left. Ilow much had I at first ? 45. f of the sum of two equal numbers is 20. What are the numbers ? 46. What number must be added to the difference between J and ^ to make ^ ? 47. The numerator of a fraction, whose value is |, is 20. What is the denominator? 48. A has $3.50, B has $1.25 more than A, and C has as much as A and B. How much money have they to- gether ? 49. If a locomotive moves f of a mile in ||- of an hour, what is its speed per hour ? 60. -f^ of a farm is in crops, and ^V of the remainder i\ woodland. What part of the farm is woodland ? 51. A man plows f of a field the first day, and ^ of it the second day. What part of it does he plow the third day, if he finishes on that day ? 52. A miller keeps as toll -\ of the corn to be ground. What is the ratio of the toll to the meal returned ? 53. How long will J of a bushel of oats last a horse, if 2^ bushels last him one week ? 54. By selling a horse for $96 I gain ^ of the cost. What did the horse cost ? 55. James had 36 cents. He lost | of it and spent ^ of it. How much has he left ? 56. If f of a cord of wood costs $3, how much will ^ of a cord cost ? 57. A man bought 5 bushels of wheat for $4, which was •J of the cost. What was the cost a bushel ? 182 SCHOOL ARITHMETIC. 66. A can do twice as much work as B. How many times B's work can both do ? 69. Daisy traveled f of lier journey by rail, f by water, and the remainder, which was 18 miles, by stage. How many miles did slie travel ? 60. Preston worked IG days, and after paying for a suit of clothes with | of his money had $24 left. How much did he receive a day ? 61. A watch and chain cost $90 ; the chain cost ^ as much as the watch. What was the cost of each ? 62. A, B, and C can paper a room in 6 hours, B and C can paper it in 10 hours. In what time can A alone paper it? 63. If J of a farm is worth $150 more than | of it, what is the whole farm worth ? 64. James being asked how many marbles he had, said he had § as many as Phil, and that both together had 155. How many had he ? WRITTEN EXERCISES. 265. 1. A man bought 20 acres of land at $50.25 an acre. He sold -^ of an acre to B, 8f acres to C, and the remainder to D. If he received $05 an acre from B and C, and $60 an acre from D, how much did he gain ? 2. How far can a man walk in 3f hours, at the rate of 3f miles an hour ? 3. If f of a pole is in the ground and 24 feet are above the ground, how long is the pole ? 4. If there are 69.16 miles in one degree of latitude, how many miles are there in 34f degrees ? 6. If 2f yards of cloth are required for a pair of pants, f of a yard for a vest, and 4 yards for a coat, how many yards will be left from a piece of 41 yards, after 5 suits have been cut off ? 6. At the rate of 3 for 10 cents, what will 75 dozen oranges cost ? REVIEW WORK. 183 7. A man paid $5 for sugai- and $5 for coffee. If sugar was 6:^ cents a pound and coffee was 25 cents a pound, how many pounds of both did he get ? 8. A man can build a fence in 16 days by working 9^ hours a day. How much longer would it take him working only 8 hours a day ? 9. Find the cost of 6 spoons when 5 dozen cost $32. 10. The quotient is 16.73, and the divisor is 8|. How much must be added to the dividend to make 150 ? 11. In a square rod there are 272^ squar6 feet. How many square rods are there in 9000 square feet ? 12. A jeweler cut a wire |q of an inch long into 11 equal parts. How many of the parts were equal to half an inch ? 13. Walter spent ^ of his money and ^ more, then ^ of the remainder and $1 more, then J of the remainder and $^ more, and then had llf left. How much had he at first ? 14. A man owning | of a store sold ^ of his share for $2250. What was the value of the store ? 15. What number is that whose | exceeds its | by 10 ? 16. I bought a horse and a buggy, paying ^ as much for the buggy as for the horse. If both cost $340, what did the buggy cost ? 17. Bought potatoes at $2.62^ and sold them for $3|^ a barrel. If my gain was $87.50, how many barrels did I handle ? 18. Seven men dug a cellar in 12f days. How long would it have taken 3 men ? 19. If 8 be added to numerator and denominator of ^, will the value of the fraction be increased or diminished, and how much ? 20. If 5 be subtracted from numerator and denominator of 4, how will the value of the fraction be affected ? Why ? 21. Mr. S has 60 hens. He sold f of them to A, and f of the remainder to B. bought what remained at $.75 a pair. How much did C pay ? 184 SCHOOL ARITHMETIC. 22. Mr. Willis sold 3 loads of hay weighing, with the wagons. If, IJ, and 2J tons respectively. The empty wagons weighed |, f, and .9 of a ton. Wliat was the value of the hay at $16.50 a ton ? 23. A man bought 20 sheep at $4.75 each, and 3 horses. If he paid $470 for the sheep and horses, what was the aver- age price of the horses ? 24. What number multiplied by ^ of 13 j will produce 1 ? 25. A can walk 60 miles in 12^ hours, and B can walk it in 15 hours. If they are 60 miles apart, and start at the same time to walk toward each other, how far apart will they be in an hour and a half ? 26. Divide (18J x 18|) by (f -f- .01), and add (| of .001) to the quotient. 27. What is the difference between 3|^ + 12^ x 2^ and (3^ + 12i) X 2i ? 28. A can fence one side of a square in 8 days, and B can fence 2 sides of it in 12|- days. In what time can both to- gether fence the field ? 29. A man left | of his estate to his first son, f to his sec- ond son, and the remainder to his daughter, whose share was $600 less than that of the first son. Find each one's share. 30. The width of a stream was measured at several points, the measurements being as follows : 42^ feet, 37^ feet, 35 feet, 41f feet, 52^ feet, and 48.875 feet. What is the average width ? 31. What number increased by f of itself is equal to 1 ? 32. Two fifths of my money is gold, I of it is silver, and the remainder is paper. I have $8 more paper money than silver. How much gold have I ? 33. A watch and chain cost $125. If the chain cost | as much as the watch, what was the cost of each ? 34. Twelve years ago, A was ^ as old as B, but now he is I as old. How old is A ? REVIEW WORK. 185 35. A mother and 3 cliildren use a pound of coffee in a week. AVheu the motlier is absent, two pounds last the chil- dren 6 weeks. How long would a pound last the mother alone ? 36. If 3 be added to | of a certain number, and J of the sum be multiplied by 3 tenths of 3, the product will be 3 times 10.8. What is the number ? 37. A man gives .lof his income to charity, .24 for edu- cating his children, .375 for other expenses, and lays by the remainder, which is $570. What is his income ? 38. A man sold a horse for 1^ times the cost, gaining $15. Find the cost. 39. I of f of what number equals -^ of 3G0 ? 40. How much will 49 men earn in 17^ days @ $2.10 t^ day? 41. If T^ of a ream of paper costs $.60, how much can be purchased for $237.60 ? 42. For what length of time will $495 pay rent at the rate of $24.75 a month ? 43. The greater jf two fractions is J| ; their difference is ■^\. What is the other fraction ? 44. How many oranges @ $.25 per dozen will pay for 36 bu. of coal worth 8-^ cents a bushel ? 45. A lady sold -^, f , and f of her fowls ; she had 30 re- maining. How many had she at first ? 46. A farmer sold 65 bu. more than ^ of his wheat, and found that the remainder was 65 bu. more than f of hi ; wheat. How many bushels luid he at first ? 47. A drover spent $50.60, wliich was ^4 of what lie ii<' for 23 sheep. What did he get apiece for them ? 48. Bought eggs at the rate of If^ each, a* d sold them at, 27^ a dozen. I gained $3.65 ; how many dozens did 1 sell ? 186 SCHOOL ARITHMETIC. 49. A merchant had 163| yd. of gingham, from which he sold 95^ yd. The remainder was made into 25 aprons of the same size. How many yards did each apron contain ? 50. A woman bought a shawl for $9.75, which was ^ of f of the price asked. At how much less than tlie price asked did she buy the shawl ? 61. John earns I24.66| per week ; James earns f as much, and Tom | as much as John and James together. If Tom gives ^ of his money to charity, how much has he left each week ? 62. -j^ of the troops engaged in a battle were killed ; f of f of the killed numbered 45. What was the original number of the troops ? 63. If one horse consumes 3| bu. of oats per week, how many bushels will 18 horses consume in 6 weeks ? 64. A can do a piece of work in 7^ days, A and B can both do the same work in 5 days. In what time cau B do the work ? 66. A lady bought a watch for $81.75, paying ^ down. How much must she pay per month to pay the remainder in 8 months ? 66. Mr. Drift sold f of his property, ^ of the remainder, and -^ of that remainder. His property was then worth $2760. What was the value of the whole property ? 67. A and B raised 320 bushels of wheat. After paying ^ of it for rent, they divided the remainder so that A received I as much as B. How many bushels did each receive ? 68. William bought 330 lemons at 30 cents a dozen, and 6^ dozen oranges at 2 cents apiece ; he sold f of the lemons at 4 for 25 cents, the remainder at 3 cents apiece ; he sold ^ of the oranges at cost, the remainder at 30 cents a dozen. How much did he gain ? 69. A jeweler sold a watch for $62.50, which was ^ more than he paid for it. How much did it cost ? 60. Mr. D purchasing a pair of horses paid ^ of the price REVIEW WORK. 187 in cash ; i the remainder he paid a month later. He still owes $310 of the debt. What was the purchase price ? 61. Two men do a piece of work in 18 days. If the first man works three times as fast as the second, in liow many days can each do the work alone ? 62. A father left a fortune to his three children ; the old- est was to have f of it, the second was to have ^ of it, and the youngest was to have the rest. The oldest received $12000 less than the youngest. What was the value of the estate ? 63. If f of a bushel of corn be worth f of a bushel of wheat, and wheat be worth $1.40 a bushel, how many bushels of corn can be bouglit for $27 ? 64. Bought 304 pounds of rice for $31. IG. One fourth of it is destroyed. I sell the remainder at a loss of 1 cent on the pound. What is my whole loss by the transaction ? 65. Y^Q of the cost of my house and barn is ?j| of the dif- ference between their costs ; and 1^ times the difference between their costs is $3600. Required the cost of each. 66. A agreed to hoe | of a field of corn while B hoed the remainder. After finishing, it was found that A had hoed 69^ rows more than one-half. How many rows in the field ? 67. The total Indian population on the reservations in 1893 was 249,366, while the area of the reservations was 134,176 square miles. What was the average quantity of land occupied by 100 Indians ? 68. The moon^s diameter is ^V ^^^^^ ^f ^^^^ earth, and the sun's diameter is 110 times that of the earth. What fraction of the sun's diameter is that of the moon ? 69. I bought 21 pigs and 15 cows for $693. Each cow cost $27 more than each pig cost. Find cost of each pig. 70. Three Indians counted their trophies. Red Cloud had twice as many as Bigfoot, and the latter had f as many as Horsehead, who had 4 less than Red Cloud. How many had each ? 188 SCHOOL ARITHMEfid. SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 206. 1. One day a man spent f of his money, and the next day 4 of the remainder. If he had ^q of a dollar left, how much did he spend the second day ? 2. A can roof a house in 6^ days, but can work only .75 of each day. If B helps him, the time required to roof the house is 2 days and f of an hour. Counting 9 hours a day, in how many days can B alone do the work ? 3. A farmer has 5 horses to each of which he gives ^ bushel of oats three times a day. When oats are worth $| a bushel, what will be the cost of the oats required in the month of September ? 4. Two men, starting from the same point, walk along a railroad track. When one has gone -^ of a mile and the other f of a mile, how many feet are they apart, there being 1G| feet in a rod, and 320 rods in a mile ? 5. If telegraph poles are ^^ of a mile apart, how many poles in a line 40.5 miles long ? 6. When is a factor, what is the product ? Why ? 7. When the multiplicand is one greater than the product, which is ^g, what is the multiplier ? 8. A man divided $10 equally among 5 boys. Each boy gave -^ of his share to the poor. How much was given to charity ? 9. A man divided $a equally among b boys. Each boy gave ^ of his share to the poor. How much was given to the poor ? 10. At one store a lady spent |- her money and $^ more ; at anotlier, ^ the remainder and $j more ; at another, | the re- mainder and $1 more, and then had a dollar left. How much had she at first ? 11. In an orchard there are a rows of trees, and between each two rows of trees are 5 rows of cabbage. What is the value of the cabbage at $b a row ? ($5ab — 5b.) COMPOUND NUMBERS. 267. A concrete number expressed in two or more de- nominations (units of measure) is called a Compound Num- ber. Thus, 2 feet 3 inches, 3 pounds 8 ounces, are compound numbers. But 3.5 pounds and $1.55 are not compound numbers. 268. A Measure is a unit by wbich quantity, such as value, length, weight, etc., is estimated. Thus, the yard is a measure, because it is a unit by which length is estimated or measured. 269. A Prime (or principal) Unit is a unit of measure from which other units of the same kind may be derived. Thus, a dollar is a prime (or principal) unit. MEASURES OF VALUE. 270. The ordinary measure of value is Money, which is sometimes called currency. Coin is metal money ; all other currency is called Paper Money. 271. United States Money is the legal currency of the United States. The prime (or principal) unit is the dollar. Table. 10 mills = 1 cent (ct. or ^). 10 cents = 1 dime (d.). 10 dimes = 1 dollar ($). $ d. ct. m. 1 = 10 = 100 = 1000 190 SCHOOL ARITHMETIC. 1. The U. S. now issues the following coins: Gold.— The 20-dollar, lO-doUar, 5-dollar, and 2^dollar pieces. Silver. — The dollar, half-dollar, quarter-dollar, and the dime. Nickel. — The 5-cent piece. Bronze. — The one-cent piece. 2. The standard weight of the gold dollar is 25.8 gr. ; it contains 23.22 grains of pure gold, but is not now coined. 3. The silver dollar weighs 412| gr. 4. The standard purity of the gold and silver coins is 9 parts pure metal and 1 part alloy (by weight). The alloy of silver coin is pure copper. The alloy of gold coins is copper, or silver and copper. If both are used, the silver is not to exceed i\, of the alloy. 5. The mill has never been coined ; it is merely a convenient name for the tenth part of a cent. 272. E]nglish Money is the currency of Great Britain. The principal unit is the pound sterling, which is called the sovereign when coined. Table. 4 farthings — 1 penny (d.). " 12 pence = 1 shilling (s.). 20 shillings = 1 pound (£). = $4.8665 in U. S. money. £ s. d. far. 1 = 20 = 240 = 960. 273. Canada has a decimal currency, and the table and denomihations are the same as those of U. S. money. 274. France also has a decimal currency. The unit is the franc. The value of a franc in U. S. money is $.193. The franc is divided into 100 centimes. 275. German Money is the legal currency of the German Empire. The unit is the mark. The value of a mark in U. S. money is $.2385. The mark is divided into \QQ pfennigs. COMPOUND NUMBERS. 191 1. How many dimes in $3 ? In $i ? In $3^ ? 2. How many cents in $10 ? How many dimes ? How many half-dollars? 3. How many pence in 3 shillings ? In 8 shillings ? In ^ shillings 2 pence ? 4. How many pence in a pound ? In £^ ? In £2 5s. ? 5. How many shillings in £5 ? In £2 10s. ? In £i ? 6. Have you changed the value of these numbers, or their form f 276. Reduction is the process of changing the denomina- tion of a number without changing its value ; that is, chang- ing the unit of measure and the number of the units. 277. Reducing a number to a lower denomination is called Reduction Descending. WRITTEN EXERCISES. 1. Reduce £5 8s. 7d. to pence. p, rt^ ^ j-^ Since there are 20s. in 1 pound, in £5 -K\f\ , *o~ -i^c there are 5 times 20s., or 100s.; 100s. + 100s. + 8s. = 108s. _ ,.„ ,, 101 • 1 u-iT 108 -i.^i _ioq/>i 8s. =r 108s. •.* there are 12a. nil shilhng, 1 -laaA X ^^'a ~— ^'ic\'iA in 108s. there are 108 times 12d., or 1296d. ; 129bd. + 7d. _ IdOdd. ^296d. + 7d. = 1303d. Reduce to farthings : 2. 6s. 4d. 2 far. 5. 9s. 6d. 1 far. 3. 10s. lOd. 3 far. 6. £2 8s. 4. 13s. lid. 7. £3 4s. 9d. 3 far. 8. Reduce £^ to pence. 9. Reduce £f to shillings and pence. £| = I of 20s. = -Lf a s. = 12is. is. = i of 12d. = 6d. .•.£§ = 12s. 6d. This may be called reducing to loiver denominations. Reduce to lower denominations : 10. £f. 12. fs. 14. £f. 16. 2is. 11. £3%. 13. fs. 15. £lf. 17. IJs. 192 SCHOOL ARITHMETIC. 18. Reduce .36 of a shilling to lower denominations. .36 of a shilling = .36 of 12d. = 4.32d. .3-3 of a penny = .32 of 4 far.= 1.28 far! J .-. .36s. = 4d. 1.28 far. Reduce to lower denominations : 19. .37s. 21. £A5 23. £.1^. 25. .56s. 20. .70S. 22. £.(JQ 24. £.|. 26. .87s. 278. 1. In 70 cents how many dimes ? 2. How many dollars in 200 cents ? In 500 ct.? 3. How many shillings in 24 pence ? In 36d.? 4. In GOs. how many pounds ? In £100 ? In £70 ? 5. Have you changed the value of these numbers ? What have you changed ? 279. Reducing a number to a higher denomination is called Reduction Ascending. WRITTEN EXERCISES. 1. Reduce 6845 farthings to pounds, shillings, etc. 4 far.)6845 far. Since in 1 penny there are 4 far., in ^2(] )1711d + 1 far. ^^^'^ ^'^^' ^^ere are as many pence as 4 far. rt^ X -, .. ^ — ~7~rn is contained times in 0845 far., or 1711, 20s. ) 142s. + 7d. .^, 1^1^ with a remainder ot 1 lar. ^^ + -^s. Since there are 12d. in 1 shilling, in 1711d. there are as many shillings as 12d. is contained times in 1711d., or 142, with a remainder of 7d. Since there are 20s. in 1 pound, in 142s. there are as many pounds as 20s. is contained times in 142s., or 7, with 2s. remaining. Therefore, 6845 far. = £7 2s. 7d. 1 far. Another Method. 1 far. = id. .-. 6845 far. = 6845 x id.= 1711id.= 1711d. + 1 far. Id. = -,Vs. .-. 1711d.=rl711 X -,\s.= 142i^2S.= 142s.+ 7d. .-. 142s. = 142 X £ iff = £"-A- = £7 + 2s. , ', 6840 far. = £7 2s. 7d, 1 far. COMPOUND NUMBERS. 1^ Eeduce to higher denominations : 2. 7500 far. 4. 8927d. 6. 13785s. 3. 6738 far. 6. 5360d. 7. 23456 far. 8. What part of a pound is ^d. ? (a) (b) Id. = -,^js. i-*-12=5V. .-. id.= i of THrS.= ^iS. 1^4 + 30 =T^. Note. — This is really reducing ^d. to a higher denomination, but, as fractions are involved, it is customary to say it is reducing id. to the fraction of a £. Eeduce to the fraction of a £ : 9. id. 11. f far. 13. .35d. 15. -^ far. 10. ifar. 12. |d. 14. .65 far. 16. ^^^d. 17. AVhat decimal part of a pound is 12s. 8d. 3 far.? ' The explanation of this solu- 8.75 -^ 12 = .72916 + ^.j^j^ jg similar to that given in 12.72916 -V- 20 = £.636458 + example 8. or The second method is simple £-^ __ 9(3Q f^j.^ and convenient. Reduce both 12s. 8d. 3 far. = 611 far. numbers to farthings, and find 611far. = £fi^ = £. 636458+ *^^ ^^^^«- 18. Reduce 9s. 6d. 1 far. to the decimal part of a £. 19. Reduce £5 to U. S. money. 20. Reduce 18s. 9d. 3 far. to shillings. 21. How many francs in $19.30 ? 22. What is the value in English money of $243,325 ? 23. Reduce 1000 francs to U. S. money. 24. How many marks in $2385 ? 25. Reduce ^ of a pound to lower denominations. 26. Reduce Is. 4d. to the fraction of a pound. 27. Express 8 dimes, 7 cents, 5 mills as the decimalpart of a dollar. . 13 194 SCHOOL ARITHMETIC. MEASURES OF CAPACITY. 280. Ijiqiiid Measure is used in measuring liquids of all kinds. The principal unit is the gallon. Table. 4 gills (gi.) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.), gal. qt. pt. gi. 1 r= 4 = 8 = 32. 1. In estimating capacity, 31| gal. are counted a barrel, and 63 gal. a hogshead ; but in commerce they are not fixed measures. 2. The gallon contains 231 cubic inches. 281. Dry Measure is used in measuring grain, fruit, vegetables, etc. The principal unit is the hushel. Table. 2 pints (pt.) = l^quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). bu. pk. qt. pt. 1 :^ 4 = 32 = 64. 1. The standard bushel of the United States contains 2150.42 cubic inches. 2. Grain, seeds, small fruits, etc., are sold by even or stricken meas- ure. Coal, corn in the ear, coarse vegetables, etc., are sold by heaped measure. 3. In dry measure a quart contains (2150.42 -^ 32), or 67.2 cubic inches. In liquid measure a quart contains (231 -?- 4), or 57.75 cu. in. ORAL EXERCISES. 282. 1. How many quarts in 16 pt.? In 28 pt.? In 19 pt.? 8. How many gallons in 20 qt. ? In 32 qt. ? In 18 qt. ? 3. What is a gallon of milk worth at 2 cents a gill ? COMPOUND NUMBERS. 195 4. When vinegar is 40 cents a gallon, how much must be paid for 2 qt. ? 1 pt. ? 5. At 5 cents a pint, how many gallons of molasses can be bought for $2 ? 6. How many pecks in 24 qt. ? In 35 qt. ? 7. In 2 pecks how many pints ? In 3^ pk.? 8. How many pecks in 5 bushels ? In 12^ bu. ? 9. A man bought 12 pk. of potatoes at 10 cents a half- peck. How much did he pay for them ? 10. Harry sold half a bushel of chestnuts at 5 cents a pint. How much did he get for them ? WRITTEN EXERCISES. 283. Reduce to lower denominations : 1. 5 gal. 3 qt. 1 pt. 6. 10 gal. 2 qt. 1 pt. 2. f gal. 6. .875 gal. 3. 5 bu. 2 pk. 3 qt. 7. 3 bu. 6 qt. 4. % bu. 8. .675 pk. 284. Reduce to higher denominations : 1. 1235 qt. (liquid measure). 4. 2457 qt. (dry measure). 2. 1323 qt. (dry measure). 6. 501 pt. (dry measure). 3. 321 pk. 6. 1620 pt. (dry measure). 7. What part of a gallon is f of a pint ? 8. Reduce 2 qt. 1 pt. to the fraction of a gallon. 9. What decimal part of a gallon is 1 pt. ? 10. Mrs. E bought 2^ gal. of milk at 4 cents a pint. How much did she pay for it ? 11. What part of a bushel is a pint and a half ? 12. Hiram feeds his horse 12 qt. of oats in a day. How long will 5 bu. last ? 13. If a bushel of salt is worth 1.40, how much must be paid for 4 qt. ? 14. When apples are worth $1.60 a bushel, how many pecks can be purchased for $2.40 ? 196 SCHOOL ARITHMETIC. MEASURES OF WEIGHT. 285. Weight is the measure of quantity estimated by the scale or balance with reference to some fixed unit. 286. Avoirdupois Weigiit is used in weighing all coarse and heavy articles, such as cattle, horses, coal, grain, grocer- ies, and all metals except gold and silver. It is the common comynercial weight. Table. 16 ounces (oz.) =: 1 pound (lb.). 100 pounds = 1 hundred- weight (cwt.). 20 hundred- weight = 1 ton (T.). T. cwt. lb. oz. 1 = 20 r= 2000 = 32000. 1. The avoirdupois pound contains 7000 grains. 2. The long ton of 2240 lb. has almost gone out of use in the United States. 3. A barrel of flour weighs 196 lb. LEGAL WEIGHT OF A BUSHEL t 5 p :d c l-H 03 o X ^ 6 1 "S § 6 "6 1-5 12; O 48 50 56 32 60 56 60 45 60 O 46 42 56 36 60 56 60 60 c 48 48 32 m 56 62 «3 > 48 52 56 32 60 56 60 45 60 i ^ Barley 50 32 54 60 48 48 5l 32 60 56 60 48 ^ 32 60 56 60 45 60 48 50 56 32 60 56 60 45 m 48 52 56 32 60 56 60 45 60 56 56 32 60 56 60 45 60 48 48 56 32 60 45 60 48 48 56 32 60 56 45 60 48 48 56 32 60 56 60 45 60 48 52 56 32 60 56 60 45 60 26 56 48 50 56 60 64 45 60 48 48 58 32 56 60 44 60 '18 Buckwheat 5>, Corn 56 Oats 3'>, Potatoes 60 Rye 56 Clover-seed 60 Timothy-seed 45 Wheat 60 COMPOUND NUMBERS. 197 ORAL EXERCISES. 287. 1. In 4000 lb. how many tons ? In 7500 lb.? In 35 cwt. ? 2. Find the cost of 2 T. 15 cwt. of hay at 116 a ton. 3. What will 2-^ lb. of cinnamon cost at 2 cents an ounce ? 4. At $.32 a pound, what will 7.5 oz. of butter cost ? 6. What part of a pound is ^ oz. ? 6. What decimal part of a pound is 12 ounces ? WRITTEN EXERCISES. 288. Reduce : 1. 3 T. 4 cwt. to oz. 4. 5| T. to lb. 2. 5 cwt. 45 lb. to oz. 5. 7f cwt. to lb. 3. 6 cwt. 90 lb. to lb. 6. 625 lb. to oz. Reduce to higher denominations : 7. 475 oz. 9. 2075 lb. 11. 150 cwt. 8. 1220 oz. 10. 1000 oz. 12. 75386 oz. 13. Reduce f of an ounce to the fraction of a pound. 14. Reduce 12 cwt. to tlie decimal part of a ton. 15. How many grains in 3 lb. ? In 5 1b.? 4oz. ? 16. How^ many barrels of flour in 686 lb.? 17. When hay is worth $15 a ton, how much must be paid for 250 lb. ? 18. How much will 3 lb. 6 oz. of butter cost, at $.32 a pound ? 19. If cloves are worth 40 cents a pound, how much must be paid for 40 ounces ? 20. A man bought 1500 lb. of wheat, at $.80 a bushel. What did it cost him ? 21. Mr. B bought a bag of oats weighing 112 lb. If oats were worth 40 ct. a bushel, how much did he. pay for them ? 289. Troy Weig:lit is used in weighing gold, silver, and jewels, and in laboratory tests. 198 SCHOOL ARITHMETIC. Table. 24 grains (gr.) = 1 pennyweight (pwt.). 20 pennyweights — 1 on nee (oz.). 12 ounces — 1 pound (lb.). lb. oz. pwt. gr. 1 = 12 = 240 = 5760. 1. The carat is a weight used in weighing diamonds and other jewels. It is equal to 3.168 troy grains. 2. The term carat is also used to indicate the fineness of gold, and means 2-4 part of the mass. Thus, gold that is 16 carats fine is ^^ gold, and 2% alloy. WRITTEN EXERCISES. 290. 1. Reduce .625 of an ounce to the fraction of a pound. 2. Reduce f gr. to the fraction of an ounce. 3. What decimal part of a pound is 5 oz, 4 pwt. ? 4. How many grains in 3 lb. troy ? In 3 lb. avoirdupois ? 5. How many rings can be made from 6 oz. of gold, if each ring contains 80 gr. ? 6. When silver is worth 1.60 an ounce, how much must be paid for 1000 gr. ? 7. When a ring is marked " 18 carats/^ what part of it is pure gold ? 8. How many knives, each weighing 3 oz., can be made from 5 lb. 9 oz. of silver ? 9. At $.80 a pwt., how much must be paid for a chain weighing 1 oz. 8 pwt. 12 gr. ? 10. What is the present market value of an ounce of gold ? What is the market value of an ounce of silver ? What is the ratio of the market value of silver to that of gold ? 291. Apothecaries' Weig'lit is used, in mixing medicines and in selling them at retail. Drugs in bulk are bought and sold by avoirdupois weight. COMPOUND NUMBERS. 199 Table. 20 grains (gr.) = 1 scruple {^). 3 scruples — 1 dram () 3 . 8 drams = 1 ounce ( 3 ). 12 ounces = 1 pound (lb). ft). 3 3 3 gi-. 1 = 12 = 96 = 288 = 5760. COMPARISON OF WEIGHTS. Avoirdupois. Tboy. Apotukcaries'. 1 lb. = 7000 gr. 1 lb. = 5760 gr. 1 lb. = 5760 gr. 1 oz. = 437i gr. 1 oz. = 480 gr. 1 oz. = 480 gr. It will be observed that the troy pound and ounce are identical with the apothecaries' pound and ounce. It will also be observed that, while the avoirdupois pound is heavier than the troy pound, the troy ounce is heavier than the avoirdupois ounce. Queries.— 1. What is the ratio of the troy pound to the avoirdupois pound ? 2. What is the ratio of the troy ounce to the avoirdupois ounce ? Note. — In compounding liquid medicines, druggists make use of the following : GO minims (tti) = 1 dram, 8 drams = 1 ounce, 16 ounces = 1 pint. MEASURES OF EXTENSION. 292. Extension has three dimensions, called length, breadth, and thickness, or height. A line has but one dimension — length. A surface has two dimensions — length and breadth. A solid has three dimensions — length, breadth, and thick- ness. 293. Measures of Extension are used in measuring lengths, surfaces, and solids. 294. Linear Measure is used in measuring length. The principal unit is the yard. 200 SCHOOL ARITHMETIC. Table. 12 inches (in.) = 1 foot (ft.). 3 feet = lyard (yd.). 5^ yards ) or V =1 rod (rd.). le^feet ) 320 rods = 1 mile (mi.), mi. rd. yd. ft. in. 1 = 320 = 1760 = 5280 = 63360. Supplementary Table. 3 feet 4 inches 18 inches 21.888 inches 6 feet 3 sizes 8 furlongs 9 inches 1 cable length 1.15 statute miles = 1 pace. = 1 hand (used to measure height of horses) = a cubit. = 1 sacred cubit. = 1 fathom. = 1 inch (used by shoemakers). = 1 mile. = 1 span. = 120 fathoms. = 1 nautical, or geographic, mile. 1 nautical mile = 1 knot = 6086 feet. 3 nautical miles = 1 league. WRITTEN EXERCISES. 295. Reduce the following : 3 yd. 2 ft. to in. 1 mi. 40 rd. to rd. ^ mi. to feet. 10 yd. 1 ft. 9 in. to in. ^ rd. to inches. 6. 2 rd. 4 yd. 6 in. to in. 7. 3520 yd. to mi. 8. 7920 ft. to mi. 9. 3000 yd. to mi., etc. 10. 720 rd. to miles. Reduce to lower denominations : 11. -|rd. 13. .8 yd. 15. 3i yd. 12. I mi. 14. .375 mi. 16. 7 K). 2 COMPOUND NUMBERS. 201 17. Reduce f of a yard to the fraction of a rod. 18. What part of a rod is 9 inches ? 19. What part of a yard is 2 ft. 8 in. ? 20. At $.55 a rod, how much must be paid for digging a ditch 10 rd. 7 ft. 6 in. long ? 21. When cloth is worth 90 cents a yard, how much must be paid for a piece 16 inches long ? 22. How many pounds (troy) in 11520 grains ? 23. How many powders of gr. V each can be made from 5 6 of calomel ? 24. A ring which weighs 120 grains is 16 carats fine. What is the value of the gold in it at $20 an ounce ? 296. Boundaries of solids are called Surfaces. A surface has length and breadth, but no thickness. Note. — We usually think of a solid as a material body — e.g., a block of wood — but the portion of space occupied by the block of wood is regarded as the solid. 297. Two lines proceeding from the same point form an Angle. The size of the angle depends upon the degree of opening between the lines. ^^^^ anglk. 298. A flat surface bounded by straight lines or by a curved line is called a Plane Figure. The distance around a plane figure is called the perimeter. 299. A plane figure that has four straight sides is called a Quadrilateral. 300. When the line CD meets AB, as in the figure, making the angles ACD and DCB equal, each of these angles is called a Right . /- Angle. ^ / D :^) B 202 SCHOOL ARITHMETIC. A RECTANGLE. 301. A quadrilateral all of whose angles are right angles is called a Rectangle. 302. An equilateral (equal-sided) rectangle is called a Square. 1. Each angle is a right angle. 2. How does the length of a square compare with its breadth ? A SQUARE. SQUARE INCH. 303. When each side of a square is one inoh long, the figure is called a square inch. 1. What is Hi square foot? K square yard ? 2. W\\2it \^ 2i square tinit 9 3. Is a square also a rectangle ? 304. The Area of a figure or surface is the number of square units of measure it contains. The area is often called the superficial contents, A rectangle 4 feet long and 3 feet wide may be divided into 3 strips, each 1 foot wide, and each strip into 4 equal parts, each part being 1 square foot (square unit of measure). Then the area of the strip AB is 4x1 sq. ft., and the total area is 3 X 4 X 1 sq. ft., or 12 sq. ft. 305. Prixciple. — The area of a rectangle is expressed hy the product of the numbers that represent its length and Ireadth. Both dimensions must be expressed in like units. 1. A board is 8 feet long and 1 foot wide. How many- square feet in its surface ? How many would there be if it were 2 feet wide ? B COMPOUND NUMBERS. 203 2. A box lid is 6 inches long and 4 inches wide. How many square inches in its surface ? What unit of measure is used here ? 3. How many square feet in the surface of a square table whose sides are 2 feet ? 3 feet ? 4. How many square inches in a square whose sides are 13 inches ? What may this square be called ? Why ? 5. How many square feet in a square whose sides are 3 feet ? What may this square be called ? Why ? 6. The sides of a square are 16^ feet. How many square feet in its surface ? What may this square be called ? Why ? SQUARE MEASURE. 306. Square Measure is used in measuring surfaces. Table. 144 square inches (sq. in.) = 1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). 30|^ square yards = 1 square rod (sq. rd.). 160 square rods = 1 acre (A.). 640 acres = 1 square mile (sq. mi.). sq. mi. A. sq. rd. sq. yd. sq. ft. sq. in. 1 = 640 = 102400 = 3097600 = 278:8400 = 4014489600. 1. In square measure the term perch is sometimes used instead of square rod. 2. In roofing, slating, etc., 100 sq. ft. is called a square. 3. When we say a surface is a " foot square." we mean that each dimension is a foot. Hence the terms "foot square," "rod square," etc., mean dimensions, while "square foot," " square rod," etc., mean area. ORAL EXERCISES. 307. 1. A board is 2 feet long and J ft. wide. What is its area ? Is it square ? 204 SCHOOL ARITHMETIC. 2. What is the area of a board 4 yards long and ^ yd. wide ? 3. A lot is 20 rods long and 8 rods wide. How many sq. rd. in the lot ? What may this rectangle be called ? What other dimensions might it have, and yet have the same area? 4. What is a square mile ? How many rods long is one side of it ? Then how can we find how many sq. rd. in a square mile ? 5. How can we find the number of acres in a square mile ? Name the unit of measure. 6. How do you know that there are 9 sq. ft. in a square yard ? Explain by a diagram. WRITTEN EXERCISES. 308. Reduce to lower denominations : 1. 2 sq. ft. 12 sq. in. 5. 2 A. 80 sq. rd. 2. 3 sq. yd. 8 sq. ft. 6. 3 yd. 2 ft. 8 in. 3. 5 sq. yd. 7 sq. in. 7. 1 A. 25 sq. yd. 4. 10 sq. rd. 5 sq. yd. 8. 1 sq. mi. 280 A. Reduce to higher denominations : 9. 1000 sq. ft. 11. 3000 sq. yd. 13. 10000 sq. rd. 10. 2000 sq. in. 12. 4000 sq. rd. 14. 33333 sq. in. 15. Reduce | sq. rd. to square yards, etc. 16. What part of an acre is | of a square rod ? 17. One square yard is what part of a square rod ? 18. AVhat part of a square rod is 5 sq. yd. 8 sq. ft. ? 19. A floor is 16 ft. long and 12 ft. wide. How many square feet does it contain ? 20. A town lot is 40 ft. wide and 120 ft. long. What is its area in square yards ? 21. How many sq. yd. in a piece of carpet 18 ft. long and a yard wide ? 22. A field 40 rods long and 25 rods wide contains how many acres ? COMPOUND NUMBERS. 205 23. At $50 an acre, liow much must be paid for'12 A. 32 sq. rd. of land ? 24. When a product and one factor are given, how may the otlier factor be found ? 25. A two-acre lot is 20 rods long. How wide is it ? 26. A field containing 10 acres is 32 rods wide. What is its length ? 27. AVhat will it cost to paint a ceiling 16 ft. by 12 ft., at $.25 a square yard ? 28. A gable barn-roof, 65 ft. long and 30 ft. wide, was covered with sheet-iron. How many sq. yd. were required ? 29. The area of a rectangular glass is 30 sq. ft., and its length is 144 inches. What is its width ? 30. A tract of prairie land is 6 miles long and 4^ miles wide. How many farms of 160 acres each could be sold from it ? 31. A lot 60 feet square has in its central part a reservoir 18 feet square. What is the distanice from the reservoir to the fence surrounding the lot ? 32. If steel rails weigh 24 lb. a foot, and can be bought for 130 a ton, what will be the cost of the rails for a mile of single-track railway ? 33. A man who owns a farm 120 rods square put 10 acres in corn, 15 in rye, 20 in oats, 15 in barley, and the re- mainder in wheat. What part of the whole farm did he put in wheat ? PARALLELOGRAMS. How does the figure ABCD differ from a A. . ^B rectangle ? Are its sides equal ? Are its opposite sides equal ? Are they parallel ? Are its angles D equal ? 206 SCHOOL ARITHMETIC. 309. A quadrilateral whose opposite sides are parallel is called a Parallelogram. Query.— Is a rectangle a parallelogram ? Why ? 310. The side of a figure on which it is supposed to stand is called the Base ; as CD. The Altitude is the perpen- dicular distance between the base (or the base extended) and the side or angle opposite. 1. Any side may be re- garded as the base. 2. What is the diago- nal of a parallelogram ? («). If we cut off the ^ ^ D end CAE, and place it at the other end, on DBF, will the size of the figure be changed ? (h). Then is the area A B of ABFE equal to that of ABDC ? Is EF equal to CD ? (c). How is the area c of the rectangle ABFE found ? Then how may the area of the parallelogram ABDC be found ? Since a parallelogram is equivalent to a rectangle having the same base and altitude, it follows that 311. Pri NCiPLE. — The area of a parallelogram is expressed hy the product of the numbers that represent its base and altitude. The base and altitude of all figures must be expressed in like units. Find the area of the following parallelograms : 1. Base 24 ft., alt. 8 ft. 4. Base 16 ft., alt. 9 in. 2. Base 35 ft., alt. 15 ft. 5. Base 42 rd., alt. 5^yd. 3. Base 72 ft., alt. 39 ft. 6. Base 132 in., alt. 30 ft. 7. A board in the form of a parallelogram is 6 ft. 6 in. E COMPOUND NUMBERS. 207 long on each side, and 1 ft. 4 in. wide. Draw the figure, and find .the area. 8. The area of a parallelogram is 518 sq. in. If its length is 37 inches, what is its altitude, or width ? 9. A ten-acre field is in tlie form of a parallelogram. Tlie shortest distance from one side to the opposite side is 25 rods. What is the length of the field ? 10. Mr. A has a rectangular field 40 rd. long and 20 rd. wide. Mr. B has a field of equal size in the form of a paral- lelogram. If its length is 32 rd., what is the shortest dis- tance across the field ? TRIANGLES. How many sides has the figure ABC ? Are they all straight ? 312. A plane figure that has three straight sides is called a Triangle. The point where two sides meet is called a Vertex ; as A. (a). Into how many equal triangles does the diagonal AB divide the parallelogram BCAD ? (b). Then the area of each triangle is what part of the area of the parallelogram ? (c). Since the area of the parallelogram is expressed by AD X BE, how is one half its area, or the area of the triangle ABD, expressed ? 313. Principle. — The area of a triangle is expressed by one half the product of the numbers that represent its base and altitude. Note. — When one angle of a triangle is a right angle, the triangle is said to be right-angled. In such a triangle one side is the altitude, or height. 208 SCHOOL ARITHMETIC. Find the area of the following triangles : 1. Base 12 ft., alt. 7J ft. 3. Base 47 rd., alt. 165 ft. 2. Base 29 ft., alt. 16 ft. 4. Base 18 in., alt. 12 ft. 6 in. 6. The base of a triangular piece of slate is 27 inches, and the altitude 33 inches. Draw the figure, and find the area in square feet. 6. How many triangles, base 5 ft. and alt. 4 ft,, are equal to a parallelogram whose base is 50 ft., and whose altitude is 40 ft. ? 7. A triangle whose base is 7 yards has an area of 819 sq. ft. What is its altitude ? 8. A house is 24 ft. wide, and the ridge of the roof is 12 ft. above the upper floor. Find the cost of painting the gables, at 1.40 a square yard. 9. What is the area of a table a feet square ? 10. A rectangle is a ft. long and J) ft. wide. What is its area ? 11. The altitude of a parallelogram is a ft., and the base is I ft. What is the area ? 12. The base of a triangle is 2^ inches, and the altitude is a inches. Find its area. 13. What is tlie area of a square whose perimeter is 4^ inches ? 14. One side of a rectangle is %b ft., and the perimeter is 6J ft. What is the area ? TRAPEZOIDS. In the figure ABCD which sides ^ ^ •• ■ — f are parallel ? How many sides / ! y / has the figure ? / i \ / 314. A quadrilateral having two sides parallel is called a Trapezoid; as ABCD. («). Cut two equal trapezoids out of paper, and place them end to end so as to form a parallelogram. COMPOUND NUMBERS. 209 (b). How does the area of one trapezoid compare with that of the parallelogram ? 315. If the trapezoid ABCD in the figure swings about the point P, to the position ECBF, then the whole figure AFED is a parallelogram. Now the area of ABCD is one- half of that of AFED ; but the area of AFED is equal to DE X BH. But DE = DC + CE = DC + AB. . •. the area of ABCD = J of (DE X BlI) = i of (AB + CD) x BH. Hence the area of a trapezoid equals half the area of a parallelogram having the same altitude and a base equal to the sum of the two parallel sides. 316. Principle. — The area of a trapezoid is expressed by one half the product of the numbers that represent its altitude and the sum of its parallel sides. Find the area of the following trapezoids : 1. Altitude 8 in., parallel sides 12 in. and 10 in. 2. Altitude 13 ft., parallel sides 19 ft. and 11 ft. 3. Altitude 25 rd., parallel sides 45 rd. and 35 rd. 4. Altitude 76 ft., parallel sides 6 rd. and 23 yd. 6. How many sq. ft. in a board 12 ft. long, 15 in. wide at one end and 9 in. at the other ? 6. The parallel sides of a trapezoidal lot are 75 ft. and 55 ft. respectively, and the shortest distance between them is 40 ft. Draw the figure and find the area of the lot. 7. One side of a farm in the form of a trapezoid is 135 rd. long, the side parallel to it is 121 rd. long, and the per- pendicular distance between them is 100 rd. What is the value of the farm at $75 an acre ? 8. A twenty-acre field is in the form of a trapezoid whose parallel sides are 72 rd. and 88 rd. respectively. What is the altitude ? 9. The distance around a triangular farm whose sides are equal is 480 rd., and the altitude is 80 rd. How many acres in the farm ? 14 210. SCHOOL ARITHMETIC. E ^c^riSES^^ CIRCLES. Is the portion of the page en- closed by ADBE a plane figure ? What kind of line bounds the figure ? Is any part of the line nearer the center than another part ? 317. A plane figure whose bound- ing line is everywhere equally dis- tant from a point within, called the center, is a Circle. 1. The curved line that bounds a circle is the Circumfer- ence ^ 2. A straight line passing from one side of a circle to the other, through the center, is a Diameter ; as AB. 3. A line from the center of a circle to the circumference is called a Radius. It is lialf a diameter ; as CD or CA. Queries. — 1. Can there be a center with- out a circumference? 2. What is the surface between the center and circumference called? 1^° Divide the diameter of a circle into 10 equal parts, and step dividers or compasses around the circumference, making each step equal to one of the divisions of the diameter. There will be a little more than 31 steps. Hence the circumference is a little more than 3.1 times the diameter. In geometry it is shown to be 3.1416 times the diameter. 318. Peinciple. — The circumfereyice of any circle is 3,14.16 times its diameter. What is the circumference of 1. A circle whose diameter is 1 ft. ? 60 ft. ? 27 in. ? 2. A circle whose radius is 6 ft. ? 12^ ft. ? 2 ft. 6 in. ? What is the diameter of a circle whose 3. Radius is 7i in.? 13fin.? 19.08ft.? COMPOUND NUMBERS. 211 4. Circumference is 6.2832 ft. ? 12.5664 yd. ? 1 ft. ? 5. What is the radius of a circular field whose circum- ference is 320 rods ? 6. What is the circumference of a 6-inch stove-pipe ? 319. To ti\ul the area of a circle. (a). May the circle AB be regarded as made up of a vast number of very small triangles ? To what is the sum of their bases equal ? Is the altitnde of each triangle equal to the radius of the circle ? (5). Is the area of all the triangles equal to the area of the circle ? (c). Since the area of a triangle is found by multiplying its base by half its altitude, how may the area of a circle be found ? 320. Principle. — TJie area of a circle is expressed ly the product of the numbers that represent its 'circumference and half its radius. This principle may be illustrated as follows in (a) and {h) : (a). In the figure ahc, the base ab is a curved line — an arc of a circle whose center is c. If this arc be pressed up or stretched until it becomes a straight line, the sides ac and be will be forced farther apart, and the figure will appear SiS, dec, the line de being equal to the arc ab, and the other sides remaining unchanged.. But co, the radius, is a little longer than the altitude ck. However, by increas- ing the number of radii the arc may be made as small as we please; and, as the arc is thus made smaller, the difference between the radius and the altitude becomes continually less. When the arc is extremely small, the radius and the altitude are regarded as equal. 212 SCHOOL ARITHMETIC. (b). Take a rubber-tired wheel with spokes, as WL. Cut it through and straighten out as in the figure. The result- ing figures 1, 2, 3, etc., are nearly triangles. The sura of all the bases is easily se^n to be the circumference of the wheel, or circle. The area of all the triangles is equal to that of the circle, and their common altitude is the radius. (See Art. 319.) (c). Regarding the circle, then, as made up of many triangles, we find its area by multiplying the su7n of all the bases (which is the circumference) by one-half the common altitude (i. e., i the radius). It is proved in geometry that this is the exact area. Find the area of the following /»\/2\/3\/4\/5\/6\/7\/8\ circles : 1. Radius 10, circumference 63.833. 62.832 X 5 = 314.16. Or, iof 62.832 X 10 = 314.16. 2. Diameter 10 in., circumference 31.416 in. 3. Diameter 15 ft., circumference 47.124 ft. 4. Diameter 7.9578 ft., circumference 25 ft. 5. Diameter 15 yd. 7. Circumference 100 ft. 6. Radius 13.5 rd. 8. Circumference 1.5708 ft. 9. At $50 an acre, what is the value of a circular field whose diameter is 100 rods ? 10. A horse is tied to an iron weight by a rope 20 feet long. Upon how many sq. ft. can he graze ? Draw the figure. 11. The fence around a circular field is 500 rods long. How many acres in the field ? For convenience, 3.1416 is usually represented by the Greek letter 7t {pi), the diameter by d, and the radius by r. Then the circumference is expressed by nd. But since d = 2r, the circumference equals 27Cr ; Compound numbers. 213 and the area equals 2'Kr x \r,or it x r x r, or nr^. Thus, if the radius of a circle is 3 ft., the area is 3.141G x 3^ x 1 sq. ft. = 28.2744 sq. ft. 12. Solve examples 2, 5, and 6 by this formula, and com- pare results with those already obtained. SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 321. 1. The perimeter of a rectangular field is 200 rods, and three times the width is twice the length. Find the area in acres. 2. Each of the three sides of a field is 255J feet long. The field is surrounded by a fence 5 boards high, the posts being 8 feet apart. If the posts cost 35 cents each, and the boards 2 cents a linear foot, what did the posts and boards cost ? 3. The distance around a circular lot is 1000 feet. If I plow around the lot until | of it is plowed, how many square feet will remain to be plowed ? 4. Find the area of a circular ring 4 feet wide, the radius of the outer circle being 32 feet. 5. Find the length of the minute-hand of a clock whose point moves 5 inches in 15 minutes. 6. A takes 924 steps in walking around a field 40 rods long. If there are 5 feet in two of his steps, how many acres in the field ? 7. A carpenter had a plank 20 in. wide, from which he wished to saw off 10 sq. ft. What will be the length of the piece sawed off ? 8. I have a rectangular farm whose perimeter is 240 rods. It is twice as long as it is wide. How many acres does it contain ? 9. A map is 4 sq. ft. 4 sq. in. in area, and is drawn on a scale of 1 inch to a mile. How many acres are represented on the map ? 10. A square lawn is bordered by a gravel drive 10 yards wide. The drive covers 4000 sq. yd. How many sq. yd. in the enclosed lawn ? 214 SCHOOL ARITHMETIC. 11. If ^ feet of fence cost $24.50, what will be the cost of fencing a square field, one side of which is 3b feet long ? 12. What is the area of a'cirfcle whose radius is /* feet ? 13. How many sq. ft. in a table which is a inches long and b inches wide ? 14. The distance around a rectangular field is p rods. If the field is q rods long, how wide is it ? 15. My storeroom is 75 ft. long, 35 ft. wide, and 20 ft. higli, the vertex of the roof being 15 ft. above the upper floor. Making no allowance for doors and windows, what will be the cost of painting at $.18 a sq. yd.? MEASURES OF VOLUME. 322. Any limited portion of space is called a Solid. A solid has three dimensions^ — length, breadth, and thickness. 323. A solid that has six rectangular sides, or faces, is called a Rectangular Solid. .1. A rectangular side or surface is one that has the form of a rectangle. 2. Is a brick a rectangular solid ? Wliy ? Name three bodies that are rectangular solids. 324. A regular solid with six square faces is called a Cube. 1. A cube has twelve edges. Where are they ? 325. A Cubic Inch is a cube whose faces are each an inch long and an incli wide. 1. What is a cubic foot? A cubic yard? A cubic unit ? A solid unit ? 326. The Volume of any solid or body is the number of solid or cubic units it contains. The term solid contents is often used instead of volume. COMPOUND NUMBERS. 215 A solid 4 ft. long, 3 ft. wide, jind 5 ft. high (or thick), may be divided into 5 slabs, each containing 3 rows of blocks, as ABCD. Each row contains 4 blocks, OY cubic feet, hence 3 rows con- tain 12 cubic feet. Since there are 12 cubic feet in 1 slab, in 5 slabs there are 5 times 12 cubic feet, or 60 cubic feet, which is the vol- ume. (4 X 3 X 5) X 1 cu. ft. = 60 cu. ft. in the solid. What is the unit of measure here ? 327. Principle.— ^Ae vol- ume of a rectangular solid is expressed hj the product of the 7iumhers that represent its length, breadth, and thickness. Since the area of one face or side of a rectangular solid 4 ft. long and 3 ft. wide is 12 sq. ft., it will be observed that the number of cubic feet in 1 foot of thickness is the same as the area of one face.* Hence, to find the volume of a rectangular solid. Rule. — Multiply the mimber of cubic units in one tmit of thickness or height by the number that represents the height. All dimensions must be expressed in like units. 1. An iron bar is 12 inches long, 1 inch wide, and 1 inch thick. How: many cubic inches does it contain ? 2. A marble slab is 12 inches square and 1 inch thick. How many cubic inches does it contain ? 3. A block of coal is 12 inches long, 12 inches wide, and 12 inches thick. How many cubic inches in it ? What may this solid be called ? Why ? 216 SCHOOL ARITHMETIC. 4. A cube of stone measures 3 feet on each side. How many cubic feet does it contain ? What may this cube be called ? Why ? CUBIC MEASURE. 328. Cubic Measure is used in measuring solids. Table. 1728 cubic inches (cu. in.) =1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.), cu. yd. cu. ft. cu. in. 1 1^ 27 = 46656 1. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high is called a cord. It contains 128 cu. ft. 2. In measuring masonry or stone, 24£ cu. ft. = 1 perch. 3. In finding the length of walls, masons measure the outside. The corners are thus counted twice. 4. A cubic foot of distilled water at its greatest density weighs 1000 oz. avoirdupois. WRITTEN EXERCISES. 329. Reduce : 1. 3 cu. ft. to cu. in. 4. 2 A. to sq. yd. 2. 5 cu. yd. to cu. in. 5. 1 cu. yd. 1 cu. ft. to cu. in. 3. 6 cu. yd. to cu. ft. 6. 108 cu*. ft. to cu. yd. 7. Reduce 594 cu. ft. to cubic yards. 8. Reduce 50000 cu. in. to higher denominations. 9. What part of a cubic yard is -^-^ of a cubic foot ? 10. Reduce ^ cu. yd. to lower denominations. 11. How many cu. in. in .75 of a cubic foot ? 12. A log 2 feet square is 12 ft. 5 in. long. How many cubic inches does it contain ? 13. A cellar is 30 ft. long, 24 ft. wide, and 8 ft. deep. How many cu. yd. of earth were removed ? 14. How many cu. ft. of air in a room 16 ft. square and 12 ft. high ? COxMPOUND NUMBERS. 217 15. My granary is 8 ft. long, 4 ft. 3 in. wide, and 5 ft. deep. How many bushels of wheat will it hold ? 16. A cubical cistern 8 ft. deep contains 2 ft. of water. How many gallons in it ? 17. Some gold miners sunk a shaft 390 feet. If it was 4 ft. by 8 ft., how many cu. yd. of material were removed ? 18. A trough 3 ft. wide and 1 ft. 3 in. deep is 8 ft. long. If a cu. ft. of water weighs 1000 oz., how many pounds of water will fill the trough ? 19. If a box car is 40 ft. long, 8 ft. wide, and 9 ft. high, how many blocks of marble, each 4 ft. by 2 ft. by 1 ft. G in., will it hold ? 20. A two-quart pail was half full of water. Several frogs jumped into the pail, and then it was full. What was the volume of the frogs ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 330. 1. At 30 cents a bushel, find the cost of a box of oats, the box being 8 feet long, 4 feet wide, and 4 feet deep. 2. A rectangular trough is f full of water. After 35 gallons are taken out, it is f full. What is the depth of the trough, the length being 10 feet and the width 2J ft. ? 3. A two-acre pond is covered with ice 6 in. thick. If a cu. ft. of ice weighs 896 ounces, how many tons of ice are on the pond ? 4. A tank 8 ft. long, IJ ft. wide, and 8 in. deep will con- tain how many pounds of water, a cu. ft. of water weighing 62^ lb. ? 6. What is the difference between a 3-foot cube and 3 cubic feet ? 6. A cistern 6 ft. long and 4^ ft. wide holds 108 cu. ft. of water. How many cu. in. of zinc will be required to line the sides and bottom, the zinc being ^ in. thick ? 7. What are the dimensions of a rectangular box whose 218 SCHOOL ARITHMETIC. capacity is 50274 cu. ft., the length, breadth, and depth being to each other as 3, 2, and 1 ? 8. A contractor received S45.50 for digging a cellar 18 ft. long and 15 ft. wide, at $.65 a cu. yd. To what depth did he dig it ? 9. A man wishes to make a bin to contain 725 bushels, the width and depth to be equal, and the length to be double the widtli. What must be its dimensions ? 10. How thin is a cu. in. of gold beaten so as to cover a space 46 ft. 10 in. by 41 ft. 8 in.? 11. How many gallons of water can be poured into a bushel measure ? 12. A vessel 3 inches square contains some water. A gold chain dropped into the water raises the fluid ^ in. What is the volume of the chain ? What is its weight if the gold weighs 19.2 times as much as water and a cu. ft. of the water weighs 1000 oz. ? SURVEYORS' MEASURE. 331. In measuring boundaries of land, in locating rail- roads, etc., and in computing the area of land, surveyors and engineers make use of measures not given in the ordinary tables under '^Measures of Extension." They are as fol- lows : • SURVEYORS' LINEAR MEASURE. Table. 7.92 inches = 1 link (1.). 25 links = 1 rod (rd.). .4 rods = 1 chain (ch.). 80 chains = 1 mile (mi.). The unit of surveyors' measure is Gu7iter's Chain, which is 100 links long. COMPOUND NUMBERS. 219 SURVEYORS" SQUARE MEASURE. Table. 16 square rods = 1 square chain (sq. ch.). 10 square chains = 1 acre (A.). 640 acres = 1 square mile (sq. mi.). 1 mile square = 1 section. 36 sections = 1 township. MEASURES OF TIME. 332. The time required for the earth to rotate once on its axis is called a day, which is the unit of time. Other divisions of time are shown in the following Table. 60 seconds (sec.) = 1 minute (min.). 60 minutes = 1 hour (hr.). 24 hours = 1 day (da.). 7 days = 1 week (wk). 365 days = 1 year (yr.). 366 days = 1 leap year (yr.). da. hr. min. sec. 1 = 24 = 1440 = 86400. 1. A period of 100 years is called a century, 10 years a decade. 2. In business transactions 30 days are usually considered a month. 3. The time from midnight to noon is caW^di forenoon — A.M. ; that from noon to midnight is called afternoon — P.M. THE CALENDAR. 333. The system of reckoning time by years and months, as found in our almanacs, is called a Calendar. 2^0 SCHOOL ARITHMETIC. 1. The time of one fevbhition of the earth around the sun^— the solar year^-^is 365 da. 5 hr. 48 rain. 46 sec, nearly. For ordinary purposes it is impol^tatit that the year contain ah exact number of days. The pres- ent calendar year secures this result. Its length is sometimes 365 and Sometimes 866 days, but its average length is almost exactly equal to that of the solar year. The year of 366 days is called a leap-year, and those years are leap-years whose date numbers are exactly divisible hy 4, except centen7iial years, whose dates must he divisible by 4OO. 2. On the supposition that the length of the year was 365^ days, Julius Caesar introduced a calendar in which every year whose date number is exactly divisible by 4 was to consist of 366 days, and all other years of 365. But as the year does not contain exactly 365^ days, Caesar's year was thus 11 minutes and about 14 seconds too long, and this would ac- cumulate in 400 years to a little over 3 days. In 1582, Pope Gregory XIII. , aiming to correct this error, arranged that only each fourth cen- tennial year {those exactly divisible by 40O) should be a leap-year. The Gregorian Calendar still leaves a very slight error, which will not amount to a day until about a.d. 5200. 3. The calendar of Caesar is known as the Julian Caleyidar, or Old Style (0. S.) ; that which Gregory substituted is called the Gregorian Calendar, or New Style (N. S.). In changing from one calendar to the other, Gregory dropped out 10 days, so that the day after Oct. 4, 1582, was called Oct. 15. England adopted the New Style in 1750, by which time it had become necessary to drop out 11 days. The difference in 1899 was 12 days. From 1900 to 2100 it will be 13 days. Example. — Columbus discovered America, October 12, 1492, 0. S. What is the date N. S. ? WRITTEN EXERCISES. 334. Reduce to lower denominations : 1. 3 hr. 10 min. 15 sec. 4. 1 wk. 1 da. 1 hr. 2. 2 da. 2 hr. 30 min. 5. 3 sq. yd. 5 sq. ft. 3. 1 wk. 7 hr. 25 min. 6. 5 cwt. 87 lb. Reduce to higher denominations : 7. 1000 sec. 9. 5050 hr. 11. 3675 da. 8. 2345 min. 10. 13258 wk. 12. 3636 cu. in. 13. What part of a day is | of an hour ? COMPOUND NUMBERS. 221 14. Reduce f of a day to hours and minutes. 15. AVhat decimal part of a day is 8 lir. 40 min. ? 16. In 1894 how many days from January 1st to March 15 ? In 1896 ? In 1900 ? 17. How many decades in one century and 15 years ? 18. John was paid $2.10 for working from 7 a.m. till 15 minutes past 12 m. How much was that an hour ? 19. George worked 5 da. 6 hr. 30 min. at the rate of $2 a day. How much did he earn ? CIRCULAR OR ANGULAR MEASURE. 335. An Arc of a circle is any part of the circumference ; as EB or BD. 336. When two lines are drawn from the circumference to the center, the arc between them is the measure of the angle formed. Thus, the arc DE is tlie measure of the angle DCE. 337. In measuring angles, the circumference is divided into 360 equal i)arts, called degrees j each degree into 60 parts, called minutes j and each minute into 60 parts, called seconds. 1. The length of a degree varies. A degree is 3^0 of a circumference, and the greater the circumference the greater the degree, or arc. 2. The angle measured by a degree does not vary. 338. Circular or Angular Measure is used in measur- ing angles, arcs of circles, in determining latitude and longitude, the location of vessels at sea, etc. Table. 60 seconds (") = 1 minute ('). 60 minutes = 1 degree (°). 360 degrees = 1 circumference (cir.). 222 SCHOOL ARITHMETIC. One fourth of a circumference, or 90°, is called a quadrant. A minute of the circumference of the earth is called a geo- graphical mile. Miscellaneous Tables. Paper. 12 things = 1 dozen. 12 dozen = 1 gross. 12 gross = I great gross. 2 things = 1 pair. 6 things = 1 set. 20 things = 1 score. 24 sheets = 1 quire. 20 quires = 1 ream. 2 reams = 1 bundle. 5 bundles — 1 bale. ORAL EXERCISES. 339. 1. Mrs. A has 30 knives. How many sets has she ? 2. Mary sold 5 doz. eggs at 2 cents apiece. How much did ghe get for them ? 3. Harry bought 36 lemons at 200 a dozen. What did they cost him ? 4. Sarah bought 5 quires of paper at 15 cents a quire, and sold it at a cent a sheet. How much did she gain ? 5. Ten years ago Mr. Smith was 3 score and 10 years of age. How old is he now ? 6. Half a dozen is what part of a gross ? 7. What is the ratio of a dozen to a score ? 8. How often is %\ contained in 1.75 ? 9. A lady sold 200 eggs at 20 cents a dozen, and took her pay in sugar at 6 cents a pound. How many pounds did she get? 10. What has the same ratio to a foot that a yard has to an inch ? ADDITION. 340. In simple numbers the scale is uniform, 10 units of any order being equal to 1 of the next higher order. In compound numbers the scale is varying. In U. S. money the scale is uniform, and it is to be observed that COMPOUND NUMBERS. 223 while $2.55 is a simple number, yet when written in the separate units $2 5(1. 5^, it may be regarded as a compound number. The addition, subtraction, multiplication, and division of compound numbers differ little in theory from the like operations in simple num- bers. Their varying scale, however, causes the numbers to be written somewhat differently, and makes a slight difference in the processes. WRITTEN EXERCISES. 1. What is the sum of $7 2d. 4^' 5m., $3 8d. 0^ 7m., and $2 7d. 5^ Im. ? M {b) $ d. m. 7 2 4 5 $7,245 3 8 6 7 3.867 2 7 5 1 2.751 13 8 6 3 1^13.863 The given numbers may be written either as in (a) or (&), since they have a decimal scale. Units of the same order should stand in the same column. The numbers in (a) are added as follows : The sum of the mills is 13, which is equal to 1 cent and 3 mills. We write the 8 in the column of mills, and add the 1 to the cents, making 160, or 1 dime and 60. Writing the 6 in the column of cents, we add the Id. to the dimes, making 18d., or $1 and 8d. We write the 8 in the column of dimes, and add the $1 to the dollars, making $13. I^^Have the pupil add the numbers in {h), and compare the process with that just explained. 2. What is the sum of 16s. 9d. 1 far., 18s. 8d. 3 far., 3 far. ? We write units of the same order in the same vertical column, regardless of the number of places they may occupy. Thus, lOd., being less than 1 shilling, must be placed in the pence column. The sum of the farthings is 7, or Id. 8 far. 2 K A '^ Writing the 8 in the column of farthings, we add the Id. to the pence, making 28d., or 2s. 4d. Writ- ing the 4 in the column of pence, we add the 2s. to the shillings, making 45s., or £2 5s., which we write in the proper columns. Query. — Why do we not *' carry " one for every tejh ? and 9s . lOd. { £ s. d. far. 16 9 1 18 8 3 9 10 3 224 SCHOOL ARITHMETIC. 3. Find the sum of 3 gal. 2 qt. 1 pt., 7 gal. 3 qt., and 8 gal. 1 qt. 1 pt. 4. Find the sum of 3 bu. 1 pk. 7 qt., 9 bu. 3 pk. 5 qt. 1 pt., 12 bu. 2 pk., and 7 bu. 1 pk. 6 qt. 5. Add 8 lb. 5 oz. 12 pwt. 16 gr., 12 lb. 9 oz. 4 pwt. 9 gr., 15 lb. 11 oz. 19 pwt. 22 gr., and 10 oz. 17 pwt. 6. What is the sum of 2 T. 15 cwt. 40 lb., 5 T. 8 cwt. 75 lb., 4 T. 9 cwt. 85 lb., and 13 T. 3 cwt. ? 7. Add 1 mi. 40 rd. 5 yd. 2 ft. 6 in., 12 mi. 185 rd. 2 yd. 1 ft. 8 in., 5 mi. 316 rd. 4 yd. 7 in., 8 mi. 1 yd. 2 ft. 3 in., and. 18 rd. 4 yd. 10 in. When a fraction occurs in the sum, it should be reduced to lower de- nominations, and added to the proper columns. 8. What is the sum of 80 sq. rd. 25 sq. yd. 5 sq. ft. 75 sq. in., 136 sq. r-d. 12 sq. yd. 8 sq. ft. 120 sq. in., 48 sq. rd. 9 sq. yd. 3 sq. ft. 56 sq. in., and 108 sq. rd. 136 sq. in. ? 9. Find the sum of 18 cu. yd. 13 cu. ft. 87 cu. in., 9 cu. yd. 21 cu. ft. 1236 cu. in., 4 cu. yd. 25 cu. ft. 1600 cu. in., and 18 cu. ft. 760 cu. in. 10. Add 12 hr. 40 min. 34 sec, 7 hr. 8 min. 50 sec, 18 hr. 25 min. 26 sec, 19 hr. 15 min. 45 sec, and 6 hr. 50 min. 12 sec 11. What is the sum of £16 12s. 7d. 3 far., £4 18s. lid. 1 far., £2 16s. 9d. 2 far., and £3 8s. 7d. 1 far. 12. Find the value of f bu. + | pk. + 5| qt. I bu. =3 pk. 4 qt. i pk. = 6 H qt. = 5 1 pt. I bu. + f pk. + 5i qt. = 3 pk. 7 qt. 1 pt. 13. What is the value of ^^ wk. + f da. + f hr. r 14. Add 3i lb. 8 oz. 13^ pwt., li lb. 3^ oz. 10 pwt. 18 gr., and 7 lb. 6 oz. 17 pwt. 15 gr. 15. Add 45 gal., 3.7 qt., and 1.5 pt. Solution same as in example 13. COMPOUND NUMBERS. 225 16. Find the sum of .25 mi., 1.15 mi., and 120 rd. 18 ft. 17. The latitude of Pittsburg is 40° 27' 36" north, and that of the Cape of Good Hope is 33° 56' 3" south. How many degrees between the two places ? Why do we add to find the diflference in the latitude of these places ? Draw figure to illustrate. 18. A farmer sold corn as follows : To A, 75 bu. 1 pk.; to B, 37 bu. 3 pk.; to 0, 110^ bu.; to D, 18 bu. 3i pk. ; to E, 42^ bu. How much did he sell to all ? SUBTRACTION. WRITTEN EXERCISES. 341. 1. Mr. A had $20 8d. 5^, and spent $12 9d. 7^. How much had he left ? (a) (b) $ d. 20 8 5 $20.85 12 9 7 12.97 7 8 8 $7.88 • The given numbers may be written either as in (a) or (b), units of the same order standing in the same column. The numbers in (a) are sub- tracted as follows : We can not take 7^ from 5^, hence we take Id. (10^) from the 8d., and add it to the 5^, making 15^. Then 7^ from 150 leaves 80, which we write in the column of cents. Having taken Id. from the 8d., only 7d. remains, from which 9d. can- not be taken. Hence we take $1 (lOd.) from the $20, and add it to the 7d., making 17d. Then 9d. from 17d. leaves 8d., which we write under dimes. Having taken $1 from the $20, only $19 remain. Then $12 from $19 leaves $7, which we write as the dollars of the remainder. m^" Have the pupil subtract the numbers in (&), and compare the process with that just explained. 2. B has £10 8s. 6d. and D has £6 5s. lOd. less than B. How much has D ? 15 226 SCHOOL ARITHMETIC. 3. From 5 bu. 1 pk. 6 qt. 1 pt., take 2 bu. 3 pk. 5 qt. 4. From 13 gal. 3 qt., take 8 gal. 2 qt. 1 pt. . 6. Take 2 T. 18 cwt. 90 lb. from 4 T. 15 cwt. 80 lb. ' 6. From 3 cu. yd. take 21 cu. ft. 1628 cu. in. 7. Mr. B had a two-acre lot from which he sold 1 A. 75 sq. rd. How much had he left ? 8. From a barrel containing 41 gal. 2 qt. 1 pt. of molasses, 10 gal. 3 qt. were sold at one time and 23 gal. 1 qt. 1 pt. at another time. How much remained in the barrel ? 9. From 10 lb. 10 oz. 10 pwt. 10 gr., take 5 lb. 11 oz. 12 pwt. 13 gr. 10. A man ate lunch at 12 o'clock 15 min. p.m., and dinner at 6 o'clock 45 min. p.m. How long was it between those meals ? 11. The latitude of Pittsburg is 40° 27' 36" north, and that of Washington is 38*=^ 53' 39" north. Find their differ- ence of latitude. 12. From f A. take 42.5 sq. rd. sq. rd. sq. yd. sq. ft. sq. in. I A. = 100 42.5 sq. rd. = 42 15 1 18 |A.-442.5sq. rd. = 57 14i 7 126 4 = 2 36 " =57 15 1 18 Query. — Why do we change i sq. yd. to 2 sq. ft. 36 sq, in. ? 13. From | mi. take 234| rd. 14. From f wk. take 2.6 da. 15. Take 1 pk. 1 qt. 1 pt. from .7 bu. 16. From f gross take 6f doz. 17. How long was it from May 12, 1892, to July 4, 1899 ? The later date expresses the greater period of time, hence it is the minuend. The later date is the 4th day of the 7th month of 1899. The other date is the 7 1 22 12th day of the 5th month of 1892. yr. mo. day. 1899 7 4" 1892 5 12 COMPOUND NUMBERS. 227 We subtract as in other compound numbers, considering 30 days as a month, as given in the table. Query. — Does the remainder express the eocact time between the given dates ? Why not ? Jt;^ To find the exact time between two dates (as, for example, Aug. 24 and Dec. 3, same year) we must proceed as follows : 7 + 80 + 31 + 30 4- 3 = 101. That is, there are 7 more days in Aug., 30 in Sept., 31 in Oct., 30 in Nov., and 3 in Dec. The sum is the difference in time expressed in days. 18. The War between the States began April 11, 1861, and ended April 9, 1865. How long did it continue ? 19. Find the time from Oct. 15, 1812, to June 3, 1912. 20. A note dated June 19, 1897, was paid Oct. 12, 1898. How long did it run ? 21. What is your age to-day ? 22. How long is it from to-day to Feb. 29, 1920 ? 23. Independence was declared July 4, 1776. How long since that, important event occurred ? MULTIPLICATION. WRITTEN EXERCISES. 342. 1. Multiply 4 gal. 3 qt. 1 pt. by 9. (*) 9 times 1 pt. = 9 pt., or 4 qt. 1 pt. 9 times 3 qt. + 4 qt. = 31 qt., or 7 gal. 3 qt. 9 times 4 gal. + 7 gal. = 43 gal. 43 3 1 .". the product is 43 gal. 3 qt. 1 pt. The calculation is conveniently made as in (a). 2. Multiply £6 12s. 8d. by 5. 3. Multiply 4 lb. 9 oz. 6 pwt. 13 gr. by 11. 4. Multiply 6 wk. 4 da. 8 hr. 20 min. by 12. 5. A grocer bought 8 bags of chestnuts, each containing 3 bu. 1 pk. 5 qt. How many did he buy ? («) gal. qt. pt. 4 3 1 9 228 SCHOOL ARITHMETIC. 6. If he sold the chestnuts at 6 cents a qt., how much did he get for them ? 7. What is the difference in time between 366 common years and 366 leap years ? 8. How far is it around a square that measures 4 yd. 2 ft. 8 in. on a side ? 9. How much calomel can be put into 20 bottles, if eacli bottle holds 3 6 3 2 gr. xv ? 10. If $20 will buy IT. 6cwt. 801b. of hay, how much hay will $240 buy ? 11. When silver is worth $.60 an ounce, how much must be paid for 5 oz. 12 pwt. 16 gr. ? 12. When vinegar is 6 cents a quart, how much must be paid for J gal. ? DIVISION. WRITTEN EXERCISES. 343. 1. A bbl. of syrup containing 43 gal. 3 qt. 1 pt. was shared equally by 9 persons. How much did each one get? (a) (h) 43 gal. -1-9 = 4 gal,, and 7 gal. (28 qt.) remaining. gal. qt. pt. 28 qt. +3qt.=81qt. Q\±^ ^ 1 31 qt. -7- 9 = 3qt., and 4 qt. (8 pt.) remaining. ^M '1 i 8 pt. +1 pt. = 9 pt. 4 3 1 9pt. -^9 = lpt. .". the quotient is 4 gal. 3 qt. 1 pt. Query. — Is the dividend here the product in example 1 in Art. 848 ? How do we find one of the two factors when the product and the other factor are given ? 2. If 5 horses eat 11 T. 16 cwt. 80 lb. of hay in a year, how much does one horse eat ? 3. The distance around a square field is 155 rd. 5 yd. 2 ft. What is the length of one side ? COMPOUND NUMBERS. 229 4. If 25 bu. 3 pk. of oats fill 9 bags of the same capacity, how much do 2 bags contain ? 5. The weight of 8 silver chains is 18 oz. 12 pwt. 16 gr. What is the average weight ? 6. A druggist made 3 2 31 gr. x of calomel into 15 powders. How much was in each powder ? 7. The distance around a rectangular field is 128 rd. 12 ft., and one side is 42 rd. 4 ft. What is the length of one end ? 8. How many barrels will contain 30 bu. 1 pk. 4 qt. of chestnuts, if each barrel holds 3 bu. 1 qt. ? Reduce botli numbers to the same denomiiifttion, ami divide, 9. If one man can earn £4 128. lOd. in a week, how many men can earn £46 8s. 4d. in the same time ? 10. How often can 1 ft. 8 in. be sawed from a board 13 ft. 4 in. long ? 11. How many dress patterns of 14f yd. each can be cut from a piece of cloth containing 44^ yd. ? 12. How many blocks 1 ft. square can be cut from a board 12 ft. long and 2 ft. wide ? 13. A two-acre lot is 8 rd. wide. How many yards is it around the lot ? 14. Six men and 2 boys weigh 1152 lb. 12 oz. If each boy weighs 70 lb. 10 oz., what is the average weight of a man ? 15. How many steel rails, each 24 ft. long, are required for a mile of sing'le-track railroad ? 16. l^ow often can a bucket that holds 2 gal. 3 qt. 1 pt. be filled from a hhd. containing 60 gal. 1 qt. 1 pt. ? 17. If each step measures 2 ft. 8 in., how many steps will a man take in walking | of a mile ? 18. A field is 165 ft. long and 66 ft. wide. What will it cost to fence it at $3 a rod ? 19. If I exchange $1023.10 for English money, how many pounds, shillings, and pence do I receive ? REVIEW WORK. ORAL EXERCISES. 344. 1. What is the difference between a square yard and 3 square feet ? 2. What is the difference between a cubic yard and 3 cubic feet? 3. What will a gallon of cream cost at 12 cents a pint ? 4. At $4 a bushel, what will be the cost of 3 pk. 4 qt. of clover seed ? 5. How many inches in 5 feet 9 in. ? 6. In 3.5 pwt. how many grains ? 7. What will 2 gallons of vinegar cost, if 2 pints cost 8 cents ? 8. A field is 40 rods long and half as wide. How many acres in it ? 9. If 9 pints of chestnuts cost 45 cents, how much will half a bushel cost ? 10. How many ounces in 6^ lb. avoirdupois ? 11. At a cent a pound, how much will a ton of iron cost ? 12. How many square feet in 5 sides of a 4-inch cube ? 13. The distance around a square surface is 48 inches. How many square feet does it contain ? 14. From } of a score take | of a dozen. 15. If you sleep 8 hours each night, how many days do you sleep in 2 Aveeks ? 16. Mr. B left home on Friday, and was gone 25 days. On what day of the week did he return ? 17. My horse eats 9 qt. of oats a day. How long will 2 bu. ^ pk. last him ? REVIEW WORK. 231 18. A rope is 10 yards long. Into how many pieces each 1^ ft. long can it be cut ? 19. If there are G ties to every rod of track, how many ties will it take for ^ of a mile of double-track raih'oad ? 20. How many pens in ^ gross and 1^ dozen ? 21. How many square inches in the surface of a rectangle 12 feet long and ^ inch wide ? 22. A lady put a half gallon of perfume into gill bottles. How many bottles were required ? 23. Mr. A's horse is 15 hands high, and Mr. B's is 16 J^ hands high. How many inches higher is Mr. B's horse ? 24. How many cubic yards of air in a room 4 yards wide, 6 yards long, and 9 feet high ? 25. How many days between Feb. 10 and March 10, 1900 ? 26. A man traveled g of the distance around a circular park. Through how many degrees did he travel ? 27. If I receive $^ for working 45 minutes, how mucli should I receive for 9 hours' labor ? 28. What is the entire surface of a cube whose edge is a feet ? What is the volume ? 29. What is the value of b gallons of milk at a cents a pint ? 30. If p bales of hay weigh q pounds, how much will a bales weigh ? WRITTEN EXERCISES. 345. 1. A lot is 120 feet deep and 60 feet front. How many square yards does it contain ? 2. A gardener put 3 bu. 1 pk. 7 qt. of berries into quart boxes. How many boxes were required ? 3. How many quarts of milk will fill a peck measure ? 4. How many ounces of quinine in 5 lb. avoirdupois ? 6. What will be the cost, at $.35 a square yard, of painting 5 sides of a cube whose edge is 4 feet ? 6. How many five-grain pills can a druggist make front 35 ^l of calomel? 232 SCHOOL ARITHMETIC. 7. How many gold dollars weigh a pound avoirdupois ? 8. How many gold dollars can be coined from a pound of gold ? 9. How many pounds of silver are required to coin 100 silver dollars ? 10. If a boy idles away ^ hr. a day, how much time will he lose in a year ? 11. A 20-acre field is 80 rods long. Find the cost of fenc- ing it at $.75 a rod. 12. The wheel of a bicycle is 7 ft. 4 in. in circumference. How often will it revolve in going a mile ? 13. At 2 J cents a pound, how many barrels of flour can be bought for $53.90? 14. The diameter of a circular field is 40 rods, and the side of a square field is the same. Which has the greater area, and how much ? 16. Find the cost of fencing the fields just mentioned, at $1.50 a rod. 16. When land is worth $80 an acre, what is the value of a field in the form of a trapezoid, whose altitude is 30 rods, and whose parallel sides are 48 and 32 rods respectively ? 17. Bought 2 lb. silver by avoirdupois weight, paying $8 a pound, and sold it by troy weight at 80 cents an ounce. How much did I gain ? 18. Bought 3 lb. quinine at $4.50 a pound avoirdupois, and sold it at $6 a pound apothecaries' weight. What did I gain ? 19. A traveler returning from Europe had £10, 10 shil- lings, 10 francs, and 10 marks, which he exchanged for U. S. money. How much did he get ? 20. A man in Boston bought 2880 lb. of wheat, at $.85 a bushel, and sold it at $.91 a bushel. How much did he gain ? 21. The altitude of a triangular field is 32 rods, and its area is 8 aeres. Find the length of the base. R?:VIEW WOKK. 233 22. What is the radius of awheel if an arc of 15° of its cir- cumference is 1 ft. 2 in. in length ? 23. Find the cost of 8 yd. 1 ft. 5 in. of i)ipe, 4 lb. to the foot, at 24^/' a pound. 24. What is the value of ^ of a square mile of land at $G an acre ? 25. How many steel rails 32 feet long are needed to build 2 miles of double-track railway ? 26. Bought 6 gross of lead pencils at $3.50, and sold them at 5 cents apiece. Find the gain. 27. My little girl is '' worth her weight in gold " dollars. If she weighs 30 pounds, what is she worth ? 28. I sold 138 eggs at 30 cents a dozen, and enough butter at 35 cents a pound to make my total receipts $4.50. llow many pounds of butter did I sell ? 29. The water in my cistern, which is G ft. long, 5 ft. wide, and 8 ft. deep, weighs 15000 pounds. Is the cistern full ? If so, what is the weight of a cubic foot of water ? 30. Mr. H has a feed-box 8 ft. long, 3 ft. wide, and 4 ft. high. AVhen it is half full of oats, how many bushels does it contain ? 31. How many square feet of zinc will be required to line the sides and bottom of a bin 12 ft. square and 5 ft. high? 32. If tobies cost $9.50 a thousand, and a man smokes 4 a day, what is the amount of his annual toby bill ? 33. How many cubes measuring 6 inches each way can be cut from a cubic yard of marble ? 34. Measure your schoolroom, and find how many cubic feet of air there are to each pupil. 35. What will be the cost of painting a box 2 feet square and 10 feet higli, at $.25 a sq. yd.? 36. How many square yards of canvas will be needed to cover two boxes, each of whose bases is 6 feet square, the height of one being 4 feet and that of the other 5 feet ? 234 SCHOOL ARITHMETIC. 37. What part of the annual revolution does the earth make in 24 da. 6 hr. 47 min. ? 38. If 1 bu. 3 pk. of seed is required to plant an acre, how much must be sowed upon 2 A. 16 sq. rd.? 39. 3960 cu. yd. of earth were removed in digging a ditch 360 rd. long and 6 ft. deep. How wide is the ditch ? 40. A druggist pays 53|^ a pound avoirdupois for 9 lb. of borax. If he sells it at the rate of 6^^ an ounce apothe- caries', how much does he gain ? 41. How many silver forks, each weighing 2.3 oz., can be made from 9 lb. 18 pwt. of silver ? 42. How many sq. yd. in the walls and ceiling of a room 24 ft. long, 18 ft. wide, and 12 ft high ? 43. How many cords of wood in a pile 336 ft. by 9 ft. by 6ft.? 44. If it takes a man | of a day to mow an acre of grass, how long will it take him to mow 3 A. 45 sq. rd. 16 sq. yd.? 45. If ^1 of a bushel of salt can be made from 54 gal. of salt water, how much salt can be made from 72 gal.? 46. A plank 37 ft. 4 in. long and 4 in. thick contains 4f cu. ft. What is the width ? 47. A man who had 2 miles to travel, walked 5 rd. 7 ft. How far had he yet to travel ? 48. If 5.3 T. of porcelain clay cost $106, what is the cost of 438 pounds ? 49. How many cubic yards of air in a room 18 ft. by 15 ft. by 9 ft. ? 50. What must be the depth of a measure 18^ in. square to. contain a bushel ? 51. If 4 bu. 3 pk. 4 qt. 1.6 pt. of wheat makes one barrel of flour, and the toll is 4 qt. a bushel, how many bushels of wheat must I take to the mill in order to get five barrels ot flour ? 52. When corn meal is selling at 80^ per cwt., how many pounds will 10^ buy ? REVIEW WORK. 235 53. A building lot 100 ft. front contains 1 acre. How deep is the lot ? 54. How many cakes of maple sugar 8 inches by 6 inches by 3 inches can be packed in a box 24 inches by 18 inches square in the clear ? 55. If a load of wood is 8 ft. long and 3 ft. wide, how high must it be to contain a cord ? 56. When each end and the middle of boards IGJ feet long are nailed to posts, how many posts are in a fence around a field 30 rods long and 20 rods wide ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 346. 1. How many secojids in the month of February, 1899 ? 2. How long will it take a body moving at the rate of a mile a minute to travel from the earth to the moon, the dis- tance being 239,000 miles ? 3. How many seconds in the circumference of a silver dollar? 4. Which is heavier, and how much — a pound and an ounce of gold, or a pound and an ounce of lead ? 5. How many pounds of gold are equal in weight to G lb. of feathers ? 6. How many bushels of corn will a receptacle contain that holds 5,000 gallons of water ? 7. How long will it take to count a million, counting 80 a minute, and 12 hours a day ? 8. A printer used 3 reams, 5 quires, 19 sheets of paper for printing half-sheet sale bills. How many did he print, allowing 1 quire to a ream for waste ? 9. There are 9 oz. of iron in the blood of one man. The blood of how many men would be required to furnish suffi- cient iron to make a kettle weighing 22^ lb.? 10. A merchant bought a barrel (42 gal.) of vinegar for 16.30. How much water must be added to reduce the first cost to 10^ a gallon ? 236 SCHOOL ARITHiMETIC. 11. In 18} carat gold, what part is alloy ? 12. How many silver dollars can be made from a bar of silver weighing 11 lb. 9 oz. avoirdupois ? 13. How many tucks ^ inch wide can be made in a strip of muslin a yard long, leaving ^ of an inch between the edge of one tuck and the stitching of the next ? LONGITUDE AND TIME. 347. 1. Does the earth rotate from west to east, or from east to west ? In how many hours does it rotate once? 2. In what time does any place on the earth's surface pass through 360° ? Through how many degrees does it pass in 1 hour ? In 1 minute ? -^^ of 15° = ( ) ? \° = how many '? 3. Since the earth moves 15' in 1 minute of time, how far does it move in 1 second of time ? ^' = how many ."? 4. How long does it take the earth to turn through 15° ? 30°? 60°?. 90°? 180°? 5. How does the number of hours compare with the number of degrees traveled ? 6. Which city has sunrise first — Baltimore or Chicago ? Why?- ^ ^ 7. If my watch shows the correct time when I leave Boston, will it be too slow or too fast when I reach Denver ? 8. A man from Cleveland arrived at another city and found his watch 20 minutes too slow. Had he traveled east or west ? How many degrees ? 348. A Meridian (mid-day line) is an imaginary line running north and south from pole to pole. All places on the same meridian have their mid-day, or noon, at the same moment ; that is, when the sun's rays are vertical on that meridian. 349. The Longitude of a place is its distance (in degrees, etc.) east or west from a given meridian. LONGITUDE AND TIME. ^37 350. The given meridian from which longitude is gen- erally reckoned is called the Prime Meridian. It passes through Greenwich, a part of London. Notes. — 1. From the prime meridian, longitude is reckoned east and west to 180°. West longitude is designated by the letter W; east lon- gitude by the letter E. 2. Clocks show later (faster) time at places east of a given place, and earlier (slower) time at places west of a given place. 351. Since the earth turns upon its axis once in 24 hours, any place on the earth's surface passes through 360° in that time. Hence we deduce the following Table of Relations. LONGITUDE. TIME. 360° corresponds to 24 hours. 15° '' '' 1 hour. 15' '' « 1 minute. 15" " " 1 second. 1° " " -^^ h\\, or 4 min. 1' '' " i^jr min., or 4 sec. 352. To find the difference In time between two places, the difference in longitude being given. WRITTEN EXERCISES. 1. Boston is 71° 3' 30" west of London. What is the dif- ference in time ? 15)71° 3' 30" It will be seen by the table that the numbers de- 7 77 ^i noting the difference in time between two places are (V of those denoting longitude. The process is the same as in division of compound numbers, and the difference in time is 4 hours, 44 minutes, and 14 seconds. 2. The difference in longitude between two places is 50° 25'. What is the difference in time ? 3. Cincinnati is 84° 29' 45" west from Greenwich. Find the difference in time. 238 SOflOOL ARtTHMETia 4. When it is noon at Cincinnati, what is the time at Greenwich ? SHORT LIST OF CITIES, WITH LONGITUDE FROM GREENWICH. CITIES. LONGITUDE. CITIES. LONGITUDE. Greenwich 0° 0' 0" St. Louis 90° 15' 15 " W. New York 73° 58' 25.5' W. San Francisco . .122° 25' 40.8 " W. Paris 2° 20' 15" E. St. Petersburg .. 30° 19' 0" E. Boston 71° 3' 30" W. Richmond 77° 26' 4" W. Chicago 87° 36' 42" W. Denver 104° 59' 33" W. Rome 12° 27' 14" E. Pittsburg 80° 2' 0" W. Cleveland 81° 40' 30" W. Honolulu 157° 51' 48" W. Washington .... 77° 3' 0" W. Jerusalem 35° 13' 25" E. Calcutta 88^ 20' 8" E. Athens. 23° 43' 55.5" E. Find the difference in time between : 6. Boston and Pittsburg. 6. Chicago and AVashington. 7. Eome and Calcutta. 8. San Francisco and Denver. 9. Philadelphia and Cleveland. 10. New York and Paris. 11. Richmond, and St. Petersburg. 12. Honolulu and Jerusalem. 13. When it is noon at New York, what is the time at Denver ? 14. Would a traveler's watch be too fast or too slow, and how much, when he goes from St. Louis to Athens ? 353. To find the difference in longitude between two places, the difference in time being given. 1. When it is noon at San Francisco, it is 8 minutes 45 seconds past 2 p.m. at St. Louis. What is their difference in longitude ? It will be seen by the table that there are 15 times as many °, ', and " of longitude between two places as there are hours, minutes, and seconds of time. The process is 32 11 15 ^^® same as in multiplication of compound numbers, and the difference in longitude is 33° 11' 15". hr. rain. sec. 2 8 45 15 LONGITUDE AND TIME. 239 a. What is the difference in longitude between two places whose difference in time is 4 hours, 12 minutes, and 23 seconds ? Find the difference in longitude between : 3. New York and Chicago. 4. Philadelphia and Cleveland. 6. Richmond and Denver. 6. Cleveland and Greenwich. 7. Boston and San Francisco. 8. Washington and Paris. 9. Pittsburg and Calcutta. 10. Rome and St. Petersburg. 11. When a traveler reached Boston he found that his watch was 4 hr. 25 min. too fast. What was the longitude of the place from which he started ? 12. When it is 12 o'clock M. at Pittsburg, it is 9 hr. 10 min. 18 sec. a.m. at Portland, Oregon. What is the longi- tude of Portland ? 13. Mr. R left Philadelphia and traveled eastward at the rate of ^° an hour. How much too slow was his watch at the end of two weeks ? 14. The difference in time between two places on the equator is 2 hr. 45 min. 30 sec. In how many hours could a railroad train run from one place to the other at the rate of 30 geographic miles an hour ? 15. Midnight comes 1 hr. 5 min. 42 sec. sooner at Quebec than at Chicago. What is the longitude of Quebec ? 16. At Richmond the sun rises 1 hr. 2 min. 52 sec. earlier than at St. Paul, and 2 hr. 59 min. 49 sec. earlier than at San Francisco. What is the difference in longitude between St. Paul and San Francisco ? 354. To find the difference in the longitude of two places : 1. If both longitudes are east, or if both are west, sub- tract ; if one is east and the other ivest, add. 240 SCHOOL ARITHMETIC. 2. If the sum of two longitudes is greater than 180°, the sum must he subtracted from 360° to obtain the correct differ- ence of longitude. STANDARD TIME. 355. The time we have been considering is called Local Time, which is determined by the rotation of the earth on its axis. To avoid the confusion and mistakes incident to such' time, the railroad companies have adopted the time of the meridians of 75°, 90°, 105°, and 120° as the standards by which to run their trains. This railroad time is called Standard Time. 356. This new plan divides the United States and Canada into 4 belts, extending north and south, each about 15° wide. All places in the same belt have the same time, regardless of their longitude. Note. — The division lines of the time belts are not exactly 7^° on either side of the hour meridians, but are somewhat irregular, passing through leading railway termini. ^^ Have the pupil turn to map in his geography showing the stand- ard time belts. Which is the widest ? 357. The first belt lies on both sides of the meridian of 75°, which passes 1° west of ^"ew York city, and all places therein have the time of that meridian, which is called East- ern time. Thus, when a man goes from Richmond to New York, Boston, or Quebec, he finds that his watch is neither too fast nor too slow. 358. Tlie second belt lies on both sides of the meridian of 90°, which passes near New Orleans and St. Louis. The time in this belt is called Central time, and is 1 hour slower than Eastern time. Query. — Why is Eastern time just 1 hour faster than Central time ? 359. The third belt lies on both sides of the meridian of LONGITUDE AND TIME. 241 105°, wliich passes near Pike's Peak. The time in this belt is called Mountain time, and is 1 hour slower than Central time, and 2 hours slower than Eastern time. Query.— A man who goes from Baltimore to Denver will find his watch how much too fast ? 360. The fourth belt lies on both sides of the meridian of 120°, which passes 9 degrees east of San Francisco. The time there is 3 hours slower than Eastern time, and 1 hour slower than Mountain time. It is called Pacific time. Query. — When it is 9 a.m. at San Francisco, what is the time at New York ? At Leadville ? At St. Paul ? COMPARISON OF TIMES. BOSTON CHICAGO DENVER PORTLAND (OR.) Standard Time, ]2 m. 11 a.m. 10 a.m. 9 a.m. Local Time, 12 m. 10:53} a.m. 9:44^ a.m. 8:34|a.m. ^^ On the 4 standard meridians, local time and standard time are the same. In other places standard time, being fixed arbitrarily, is not the correct or local time. WRITTEN EXERCISES. 361. 1. At Pittsburg, what is the difference between local time and standard time ? Longitude of Pittsburg, 80° 2' 0" W. Longitude of the meridian of 75°, 75° 0' 0" W. 15) 5° 2' 0" 20 8 Pittsburg has the time of the meridian of 75" instead of that of her own meridian. The difference in longitude between the two meridians is readily found, and the difference in time is ^ as great. Hence, stand- ard time at Pittsburg is 20 min. 8 sec. faster than local time. 2. What is the difference at New York between standard and local time ? 3. When it is 3 p.m., standard time, at San Francisco, what is the local time there ? 10 242 SCHOOL ARITHMETIC. 4. The longitude of Staunton, Va., is 79° 4' 15". When it is 5 A.M. there, standard time, what is the local time ? 6. When it is 12 m. at Paris, Avhat is the standard time at New York ? The local time ? 6. What is the local time at Denver when the standard time at Cleveland is 9 a.m. ? I>ATE LINE. 362. The boundary line between adjoining regions in which the calendar day is different is called a Date Line. This line as now agreed upon nearly coincides with the me- ridian of 180°. Note. — Suppose a man to leave London at noon on Friday and travel westward just as fast as the earth rotates. He keeps the sun directly overhead all the time, and it seems to him that it is still Friday noon when he reaches London again 24 hours after starting, when in reality it is Saturday noon. He has lost a day in his reckoning. Had he trav- eled eastward he would have gained a day. There is the same loss and gain wheii traveling is done at ordinary speed. Hence is seen the necessity for a fixed place at which each new day shall begin. This place is the date line ; there a new day begins every midnight. Thus, July 4th begins on the meridian of 180° at midnight following the 3d of July. At that time it is midday, July 3, at London (Greenwich) ; 12 hours later it is midnight at London, and that city enters upon the new date — July 4 ; it is then noon of the 4th on the date line, while at Chicago it is about 6 p.m., July 3. Each day has the same date all around the earth. Thus, Sunday, May 1, 1899, began at midnight on the date line, and thereafter as each place on the earth had midnight it began to record the date as "Sun- day, May 1." It is a day later on one side of the date line than it is on the other. Thus, when it is Monday in San Francisco it is Tuesday in Hong Kong. Hence navigators change their calendar one day in crossing this date line. Going east, they count the same day twice ; going west, they skip a day. Queries. — 1. When it is noon, Jan. 1, 1900, at New York, what is the date and time on the island of Luzon, longitude 120° E. ? LONGITUDE AND TIME. 243 2. When it is Saturday noon at Athens, what is the day and local time at Chicago ? 3. When it is 2 a.m. on Friday at Calcutta, what is the day and hour at Boston ? SUPPLEMZNTARY EXERCISES (FOR ADVANCED CLASSES). 363. 1. If A leaves home at 12 m. on Monday, and on Saturday finds his watch 1 hr. 15 min. slow, in what direc- tion and how far has he traveled ? 2. A mail starts from Philadelphia and travels westward. When he stops, he ascertains that his watch is 6 hr. slow. In what longitude did he stop ? Through how many degrees did he travel ? 3. A man traveling eastward from a point on the equator stops when his watch has become 2 hr. 40 min. slow. How many miles has he traveled ? 4. If a man starts westward from Paris and travels 181°, will his watch be too fast or too slow, and how much ? 5. A and B start from different points and travel towards each other. When they meet, A's watch is 40 minutes slow, and B's is 1 hour fast. How far apart are the starting-points ? In what direction did each travel ? 6. When it is noon, local time, at New York, on what meridian is it midnight ? 7. When it is Sunday noon, local time, at Chicago, what is the day and time at Jerusalem ? 8. A vessel had sailed a certain distance on a parallel of latitude when the captain found that, although the sun was on the meridian, his Greenwich chronometer indicated 1:47 P.M. What was the ship's longitude ? 9. The battle of Manila began at 6 a.m.. May 1. Had Dewey cabled at once to Washington, at what hour and on what date would the message have reached the President, allowing 1 hour for transmission, and considering the longi- tude of Manila to be 120° E. ? PRACTICAL MEASUREMENTS. painti:ng and plastering. 364. Painting and plastering are usually estimated by the square yard. Allowances for openings ar& sometimes made, but there is no fixed rule as to how much should be deducted. 1. Find the cost of painting the outside of a house 36 ft. long, 28 ft. wide, and 24 ft. high, at $.25 a sq. yd. 2. A room 18 ft. long, 15 ft. wide, and 12 ft. high has 2 doors, each 8 ft. by 4 ft. 6 in. Find the cost of plastering the walls and ceiling at $.35 a sq. yd., making allowance for the doors. 3. A parlor is 15 ft. 9 in. square, and a reception-room is 12 ft. by 16 ft. The height of each is 11 ft. 6 in. What will 1)6 the cost of plastering the walls and ceilings at $.33 a sq. yd. ? 4. How much would be saved by having the ceilings kal- somined at a cost of $.18 a sq. yd. ? 5. Measure your schoolhouse, and prepare a problem in painting for your class. 6. Find the cost of wainscoting a hall 30 ft. long, 8 ft. wide, and 12 ft. 8 in. high, at $.40 a sq. yd. 7. Measure your schoolroom, and prepare a problem in plastering for your class. 8. Find the cost of painting a gable roof 45 ft. long and 24 ft. wide, at 28 ct. a sq. yd. MtiAsiJRtiM^NT <^P LiltMBElt. 365. When boards are 1 inch thick or less, they are esti- mated by the square foot of surface, the thickness not being considered. Thus, a board 8 ft. long, 1 ft. wide, and 1 inch (or less) thick con- tains 8 square feet. Its surface is a rectangle. 366. A board 1 foot square and 1 inch tliick is called a board foot, or a foot board measure. When lumber is more than 1 inch thick, the number of board feet depends upon the thickness. Thus, a. board 8 feet long, 1 foot wide, and 2 inches thick contains 16 board feet, or twice as many as if only 1 inch thick. 1. How many square feet in a board 12 ft. long and 12 in. wide ? How many board feet in it if it is 1 inch thick ? 2. How many feet board measure in a board 16 ft. long and 15 in. wide if it is 1 inch thick ? How many if it is ^ inch thick ? 3. Find the cost of a board 14 ft. long, 6 in. wide, and an inch thick, at 3^ a board foot. 4. An inch board 18 ft. long is 16 in. wide at one end and 8 in. wide at the other. How many board feet does it contain ? The average width of a board that tapers uniformly is one-half the sum of the end widths. A board like this has the form of a trapezoid. 5. A half-inch board 16 ft. long is 16 in. wide at one end and tapers to a point at the other. What is it worth at 3^^ a foot board measure ? What is the form of this board ? 6. How many feet board measure in a beam 24 ft. long, 15 in. wide, and 3 in. thick ? 24 X IJ = 30, the number of sq. ft. in the surface, or the number of board feet in 1 inch of the thickness. Hence, in 3 inches of thickness there are 3 x 30 board feet, or 90 board feet. Jn any board or timber the number of board feet in 1 inch 246 SCHOOL ARITHMETIC. of thichness is equal to the number of square feet in the sur- face of one side. Hence, to find the number of feet boiird measure, when lumber is more than 1 inch thick, EuLE. — Multiply the number of square feet in the surface of one side by the number representing the thickness in inches. 7. How many feet board measure in a plank 14 ft. long, 16 in. wide, and 2 in. thick ? 8. How many board feet in a log 18 ft. long, 15 in. wide, and 12 in. thick ? 9. Mr. H bought 30 joists, each 20 ft. long, 8 in. wide, and 3 in. thick, at 118.50 per M. Find the cost. • It^f^In lumber measure " per M " means "by the thousaml" (board feet). 10. What will it cost to floor a two-story warehouse, 24 ft. by 36 ft., with two-inch planks, at $25 per M ? 11. How many feet board measure in a piece of timber 32 ft. long, 18 in. wide at one end and 6 in. wide at the other, the thickness being 14 inches ? 12. Two men bought a piece of timber 10 ft. long and 15 in. square, paying at the rate of $24 per M. How much should each man pay ? 13. A farmer used inch boards to make a feed-box which measured on the outside 8 ft. in length, 3 ft. 2 in. in width, and 4 ft. 9 in. in height. What was the cost of the boards, at $18 per M ? 14. James McKnight bought from Jas. Laird, Charleston, S. C, as follows : 40 joists, 2x6, 18 ft. long, @ $25, 16 beams, 6x9, 20 " " '' 30, 72 scantling, 2x4, 12 '' '' " 24, 240 boards, 1 x 10, 12 '' '' '' 18, 24 planks, 2 x 14, 16 '' '' '' 17.50. Make out a complete bill, and find the amount due Laird. PRACTICAL MEASUREMENTS. 247 BRICK WORK AND STONE WORK. 367. Brick work is commonly estimated by the thousand bricks ; stone work by i\\Q perch , which contains 24J cu. ft. The number of bricks in a cubic foot of wall depends upon the size of the bricks. Common bricks are 8 in. x 4 in. x 2 in., and 22 of these are assumed to build 1 cubic foot of wall. ■ In measuring walls of buildings, masons and bricklayers take the entire outside length, thus measuring the corners twice. This must not be done in estimating material. In estimating the tvorky allowance for openings in the walls is sometimes made, but the amount to be deducted, if any, should be specified in a written contract. In estimating the material, all openings should be deducted. In stone work \ is allowed for mortar and filling. In a perch of masonry, without openings, there are only 22 cu. ft. of stone. 1. If 22 common bricks build a cubic foot of wall, how many bricks will be required to build a wall containing 1000 cu. ft. ? 2. How many common bricks in a wall 36 ft. long, 7 ft. high, and 16 in. thick ? 3. How many perches of masonry in the walls of a cellar that is 40 ft. long and 24 ft. wide, the walls being 9 ft. high and 18 in. thick ? 4. How many perches of stone in the walls of the above cellar ? 5. Find the cost of building said walls at $1.50 a perch. €. Had the walls been built at 10^ a cu. ft., how much more or less would the mason have received ? 7. The inside length of a storeroom is 56 ft., the width 44 ft. The outside length is 60 ft., the width 48 ft. If the walls are 22 ft. high, how many cubic feet do they contain, allowing nothing for openings, and counting corners once ? 248 SCHOOL ARITHMETIC. 8. Mr. S built a house 38 ft. long, 32 ft. wide, and 26 ft. high. The front wall is built of stone, and is 2 ft. thick. The others are built of common bricks, and are 16 in. thick. There are 16 windows, each 6 by 3 ft., and 4 doors, each 7 by 3^ ft. In the front wall are 4 windows and one door. How many bricks and perches of stone were used ? CARPETING. 368. The amount of carpet that must be bought for a room depends upon the length and number of strips, and the waste in matching the patterns. The number of strips often depends upon whether they are laid lengthwise or crosswise. Thus, in a room 15 by 17, 5 strips, each a yard wide, will be sufficient, if laid lengthwise. If laid the other way, 6 strips will be required, and one foot of width may be turned under or cut off. 5 X 17 ft. = 85 ft., or 28i yd. when laid lengthwise. 6 X 15 ft. = 90 ft., or 30 yd. " " crosswise. The waste in matching tlie patterns cannot be estimated except by actual measurement of the carpet. In the following problems waste in matching is not considered. 1. A schoolroom is 30 ft. long and 26 ft. wide. How much matting a yard wide should be bought to cover the floor if the strips are to be laid lengthwise ? Suggestions. — 1. How wide is the room ? Then how many strips will be needed ? 2. Can we buy § of a strip (in width) ? Then how many strips must be bought ? What may we do with the surplus width ? 3. How long must each strip be ? Then how many yards must be bought ? 2. A dining-room is 18 ft. 6 in. long and 15 ft. wide. How much ingrain carpet a yard wide must be bought to cover it if the strips are to be laid lengthwise ? 3. What will be the cost of carpeting a room 14 ft. by 18 ft. with Brussels carpet 27 in. wide, laid lengthwise, if 3 yards cost $4.50 ? 4. A barber shop 15 ft. by 17 ft. 8 in. is covered with oil- PRACTICAL MEASUREMENTS. 249 cloth 2 yd. wide, laid across the room. What was the cost of the oilcloth if 2^ yd-, cost $2.25 ? 5. In a parlor 14 ft. by 20 ft. the carpet is J yd. wide, and is laid crosswise. Find its cost at $.85 a yard. 6. My library, which is 12 ft. square, is covered with Cliina matting 36 in. wide, for which I paid $.05 a yard. IIow much did the matting cost me ? 7. Find the cost of a rug for a room 3 yd. 2 ft. 8 in. by 4 yd. @ $1.25 a sq. yd. 8. How many tiles 8 in. square will be required to lay a floor 12 ft. by 32 ft. ? 9. Brussels carpet 27 in. wide is laid lengthwise on the floor of Mr. B's reception-room, which is 13 ft. wide. One yard cost $1.75, and the entire cost was $52.50. What was the length of the room ? PAPERING. 369. The amount of wall paper required to paper a room depends upon the area of the walls and ceiling and the waste in matching. (a). Wall paper is put up in double rolls, but the prices quoted are for single rolls. (b). A single roll contains 8 yards, 18 inches wide, its area being 36 square feet. Allowing for all waste, this will cover 30 square feet of wall. (c). In estimating the number of rolls required for a room, some dealers deduct the exact area of the doors and windows, while others deduct an approximate area, allowing 20 square feet for each. (d). Borders vary in width, and are sold by the yard. (e). The number of single rolls required for the walls and that for the ceiling must be estimated separately, and can be found only approximately. Two methods employed by dealers are shown in (a) and (J) below. 250 SCHOOL ARITHMETIC. . Example. — Room 12 x 14, 10 feet liigh, one window 6x4, one door 7x4. (&) Area of walls 520 sq. ft. Two openings, 20 sq. ft. each J^ " " Difference : 480 " " 480 -r- 30 =: 16, number of single rolls needed. This method allows for waste and matching. («) Area of walls 520 sq. ft. Area of openings 52 " " Difference 468 " " 468 -f- 36 = 13, number of single rglls required. No allowance is here made for matching and waste. The area of the ceiling divided by 36 (or by 30 to allow for waste) gives the number of rolls required for that ceiling. The number of yards of border required is the same as the perimeter of the room in yards. 1. How many single rolls of wall paper will be required to paper the walls and ceiling of a room 17 ft. long and 15 ft. wide, not allowing for waste, the height from baseboard to ceiling being 9 ft. ? 2. Making no allowance for waste, what will be the cost of paper and border for the walls of a room 22 ft., long, 16 ft. 6 in. wide, and 14 ft. high, if paper costs $.75 a roll and border $. 60 a yard ? 3. Measure your sitting-room at home, and. prepare a problem in papering for your class. 4. Estimate the cost of paper for your parlor at $.75 a roll. 5. My dining-room is 16 ft. long, 13 ft. 9 in. wide, and 10 ft. high. It has two windows, each 7 ft. by 4 ft., and a door 8 ft. by 4J ft. Estimate the number of single rolls required to paper walls and ceiling, allowing for waste. BINS, CISTERNS, ETC. 370. The method of computing the contents of a rect- angular solid was learned in Art. 327. The number of cubic units in a box or cistern is found in the same manner. Note. — Cubic inches can readily be reduced to bushels or gallons if we bear in mind that 2150.42 cu. in. = 1 bushel. 231 cu. in.- = 1 gallon. PRACTICAL MEASUREMENTS. 251 1. How many bushels of oats will be required to fill a feed-box 6 ft. long, 3 ft. wide, and 4 ft. deep ? Find the contents in bushels : 2. A box 4 ft. long, 3 ft. wide, and 2 ft. deep. 3. A bin 7 ft. long, 4 ft. wide, and 5 ft. high. 4. A granary 18 ft. long, 6 ft. 9 in. wide, and 5 ft. G in. high. Find contents in gallons : 6. A cistern 5 ft. long, 4 ft. wide, and 6 ft. deep. 6. A trough 8 ft. long, 12 in. wide, and 8 in. deep. 7. A tank 6 yd. long, 6 ft. wide, and 6 in. deep. 8. A bin 8 ft. long, 4^ ft. wide, and 5 ft. high is half full of wheat. How much is the wheat worth at $.80 a bushel ? 9. How many gallons will fill a tank 4 ft. square and 6 ft. deep ? 10. My cistern, which is 6 ft. long, 5 ft. wide, and 8 ft. deep, is ^ full of water. If we use 2 bbl. of the water, how many gallons will remain in the cistern ? 11. Fifty bushels of rye are in a bin 4 ft. 6 in. square. If the bin is 6 ft. high, how much more rye will it hold ? 12. A cistern 6 ft. by 4 ft. will contain 1000 gallons. How deep is it ? 13. A wagon-box is 11| ft. long and ^ feet wide. When even full of oats it contains 64.686 bushels. What is its depth ? 14. A shed is 8 ft. long, 6 ft. wide, and 6 ft. 4 in. high. How many bushels of coal will it hold ? How many of oats ? 15. A box 3 ft. long, 2 ft. 8 in. wide, and 2 ft. high is j full of wheat. What is the weight of the wheat ? 16. A tin box is 1 foot square and 1 foot deep. Find the number of gallons it will contain, correct to 3 decimal places, and reduce the decimal to qt. and pt. 17. A tank 12 ft. long by 8 ft. 8 in. wide is full of oil. How many gallons must be drawn off to lower the surface 3 ft. ? How many barrels ? THE METKIC SYSTEM. 371. The uniform system of measures expressed in the decimal scale, which was first adopted in France in 1795, is called the Metric System. It is used in nearly all countries of continental Europe and South America, especially in scientific work. An act of Congress authorizes its use in the United States. 372. The system is based on the principal unit of length, called the Meter (meaning measure). The original standard meter is a rod of platinum which is preserved at Paris by the French government. It was intended to be .0000001 of the distance from the equator to the pole, but more careful meas- urements show this distance to be 10,001,887 meters. 373. The names of the higher and lower units are formed by attaching certain prefixes to the names of the princip;il units. These prefixes are, in part, as follows : (Greek) (Latin) deka, meaning 10 deci, meaning .1 hekto, " 100 centi, '' .01 kilo, '' 1000 milli, " .001 myria, '' 10000 Table of Length. A myriameter = 10,000 meters. A Mlometer (Km.) = 1000 t( A hektometer (Hm. ) = 100 a A dekameter (Dm.) = 10 Si Meter (m.) A decimeter (dm.) = .1 of i a, meter. A centimeter (cm.) = .01 a A millimeter (mm.) — .001 a THE METRIC SYSTEM. 253 This table may be arranged thus : 10 millimeters = 1 centimeter. 10 centimeters = 1 decimeter. 10 decimeters = 1 meter. 10 meters = 1 dekameter. 10 dekameters = 1 hektometer. 10 hektometers = 1 kilometer. 10 kilometers = 1 myriameter. Notes. — 1. As in U. S. money we seldom speak of anything except dollars and cents, so in the metric system only the units printed in italics are commonly used. 2. In practice, length values are read in three denominations. Thus, 1 dm. 5 cm. is read fifteen centimeters. Values inconveniently large to be expressed in meters are read as kilometers. 3. A length given in one unit may be changed to another by simply moving the decimal point the requisite number of places. Thus, 75 dm. = 7.5 m., and 75 cm. = .75 m. 374. The annexed scale shows the decimeter divided into centimeters, and the latter into millimeters. It also com- pares the decimeter with four inches. The teacher should by all means have a metric stick for reference. FOU R INCHES IN SIX! PEENTHS , OF AN INCH Illllll III III mm: III III l|i|Mi l|l|iM l|l|l|l i|i|)!l 1 2 3 4- 1 2 3 4 5 6 7 8 9 10 iliiliiil lllllllll, JJillllll lllllllll lllllllll lllllllll iiuiuiJ ULlllll lllllllll lllllllll one decimeter in millimeters Equivalents. 1 meter = 39.37 inches. 1 decimeter = 3.937 inches. 1 centimeter = .3937 inch. 1 inch = 2.54 centimeters. 1 mile = 1.6093 kilometers. 1 kilometer = .6214 of a mile. 254 SCHOOL ARITHMETIC. 375. In surface measures the principal unit is the Square Meter. Table of Square Measure. 100 square millimeters (qmm.) = 1 square centimeter (qcm.). 100 square centimeters = 1 square decimeter (qdm.). 100 square decimeters = 1 square meter (qm.). 100 square meters = 1 square dekameter (qDm.). 100 square dekameters = 1 square hektometer (qHm. ). 100 square hektometers = 1 square kilometer (qKm.). Note.— The square dekameter is also called an are (a.), pronounced like the word air, and the square hektometer is called a hektare (ha.). The square meter is sometimes called a centare (ca.). These are used in measuring land. The area of a farm is expressed in hektares ; that of a country in square kilometers. Equivalents. 1 square inch = 6.452 sq. centimeters. 1 square foot — .0929 sq. meter. 1 square yard = .8361 sq. meter. 1 square mile = 2.59 sq. kilometers. 1 acre = .4047 hektare. 1 square meter — 1.196 sq. yards. 1 hektare = 2.471 acres. 376. In measures of volume theprincipalunit is the Cubic Meter. Table of Cubic Measure. 1000 cubic millimeters (cmm). — 1 cubic centimeter (ccm.). 1000 cubic centimeters = 1 cubic decimeter (cdm.). 1000 cubic decimeters = 1 cubic meter (cu m.). Note. — When used in measuring wood, the cubic meter is called a stere (st.), pronounced steer. THE METKIC SYSTEM. 255 EqU IV ALEUTS. 1 cubic inch =16.387 cubic centimeters. 1 cubic foot = .02832 cubic meter. 1 cubic yard = .7045 cubic meter. 1 cubic meter = 1.308 cubic yards. 377. The principal unit of weiglit is the Gram. It is the weight of a cubic centimeter of distilled water at its maximum density. Table of Weight. 10 milligrams (mg.) = 1 centigram (eg.). 10 centigrams = 1 decigram (dg.). 10 decigrams = 1 gram (g.). 10 grams = 1 dekagram (Dg.). 10 dekagrams = 1 hektogram (Hg.). 10 hektograms = 1 kilogram (Kg.). 1000 kilograms = 1 metric ton (T.). A quintal (Q.) = 100,000 grams. A myriagram = 10,000 grams. Note. — The metric ton is the weight of a cubic meter of water ; the kilogram of a cubic decimeter or a liter of water, which is about 2.2 lb. The kilogram is sometimes called a kilo, and is the unit used in weighing ordinary articles. Equivalents. 1 pound avoir. =r .4536 kilo. 1 pound troy = .3732 kilo. 1 ton avoir. = .9072 metric ton. A gram = 15.432 grains. 378. The principal unit of capacity is the Liter {lee'ter). It is the capacity of a cube whose edge is .1 of a meter. 256 SCHOOL ARTIHMETIC. Table of Capacity. 10 milliliters (ml.) = 1 centiliter (cl.). 10 centiliters = 1 deciliter (dl.). 10 deciliters = 1 liter (1.). 10 liters = 1 dekaliter (DL). 10 dekaliters = 1 hektoliter (HI.). 10 hektoliters = 1 kiloliter (Kl.). Note. — The hektoliter is used in measuring grain, vegetables, etc. ; the liter in measuring liquids and small fruits. Equivalents. 1 gallon = 3.786 liters. 1 bushel = .3524 hektoliter. 1 liter = 1.0567 liquid qt. 1 hektoliter = 2f bushels, nearly. ORAL EXERCISES. 379. 1. How are the decimal values of the principal units named ? 2. Multiples of the principal units use what prefixes before the name of the unit ? 3. What is the prefix which means 10 ? 100 ? 1000 ? .1 ? .01? .001? 4. Can you mention any advantages in the use of a metric system of weights and measures ? 5. Since metric measures have a decimal scale, how may units of one denomination be reduced to another ? How should they be written before adding or subtracting ? 6. How many cm. long is this book ? Your desk ? 7. Cut a qdm. out of paper, and draw a sq. ft. How many of the former does the latter equal ? 8. How many mm. in 5 Km.? In 1 Hm.? How many cm. in 25 m.? 9. How many qcm. in a qm. ? How many cmm. in a ccm. ? In a liter ? THE METRIC SYSTEM. 257 10. Explain why in measures of surface each unit is 100 times as hirge as the next smaller unit. Why 1000 times as large in measures of volume. 11. What part of a square meter is a square decimeter ? 12. For what do we use the inch ? The yard ? The mile ? For what are the mm., the cm., the m., and the Km. respectively used ? 13. For what purposes is the liter used ? 14. What part of a liter is 100 g. of water ? 15. What is the weight of a cubic meter of water ? 16. If a train travels at the ratQ of 20 m. a second, what is the rate in Km. an hour ? WRITTEN EXERCISES. 380. 1. What part of a cubic meter is a cubic foot ? 2. A girFs hoop is 6 m. in circumference. How many times will it turn in rolling a distance of 1.08 Km.? 3. How many kilograms does a barrel of flour weigh ? 4. How many square meters in a circle whose diameter is 15 meters ? 5. If sulphuric acid is 1.84 times as heavy as water, what is the weight in dekagrams of 26 1. of the acid ? 6. What is the area of Virginia in square kilometers ? 7. At 16 cents a liter, what is the cost of 52.4 HI. of olive oil ? 8. How many kilos will a hektoliter of water weigh ? 9. Mt. Blanc is 4800 m. high. How many feet high is it ? 10. Find the weight in kilograms of a cubic foot of gold, if gold is 19.5 times as heavy as water. 11. A liter of mercury weighs 13.596 Kg. How many cu m. of mercury weigh 1 g. ? 12. If marble is 2.7 times as heavy as water, what is the weight of a block 2 m. by 1 m. by 1 m.? 13. What will be the duty on 150 liters of wine at 50^ a gallon ? 17 ^58 SCHOOL ARITHMETIC. 14. A lot of land containing 62.5 ares is sold for 25 cents a square meter. For how much does the lot sell ? 16. The capacity of a tank is 60 cu m. How long will it take a pipe to fill it at the rate of 3.8 Dl. a minute ? 16. How many steps 2.5 ft. long will a man take in walk- ing a kilometer ? 17. A silver five-franc piece weighs 25 g., and is composed of 9 parts of pure silver and 1 part of pure copper. AVhat is the total weight of silver in 200 five-franc pieces ? 18. My cistern is 2 m. 5 dm. long, 2^ m. wide, and 2 m. deep. How many gallons of water will it hold ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 381. 1. If a rainfall is 700 liters per hectare, it is how many inches an acre ? 2. If a ream of paper is .7872 Dm. in thickness, what is the thickness in millimeters of a single sheet ? 3. At $7 a metric ton for coal, what will the coal for a week cost if 30 kilos are burned each day ? 4. Air being .001276 as heavy as an equal volume of water, what is the weight of the air in a room 6 m. long, 4 m. wide, and 3 m. high ? 5. A certain vessel when empty weighs 2.7 Kg., and when full of water weighs 4235 Dg. What does it weigh when full of petroleum, which is .7 as heavy as an equal volume of water ? 6. If it costs $5 to travel 384 Km. by rail, what is the rate of fare in cents per mile ? 7. A hektoliter of potatoes to the are is equivalent to how many bushels to the acre ? 8. What is the distance in miles around the earth through the poles, if the distance from the equator to the pole is 10,001,887 meters ? 9. The distance between two places on a map is 12.5 centi- meters. What is the actual distance in miles, if the scale of the map is 1 to 6000 ? GENERAL REVIEW WORK. ORAL EXERCISES. 382. 1. A man built 11 rods of fence in 3 days. How much did he build in 2 days ? 2. How much will a dozen lemons cost, at the rate, of 2 for 5 cents ? 3. If 4 apples cost 5 cents, how many apples can be bought for a quarter ? 4. Bought eggs at the rate of 2 for 3 cents, and sold them at the rate of 2 for 5 cents, gaining 66 cents. How many dozen did I buy ? 5. A lady gave ^ of lier money for a shawl, and ^ of it for a dress, and had $5 left. How much had she at first ? 6. How many pigs can be bought for 130 when 6 pigs cost $20 ? 7. If 7 bats cost $3.50, what will a dozen cost at the same rate ? 8. I gave 136 for shoes, at the rate of $9 for 3 pairs. How many pairs did I get ? 9. How many books can be bought for $12.50, at the rate of 3 books for $3.75? 10. If f of a barrel of sugar costs $8, what will 5 barrels cost ? 11. If I of the pupils in a school are girls, and there are 20 boys, how many pupils in the school ? 12. A is 30 years old, and f of his age is | of his wife's age. How old is his wife ? 13. B has 8| acres of land, and A has ^ as many. How many acres have both ? 260 SCHOOL ARITHMETIC. 14. If 12 cows are worth 3 horses, and 5 horses are worth 10 yoke of oxen, how many oxen are worth 4 cows ? 15. If 4 bbl. of flour cost $14.40, what will } of a barrel cost? 16. Tom spent f of his money for a cap, and with the re- mainder bought a dozen apples at the rate of 2 for five cents. How much money had he at first ? 17. If a 10-foot pole casts a shadow 18 feet long, what is the length of a pole that casts a shadow 72 feet long ? 18. D has $50 more than E, and $30 is | of what they both have. How much has E ? 19. One half of a certain number is 12 more than ^ of it. What is the number ? 20. Two thirds of A's money equals f of B's, and both have $8.80. How much has B ? 21. How many square inches in one side of a two-foot cube ? 22. If 5 men can build a wall in Gf days, how long would it take 11 men ? 23. Fifteen men can build a house in 10 days. How many men can build it in 3^ days ? 24. A can cut a field of grass in 6 days, B in 8 days, and C in 12 days. In what time can they together cut it ? 25. A man bought 2 hats for $5, and one cost f as much as the other. What was the cost of each ? 26. A suit of clothes cost $35. The pants cost ^ as much as the coat, and the vest ^ as much as the coat. What was the cost of each ? 27. Two fifths of Kobert's money equals f of Andrew's, and Andrew has $2 more than Robert. How much has each ? 28. Two men can husk a field of corn in 6 days. If one of them alone can husk it in 10 days, how long would it take the other ? 29. What will be the cost of painting a sign-board 20 feet long and 9 feet wide, at the rate of 3 square yards for $1 ? REVIEW WORK. 261 30. By selling a cow for $45 I gained as much as she cost me. Find the cost. 31. Two pipes can fill a cistern in 8 hours. If one carries twice as much water as the other, in how many hours can each alone fill it ? 32. Harry has 3 times as many marbles as Warren, and Earl has 5 times as many as Warren. If all have 30, how many has each ? 33. A man having a lot 8 rods square, divided it into 4 equal lots. After selling one of the lots, what part of an acre did he have left ? 34. A is 20 yards behind B, and runs 9 yards while B runs 8. How far will B run before he is overtaken ? 35. By selling a stove for $34, a merchant gained ^V ^^ what it cost. How much did it cost ? 36. Mr. S has 20 lots, each 8 rods long and 2 rods wide. How many acres has he ? 37. If 36 men can earn $a in 6 days, how many men can earn $Sa in 24 days ? . 38. The sum of two numbers is 25, and their difEerence is 13. What are the numbers ? WRITTEN EXERCISES. 383. 1. Sold 42 horses for $5600, thereby gaining $13.33^ on each. What was the cost a head ? 2. If a horse can trot 64 rods in half a minute, in what time can he trot 2^ miles ? 3. A telegraph line is 200 miles long. If the poles are 150 feet apart, what is their value at $1.33^ each ? 4. How many boards 14 ft. long and 15 in. wide would be required to cover the sides of a shed 28 ft. long, 21 ft. wide, and 10 ft. high ? 5. A street one fourth of a mile long and 48 feet wide is to be graded down half a yard. What will be the cost of excavating at $.12^ a cubic yard ? 262 SCHOOL ARITHMETIC. 6. How many boards 12 ft. long, 1 ft. wide, and one inch thick, can be made from a square log whose length is 24 feet, and whose ends are 2 ft. square ? 7. A cistern is 22 ft. long, 14 ft. wide, and 8 ft. deep. How many gallons in it when it is | full ? 8. A log 2 ft. square is 24 ft. long. How many planks 12 ft. long, 12 in. wide, and 2 in. thick can be sawed from it? 9. An encyclopedia averages 764 pages to the volume, and 126 lines to the page. If the entire work contains 1443360 lines, how many volumes are there ? 10. How many gold dollars weigh as much as a silver dol- lar ? 11. A silversmith paid $.60 an ounce for 5 lb. of silver, and made it into chains weighing 1 oz. 4 pwt. each, which he sold at $1.50 apiece. How much did he receive for his labor ? 12. When it is 9 a.m. on the meridian of Greenwich, what is the time on the 180th meridian ? 13. Divide the product of 3^^ and 7^ by their sum. 14. A druggist bought a pound of calomel for $6, and sold it at the rate of 5 grains for a cent. What was his profit ? 15. 440 lb. of copper was made into wire, a yard of which weighed 4 oz. What was the length of the wire ? 16. If a cubic foot of granite weighs 250 lb., what is the weight, in tons, of a block 6 ft. long, 4 ft. wide, and 3 ft. thick ? 17. If steel rails weigh 180 lb. per yard, how many tons will be required to lay 2 miles of railway, one of which has a double track ? 18. If a wheel is 4 ft. in diameter, how many revolutions will it make in going a mile ? 19. How many horses can be supplied with shoes from 10 lb. of iron, if 8 oz. make one shoe ? go, The distance over a hill is 60 rods, and the distance REVIEW WORK. 263 through on a level is 40 rods. If 81 posts are required for a fence from one side to the other on the level, how many would be needed to build a fence over the hill ? 21. Explain the effect of removing the cipher in each of the following : 750, 025, .250, .025. 22. llow often can a 3-bushel bag be filled from a bin con- tainiDg 181 bu. 16 qt. ? 23. A man bought 500 fence-boards, each 16 ft. long and 6 in. wide, at $14.50 per M, and 50 posts at a quarter apiece. Find the total cost. 24. How many perches in a pile of stone 45 ft. long, 30 ft. wide, and 4 ft. high ? 25. A rectangular solid standing on a base 6 in. square is 5J ft. high. How many cubic feet does it contain ? 26. A trough 4 ft. in length and 2 ft. square is full of water. What is the weight of the water, if a cubic foot of it weighs 1000 ounces ? 27. A wheel of a bicycle travels 235.62 yd. in making 50 revolutions. Wlnit is the radius ? 28. The area of a triangular field is 9 A. 65 sq. rd., and the length of its base is 70 rods. What is the altitude ? 29. A floor is 24 feet wide at one end, 16 ft. at the other, and its area is 40 sq. rd. What is the length ? 30. A roof is 50 ft. long and 20 ft. wide on each side. What will be the cost of roofing it at $8.75 per hundred sq. ft.? 31. How many yards of lining | of a yard wide will be required to line 5 coats, each containing 4| yards of material 1} yd. wide ? 32. Willie lost y^ of his marbles less 15 ; he then gave f of the remainder and 8 more to John ; he had 32 remaining. How many had he at first ? 33. The point of a minute hand moves 4 inches in two Jipurs. What is its length ? 264 SCHOOL ARITHMETIC. 34. Mrs. Brown sold ^g- of her turkeys less 17 ; | of the remainder less 3, and then had 39 remaining. How many had she at first ? 35. How many quart, pint, half pint, and gill bottles, of each an equal number, can be filled from a vessel containing 5 gal. 2 qt. 1 pt. ? 36. I have a lot the length of whose sides is 56, 84, 98, and 112 ft. respectively. I desire to enclose it with a fence four boards high, using boards 14 ft. long. How many will it take ? 37. A passenger train 65 yd. long, and running at the rate of I of a mile a minute, met a freight train moving ^ as fast. They passed each other in 5 seconds. How long was the freight train ? 38. If 9 men can do as much work as 15 women, and 70 women do as much as 25 boys, and 12 boys do as much as 36 girls, how many men would it take to do the work of 50 girls ? 39. Find the difference between the largest fractional unit in decimals and the largest fractional unit in common frac- tions. 40. If the strips are laid lengthwise, how many yards of carpet 27 inches wide must be bought for a room 18 feet long and 16 feet wide ? 41. Five minutes after two ships pass each other the distance between them is 2160 rods. One of them sails at the rate of 35 miles an hour. What is the hourly speed of the other ? 42. A bin is 25 ft. long and 20 ft. wide. The oats in the bin is an inch deep, and a bushel of it weighs 32 pounds. How long will it last 5 horses, if one horse eats 18 pounds a day? 43. From 11 a.m. to 1.30 p.m. my watch gained 10 seconds. In how many days did it gain 10 minutes ? REVIEW WORK. 265 SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 384. 1. When a certain number is divided by 7, the quo- tient is equal to tlie ratio of 5 to 2. What is the number ? 2. If ^ of an acre is worth f of $100, how many thirds of an acre can be bought for f of 175 ? 3. How many fractional units in .375 ? Give three answers. 4. If 8 yards of cloth 1^ yd. wide will make Lucy a dress, how many yards of 48-inch goods will make her a dress ? 5. How many fractional units equal to .25 are there in 2| ? 6. In an orchard there are 18 rows of trees. Between every two rows of trees there are 8 rows of potatoes. If the average yield of a row is 12 bushels, what is the value of the potato crop at $.75 a bushel ? 7. Is I a fraction, or a number, or a unit ? May it be all three ? Why ? 8. If marble weighs 2.8 times as much as water, bulk for bulk, what is the weight of a block of marble 12 ft. 9 in. long, 4.5 ft. wide, and 3.2 ft. thick ? 9. The average depth of a certain rainfall was .25 of an inch. What weight of water fell on .a lot 40 ft. by 60 ft., if 1000 oz. of water measures a cubic foot ? 10. If a train loses f of an hour in running 80 miles at 18 miles an hour, in how many hours does it run 360 miles when running at the regular speed ? 11. If 4 horses can draw 80 bushels of wheat, 00 lb. to the bushel, on a wagon whose weight is 000 lb., how many bushels can 2 horses draw on a wagon that weighs 600 lb. ? 12. Does the expression .^ mean anything ? If so, what ? 13. A cubical box is 4.8 m. on an edge. How many hektoliters of oats will it hold ? How many bushels ? 14. How many bricks 20 cm. long and 10 cm. wide will it take to pave a sidewalk 2.4.m. wide and 1.4 Km. long ? 15. At 43^ a cubic meter, what will it cost to macadamize a road 1 Km. long and 7 m. wide, to the depth of 46 cm. ? PERCENTAGE. 385. 1. Warren had 100 cents and spent 1 cent. What part of his money did he spend ? 2. Boyd had $1 and spent 2 cents. AVhat part of his dollar did he spend ? 3. Emma misses 5 words in a hundred. What part does she miss ? '4. A man had 400 sheep, and 10 of every hundred were killed by dogs. How many were killed ? 6. A farmer having 50 hogs lost 10 hundredths, or 10 per cent of them. How many did he lose ? 6. If yJ^, or 7 per cent, of the pupils in a school of 100 are absent, how many are absent ? How many are- present ? How many per cent are present ? 7. WhAt is 6 per cent, or .06, of 1100 ? Of 1300 ? Of 1500 ? 386. Per cent means hundredths. Thus, 5 per cent of a number means 5 hundredths of it. Note. — The phrase " per cent " is from the Latin per centum, by the hundred. 387. The symbol ^ stands for the words per cent, and means either per cent or hundredths. Thus, %% is .06, and is read 6 per cent or 6 hundredths. 1. How do we express cents ? Hundredths ? 2. Since per cent is so many hundredths, how may we express per cent 9 3. Is there any difference in value between 20^, .20, and^ ? 4. Express VZ^ per cent in three ways. ^ per cent in 3 ways. $. Explain how 5 per cent, or 5^ = .05, or -^-q, or ^. PERCENTAGE. 267 6. Express 225 per cent in two ways ; 90 per cent ; 100 per cent. 388. Change to per cent : 1. .06 = 2. .15 = 3. .25 = 6^. 7. Th = 8. ^\ = 9. m = 7^. 13. i = 14. f = 15. i = 50^. 19. 20. 21. .OOi = .00| =f .005 = 4. .39 = 10- 2 = 16.* = 22. .0005 = 5. 1.35 = 6. 3. = 11. 1.375 = 12. .01 = 17. t = 18. A = 23. 24. .2775 = .0325 = i^. 25. 226 hnndredtlis = liow many per cent ? 389. Change to decimal fractions : 1. 6^ =.06 6. i^ =.005 11. 100^ =1.00 16. 162^^ = 1.625 17. 237^^ = 2. 1 3.2 4. 5. 99J^ = 7.J^ = 8. :J^ = 9. f^ = 10. i|^= 12. 227^ = 13. 3000^= 14.^^^ = 15. 6i^ = 18. 266f^ = 19. 267i?^= 390. Change to common fractions : 161 3^^=i 6. 16|^ =100=* ^- ^'^^=i 13. 87J^ =1 35^= 6. 37^^ = 10. 150^= 14. .283 = 7. 83^^ = 11. 180^= 15. .375 = 8. 116|^= 12. 225^= 16. 233^^ = 75^ = 391. The following per cents and their equivalents are so often used that pupils should be able to give their values in the different forms at sight. Drill Table. 1 = 100^. 1=75^. i = 12J^. A = 6|! i= 50^. i = 20^. f = 37^^. jV = 5^ i = 33^^. 1 = 60^. f = >->H^. A = 4i^ 1 = c^Hfc. i = mfo. 1 = 871^. A = 3^ i= 25^. i = m^- 1 IT = 10 f. iijs = 1^ 268 SCHOOL ARITHMETIC. 392. The result obtained by taking a certain per cent (meaning a stated number of hundredths) of a number is called the Percentage. The name Percentage is also applied to that portion of arithmetic which involves the taking of per cents. 393. The number of which the per cent is taken is called the Base. 394. The per cent taken is called the Rate. Thus, in 6^ of 50 = 3, the Q% is the rate. The 6 alone is usually called the rate per cent. 395. Using ih^ first letters of the vford.^ percentage, base, and rate to represent the numbers called by these names, we readily express in the form of equations the relations that these numbers bear to each other. From the definition of percentage, p = hr. (1) Dividing both members of (1) by b, | = r, orr=f. (2) Dividing both members of (1) by r, 6 = 1. (3) r ^ ' In (2) we have a 'product {percentage) and one of the factors (base) to find the other /ac/or (ra^e). What have we in (1) ? In (3) ? What rela- tion, then, do the base, rate, and percentage bear to each other ? 396. One hundred per cent of a number is the number itself. Thus, 100^ of 50 is 50. 397. To fiud a given per cent of any number. 1. What part of a number is 10 hundredths of it ? 10^ of it? PERCENTAGE. 269 2. How many hundredths in 1 ? How many cents in $1 ? How many per cent in 1 ? 3. What per cent of a number is ^^ of it ? J of it ? All of it? 4. Since Ifo of a number = j^^ of it, 10^ = -^jPq or Vtt of it, and 50^ = ^%- or ^ of it, what does 100^ equal ? 6. Since any number is 100^ of itself, and the base is the number of which the per cent is taken, the base equals what per cent ? 6. If you lose 4^ of your money, what per cent do you have left ? 100^ - 4^ = ( ). What is : 7. 1^ of 50 ? 13. 5^ of 50 ? 8. 10^ of 50^ ? 14. 100^ of 50 ? 9. 12^ of 40 ? 15. 50^ of 100 sheep ? 10. 161^ of 66 ? 16. 75^ of 120 horses ? 11. 33J^ of 60 ? 17. 83i^ of 24 quarts ? 12. 66f^ of 72 ? 18. 100^ of 100 bushels ? WRITTEN EXERCISES. 398. 1. Find 20^ of I960. (a) (b) .20^ = .20 or i 100^ of $960 = m of $960. .20 of $960 = $192. .-. 1% of $960 = yio of $960, or $9.60, Or and 20^ of $960 = 20 x $9.60, or $192. i of $960 = $192. Have the pupil solve the above problem by substituting in the equation p = ir. 2. Find j^ of $800. - (a) (b) 100$^ of $800 = $800. 1% of $800 = $8.00. .-. Ifc of $800 = T^TT of $800, or $8, 1% of $800 = f x $8.00,or $6. and i% of $800 = ^ x $8, or $6. 2Y0 SCHOOL ARITHMETIC. Suggestion. -=-The pupil should be tauglit to select and apply the method that is most convenient in each particular problem. 3. 12fc of $240.50. 7. 325^ of 55.2 rods. 4. 7bfo of 11286.45. 8. 133^ of 17824. 6. 16f^ of 120 sheep. 9. 90 fo of .0577. 6. 1^ of 1200. 10. li^of^V 11. What is tlie difference between ofo of 1120, and 120^ of $5? Query. — Is the percentage a factor or a product ? 12. How much had I left after paying out 15^ of my $3000 ? 13. The owner of a threshing machine charges 2^^ for threshing a crop of 275 bushels. How much does he get ? 14. A man who owed $1750 was able to pay only 39 per cent. How many dollars could he pay ? 15. If a foot of rope shrinks 4|^^ when wet, how much would 500 feet of rope shrink ? 16. If A^s income is $1000 a year and he saves 25^ of it, how much Avill he save in 25 years ? 17. I bought 40 head of cattle for $166()|, and sold them at a profit of 3^^. What did I make ? 18. Dickson and Tribby engaged in business, each with $1250. Tribby gained 33|^^ of his capital, and Dickson 37^^ of his capital. How much did Dickson gain more than Tribby ? 19. A farmer raises 500 bushels of grain, of which 29^ was wheat, 47^ rye, and 22^ oats. How many bushels of each did he raise ? 20. I own } of a mill and sell 33^^ of my share. What part of the mill do I sell ? 21. A barrel that will hold 42 gallons is 66|^ full. How many gallons does it contain ? 22. A man has $1500. He spends 6Gf^ of it, and gives away 6^ as much as he spends. How much has he left ? PERCENTAGE. 271 ft3. If pure air consists of 20.0265^ of oxygen gas and 79.9735^ of nitrogen, how much oxygen in 1500 cu. ft. of air ? How much nitrogen ? 24. If 25^ of a certain ore is melted, and If^ of the metal is silver, how much silver in a ton of the ore ? 25. A certain lot of cane has 89^^ juice, and the juice con- tains 11.4^ sucrose. How much sucrose in 5 tons of cane ? 26. A man has a library of 1600 volumes. 14^ are biog- raphy, 62^ are history, and 83^^ of the remainder are fic- tion. How many volumes of fiction in his library ? 27. A maltman malts 1500 bushels of barley, which in the process increases 12|^^. How many bushels of malt has he ? 28. An agent sells 25 bicycles at $60 each, and is allowed 15^ of the receipts. How much does he make ? 29. Water is composed of 88.9^ of oxygen and 11. Ij^ of hydrogen. How many pounds are there of each in 1 cu. ft. of water ? 399. To find what per cent one number is of anotlier. 1. 12 is what part of 24 ? How many hundredths of 24 ? What per cent of 24 ? 2. 15 is what part of 45 ? How many hundredths or per cent of 45 ? 3. $5 is what part of $25 ? What per cent of $25 ? What is the ratio of $5 to $25 ? What per cent of : 4. 24 is 12 ? 8. $25 is $10 ? 12. 1 is J ? 5. 7 is 21 ? 9. 30 yd. is 20 yd. ? 13. 9 is f ? 6. 8 is 18 ? 10. 100^ is 50^ ? 14. f is i ? 7. 30 is 5 ? 11. 5 is 5 ? 16. f is f ? 16. A boy having 10 cents gave his sister 5 cents ? What .per cent of his money did he give away ? 17. A teacher whose salary is $1250 spends $1000. What per cent of his salary does he spend ? 272 SCHOOL ARITHMETIC. 18. A's money is twice B's. What per cent of A's money is B^s ? What per cent of B's is A's ? WRITTEN EXERCISES. 400. 1. 9 is what per cent of 30 ? 30: .-. 1 . and 9 : (a) = 100^ of 30. = ^h of 100^, = 9 X ^1%, or r or ^H of 30, ■ 30^ of 30. (c) = !=,% = .30, or 9 is and 30^. (b) = .30, of 30, or 30^. 2. If a miller takes 4 qt. for toll from every bushel, what per cent does he take for toll ? 3. 3^ is what per cent of 20 ? 4. Edward bought 2 lb. of candy. He ate 4 oz. and gave away ^ lb. What per cent of his candy did he have left ? 5. From a cask containing 66^ gal., 26.6 gallons were drawn. What per cent of the whole remained in the cask ? 6. If gold coin is 9 parts pure and 1 part alloy, what per cent is pure ? 7. An attorney charges $68.75 for collecting $550. What per cent does he charge ? 8. Mr. S paid $45 for the use of $750. What per cent did he pay ? 9. What per cent is a pound avoirdupois of a pound troy ? 10. 25^ of I of a number is what per cent of f of it ? 11. A merchant buys 5 gross of pens, and sells 5 dozen. What per cent of them does he sell ? 12. Frank has $10 and Kay $4. What per cent of Ray's money is equal to Frank's, and what per cent of Frank's money equals Eay's ? 13. What per cent of his time does a man sleep who sleeps 7 J hr. out of 24 ? 14. If 7 lb. of a certain article lost 4 oz. by drying, what per cent of its original weight was water ? PERCENTAGE. 273 15. In a mixture of copper and zinc, the copper is to the zinc as 3|- to 2^. Express the percentage of each ingredient in the mixture. 16. If carpeting, which should be one yard wide, is only 34J inches wide, what per cent should be deducted from the price ? 17. If I sell ^ of my interest in a business to one man and i of it to another, what per cent have I remaining ? 18. If to 23 gallons of alcohol 2 gallons of water are added, what per cent of the mixture is water ? 19. In an examination 50 questions were asked, of which A answered 45, B 35, and C 18. What per cent did each make ? 20. If B's age is 33Jj^ more than A^s, A's age is what per cent less than B^s ? 21. If a gold ring is 18 carats fine, what per cent of it is gold? 22. If a piece of bronze weighing 7f pounds contains 6.5 pounds of copper, what per cent of the bronze is copper ? 23. Of 25320 votes cast in a certain city, A received 11394 and B the remainder. What per cent did B receive ? 24. An army paymaster receives $125000, but embezzles 15000 of it. What per cent of the money does the govern- ment lose ? 25. In a certain year (1898) 501,066,681 passengers were carried on railways in the United States, and 221 were killed. What per cent were not killed ? 26. The area of North America is 9,350,000 sq. mi., and the area of the Missouri-Mississippi basin is 1,250,000 sq. mi. What per cent of the area of the continent is drained by these rivers ? 401. To find a number ivhen a certain per cent of it is given. 1. 15 is 3 times what ? | of what ? .03 of what ? 3 per cent of what ? 18 274 SCHOOL ARITHMETIC. 2. 24 is f of what ? .06 of what ? 6 per cent of what ? Find the Dumber of which : 3. 10 is 10^. 6. 30 is 12^^. 9. f is 50^. 4. 60 is 25^. 7. $150 is 50^. 10. 50 is Ifo. 5. 36 is 33i^. 8. 1 is 100^. 11. f is 5^. 12. Of passengers aboard a ship 16f^ of the number, or 800 persons, were lost. What was the number of persons aboard ? 13. ^ of f of a yard is 20^ of what ? 14. A bin holds 60 bushels of wheat, which is 3^ of a farmer's crop. How many bushels did he have ? WRITTEN EXERCISES. 402. 1. 30 is 5^ of what number ? (a) (b) 5^ of a number = 30. 5^, or ichmond, %, Value received. Due, J'o/m 3)oe. When Richard Roe indorses this note, John Doe takes it to the bank, which loans him 1100 less the interest (dis- count) for 61 days at the legal rate. This discount is $1.02, hence Mr. Doe gets 198.98 ; but in two months he must pay the bank 1100. 1. By indorsing the note Richard Roe binds himself to pay it if John Doe does not ; and since he indorses tiie note to enable the latter to secure the loan, that is, to accommodate the maker, he is called an accommodation indorser. 2. If this note is not paid May 15, it goes to protest. 3. No interest is specified in notes of this kind, but the legal rate is charged. WRITTEN EXERCISES. 477. 1. $1000. New York, Dec. 31, 1900. Two months after date I promise to pay J. A. Greene, or order, one thousand dollars, value received. J. B. MUKDOCK. This note was discounted Dec. 31 at 6^, the legal rate. Find the proceeds. BANKS AND BANK DISCOUNT. 319 The date of maturity = Feb. 28, 1901. The term of discount = 59 days. The discount = interest on $1000 for 59 da. = $9.83. The proceeds = $1000 - $9.83 = $990.17. j^OTE. — In Pennsylvania, and in other states where the day of discount is included, the term of discount in the above note would be 60 days ; where grace is allowed, it would be 3 days more. In Boston, and in some other places, when the time a note has to run is expressed in months, the term of discount is computed for this number of months, and not for the exact number of days. $000. Philadelphia, Pa., Feb. 28, 1898. Three months after date I promise to pay to William Post, or order, six hundred dollars, value received. W. E. Sankey. Discounted Feb. 28. Find proceeds. Day of maturity = May 31 (the 28th being Saturday, and the 30th a legal holiday). Term of discount = 93 days. 3. 14000. Richmond, Va., June 30, 1899. Sixty days after date I promise to pay to J. F. Guffey, or order, four thousand dollars, value received. W. R. Ford. Discounted June 30, at 6^. Find proceeds. 4. $2500. Columbia, S. C, Aug. 31, 1901. Three months after date I promise to pay to the order of J. F. Bunn, two thousand five hundred dollars, value re- ceived. AV. H. McKelvey. Discounted Aug, 31^ at 6^. Find the proceeds. (Grace allowed.) 320 SCHOOL ARITHMETIC. 5. $3000. Hartford, Conn., May 4, 1903. Sixty days after date I promise to pay J. M. Clark, or order, three thousand dollars, value received. R. J. Stoney, Jr. Discounted May 4. 6. $750. Frankfort, Ky., April 7, 1900. Three months after date I promise to pay to the order of W. M. Gill, seven hundred fifty dollars, value rec'd. Geo. H. Welshons. Discounted April 7. (Grace allowed, and day of discount included.) 7. $1500. Baltimore, Md., March 30, 1902. Two months after date I promise to pay to Howard Welsh, or order, one thousand five hundred dollars, value rec'd. W. S. Finney. Discounted Mar. 30. (Day of discount included.) 8. $1000. Wilmington, Del., May 18, 1901. Sixty days after date I promise to pay William Pollock, or order, one thousand dollars, value received. H. W. Walker. Discounted May 18. 9. $500. Charlotte, N. C, April 15, 1900. Three months after date I promise to pay to the order of Anna Bamford, five hundred dollars, value received. George Dewey. Discounted April 15. (Grace allowed.) BANKS AND BANK DISCOUNT. 321 10. $10,000. Savannah, Ga., May 25, 1901. Two months after date I promise to pay to the order of W. H. McCleary, ten tliousand dollars, value received. John D. Miller. Discounted May 25. (Grace allowed.) 11. $400. Nashville, Tenn., May 4, 1900. Sixty days after date I promise to pay to Wm. H. McGary, or order, four hundred dollars, value received. Samuel Harper. Discounted May 4. (Grace allowed.) 12. $2000. Austin, Tex., Jan. 15, 1901. Sixty days after date I promise to pay to the order of H. M. Jones, two thousand dollars, value received. W. W. Ulerich. Discounted Jan. 15, 1901. (Grace allowed.) 13. $6000. Little Rock, Ark., May 9, 1901. Ninety days after date I promise to pay to the order of R. S. Latham, six thousand dollars, value received. J. D. Anderson. Discounted May 9, 1901. (Grace allowed.) DISCOUNTING NOTES. 478. John Mason bought a lot from Richard Adams for $2000, but not having the ready money agreed to pay the $2000 in sixty days, together with interest at 6^. He gave the following note : 21 322 SCHOOL ARITHMETIC. $2000. ^^Ew York, March 16, 1900. Sixty days after date., for value received, I promise to pay to the order of Richard Adams, two thousand dollars, with interest at 6^. JoHN" Mason. Needing money, Richard Adams took the note to a bank and had it discounted the same day, March 16, transferring the note to the bank by indorsement. The bank paid to Mr. Adams the maturity value, less Gfo interest thereon for 60 days. 1. The note matures May 15, at which time John Mason must pay the bank $2020, the maturity value. The bank retains the interest on this at 6% for 60 days, or $20.20 ; hence the owner of the note, Mr. Adams, gets $2020 - $20.20 = $1999.80. 3. So far as the bank is concerned, this process of discounting notes purchased by way of discount — being in effect a mode of lending money — is essentially the same as that of lending money on an indorsed promissory note ; but there is the important distinction that the indorser is not now an accommodation indorser ; he is the oivner of the note and is the party who receives the money from the bank. 3. Notice, also, that the note in the latter case usually draws interest, and is frequently not discounted on the day of its making. When it is discounted at a subsequent date, the discount is reckoned on the maturity value for the time from the day of discount to the day of maturity. , _ WRITTEN EXERCISES. 479. 1. $800. Richmond, Va, June 17, 1900. Three months after date I promise to pay to the order of L. F. Graham, eight hundred dollars, value received, at the First National Bank, with interest at 6^. Jno. B. Head. Discounted July 12, at 6^. Find the proceeds. Date of maturity = Sept. 17. Maturity value = $800 + interest for 3 mo.== $812. Term of discount = 67 days {i.e., 19 in July, 31 in Aug., 17 in Sept.). The discount = the interest on $812 for 67 days at 6% ~ $9.07. The proceeds = $812 - $9.07 = $802.93. (Many banks in Virginia charge for day of discount.) BANKS AND BANK DISCOUNT. 323 2. $300. Pittsburg, Pa., July 10, 1900. Sixty (lays after date I promise to pay to J. F. Miller, or order, three hundred dollars, with interest at 0^, value received. Lewis Moran. Discounted Aug. 6, at 6^. Find the proceeds. (Day of discount included.) 3. 1500. Dayton, 0., Sept. 29, 1901. Two montlis after date I promise to pay to James II. Piatt, or order, five hundred dollars, value received. C. IIORNUNG. Discounted Sept. 29, at iWc. Find proceeds. 4. $1000. Jackson, Miss., Dec. 31, 1900. Two months after date I promise to pay to Thomas E. Boyd, or order, one thousand dollars, witli interest at (j^/>, value received. S. C. Hepler. Discounted Dec. 31, at 6i. Find proceeds. (Grace allowed.) 5. James 11. Ljifferty wishing to borrow some money for two montlis from the Atlas National Bank of Boston gives a promissory note for 11800, George Gosser being the accom- modation indorser. Draw the note, dating it May 4, 1900, and find the proceeds. 6. Prepare a 90-day note for $250 on which you can obtain a loan from a bank in your locality. Compute the bank's charge for discounting it in accordance with local practice. 7. , 1 have a note for $5000 dated June 5, and payable three montlis after date, with 6^ interest. How much would a bank in your locality give me for the note on July 3 ? 324 SCHOOL ARITHMETIC. 8. For what sum is a 60-day note given wlien a bank dis- counting it at 8^ gives the maker $725, allowing days of grace ? The discount on |1 for 63 days at 8^ - |.014. The proceeds of a $1 note = $1 - $.014 — $.986. The face required = $725 -f- .986 = $735.29. 9. If yon wish to procure $1000 from a bank for ninety days at 6^, for what sum must you write the note ? 10. The National Union Bank of New York (which counts neither grace nor day of discount) loans J. 0. Brown $7500 on a 3-month note dated April 4, J. W. Lee being the indorser. Write the note. PRESENT WORTH AND TRUE DISCOUNT. 480. The Present Worth of a debt is the sum which, if put at simple interest, would amount to tlie debt when due. 481. The difference between the amount of the debt and its present worth is called the True Discount. WRITTEN EXERCISES. 482. 1. Find the present worth and true discount of $6^1 due in 2 yr. 6 mo., if money is worth 6^. (a) Amount of $x for 2i yr. = $621. Amount of $1 for 2^ yr. = $1.15. ^x - $621 -r- 1.15 = $540, the present worth. The true discount = $621 - $540 = $81. (b) In equation (5), Art. 459, a ^ $621 ; r = .06 ; ^ = 2^ ; 1 + r^ = 1.15. Hence p = $621 -f- 1.15 = $540. STOCKS AND BONDS. 325 2. Find the present worth of $590 due in 1 yr. 6 mo., the current rate of interest being 6^. 3. Find the true discount on i\ debt due in 4 mo. 10 da., the debt being $450, and money being worth 5^. 4. Find the present worth and true discount of $1235 due in 1 yr. 7 mo. 12 da., the current rate being 6^. 5. A man buys flour for $2840 on six months' time. If payment is made at the time of purcliase, how much should be deducted from tlie bill, money being worth 6^? 6. Wluit is the difference between the true discount of $G40, due in 1 yr. 3 mo. 15 da., and the interest on the same amount for the same time, money being worth Q^ ? 7. Wiiat sum must I put at interest at 8^ to liquidate a debt of $1250 due 2 yr. 6 mo. hence ? 8. I am offered $8250 cash for my farm by one man, and another offers me $8580 in 4 months, without interest. Which offer is the better, if money is worth 6^ ? ■ 9. What is the difference between the bank discount and the true discount, each at 5i, on a note for $654 due in 90 days ? Show that the bank discount equals the true dis- count plus the interest on the true discount. STOCKS AND BONDS. 483. When a numt?er of persons wish to engage in any extensive business, they usually form themselves into an association called a Stock Company. 484. The sum of money subscribed by the members of the company to inaugurate the business is called the Capital Stock. 485. The capital stock is usually divided into a definite number of shares of a specified value, and is issued in the form of certificates of stock, each stating that the person named therein owns so many shares, 326 SCHOOL ARITHxMETIC. 486. The value of a share named in the certificate of stock is called the Par Value. The stock is usually divided into shares of the face or par value of $100 each. 1. If the value of each share is $100, what is the par value of 10 shares ? 100 shares ? 487. The price at which stocks are selling in the market is called their Market Value. 1. When stock sells at 5^ above par value, what is the market value of a $100 share ? 488. A stock is said to be at a premium, or ahove par, when it sells for more than its face value; it is said to be at a discount, or below par, when it sells for less than its face value. Thus, if a stock is quoted at 107, $100 stock sells for $107, and the stock is at 1% ijremium ; if it is quoted at 93, $100 stock sells for $93, and the stock is at 1% discount. 1. If stock is quoted at 106, what is the rate of premium ? Why do stocks vary in price ? 489. If the company makes more than its expenses, part or all of the surplus is divided among the stockholders as Dividends. The dividends are usually expressed as a cer- tain per cent of the par value, but sometimes as a certain number of dollars a share. 1. If a company declares a dividend of 10^, how much does the owner of a $100 sliare get ? 490. A written obligation under seal securing the pay- ment of a sum of money at a specified time, and bearing a certain rate of interest, is called a Bond. 1. Which would be preferable to owu, $1000 of stock in a company, or one of its $1000 bonds, if each pays 5^? 491. Bonds are issued by governments (local, state, or national), or by stock companies, for the purpose of effecting loans. They are of two kinds — registered bonds and coupon bonds. STOCKS AND BONDS. 327 492. Registered bonds are recorded by their numbers and the names of the persons owning them, and cannot be trans- ferred without a change in the record kept by tlie party issuing them. Coupon bonds have attached to them coupons, or certificates of interest, which are detached as interest becomes due, and presented for payment. 493. Bonds may be bought and sold in the market in tlie same manner as stocks, and are designated in quotations by the name of tlie company or government issuing them, and the rate of interest they bear, with the date of maturity, and whether registered or coupon. Thus, "U. S. 4's, coup., 1925" means United States coupon bonds bearing 4 per cent interest, the principal payable in 1925. Note. — Bonds pay interest on their /ace value at a fixed rate, hence thkir market price does not affect tlie interest they yield. The income from stocks is variable, as it depends upon the prosperity of the busi- ness. 494. Persons who buy and sell stocks and bonds are called Stock Brokers, and their commission is called Brokerage, The brokerage is usually ^fo of the par value. This is cliarged for huying and also for selling. 1. If a broker charges |^ for selling 10 shares of stock, how much w^U be the brokerage ? 495. The market values of stocks and bonds as given daily in the newspapers are called Stock Quotations. A quotation of 127 means that $100 of stock is selling for $127. In this case the seller receives $127 — $^, or $126| for each share, and the buyer pays $127 + $-«, or $127g, for each share, provided the deal is made through a broker. 1. What will a seller receive from his broker for one share of stock sold at 11^, brokerage ^^o ? What for 10 shares ? 2. What will a buyer have to pay for one share of stock purchased at 97J, brokerage ^^ ? What for 10 shares ? 328 SCHOOL ARITHMETIC. 496. The following are quotations of U. S. bonds in the market of June 7, 1900 : BID. ASKED. U. S. 3's, reg 109^ 109| U. S. 3's, coup 109i 109f 4'8, reg., 1907 114^ 115 4's. coup., 1907 115i 116 4's, reg., 1925 134^ 135 4's, coup., 1925 134i 135 5's, reg., 1904 113^ 114 5's, coup., 1904 113i 114 497. The following are from the stock quotations of the same day : STOCKS. HIGHEST. LOWEST. CLOSING. Am. Sugar Ref 116i 114| 1141 Am. Tobacco Co 128 128 128' Brooklyn Rap. Tr 69f 68f 68i C. B. & Q 129i 128i 128i Del. & Hud 113^ 113 113 Ches. & Ohio 27f 27| 27f N. Y. Central 131i 130i 130^ Pennsylvania R.R 130| 129| 128| Northern Pacific 60 59| 59f Southern Railway 12 12 12 1. What would I receive for one share of N. Y. Central at the highest quotation, brokerage ^^ ? What for 10 shares ? 2. What would one share of Pennsylvania R.R. stock cost at the closing quotation, brokerage ^^ ? 10 shares ? How many shares could be bought for $2580 ? 3. What is the difference between the highest and the lowest quotations of Brooklyn Rapid Transit stock on the given date ? WRITTEN EXERCISES. 498. 1. What is the cost of 84 shares of bank stock at 95^, brokerage ^^ ? 1 share costs $95^ + $i = $951. 84 shares cost 84 x |95i = $8011.50. STOCKS AND BONDS. 329 2. What will 40 shares of Northern Pacific stock cost at 59f, brokerage | per cent ? 3. How much must be paid for 125 shares of Delaware and Hudson stock at 113, brokerage ^ per cent ? 4. My broker bought for me 75 shares of C. B. & Q. stock at 129^, charging ^fo brokerage. Find the cost. 5. How many shares of railroad stock at 104|- can be bought for $9450, brokerage ^ per cent ? 1 share costs 1104^ + $i = $105. .-. the number of shares = $9450 -4- $105 = 90. 6. How many shares of Brooklyn Rapid Transit stock can be bought for $3475 if the quotation is 69f , and brokerage |^ ? 7. When Federal Steel is quoted at 33f , how many shares can be bought for $1355, brokerage ^ per cent ? 8. I sent my broker $7938 with which to buy Canadian Pa- cific stock at 94f, brokerage ^^. How many shares did I get ? 9. What annual income will be realized from $4982 in- vested in 4^ stock at 105^, brokerage ^ per cent ? 1 share costs $105^ + $i = $106. The number of shares = $4982 -f- $106 = 47. .-. the income = 47 x $4 = $188. 10. What income will be derived from $4565 invested in railroad stock at 114, brokerage -^^, if the stock pays 6^ dividends annually ? 11. What annual income will a man receive if he invests $12830 in bank stock paying quarterly dividends of 3^, pro- vided the stock is bought at 160f, and no brokerage paid ? 12. I invested $4535 in U. S. 4's at 113^, brokerage i^. What does the government pay me annually ? 13. What sum of money must be invested in 3^ stock at 90, brokerage ^^, to realize an annual income of $1800 ? 1 share yields an income of $3. The number of shares = $1800 -^ $3 = 600. 1 share costs $90 + $i = |90i. 600 shares cost 600 x $90^ = $54075, the investment. 330 SCHOOL ARITHMETIC. 14. A man has 4^ U. S. bonds on which the quarterly interest amounts to $480. He bought the bonds at 105. How much did he pay for them ? 15. A certain bank pays 4^ semi-annual dividends, my share of which amounts to $360. If the stock cost me 152^ and ^^ brokerage, how much have I invested in that se- curity ? 16. A man sold 144 shares of railroad stock at 2^^ below par, paying 1% brokerage. With the proceeds he bought Cotton Oil stock at 36f, brokerage ^fo. How many shares did he get, and how many dollars were left over ? 17. If 250 shares of Southern Railway are bought at 12^ and sold at 13^, brokerage ^^ in each case, what is the profit ? 18. A speculator sold through his broker 240 shares of Am. Sugar stock at 152f. It cost him $27720. What was his gain? 19. Mr. Jones bought 75 shares of Am. Tobacco at 142^, held it a year, and sold it at 210. If he received a 15^ divi- dend and paid ^, L, and G, who were engaged in business three years, made an annual profit of $7200. During the first year D owned -I, L ^, and G ^ of the stock ; during the second year each owned ^ of it ; and during the last year G owned ^ of the stock, while the other half was equally divided between L and D. What was each partner's share. of the total profits? 340 SCHOOL ARITHMETIC. 8. A and B bought a lot for 11500, agreeing to pay 1500 cash and $1000 in 3 months. A pays the $500 cash, and B pays the $1000 three months later. Nine months after the purchase they sold the lot for $1750. What was each one's share of the gain ? The use of $500 for 9 mo. = the use of 9 x $500, or $4500, for 1 mo. The use of $1000 for G mo. = the use of 6 x $1000, or $6000, for 1 mo. Hence the respective gains are proportional to 4500 and 6000. 9. D, E, and F gain in trade $8000. D furnishes $12000 for 6 mo.; E, $10000 for 8 mo.; and F, $8000 for 11 mo. What is each man's share of the gain ? 10. X, Y, and Z hired a pasture for $420. X put in 6 horses for 9 weeks, Y 9 horses for 8 weeks, and Z 12 horses for 7 weeks. How much should each pay ? 11. A, B, and C contribute capital to a business as follows: A $3000 for 12 months, B $4000 for 10 months, and C $5000 for 8 months. Their profits are $900. AYhat is the gain of each ? 12. A, B, and C hired a pasture for $452. A put in 12 horses for 15 weeks, B 80 sheep for 8 weeks, and C 18.cows for 20 weeks. How much should each pay if a cow eats as much as 3 sheep, and 5 sheep eat as much as a horse and 2 sheep ? 13. K rented a house for $720 a year. After 3 months B moved in with him, agreeing to pay his share of the rent. Five months later C also moved in on the same conditions. How much of the $720 did each pay ? 14. In a certain company B has 3 times as much capital as A, and C has | as much as the other two. What is each one's share of a profit of $393 ? 15. M hired a rig for $10 to drive from Salem to Manor, a distance of 10 miles, and back again. At Derby, midway between the two places, he took in L, who agreed to pay his proportional share of the expense if allowed to ride to Manor and back to Derby. How much should L pay ? GENERAL REVIEW WORK. ORAL EXERCISES. 609. 1. If f of an acre of land costs ^ of $120, what will 5 acres cost ? 2. What will 3 ounces of silver cost if half a pound costs $4.20 ? 3. How many 3-inch squares are there in a piece of paper 2 yards long and 3 feet wide ? 4. At two cents a foot, how much will 8 rods of wire cost ? 5. If .3 of John's money equals f of Kate's, and both have 27 cents, how much has each ? 6. How many cubic inches are there in a piece of scantling ^ of yard long and a foot square at the ends ? 7. After spending 4 of his money, Henry had $2 less than half his money left. How much had he at first ? 8. What is the average cost of 25 cows, if 13 of them are bought at $50 a head, and the others at $75 a head ? 9. If 4 of one number is 135, and f of another is 7^, what is the sum of the two numbers ? 10. What is the number whose half exceeds its third by 126? 11. Forty per cent of George's marbles equals ^ of Tom's, and both have 54. How many has each ? 12. A man sold 50 acres of his land, and had 37^^ of it left. How many acres had he at first ? 13. In a mixture of grain there are 50 bushels of oats, 40 of corn, and 15 of wheat. What part of the mixture is each? 14. A is 5 miles ahead of B, and walks 3^ miles while B walks 4. How many miles will B walk before overtaking A ? 34:2 SCHOOL AUITHMETIC. 15. Divide 42 apples between Edna and May so that Edna will have -^ of |^ as many as May. 16. When money is worth 5^, what must I pay for the use of 130 for 3 years 8 months ? 17. If 7 be added to both numerator and denominator of the fraction |, how much will the value of the fraction be increased or diminished ? 18. When money was worth 6^ a year, I paid $32 for the use of 1200. How long did I have it ? 19. At $1.50 a cord, what is it worth to saw a cubical pile of wood 16 feet long ? 20. Ten years ago A was half as old as B. Ten years hence B will be three score years of age. How old is each now ? 21. Ten years ago Mr. H was ^ as old as he will be 30 years hence. What is his age ? 22. Bought shirts at 115 a dozen, and marked each to be sold at a profit of 20,^^. What was the marked price ? 23. What rate of income do I receive when I buy 6^ stocks at 50^ premium ? 24. A fruit dealer paid 15 cents a dozen for oranges, and sold them at the rate of five cents for two. What per cent did he gain ? 25. What is the difference between .1 and 1^ ? 26. Sixty per cent of Nell's money is 75^ of Ada's, and both have $18. How much has each ? 27. By selling eggs at 4 cents a dozen more than cost, a grocer made 25,^^. At what price did he sell them ? 28. A man gained 80 cents on a bushel of berries sold at the rate of 25 cents for two quarts. What was the cost a quart ? 29. Divide $112 among 2 men and 3 women, giving each man twice as much as each woman. 30. What per cent of a yard is a foot and three inches ? 31. What is the least sum of money with which a trader can buy sheep at $6 apiece or cows at $26 ? GENERAL REVIEW WORK. 3-|.3 32. IIow many tiles each G inches square will be required to cover a space 6 feet square ? 33. A boy sold papers at the rate of 2^ for 5 cents, 50^ of which is profit. How many papers could he buy for a quarter ? 34. IIow many square yards are there in the surface of two cubes whose edges are each 2 feet 6 inches ? 35. A room is ^ as long as it is wide, and its perimeter is 84 feet. What are its dimensions ? 36. Sugar worth $1.55 is weighed in a false balance which gives only 15Joz. to the pound. What is the selling price of the sugar ? 37. What per cent of a score is a dozen ? What per cent of a dozen is a score ? 38. If I charge $1.50 a cord for sawing wood into three pieces, how much should I charge for sawing it into five pieces ? 39. One square rug is li yd. on a side, and another is 1^ yd. on a side. If the larger rug costs $1.44, what will the smaller cost at the same rate a square yard ? 40. A, B, and C bought a horse for $100, A paying $20, B $30, and C $50. They sold him for $175. How much did each gain ? 41. F, G, and H have $510. If G has | as much as F, and H has f as much as F, how much has each ? 42. If 2^ yards of cloth make a pair of pants, how many pairs can be made from a piece of cloth containing 40 yards ? 43. What number increased by 6, the sum multiplied by 5, and the product divided by 10, gives 3 as a product ? 44. A man started westward from London and traveled through 360°. Did his watch then indicate the correct time ? Why ? 45. How many sheep are worth as much as a cow, if 4 cows are worth one horse, and 2 horses are worth 48 sheep ? 844 SCHOOL ARITHMETIC. 46. James, who lives If miles from the schoolhouse, goes to school 5 days each week. If he goes home for lunch every other day, in how many days does he walk 21 miles ? 47. A can do as much work in | of a day as B can do in f of a day. How long will it take B to paint a house that A can paint in 18 days ? 48. A and B ran a mile, A beating B by 40 rods. In what time can B run a mile, if A^s time in the race was 7 minutes ? 49. W can build 30 rods of fence in 4 days, and R can build as much in 6 days as W can in 8 days. In what time can R build 75 rods of fence ? 50. A man who had 3b sheep bought three times as many as he had, and then sold ^ of all. How many had lie left ? 51. Harry has 6a cents, which is f as many as Marie has. How many cents have they both ? 52. How many square feet are there in a board a feet long and b inches wide ? 53. The dimensions of a cube are a inches. How many square inches are there in 5 of its sides ? 54. In a school there 'drep pupils, and q of them are girls. How many boys are there in the school ? 55. If a yards of cloth cost b dollars, what will 3 yards cost ? 56. What will b yards of cloth cost if c yards cost d cents ? 57. The area of a field is ab square rods, and the length is a rods. What is the distance around the field ? 58. What is the volume of a cube whose edge is b feet ? 59. A living-room 12 feet square and 10 feet high is occu- pied by 5 persons. How many cubic feet of air are there to each person ? 60. A lot two rods wide is planted in corn, the rows being a yard apart. How many rows are there, no row being nearer the fence than 1 foot 6 inches ? 61. I have work for either 8 men or 12 boys. If I employ 6 men, to how many boys can I give employment ? GENERAL REVIEW WORK. 345 62. B sold a bnggy to C, gaining 20^, and C sold it to D at a loss of 20fo. If D paid $150 for it, what did B gain ? 63. After a rain it was found that there was half an inch of water in a box 3 feet square. What was the volume of the water in the box ? 64. Said A to B, *' I have as many quarters as you have half-dollars, and we together have $9." How much had each ? 65. How many ounces in p pounds and q ounces ? 66. What is the quotient of a fourths -^ a eighths ? 67. There are 946 pupils in a school. If f of the number of girls is equal to f of the number of boys, how many of each are there in the school ? 68. What per cent does a merchant gain on his investment if 20^ of his sales is profit ? 69. I sold a piece of land so that ^ of the profit equalled J of the cost. Find the gain per cent. 70. If 8 men can do f of a piece of work in 9 days, how many men can do the whole of it in 4 days ? 71. How many hours a day must 4 men work to do half as much work in 10 days as 15 men can do in 4 days, working 10 hours a day ? 72. A can hoe -^^ of a row of corn in 1 minute, and B can hoe 2f rows in an hour. In what time can they together hoe a row ? 73. Had a bin contained twice as much oats, and the oats been used one-fourth as fast, the oats would have lasted 48 weeks. How many days did they last ? 74. Mv pony is 13 hands high. How many feet is that ? 75. A lady gave f of her money to the poor, and then found f as much as she had given away, and then had $30. How much had she at first ? WRITTEN EXERCISES. 510. 1. A wagon box is 8 ft. long, 3 ft. 6 in. wide, and 2 ft. deep. How many bushels of coal will it hold ? 346 SCHOOL ARITHMETIC. 2. A and B can do a piece of work in 5 days, B and C in 6 days, and A and C in 10 days. How long would it take each alone ? 3. A saddle cost ^ as much as a horse, and the horse cost J as much as a buggy. If all cost $500, what was the cost of each ? 4. What number is that from which if '7-^ is subtracted, f of the remainder is 25 ? 5. When the gold dollar was worth 7^ more than the greenback dollar, how much gold was $371.29 in greenbacks worth ? 6. Sold 3 acres of land for $100 more than 5 acres cost, and thus gained 100^ on -the amount sold. What was the cost an acre ? 7. After losing -f of his money, A found $15, and then lacked -^j of having his original amount. How much did he lose ? 8. Divide and multiply 1 by .001, and to the sum of the quotient and product add the quotient of .01 -f- 50. 9. A man bought 5 shares of 10^ bank stock ($100), which yielded him Sfo. What did it cost him a share ? 10. A's money is to B's as 7 : 11, but if each had $9 more, A's would be to B's as 5 : 7. How much has each ? 11. A ship consumes 3^5- of its coal supply each day. It starts with its bunkers f full, and when it reaches port has only -^~ of its supply left. How many days were occupied on the voyage ? 12. Two men hired a pasture for $56. A puts in 10 cows and B puts in 36 horses. If a cow eats twice as much as a horse, how much should each pay ? 13. Four pipes, each 2 inches in diameter, empty a tank in 9 hours. What must be the diameter of a single pipe that will empty the tank in the same time ? 14. A, B, and C start together to walk around a race- track. A goes once around in 2 hours, B in 3 hours, and C GENERAL REVIEW WORK. 347 in 4 hours. In how many liours will all he together again at the starting point ? 16. A's money added to ^ of B's equals $2000. How much has each, if A's money is to B's as 3 to 4 ? 16. Bought 5^ stock at 124f, brokerage i^. What rate of interest do I receive on my investment ? 17. If 24 sheep are worth 6 cows, 8 cows worth 2 horses, and 3 horses worth 90 pigs, how many pigs are worth a dozen sheep ? 18. Sold a lot for $G00, payable in 90 days without inter- est. Bought it back the same day for $500, payable in 60 days without interest. How much did I gain, money being worth 8^ ? 19. The assessed valuation of a town is $760,000. What rate must be levied to raise $3610, exclusive of the collector's commission of 5^ ? 20. A offers $300 cash for a lot, and B offers $325, payable in 9 months without interest. Which is the better offer, and how much, money being worth 6^ ? 21. Bought a horse for $150 on a year'*s credit, without interest, and sold him at once for $150 cash. How much did I make, money being worth 5^ ? 22. The shadow of a man 6 feet tall is 8 ft. 6 in. long. Another man's shadow is 7 ft. 9 in. long. How tall is the latter ? 23. What is the least quantity of milk from which if 1 quart be taken the remainder can be exactly measured by either a 2-quart, a 4-quart, a 6-quart, or an 8-quart meas- ure ? 24. B walked twice as far as C ; but if he had walked 4 miles less, and C 6 miles more, he would have walked -J farther than C. How far did each walk ? 25. If Tom gives May a penny, each will have the same sum ; but if May gives Tom a dollar, he will have twice as much as she has left. How much has each ? 348 SCHOOL ARITHMETIC. 26. A and B run a race, their rates of running being as 17 : 18. A runs 2^ miles in 16 minutes, 48 seconds. B the whole distance in 34 minutes. What is the distance run ? 27. A owned f and B f of a store, but they took C into the firm and reorganized as equal partners. If C paid them 14000, what was A's share of it ? 28. If 16 yd. of cloth cost $56 when wool is $.75 a pound and labor $.20 an hoar, what would it cost when wool is $.60 a pound and labor $.25 an hour, if 24 lb. of wool and 60 hr. of labor are required to make it ? 29. Two houses cost $8100, and f of the cost of one is equal to -^ of the cost of the other. What is the cost of each ? 30. A sold B a horse for ^ more than it cost, and B sold it for $80, losing I of its cost. How much did A pay for the horse ? 31. I paid $214.20 for a piano after discounts of 20^, 15^, and 10^ had been allowed. What was the list price ? 32. Schley and Sampson were partners for two years, making an annual profit of $5460. During the first year Sampson owned f of the stock, but during the second year Schley owned J of the stock. What was each one's share of the profit ? 33. A steamer sails a mile down stream in five minutes, and a mile up stream in 7 minutes. How far down stream can she go and return in one hour ? 34. A stable 30 ft. long, 20 ft. wide, and 18 ft. high has two gables each 12 ft. high. Find cost of painting the outside at 50^' a sq. yd. 35. How many yards of carpeting 27 inches wide will cover a hall 45 ft. long and 32 ft. wide, the strips running lengthwise, and there being a waste of ^ yard in matching the pattern ? 36. What will it cost to plaster the walls of a room 18|^ ft. long, 16^1 ft. wide, 12 ft. high, at 11^ a square yard, allow- ing nothing for openings ? GENERAL REVIEW WORK. 34,9 37. How many board feet of siding 5 in. wide will be re- quired to cover tlie sides of a house 40 ft. long, 28 ft. wide, 20 ft. high, if they are laid 4 inches to the weather, and 150 sq. ft. are deducted for doors and windows? 38. If as many silver dollars as possible are laid on the bottom of a box 18 inches long by 12 inches wide, how much space will be left uncovered ? 39. The fence around a circular field is 1.19 miles in length. How many acres inside the fence? 40. Home is 20° 27' 14" E., and Washington IT 3' W. When it is 9 a.m. at Washington, what is the time at Rome ? 41. How many ounces of gold in a IG-carat chain that weighs 3^ ounces ? 42. At one point an eclipse of the moon was seen at 9 A.M., at another point at 11:30 a.m. What is the differ- ence in the longitude of the two places ? 43. A town has a water supply of 104 gal. a day for every house. If the number of houses increases ^, and the total supply diminishes -j^, what will be the daily supply to a house ? 44. Either 30 pears and 20 apples or 14 apples and 42 pears will just till a basket. How many of either will fill it ? 45. A and B receive $1000 for grading a street. A fur- nishes 3 teams 20 days, B 5 teams 30 days. If A receives 1100 for overseeing the work, what does each receive of the $1000 ? 46. A ten-acre field was divided into lots, each containing f of an acre. The partial lot was sold at the rate of $300 an acre, and the others at $150 each. What was received for the field ? 47. In a spelling contest there were 75 words given ; 6 contestants spelled 74 words each, and 13 spelled 70 words each. Find the average per cent made by these contestants. 350 SCHOOL ARITHMETIC. 48. City lots, 200 feet deep, sell for 180 a front foot. What is the value of an acre at that rate ? 49. A train runs ^5 miles an hour. How far can I ride on it and walli: back at the rate of 3| miles an hour, and be gone just 5 hours ? 60. A room 16 ft. by 18 ft. is covered with carpet 27 inches wide, and the smallest possible number of yards of the carpet is in use. How many yards? 51. Find the cost of a bushel of ground feed, the ingre- dients of which are 60 bushels of corn at 45^, 90 bushels of oats at 32^, and 26 bushels of rye at 64^, the cost of grinding being $6.20. 52. Two men together received $97.75, but one received $18.25 more than the other. How much did each receive ? 53. An agent, having in his hands $3150 of his principal's funds, is instructed to invest it in barley at $.48 a bushel, after retaining his commission of 5^. How many bushels should he buy ? 54. Two districts buy a road machine for $285, and pay the freight from the factory, one district paying -| and the other 4^ of the entire cost. The cost of the first district be- ing $127.50, how mucli was charged for freight ? 55. A commission merchant sold 1014 bushels of oats at 41 cents a bushel, paid $33.74 freight charges, and' retained 3^^ commission. How much should he remit to the con- signor ? 56. The Columbian souvenir half-dollar weighs 192.9 grains. How many of them weigh as much as 50 ordinary silver dollars ? 57. An upright pole 16 ft. long casts a shadow 5 ft. 4 in. long, and at the same time the shadow of a tree is found to be 26 ft. 9 in. How high is the tree ? "58. If one fifth be allowed for matching and waste, how many board feet of inch lumber will be required for flooring and ceiling a porch 17 ft. 4 in. by 7 ft. 6 in.? . aENERAL REVIEW WORK. 351 59. By the introduction of improved machinery in a cer- tain factory it was found that 7 men could do the work formerly done by 11 men. What per cent of the labor re- quired to turn out the same product was saved by using the improved machinery ? 60. If Tennessee 3^ bonds are selling at 95, how much money must be invested in them to secure an annual income of $750 ? 61. When the grade on a road is 1320 feet to the mile, what is the per cent of grade ? 62. If the interest is $19.07, the time 8 mo. 2 da., and the rate 5^^^, what is the principal ? 63. If it costs $110 to dig a cellar 40 ft. long, 27 ft. wide, and 4 ft. deep, how much will it cost to dig a cellar 36 ft. long, 30 ft. wide, and 5 ft. deep ? 64. The running time of a train from New York to Buf- falo is 8^ hours, and the distance is 440 miles. If stops of 5 minutes each are made at Albany, Utica, Syracuse, and Rochester, what is the average speed an hour ? 65. At what price is 4J^ stock equal as an investment to 3^fo stock at $87.50 a share ? 66. A man sells 22^ shares of 5 per cent stock at 105, and buys 4 per cent stock at 94|. How much is his income diminished ? 67. A cooper paid $78.32 for 16488 barrel staves. Required the price per M. 68. Bought stock at 8fo below par and sold it 12^^ below par, thereby losing $99. How much did I invest ? 69. Bought 4 loads of hay, 2750 lb. each, at $20 a ton, and paid for it with a GO-day note without interest. What will be the proceeds of the note if discounted at bank im- mediately at 6^ ? 70. Railroad stock that cost $121.75 a share pays a semi- annual dividend of 4^. Required the rate per annum of in- come on the investment. 352 SCHOOL ARITHMETIC. 71. Divide $744 among A, B, and C so that f of A's money will equal f of B's or | of C's. 72. The sum of three numbers is 940. The first equals -f of the second, and the second equals j^q- of the third. Find the numbers. 73. Write a 30-day note the proceeds of which, when dis- counted at a New York bank on the day of making, shall be $514. 74. Dewit purchased a house and lot for $3300 ; paid $975 for repairs, and now rents the premises for $30 a month. If he expends annually for taxes $48.70, and for incidental re- pairs $35, what is his per cent of annual income on his in- vestment ? 75. What is the difference between the true and the bank discount of $200 for 60 days, at Gfo? (No grace.) 76. A farm is worth 10^ less than a store, and the store 20^ more than a lot. The owner of the lot exchanges it for 80^ of the farm, thereby losing $850. What is the farm worth ? 77. In a proportion whose ratio is 12^, the first number is 25 and the last number is 8. What is the third number ? 78. B owns a square mile of land, and D owns a farm of equal area whose width is 128 rods. At $1.25 a rod, how much more will it cost to fence D's land than B's ? 79. A farmer agreed to give his hired man $100 and two cows for a year's labor. The man quit work at the end of 10 months, receiving the cows and $70 as a fair settlement. At how much were the cows valued ? 80. When a railroad company is declaring quarterly divi- dends of 1|^, and its stock is quoted at 125, what annual rate of income does an owner of that stock receive ? 81. A young man puts $10 in a savings bank each month, making his first deposit Jan. 1, 1901. How much will there be to his credit Jan. 1, 1902, if the bank pays 4^ per annum, and adds the interest at the end of each quarter ? GENERAL REVIEW WORK. 353 SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 611. 1. If postage stamps are ^ of an inch long and f of an inch wide, how many will be required to cover a ceiling 12 ft. 3 in.' by 13 ft. G in. ? 2. How must I invest in 3^ stock at 90 so as to get the same income as if I invested $4950 in the same stock when it is quoted at 99 ? 3. Divide $14600 among 3 boys, aged 9, 13, and 17 years respectively, so that if invested at 5^ simple interest each will receive the same amount at the age of 21. 4. A owns a mine worth $11000, which pays 6^ on his in- vestment. Paying $100 brokerage, he exchanges the mine for bank stock at 109, thus increasing his annual income $340, What dividend does the bank stock pay ? 5. A pole was f under water. The water rose 8 feet, and then there was as much of the pole above the water as was previously under it. Find the length of the pole. 6. Two equal annual payments have been made on an 8^ note for $200, dated two years ago to-day. The balance due is $44. What was the annual payment ? 7. A cubic foot of water weighs 62.5 lb., and lead is 11.44 times as heavy as water. How many cubic inches are there in a piece of lead weighing 35 lb. 6 oz. ? 8. M and K, equal partners, found on settlement that M owed the firm $240, and that the firm owed N $260. How much should M have given N to square the account ? 9. Snow has fallen to the depth of 25 cm. If 12 cu m. of snow produces 1 cu m. of water, find the volume of water produced by this snow on one acre of land. 10. A rented a farm from B, agreeing to give B | of all the produce. During the year A used 45 bu. of wheat, and at settlement first gave B 18 bu. to balance the 45 bu., and then divided the remainder as if neither had received any. How much did B lose ? 23 POWERS AND ROOTS. INVOLUTION. 612. 1. What is the product of 2 multiplied by itself, or used twice as a factor ? 2. How often must 3 be used as a factor to produce 9 ? To produce 27 ? 3. What is the product of a used twice as a factor ? Used three times as a factor ? 513. The product of two or more equal factors is called a Power. The product of two equal factors is called the second yoiver ; of three equal factors the iJdrd power, and so on. The second power of a number is also called the square of the number, because the area of a square is expressed by the product of two equal factors. Why is the third power of a number also called its cube 9 1. What is the square of 5 ? Of 7 ? Of 8 ? Of 10 ? Of i? Off? 2. Find the third power of 2. Of 4. Of 7. Of \. Of |. Of .3. 514. The number of times a number is to be used as a factor is indicated by a small figure placed at the right of the number. This figure is called an Exponent. Thus, a^ is read "a square," or *'a to the second power," and means a X a; d^ is read "a cube," or "a to the third power," and means a y. a X a ; a* is read "a to the fourth power," and so on. Write the following products as powers : 1. 3 X 3. 4. 5 X 5. 7. 9 X 9. 2. 2 X 2 X 2. 5. 23 x 23. 8. 7 x 7 x 7. 3. « X «. Q. .a X a X a. 9. bxbxbxbxb. POWERS AND ROOTS. 355 Find the following indicated powers : 10. 8' = ( ). ir = ( ). ay = ( ) (HY = ( ). 11. 5' = ' ). 6' = ( ). ay = ( ) .3' = ( ). 12. 2- = ^ ). 3- =. ^ ). (i)' = ( ) .3' = ( ). 515. Involution is the process of finding a power of a number. WRITTEN E/ERCISES. 516. 1. Find the third power of 12. 2. Find the square of 10, 13, 15, 25, 36. 3. What is the cube of 8 ? 12 ? 20 ? 44 ? 4. Find the fourth power of 6, 10, .5, j. The second power of a number multipHed by the second power equals the fourth power, a' x a' — a*. What is the product of a' x a' ? a* X a* =.{ ). 5. Siiice 2" X 2' = 2*, and 2* x 2* = 2", what is the shortest method of finding the IGth power of 2 ? 6. Find tlie third power of |, }, -f, f, .7. 7. What is the square of 1 ? .1 ? 100 ? .01 ? 2.5 ? 8. Can a number ending in 2, 3, 7, or 8 be a perfect square ? Why not ? 9. How many figures are used to express the square of a num- ber of two figures ? Tlie square of a number of one figure ? 10. What is the difference between the square of 48 and the cube of 24 ? 11. How much does the cube of 15 exceed twice its square ? 12. Which is greater and how much — the cube of ^ or its square ? Find the value of : 13. 431 17. 6.25^ 21. (f)*. 25. 14=" - U\ 14. 46^ 18. .00b\ 22. (2|)^ 26. .500'. 15. 14*. 19. {{iY. 23. 5* - 2\ 27. d9\ 16. 3.75'. 20. (if)'. 24. (f)' - (f)^ 28. 3' + 2 x 5\ 517. Since 53 = 50 + 3, the square of 53 maybe obtained as follows : 356 SCHOOL ARITHMETIC. (a) (50 + 3) X 3 = 50 X 3 + 3 (50 + 3) X 50 = 50' + 50 X 3 X 3 .-. 53' =: 50' + 'A (50 X 3) + 'i<' That is, 53' = square of tens + twice (tens x ones) + square of ones. (b) 53 53 53 53 159 265 9 rz: 3-^ 150 = 50 X 3 150 1= 50 X 3 2500 = 50' 2809 ■= 2809 = 50' -h 2 (50 x 3) + 3'. 518. Since any integral num- ber expressed by two or more figures may be regarded as com- posed of tens and ones, if we represent the number of tens by t and the number of ones by o, we have (t + of = f + Mo + o'. Hence, The square of a numher is equal to the square of the tens, ^ plus twice the tens multiplied hy the ones, plus the square of t + o the ones. We may write the formula thus: {t + o)' = f + {2t + o) X 0. Show why. Show by the diagrams that (a). 9' = (5 + 4)' zrr 5' + 4' + 2 X (5' X 4). (b). (t + o)' = f + o'-i- 2to. 5 4 i' . txo txo 0^ POWERS AND ROOTS. 357 Square the following by the above method : 1. 25. 4. 56. 7. 87. 10. (4 + 3). 2. 32. 6. 64. 8. 33. 11. {a-^b). 3. 43. 6. 71. 9. 98. 12. (a + 1). 619. The cube of a number may also be found by the above method. 1. Raise 25 to the third power. By Art. 517, 25' = 20' + 2 (20 x 5) + 5', which must be multiplied by 25, or 20 + 5. 25' X 5 = 20' X 5 + 2 (20 X 5') + 5' 25' X 20 = 20' + 2 (20' x 5) + 20 x 5' 25' = 20' + 3 (20' X 5) + 3 (20 x 5') + 5*. If we represent the number of tens by t and of ones by o, we have (t + oy = f+ 3t'o + 3(0' + o\ Hence, The cube of a ?iumber is equal to the cube of the tens, plus three times the product of the square of the tens by the ones, plus three times the product of the tens by the square of the ones, plus the cube of the ones. We may write the above formula thus : {t + oy = f + (3f + 3to + o') x 0. Show why. Cube by the above method : 2. 12. 5. 45. 8. 76. 11. {a + b). 3. 21. 6. 54. 9. 89. 12. (2 + 3). 4. 33. 7. 67. 10. 98. 13. {a + 1). EVOLUTION. 520. 1. What is one of the two equal factors of 16 ? Of 36 ? Of 64 ? 2. What is one of the three equal factors of 8 ? Of 27 ? Of 125 ? 358 SCHOOL ARITHMETIC. 521. One of the equal factors of a number is called a Root of tlie number. The square root is one of the two equal factors ; the cube root one of the three equal factors, and so on. 522. The process of finding a root of a number is called Evolution. The symbol ^ is called the Radical Sig'ii, and calls for the square root. The radical sign with index 3 calls for the cube root ; with index 4 it calls for one of the four equal factors, and so on. Notes, — 1. The symbol V was first used in this form by Rudolff in 1525. 2. The square root is also indicated by the fractional exponent I ; the cube root by the exponent i, etc. Find the root called for : 1. V^5. 4. ^27. 7. Vi lO. A^lG. 13. 9*. 2. v/400. 5. ^I. 8. V^. 11. ^^1. 14. 64*. 3. Vol. 6. ^105. 9. ^l\ 12. ^c\ 15. ^{a + x)\ SQUARE EOOT. 523. 1. Since 1 = 1", 100 = 10^ 10,000 = 100^ and so on, the square root of any integral number between 1 and 100 lies between what two numbers ? By how many figures is it expressed ? 2. The square root of any integral number between 100 and 10,000 lies between what two numbers ? By how many figures is the root expressed ? 524. 1. The square root of any integral number expressed by one or two figures is a number of one figure ; expressed by three or four figures is a number of two figures, and so on. ^. If an integral number is divided into groups of two POWERS AND ROOTS. 359 figures each, from right to left, the number of groups will be equal to the number of figures in the root. Query. — May the left-hand group have but one figure ? Why ? 525, The method of finding the square root of numbers is derived from the identity, {t + oV = f + no ^ d'^f ^ {2i +o)x 0. (Art. 518.) WRITTEN EXERCISES. 526. Find the square root of 1369. (a) (b) 13 69 t + 30 4- 7 = 37 13 69 [37 9 00 = (2t + o)x 0. 9 2t = 2 x30 = G0 0= 7 4 69 = 4 69 67 4 69 4 69 2t + = 07 The four figures of the number show that the root is expressed by two figures. In 1300 the greatest tens-square is 900, and its square root is 30, which is the tens of the root. The square of the tens, f, is subtracted, and the remainder, 469, con- tains twice the tens x the ones + the square of the ones. This remainder is largely the product of two factors — twice the tens x the ones, of which twice the tens, or 60, is so much the larger that it maybe used as a trial divisor ; using it thus, we find 7 to be the ones' figure of the root. Since 2to + o^ is equal to (2^ + o) x o, the seven units are added to twice the tens, and the sum, 07, is multiplied by 7. In the above, 2t, or 60, is called the trial divisor, while 2t + o, or 67, is the complete divisor. In using the trial divisor, if the quotient is found to be too great, it must be diminished. In practice the operation is conveniently performed as in (b), omitting unnecessary ciphers. The first group, 13, contains the square of the tens' number of the root. The greatest square in 13 is 9, and its square root is 3. The square of the tens is subtracted. Twice the 3 tens is 6 tens, and 6 tens is contained in the 46 tens of the remainder 7 times, giving the ones' figure of the root, 360 SCHOOL ARITHMETIC. 527. We may illustrate square root by the problem of finding the side of the square whose area is 1369 square inches. The square root of 1369 is 37. The square of 37 = (30 + If =: 30^ + 2 (30 X 7) + T. The 30 may be represented by a square ^ = 30 in. on a side. The 2 (30 x 7) may be represented by two strips each ^ = 30 in. long and o = 7 in. wide, while the 7 may be represented by the small square o = 7 in. on a side (Fig. 1). f + 2to + 0" = 1369 f = 900 t = 50 0=7 6-30 FiO. 2 t=r30 to=30\7 Q* Fi$.l 2^0 + = 469 That is, in extract- ing the square root of 1369, the large square, which is ^ = 30 in. on a side, is first removed, and a surface of 469 sq. in. remains. What is this remainder the area of ? The two rectangles and the small square have one dimension, o = 7, in common. If placed as in Fig. 2, they form one rect- angle whose width is o = 7, and whose length is 2^ + o = 60 -i- 7. The area is therefore expressed by (2/ -\- o) x o, or (60 + 7) x 7. In finding the width o, we are obliged to use 2t, or 60, as a trial divisor, since the whole length is as yet unknown. 469 -V- 2^ = 469 -r- 60 = 7. This gives (2^ + o) x o = (60 + 7) x 7 = 469. .-. V^Vdi^ = 30 + 7 = 37, the number of inches in the side. 628. We may apply the method given above to numbers of more than two groups of figures, by always regarding the part of the root already found as so many tens with respect to the next figure of the root. 1. What is the square root of 54756 ? (a) 234 •5 47 56 4 2 X 20 =: 40 1 47 40 + 'd = 43 1 29 2 X 230 = 46 18 56 460 + 4 = 46 4 18 56 43 (b) 5 47 56 4 147 1 29 4 18 56 18 56 234 POWERS AND ROOTS. 361 The first trial and complete divisors are obtained as they would be if the given number were 547 ; that is ^ = 20 and o = 3. For the second divisors, t = 230 and o = 4. 529. Rule. — Beginning at ones, separate the number into groups of two figures each. Find the greatest square in the left-hand group, and ivrite its root for the first part of the required root. Subtract the square of this root from the left-hand group, and to the remainder annex the next group for a dividend. Divide this dividend by twice the root already found, con- sidered as tens. The quotient {or the qtiotient diminished) will be the next figure of the root. To the last trial divisor add the part of the root last found for a complete divisor. Multiply this complete divisor by the part of the root last found, subtract the product from the dividend, to the remainder annex the next group for a new dividend, and proceed as before until all of the groups have been thus aimexed. 1. A decimal is separated into groups of two figures each by begin- ning at the decimal point, and its root is found precisely as the root of an integer is found. 2. The square root of a common fraction is found by extracting the square root of numerator and denominator separately, or by reducing it to a decimal and then finding the root. 3. If a number is not a perfect square, ciphers may be annexed, and the value of the root found to any required degree of approximation. Find one of the two equal factors of : 1. 256. 4. 1024. 7. 4356. 2. 441. 5. 2401. 8. 6080. 3. 625. 6. 2809. 9. 7225. Find the square root of the following : 13. 1225. 17. 13225. 21. 143641. 14. 1849. 18. 15625. 22. 173056. 15. 4480. 19. 26001. 23. 499849. 16. 9216. 20. 60516. 24. 801025. 10. 2025. 11. .3249. 12. .000144. 25. 1234321. 26. 5416.96. 27. 97.8121. 28. 31.4721. 362 SCHOOL ARITHMETIC. Extract the square root of : 29. m 32. m. 35. 6J. 30. ^\. 33. T^iWr- 36. 3^. 31. im- 34. ma. 37. 169AV- Find the value to four decimal places : 38. VU. 40. V3. 42. VX 44. V^T 39. 1/2". 41. V20. 43. V3r4. 45. V52.321. APPLICATIONS OF SQUARE ROOT. 530. 1. A square field contains 40 acres. Find the length of a side. 2. A square court is paved Avith 3844 marble slabs 8 inches square. What is the distance around the court ? 3. A 40-acre field is three times as long as wide. Find its length. 4. A rectangular field is 60 rods long and 40 rods wide. What is the side of a square field of equal area ? 6. How many rods of fence will be required to enclose a square farm of 160 acres ? 6. A rectangular farm 80 rods wide contains 160 acres. At $1.45 a rod, how much will it cost to fence it ? 7. A farmer has a square ten-acre field of grass. How many times will he have to mow around it to cut the grass, each swath being 5 feet wide ? 8. A rectangular field, the sides of which are in the ratio of 4 to 7, contains 4032 sq. rd. Find the cost of fencing it at $2 a rod. 9. What is the difference between the perimeters of two fields, one of which is 20.25 rd. square, and the other 20.25 sq. rd. ? 10. An army of 7056 men is arranged in a solid square. How many men in rank and file ? How many soldiers would be required to make another row around the square ? POWERS AND ROOTS. 363 H 531. In a right-angled triangle, the side opposite the right angle is called the Hypotenuse. 1. Cut from a cardboard a right triangle with a base 3 inches in length and an altitude of 4 inches. Find by actual measurement tlie length of the hypotenuse. 2. Square the nunibers representing the lengths of the three sides, and find whether one of the squares is equal to the sum of the other two. Which one ? 3. In the figure, II is the square on the hypotenuse, B the square on the base, and A the square on the altitude. How many small squares in H ? How many in A and B together ? 4.' Explain by the figure that 5' = 3' + 4'. + B^ 532. The square on the hypotenuse of a right-angled tri- angle is in area equal to the sum of the squares on the other two sides. Note. — This is known in geometry as the Pythagorean theorem, be- cause it is supposed to have been first proved by Pythagoras (about 500 B.C.). 533. The alcove relation is expressed in the equation h^ = V' + «^ where h and a represent the base and altitude re- spectively, and h the hypotenuse. 1. Cut a cardboard as in figure 1. If the triangles 1, 2, 3, 4 are taken away, the square on the hypotenuse of a right-angled triangle remains ; and if the two rectangles AP, PB, are taken away from the whole figure, the sura of the squares on the two sides of the triangle re- mains. Do the four triangles together equal the two rectangles ? Does this prove the relation Ih^ = i^ + a'' ? B Also, H' = A^ B a+b Fig.l \ \: \ 2 h Fig.3 a^ axb apto r^ 3 X 30 X 6 == 540 o' = 6' = 36 3^ + dto -h o' = 3276 19 656 27 000 = r 19 656 = (Sr + 3to + o') X o. (b) 46 656 136 27 3 X 30' = 2700 3 X 30 X 6 = 540 6' = 36 19 656 3272 19 656 The five figures of the number show that the root is expressed by tivo figures. In 46000 the greatest tens-cube is 27000, and its cube root is 30, which is the tens of the root. The cube of the tens, P, is subtracted, and the remainder, 19656, con- tains three times the product of the square of the tens by the ones + three times the product of the tens by the square of the ones + the cube of the ones. Each of these parts contains the ones' number as a factor. POWERS AND ROOTS. 36T Hence the 19656 consists of two factors, one of which is the ones* num- ber of the root ; the other is three times the square of the tens + three times the product of the tens by the ones + the square of the ones. Of this last factor, three times the square of the tens, or 2700, is so much the larger part that it may be used as a trial divisor ; using it thus, we find 7 to be probably the ones' figure of the root. But by trial we find this value too large, and we must take one less, or 0, for the ones' figure of the root. Since Zf^o + Zto^ + o' is equal to (3^' + ^to + o') x o, the trial divisor is completed by adding to the 2700 the 3 x (30 x 6) = 540, and e'' = 36; the sum, 3276, is multiplied by 6. Notes. — 1. In finding the ones' figure, we have given a product and the greater portion of one factor to find the other factor. 2. In practice, ciphers are omitted for convenience, as in (b). The first group, 46, contains the cube of the tens' number of the root. The greatest cube in 46 is 27, and the cube root of 27 is 3. Hence 3 is the- tens' figure of the root. We then divide the 196 hundreds of the re- mainder by the 3 x 30^^ = 27 hundreds to get the ones' number of the root. 538. We may illustrate cube root by the problem of find- ing the edge of a cube whose volume is 46656 cubic inches. Here t^ + 3^'o + 3^0" + o' = 46656, whose cube root is X 6') + 6=" The cube of 36 = (30 + 6)" = 30' + 3 (30' x 6) + 3 (30 x 6') The 30^ may be represented by a cube whose edge is 30 inches. The 3 (30'^ x 6) may be represented by three rectangular solids, each 30 in. long, 30 in. wide, and 6 in. thick, to be added to three adjacent faces of Fig. 1. The 3 (30 x Q"^) may be represented by three equal rectangular solids 30 in. long, 6 in. wide, and 6 in. thick, to be added to Fig. 2. The 6^ may be represented by the small cube required to complete the cube of Fig. 3. f + ^fo + Ma' t' + 0- = 46656 = 27000 ^^ V ZL ^ / W'o + 3^0" + o' = 19656 That is, in extracting the cube root of 46656, the large Fig. l. Fig. 2. Fig. 3. Fig. 4. cube, whose edge is ^ = 30 in., is first removed. There remain 19656 cu. in. Of what is this the volume ? 368 SCHOOL ARITHMETIC. The seven additions to the cube of the tens have one dimension, = 6, in common. If these seven solids were laid side by side so as to form one solid, the area of its base would be 3^^ + Bfo + o"^, or 3 x 30'^' -h 3 (30 X 6) + C^ ; its height would be = 6 ; and its volume would be the product of these factors. In finding the height 0, we are obliged to use 3/^ or 2700 (trial divisor), as the area of the base, since the whole area is as yet unknown. If this gives too large a value for 0, we must take one less. 19656 -f- 3f = 19656 -4- (3 x 30^^) = 7 + . By trial we find this too large ; hence we must take = 6. This gives {Sf + Sio + o'^) X = (3 x SO'' + 3 X 30 X 6 + 6') X 6 = 19656. s •*• V46656 r= 30 + 6 = 36, the number of inches in the edge of the cube. 539. The methods given above will apply to numbers of more than two groups of figures if we regard the pari of the root already found as so many tens with respect to the next figure of the root. 1. What is the cube root of 1906G24 ? 3 X 1 906 624 1124 1 3 X lO'' =r 300 906 (10 X 2) = 60 22 _- 4 364 728 3 X 120^ =: 43200 . 178 624 120 X 4) =. 1440 4'" = 16 44656 178 024 540. Rule. — Beginning at ones, separate the number into groups of three figures each. Find the greatest cuhe in the left-hand group, and write its root for the first part of the required root. Subtract the cube of this root from the left-hand group, and to the remainder annex the next group for a dividend. Divide this dividend by three times the square of the root POWERS AND ROOTS. 369 already found, considered as tens. The quotient (or the quo- tient diminished) will be the next figure of the root. To the last {trial) divisor add three times the product of tlie first part of the root, considered as tens, by the part last found, and also the square of the last part, for a complete divisor. Multiply the complete divisor by the part of the root last found, subtract the product from the dividend, to the remain- der annex the next group for a netv dividend, and proceed as before, uiitil all of the groups have been thus annexed. Notes. — 1. A decimal is separated into groups of three figures each by beginning at the decimal point. 2. To find the cube root of a common fraction, extract the cube root of numerator and denominator separately ; or change to a decimal and then extract the root. 3. If a number is not a perfect square, ciphers may be annexed, and the value of the root found to any required degree of approxima- tion. Find the cube root of the following : 1. 2744. 7. 74088. 2. 4096. 8. 140608. 3. 8000. 9. 226981. 4. 24389. 10. 1860867. 5. 10648. 11. 12167000. 6. 42875. 12. 926859375. APPLICATIONS OF CUBE ROOT. 641. 1. A cubical block of marble contains 13824 cubic inches. What is the length of a side ? 2. A cubical cistern holds 1000 gallons of water. How deep is it ? 3. How many square inches in one face of a cube of granite whose contents are 5832 cubic inches ? 4. Find the length of the diagonal of a cube whose volume is 8000 cubic inches. 13. 1%. . 14. mi. 15. mh 16. 140.608. 17. 250.047. 18. .970299. 370 SCHOOL ARITHMETIC. 6. A cubical bin contains 500 bushels of wheat. How deep is it if it is half full of wheat ? 6. A square box 16 in. deep will hold 9.64276 bushels of grain. What is the length of its side (inside measure) ? 7. A miller wishes to make a cubical bin that will hold 200 bushels of grain. What must be its depth ? 8. What is the length of the edge of a cubical box that will hold one half as much as one whose edge is 4 feet ? 9. Two thousand gallons just fill a vat whose length is twice its width, and whose height is J of its length. Find the length. 10. Assuming that cast iron weighs 7.15 times as much as water, find the edge of a cube of such iron that would weigh a ton. 11. The width and depth of a cistern are equal, its length is twice its width, and it will hold 24891 gallons of water. What are its dimensions ? 12. By cooling a red-hot cube of iron, the length of each of its edges was diminished by 6^. Find correct to three decimals the ratio of decrease in the volume of the cube. MISCELLANEOUS PROBLEMS. ORAL EXERCISES. 542. 1. The sum of two numbers is 60, and their differ- ence is 12. What are the numbers ? 2. If a man can build .3 of a fence in a day, in what time can he build ^ of it ? 3. How many pies will be needed to give each of 48 boys f of a pie ? 4. If a man can do f of a piece of work in a day, how long will it take 2 men to do .5 of it ? 5. If a bushel and 3 pecks of potatoes last a family 5 weeks, how many days will a peck and a half last them ? 6. A man lost 12^^ by selling a cow for $35. What per cent would he have gained by selling her for 150 ? 7. The sum of two numbers is 45, and tlieir difference is 25^ of the smaller number. What are the numbers ? 8. I bought eggs at 20 cents a dozen. Had I paid a quarter a dozen they would have cost a dollar more. How many did I buy ? 9. A man spent f of his money for a sleigh and ^ of it for a book-case. What per cent of his money had he left ? 10. By selling books at a profit of 40^ an agent gained $250. What did the books cost him ? 11. What number increased by its half, third, and fourth equals 50 ? 12. If a man can earn a dollar in f of a day, how much can he earn in f of a day ? 13. What number diminished by 12 is equal to 4 times -f^ of the number ? 372 SCHOOL ARITHMETIC. 14. A man divided his farm of 160 acres between his two sons in the ratio of ^ to -J. How many acres did each receive ? 15. One half of the money in my purse is quarters, ^ is nickels, ^ is dimes, and the remainder is 5 pennies. What sura is in my purse ? 16. B and C together have $50, and B has a dollar more than C. How much has each ? 17. What is the diagonal of a table whose length is 4 feet and whose width is 75^ of the length ? 18. I borrowed a sum of money for 3 years, at the end of which time I repaid the loan by a check for $22.50 more than I borrowed. If the rate M'as 5^, what was the sum bor- rowed ? 19. A, B, and C shared a tract of land in the ratio of 1, ^, and .2. C received 4|- acres less than B. What was the share of each ? 20. Wliat is the gain per cent when half a yard of cloth is sold for what f of a yard cost ? 21. A man engaged to work 10 days for $30, agreeing to forfeit $2 for every day he failed to work. If he received $22.50, how many days was he idle ? 22. A lady, being asked how many children she had, re- plied, '' If I had twice as many and 6 more I would have a dozen." How many had she ? 23. A fox is 80 rods ahead of a hound, and runs 20 rods while the hound runs 25. How far will the hound run before he catches the fox ? 24. A man paid $45 for some sheep. Three of them died and I of the remainder were sold for cost, which was $30. How many were sold ? 25. Either 8 turkeys or 12 ducks are needed for a dinner. If only 3 ducks can be had, how many turkeys must be taken ? 26. A starts on a journey and travels 27 miles a day ; 7 days later B starts, and travels 36 miles a day. In how many days will B overtake A ? MISCELLANEOUS PROBLEMS. 373 27. A farmer raised 60 bushels of potatoes, and the crop was 1500^ of the seed. How many bushels were planted ? 28. A man sold f of his sheep to A, -^ of the remainder to B, and then had 33 left. How many did he sell to B? 29. If f of A's age is f of B^s, and the sum of their ages is 51 years, what is the age of each ? 30. Three men hired a pasture for $114. A put in 3 horses 16 weeks ; B, 5 horses 12 weeks ; C, 8 horses 15 weeks. How much should A pay ? 31. If an orange and a half cost a cent and a half, how many oranges may be bought for a dime ? 32. I sold a watch to A for -^ more than it cost me; he sold it for $18, thereby losing § of what it cost him. How much did I pay for it ? 33. A is f as old as B, but if he were 6 years older, he would be . 9 as old as B. How old is each ? 34. D's money is $3 more than | of B's, and $5 less than f of B's. How much has each ? 35. If 3 boys do a piece of work in 9 hours, how long will it take a man who works 4^ times as fast as a boy ? 36. If 6 men can dig a ditch in 3^ days, how much time will be saved by employing 2 more men ? 37. A boat goes ip miles an hour up stream, and 15 miles an hour down stream. How far can she go and return in 10 hours ? 38. A party of 6 hired a coach. If there had been 2 more, the expense would have been $1 less for each person. How much was paid for the coach ? 39. A can do a piece of work in 12 days, or in 8 days with B's assistance. After they work together 6 days, B finishes the work, for which he receives $10. How much should A receive ? 40. AVhat time after midnight are the hour and minute hands of a clock first together ? 374 SCHOOL ARITHMETIC. WRITTEN EXERCISES. 543. 1. What is the value of a lot .625 of which is worth $1250 ? 2. Of the people in a building f are boys, .375 are girls, and the remainder, which is 22, are men. How many girls in the building ? 3. Tiie quotient is 3, the remainder -fj, and the divisor 3'\. What is the dividend ? 4. How often can .125 be subtracted from the sum of 125 tenths and 125 hundredths ? 6. If -| of an acre of land costs $41f, what will 4| acres cost ? 6. Find V^ x ^/'d correct to two decimal places. 7. The minuend is .875, and the remainder is one less than 1500 thousandths. What is the subtrahend ? 8. Explain the short process of dividing by 33^ ; by 125, 9. What will be the cost of 6160 lb. of coal, at $5.50 a ton ? 10. Divide, by using factors, 2875 by 48, and explain the process. 11. Between the lightning and the thunder I noted 12f seconds. How far away was the thunder, if sound traveled 1140 feet a second ? 12. What number divided by either 3, 4, 8, 9, 12, 18, 24, or 36 leaves a remainder of 3 ? 13. Making no allowance for mortar, how many bricks 8 in. long and 4 in. wide will be required to pave a walk 40 yards in length and 5 feet in width ? 14. If the velocity of electricity is 288,000 mi. a second, how long would it take electricity to travel around the earth, considering the circumference to be 24,900 mi.? 15. A and B hirea pasture. A puts in 21 cows and B puts in 35. If B's part of the rent is $185, how much is A^s ?' 16. If 40 pupils use 6 boxes of crayons, 200 in a box, in MISCELLANEOUS PROBLEMS. 375 3 mo., how many boxes, 150 in a box, will be required, at the same rate, to supply 75 pupils for 2 mo. ? 17. How many books the size of this would be required to cover the floor of your schoolroom ? 18. How many cubic feet of ice will an ice-house hold whose dimensions are 50 feet by 30 feet, and 18 ft. high, allowing 2 ft. above and below and on each side for saw- dust? 19. What is the value of 7^ + .3 + 18 4- 4.5 -2^ x If ? 20. A lot 40 ft. by 120 ft. is enclosed by a wire fence 3 wires high. If 25 feet of wire weighs a pound, and a pound costs 5 cents, what did the wire for the fence cost ? 21. A party of 60 hired a boat. Had there been 20 more, the expense of each would have been reduced i^l. How much was paid for the boat ? 22. What is the area of a square field whose perimeter is 160 rods ? 23. The perimeter of a rectangular field is 240 rods, and the width is |- the length. How many acres in the field ? 24. One half the diagonal of a rectangular field is 25 rods, and the width is 30 rods. What is the area of the field ? 25. What is the volume of a rectangular solid a feet square and b feet long ? 26. One evening a tree 45 feet high cast a shadow 75 feet long. At the same time a shadow of another tree was 160 feet. How high was the other tree ? 27. If 10 ounces of cotton make 6'f yards of cloth a yard wide, how much will be required to make 12 yards 48 inches wide ? 28. The rafters of a barn are 25 feet long, and their ends are 40 feet apart. What is the height of the gable ? 29. If 29 cows average 9 quarts of milk each per day throughout the year, and the milk is sold at an average of 7 cents a quart, what is the total annual profit if the expenses are 178 a head ? 376 SCHOOL ARITHMETIC. 30. How many feet of lumber in 12 planks, each 18 feet long, 10 in. wide, and 3 in. thick ? 31. The extreme end of the minute hand of a town clock moves 19 inches in 12 minutes. What is the length of the minute hand ? 32. Each board in a floor 56 feet long and 28 feet wide is 14 feet long and 6 inches wide, and it is held in place by 8 nails. If 68 nails weigh a pound, what is the weight of the nails in the floor ? 33. If 36 yards of cloth cost $54 when wool is 25{Z5 a pound, what will 25 yards cost when wool is 20^ a pound ? 34. A carpenter, a mason, and a painter built a house by contract for $3000. The carpenter worked 108 days, the mason 72 days, and the painter 45 days. The material used cost $1775. How much did each man earn if carpenter's wages were $3 a day, mason's $4, and painter's $2.40 ? 35. A load of 120 bushels consists of corn and rye in the ratio of 7 bu. of corn to 3 bu. of rye. How much rye must be taken away that the corn may be to the rye as 10 is to 4? 36. A garden 40 yards square is surrounded by a walk 24 feet wide. What part of an acre does the walk contain ? 37. What is the diameter of a circular field containing 10 acres ? 38. An orchard contains 7200 square rods, and its length is to its breadth as f to f . What is its length ? 39. A, B, and C can build a fence in 10 days. A can build twice as much as B, and C f as much as A. In what time can each alone build the fence ? 40. The property of an insolvent debtor amounts to $3560, and his liabilities to $8900. How many cents on the dollar will his creditors receive ? 41. What is the distance from a lower corner to the opposite upper corner of a room 16 feet long, 12 feet wide, and 10 feet high ? MISCELLANEOUS PROBLEMS. 377 42. The entire surface of a cube is 1014 square inches. How many cubic inches does it contain ? 43. Required the cost of fencing a square 40-acre field at $1.50 a rod. 44. Measure your schoolroom and calculate the distance from tlie center of the ceiling to a lower corner. 45. If steel rails weigli 180 lb. to the yard and cost $12 a ton, what will be the cost of rails for 2 miles of railroad, one of which has double track ? 46. A cow is tied by a rope 20 feet long. Upon how many square yards can she graze ? 47. An iron slab is 4 feet long, 3 feet wide, and an inch thick. If drawn out until only ^ of an inch thick, how many square feet will its sides contain ? 48. A square park is surrounded by a walk 2 rods wide. The area of the walk is 2 acres. What is the area of the square ? 49. Find the value of the following lumber at $24 per M : Six 6 X 9 sills, 18 ft. long. Thirty-two 2x8 joists, 18 ft. long. Thirty-four 2x6 rafters, 22 ft. long. 50. Bought apples at $3 a barrel. Half of them rotted. At what price must I sell the remainder in order to gain 33^^ on the amount bought ? 51. If a man, starting at noon from Pittsburg, could travel at the rate of 15 degrees an hour, where would he be in 24 hours ? Would his watch be too fast ? 52. How many revolutions will a wheel whose radius is 2 ft. 3 in. make in rolling a mile and a half ? 53. The troy pound is what per cent of the avoirdupois pound ? 54. The base of an isosceles (two* sides equal) triangle is 32 feet, and the altitude is 12 feet. What is the perimeter ? Draw the figure. 378 SCHOOL ARITHMETIC. 55. A vessel can sail 25 miles an hour with the current, and 15 miles against it. What is her rate of sailing in still water ? 66. A ship, whose rate of sailing in still water is 10 miles an hour, sails 40 miles up a stream in 5 hours. What is the rate of the current ? 67. Last year my rent was $350, which is 12|^ less than it is this year. How much has my rent been increased this year ? 68. If the water is 25^ of the milk, how many gallons of each in a mixture containing 62^ gallons ? 59. In a company of 120 persons, the children are 33-^^ of the men, who are 75^ of the women. How many children are there ? 60. Each of two dealers, A and B, wishes to sell an organ to C. A ask-ed a certain price, and B 33^^ more. A. then reduced his price 25^, and B his price 30^, and C took both organs, paying $302. What was B's price ? 61. A druggist pays $3.50 a pound avoirdupois for 3 lb. of a certain article. If he sells it at the rate of 60 cents a troy ounce, how much does he gain ? 62. Find the cost, at $24 per M, of 225 2-inch planks 18 ft. long and 8 inches wide. 63. The rainfall one day was .08 in. How many cubic feet of water fell on an acre ? 64. The cost of polishing 5 sides of a cubical block of mar- ble, at 20 cents a square foot, was $144. What are its solid contents ? 65. A man sold two lots for $3600 each. On one he gained 33^^ and on the other he lost 33^^. Did he gain or lose, and how much ? 66. In a mixture of silver and copper weighing 55 ounces, there are 15 ounces of silver. How much copper must there be added that tliere may be If ounces of silver in 10 ounces of the mixture ? MISCELLANEOUS PROBLEMS. 379 67. Forrest and John receive the same salary. Forrest saves ^ of his, but Jolin spends $200 a year more than For- rest, and at the end of three years finds himself $150 in debt. Required tlie salary of each. 68. A 2-inch plank is 16 feet long, 18 inches wide at one end, and tapers regularly to a point. How many feet board measure does it contain ? 69. How long will it take a men to do a piece of work that b men can do in c days ? 70. A man receives a salary of $1250 a year, and 3^^ of his salary equals 16f^ of his savings. What sum does he save per annum ? 71. How shall I mark goods which cost me $240 that I may fall 14f^ and still gain 20 per cent ? 72. From an excavation whose length, width, and height are equal, 2197 cu. ft. of earth were taken. How many feet of boards will be required to cover the sides and bottom of the cavity ? 73. A family consumes 47^ lb. of meat a week. If each member of the family consumes J lb. daily, how many in the family ? 74. A grocer sold 18 dozen and 9 boxes of matches from 2 gross. What part of the whole did he sell ? 75. How many sheets of tin, each 18 x 30 in., will it take to cover a roof, each side being 24 ft. long and 16 ft. 3 in. wide ? 76. A commission merchant after paying $1605 for various expenses found that he had cleared $2494. What amount did he collect if. his rate of commission was 5 per cent ? 77. A, B, and invested $10,800 in business, B paying f as much as A, and C -^^ as much as A and B together. The profit the first year was 33^^ of the capital. How much should each •receive ? 78. Three men rent a room for one year, four months, at the rate of $210 a year. The first man paid $122.50 for the 380 SCHOOL ARITHMETIC. time he occupied the room ; the second man occupied the room for 4 mo., and the third for the remaining time. If each paid according to the time he occupied the room, how much should the last two pay ? 79. A man drew out of the bank \ of his money and $16 more ; then ^ of the remainder less $20. He then had $182 remaining in the bank. How much had he at first ? 80. How many times will a wagon wheel 12 ft. 6 in. in cir- cumference turn round in going 10 mi. 24 rd. 4 ft. ? 81. In a pile of wood 16 feet long and 16 ft. wide, there are 12 cords. How high is it ? 82. How many times can .013 be subtracted from 125.78, and by what must the remainder be divided to give 350,000 as a quotient ? 83. What length of a road 32 feet wide will have an area of half an acre ? 84. Find the weight in tons per acre of a rainfall of an inch, one cubic foot of water weighing 62.5 pounds. 85. If the gas for 5 burners, 5 hours each evening for 10 days, costs $1, what will be the cost of gas for 75 burners which are lighted 4 hours every evening for 15 evenings ? 86. The average age of 200 boys is 14.75 years ; what will be the average if 10 boys are added whose average age is 11.6 years ? 87. The combined weight of 2 bars of silver is 271b. 3 pwt. 5 gr. The larger one w^eighs 12 lb. 19 cwt. 21 gr. more than the smaller one. Required the weight of each. 88. A and B each have $10 so invested that A receives 4 per cent and B 5 per cent interest. What per cent of A^s in- terest is B's ? 89. At the rate of a ton of coal every 21 days, what will be the cost of the coal used by a family from Oct. 17, 1904, to April 25, 1905, excluding both of the days named, at $4.50 a ton ? 90. A publisher sells to the wholesale trade 40 copies of a MISCELLANEOUS PROBLEMS. 381 book lit the retail price of 24 copies. What does he receive wholesale for a book which retails at $1.50 ? 91. A^s weight is f of B's, and C's is as much as A^s and B^s together ; the sum of their weights is 490 lb. How mucli does each weigh ? 92. The duty on a shipment of blankets being 33 cents a pound and 40^ ad valorem, what is the invoice price, if they cost the importer $3874, including the duty, the total weight of the blankets being 5800 lb. ? 93. If a man buys 100 shares of railroad stock at 107^, and sells it a month later when quoted at llof, what is the gain, if in the meanwhile the stock "paid a 2^ dividend ? 94. The first of four cog wheels which work together has 21 cogs, the second has 18, the third 15 ; and the number of cogs that the fourth has is to the number the third has as 2 to 3. After how many revolutions of the smallest wheel will they all be in their position at starting ? 95. What is the area of a circle 20 ft. in diameter ? 96. Divide $34.50 among 6 men and 11 boys, giving each boy .5 of a man's share. 97. The great pyramid, whose base is square, measures 763 ft. on each side. How many acres does it cover ? 98. To cover the floor of a gentleman's carriage-house, it took 170 planks 16 ft. long, 9 inches wide, and 2 inches thick. How much did the planks cost at $12 a thousand feet ? 99. How many yards of carpet, 28 inches wide, laid length- wise, will be required to cover a floor 9 ft. 4 in. wide and 18^ ft. in length ? 100. A house which cost $4800 rents for $300. If the taxes are 2 per cent, insurance f per cent, and repairs $5.50 per annum, what per cent is the net income ? 101. There are in the library of a certain school 683 books, which number will give 23 books to each pupil, and 16 books over. What is the number of pupils ? ^R2 SCHOOL ARITHMETIC. 102. If one pound of zinc covers a square yard, and it is worth 45^ a pound, what will it cost to line a tank 10 ft. square and 5 ft. deep ? 103. \yhen it is 13 o'clock M. at St. Louis, 90° 15' 15" W., what is the time at Kichmond, 77° 20' 4" W. ? 104. How many shingles will be required to cover the roof of a building 54 feet long, the two sides each 16|- feet wide, if one shingle covers a space 6 in. square ? 105. What is the amount of the following bill ? 32|- yd. of muslin at 6f ct. I5J lb. of lard at 12 lb. for $1.00. 23 lb. 3 oz. of butter, at 25^ a lb. 13^ lb. of sugar at 16 lb. for $1. 2 doz. bananas at 4 for a dime. 106. Find the cost of 25 joists 10 in. wide, 18 ft. long, and 4 in. thick at $35 per M. 107. Edward and Peter hire a pasture for $14. Edward puts in 8 horses ; Peter puts in 50 sheep. If 21 sheep eat as much as 2 horses, what must Edward pay ? 108. If a miller takes j\ for toll, and a bushel of wheat produces 40 lb. of flour, how many bushels of wheat must be taken to the mill to obtain a barrel of flour ? 109. The valuation of the real property of a village is $125000. How many mills on the dollar must be levied to net $475 after paying the collector 5 per cent ? 110. How much money must I put in a bank which allows 4^ interest on deposits in order to receive $100 at the end of 9 months ? 111. Ten horses and 12 cows cost $1160 ; 4 horses and 7 cows cost $530. What is the value of a horse ? 112. My shoemaker sends me a bill of $18 for a pair of boots and two pairs of shoes. Some months afterwards he sends me a bill of $32 for two pairs of boots and three pairs of shoes. What do the boots cost a pair ? MISCELLANEOUS PROBLEMS. 383 113. In a lot of eggs 7 of the largest, or 10 of the smallest, weigh a pound. When the largest are worth 15^ a dozen, what are the smallest worth ? 114. When I was married I was 3 times as old as my wi>e, but 15 years after our marriage I was only twice her age. Find my age at marriage. 115. The sum of f of A's money and ^ of BV being on in- terest for 8 years at 6 per cent, gives I960 interest. How much money has each if J of B's is 3 times f of A's ? 116. Take the proportion 8 : 3 = 5 : 1|^. If the second and third numbers each be increased by 7, what multiplier will be needed by the first to make a proportion ? 117. Find the diagonal of a room 40 ft. long, 30 ft. wide, and 12 ft. high. 118. The end of the minute hand of a town clock passes over 30 inches in 12 minutes. What is the length of the hand ? 119. What length of rope will enable a horse to graze on 2 acres of grass ? 120. Out of a piece of paper 5 ft. 10 in. square is cut the greatest possible circle. How many square inches of paper are cut away ? 121. How many pounds of flour should I give for $42.30, at the rate of $4.90 a barrel ? 122. How many acres are in a square the diagonal of which is 20 rods more than either side ? 123. A bin 6 ft. long, 2 ft. wide, and 1^ ft. high is filled with oats worth 40^ a bushel. What is the value of the oats ? 124. Find the cost of paving and curbing one mile of street, the paving being 30 feet wide and costing $2.75 a square yard, and each line of curbing costing 30^ a linear foot. 125. Two men enter into a partnership, one putting in $5000, the other $2000. The partner that puts in the less sum is to receive $300 extra from the proceeds for his ex- 384 SCHOOL ARITHMETIC. perience in the business. They gain 14725. What is the share of each ? 126. Kequired the area in acres of a piece of land .5 of a mile long and .3 of a mile broad. 127. A wine merchant mixes 12 gallons of wine worth $1 a gallon with 5 gallons of brandy worth $1.50 a gallon, and 3 gallons of water of no value. What is the value of one gallon of the mixture ? 128. How much clover seed at 15.50 a bushel could a man buy for $1017, after deducting his commission of 2^^ on amount paid for seed, and drayage at the rate of one and a fourth cents a bushel ? 129. A man has 14400. How much must he borrow at 4^ and put with it so that the two sums invested in a business that pays 12 per cent per annum may net him a gain of $600 a year ? 130. John can write a page in a minutes, and Sam can do the same in r minutes. How much can they both write in 5 minutes ? 131. In the A class there are twice as many girls as boys. Each girl makes a bow to every other girl, to every boy, and to the teacher. Each boy makes a bow to every other boy, to every girl, and to the teacher. In all there are 900 bows made. How many boys in the class ? 132. Sold wheat on commission at 6^, and invested the net proceeds in flour at 4^ commission, my whole commission being $625. What was the value of the wheat and flour ? 133. A man invested a certain sum of money in 5^ stock at 80, and twice as much in 4^ stock. If his income from the former is $300, and from the latter 1^ times as much, what was the price of a share in the latter investment ? 134. If water expands -^-^ in volume in being heated from the freezing point to the boiling point, find the weight of a cubic foot of boiling water, the weight at freezing point being 62.5 poundg, MISCELLANEOUS PROBLEMS. 385 136. If the pressure of the air on the surface of water is 15 lb. per square inch, and if 1 cu. ft. of water weighs 1000 oz., find the pressure per square foot of a column of water 100 ft. deep. 136. If each person in breathing spoils the air of a closed room at the rate of 8 cu. ft. a minute, how long can the windows and doors of a schoolroom be safely kept closed when occupied by 50 pupils, if the room is 25 ft. by 20 ft., and 10 ft. high ? 137. By raising the temperature of a cube of iron, the length of the edge was increased 6^. Find the ratio of in- crease in the volume of the cube. 138. Each side of a hexagonal (six-sided) field is 20 rd., and the distance from the center of the field to the middle point of each side is also 20 rd. What is the area of the field? 139. During a heavy rainstorm, a circular pond is formed in a circular field. If the diameter of the field is 250 rd. and that of the pond 125 rd., what is the ratio of the land area to the water area of the field ? 140. If the average velocity of a bullet is 1342 feet a sec- ond and that of sound 1122 feet a second, how much time elapses, on a range of 3000 feet, between the time the bullet strikes and the time the sound reaches the target ? SUPPLEMENTARY EXERCISES (FOR ADVANCED CLASSES). 544. 1. Is a billion a million million ? Explain. 2. Multiply 789627 by 834, beginning at the left to multiply. 3. Reduce f to a fraction whose denominator is 11. 4. Change |, f, -j^, and .7 to fractions having a common numerator. fofi 5. Find the value of -; — rf -^ .125. 25 386 SCHOOL ARITHMETIC. 6. If I receive a discount of 20^, 10,^, and bfo, and sell at a discount of 10^, 6fo, 2^^, what is my per cent of gain? 7. At noon the three hands — hour, minute, and second — of a clock are together. At what time will they first be to- gether again ? 8. A merchant bought cloth at $3.25 a yard, and after keeping it 6 months sold it at $3.75 a yard. What was his gain per cent, counting 6^ per annum for the use of money ? 9. Bought 10 bushels of corn and 20 bushels of turnips for $11 ; at another time 20 bushels of corn and 10 bushels of turnips for 113. What did the corn cost a bushel ? 10. A man wishing to sell a horse and a cow asked 3 times as much for the horse as for the cow ; but finding no pur- chaser, he reduced the price of the horse 20^, and the price of the cow 10^, and sold both for 1165. How much did he get for the cow ? 11. If J of an article is sold for the cost of -J of it, what is the rate of loss ? 12. A man sold a pig and a sheep for $18, gaining 25^ on the cost of the sheep, and 20^ on that of the pig. If f of the cost of the pig equaled f of the cost of the sheep, what was the cost of each ? 13. Mr. G spent $260 for apples at $1.30 a bushel. Re- taining a part for his own use, he sold the rest at a profit of 40^, clearing $13 on the entire cost. How many bushels did he keep ? 14. If 32 men have food for 5 days, how many men must leave, so that the food may last the remaining men 70 days ? 15. What is the difference in area between a square whose diagonal is one foot and a circle whose diameter is one foot ? 16. A dealer bought 3000 bushels of oats at $.30 a bushel, and sold them out by the bushel at 10^ above cost. When all had been sold, it was "^'ound that the quantity of oats had shrunk 2^. What per cent dii the dealer make on the investment ? MISCELLANEOUS PROBLEMS. 387 17. After measuring f of a mile with a chain 100 feet long, the surveyor discovers a kink in the chain which shortens its length ^ inch. How much less than f of a mile was the distance measured ? 18. A man sold two horses for $210 ; on one he gained 25^, on the other he lost 25^. How much did he gain, supposing the second horse cost f as much as the first ? 19. A merchant sold goods at 20^ gain, but had their cost been $49 more, he would have lost 15^ by selling at the same price. How much did the goods cost him ? 20. Had an article cost 20^ more, the gain would have been 25^ less. What was the gain per cent ? 21. A, B, and C, having 4 loaves for which A paid 5^, B 8^, and C 11^, eat 3 loaves, and sell the fourth to D for 24^. How many cents should each receive ? 22. The distance from the center of a circle to the middle of a chord 10 inches long is one foot. What is the area of the circle ? 23. A man bequeathed $9000 to his three sons, aged 13 yr., 15 yr., and 17 yr., in such a manner that the share of each, placed at compound interest at 6 per cent until he arrived at the age of 21 years, should amount to the same sum. Find the share of each. 24. What is the distance from the lower corner to the op- posite upper corner of a room 15 ft. by 12 ft., and 10 ft. high? 25. If f of the time past noon equals f of the time to mid- night, what time is it ? 26. Between 2 and 3 o'clock I mistook the minute hand for the hour hand, and consequently thought the time 55 minutes earlier than it was. What was the correct time ? 27. If sound travels in still air 1090 ft. a second when the temperature is 32° Fahrenheit, and if the velocity increases 1.1 ft. for every degree of increase in temperature, how far off is an explosion when the report follows in 8 seconds, the temperature being 70° ? 388 SCHOOL ARITHMETIC. 28. If it takes 75 Kg. of saltpeter, 12.5 Kg. of charcoal, and 12.5 Kg. of sulphur to make 100 Kg. of powder, how many kilograms of each will be required to make 10,000,000 cartridges, each containing 5 g. of powder ? 29. In digging a cellar, 15625 cubic feet of earth were removed. The length is twice the width, and the width is twice the depth. What are the dimensions ? 30. What is tlie width of a doorway 8 feet high that is just wide enough to allow a circular mirror 31.416 feet in circum- ference to pass through ? 31. B has a circular garden containing 75 sq. rd. What is the area of the largest square garden he can make in it ? 32. What is the area of a square whose diagonal is 100 feet? 33. A boy weighing 96 lb. is seated on one end of a see- saw 16 ft. long, and a boy weighing 120 lb. is seated on the other end. Find the distance of each boy from the point of support, the lengths of the two arms of the plank being in- versely proportional to the weights at their ends. 34. How many apple trees can be planted in an orchard 15 rods square, allowing no two to be nearer each other than 1:1^ rods ? 35. A cube of water 1.8 dm. on an edge weighs how many kilograms ? 36. B lends A $150 for 6 months, and a year later A lends B $100. How long may B keep the $100 to balance the use of his loan to A ? 37. Starting from Dayton, I go 200 miles due east, then 150 miles due south, then 100 miles due west, then 350 miles due north. How far and in what direction from Dayton am I ? 38. How many rods is it from the center to one corner of a square field containing 20 acres ? 39. What is the difference between half a cubic foot and a cubic half foot ? MISCELLANEOUS PROBLEMS. 389 40. If my horse had cost me 20^ less, my rate of gain in selling him would have been 30^ greater. What was my gain per cent ? 100^ of the cost = the cost. S0% of 80^ = 24^ of cost, yielded by 20^ of cost. 100^ of cost yields 5 x 24^ = 120^ of cost. .-. the gain = 20^. 41. A squirrel goes spirally up a cylindrical post, making a circuit in each 5 feet. How many feet does it travel if the post is 20 feet high and 6 feet in circumference ? 42. A man agrees to pay $6000 for a lot in three equal payments, including 6^ interest on unpaid money. What is the yearly payment ? Amount of $6000 at 6^ compound interest for 3 yr. = $7146.090. $1.00 + $1.06 + $1.1236 = $3.1836, or 3.1836 x $1. $7146.096 -5- 3.1836 = $2244.66, the payment. 43. Two-fifths of a mixture of wine and water is wine ; but if 10 gallons of water be added to it, then only ^'^^ of the mixture will be wine. How many gallons of each liquid in the mixture ? 44. In the center of a circular island 100 feet in diameter stands a tree 140 feet high. A line 500 feet long will reach from the top of the tree to the farther shore. What is the width of the river, the island being in the middle ? 45. What is the length of the shortest possible route by which a fly can crawl from a lower corner to the opposite upper corner of a room 16 feet long, 12 feet wide, and 8 feet higli ? 46. In a certain game A can make 20 points while B makes 30 ; B can make 20 points while C makes 18. How many points can C make while A makes 100 ? 47. Two trains start at the same time, one from Jackson- ville to Savannah, the other from Savannah to Jacksonville. If they arrive at destinations 1 hour and 4 hours after pass- ing, what are their relative rates of running ? APPENDIX SUPPLEMENTARY WORK. MENSURATION. 545. The process of measuring lines, surfaces, and solids is called Mensuration. The principles of mensuration that apply to rectangles, parallelograms, triangles, trapezoids, circles, and rectangular solids have already been given. PRISMS. 546. A Polygon is a plane figure bounded by straight lines. Thus, the ends of the solids A, B, and C are polygons, 1. Are the two ends of each solid equal ? Are they parallel ? What is the form of the sides f 2. What is the form of the ends of the solid A ? Of the solid B ? 547. A solid whose ends are equal and parallel polygons and whose sides are parallelograms is called a Prism. Thus, A, B, and C are prisms. The solid B is also a rectangular solid. MENSURATION. 391 1. The polygons are called bases, and the prisms are named from the form of the bases. Thus, A is called a triangular prism; B, a quadrangular prism. 2. What is the prism whose bases are square called ? 3. When the bases are parallelograms, the prism is called a Parallelopiped; as B. 4. Is any triangular prism equal to half of a parallelopiped of the same altitude and of double the base ? Is it therefore equal to a parallelopiped of the same altitude and equal base ? (Any prism can be cut into triangular prisms, as in C.) 548. To find the volume of a prism. How to find the volume of a rectangular solid or prism was shown in Art. 327. The volume of any prism is found in the same way, viz. : Multiply the number of cubic units in 1 unit of length by the number of units of length. Or, multiply the area of the base by the altitude. It should be kept constantly in mind that the number of cubic units in 1 unit of length is the same as the number of square units in the base. 1. What is the volume of a prism whose length is 4 feet, and whose base is a rectangle 4 inches by 9 inches ? 2. The base of a prism 9 feet long is a triangle, each of whose sides is 3 feet, and whose altitude is 2.598 feet. How many cubic feet does it contain ? 3. A cord of Virginia pine weighs 2700 lb. What is the weight of a single piece of timber 18 inches square and 16 feet long ? 4. If a cubic foot of rolled steel weighs 489.6 lb., what is the weight of a piece 4 inches square and 30 feet long ? 6. Find the lateral surface of a prism whose length is 12 feet, and whose base is a triangle, each of whose sides is 2 feet. The prism has 3 equal sides, each 12 feet long and 2 feet wide. The 392 SCHOOL ARITHMETIC. area of one side is 24 sq. ft. ; hence the surface of the 3 sides is 3 times 24 sq. ft., or 72 sq. ft., the lateral surface. The 3 sides together are equal to one rectangle 12 ft. long and 6 ft. wide. Make a rule for finding the lateral surface of prisms. Note. — The lateral surface is the entire surface except that of the ends. 6. Find the lateral surface of a prism 3 feet square and 8 feet long. 7. The sides of a triangular prism are each 2^ feet, and its height is 6 feet. What is the lateral surface ? 8. What is the lateral surface of a pentagonal (five-sided) prism whose sides are each 18 inches, and whose altitude is 14 feet ? THE CYLINDER. What is the form of the solid C ? Of its ends ? Are the ends equal ? Are they parallel ? 549. A solid whose ends (bases) are two equal and parallel circles, and whose lateral surface is a uniformly curved surface, is called a Cylinder. It is described by revolving a rectangle about one of its sides as an axis. 1. The circles are called bases. 2. Name four objects that are cylinders. How does the number of square units in the base of the cylinder C compare with the number of cubic units in 1 unit of length ? Then may the volume of a cylinder be found in the same manner as the volume of a prism ? 1. Find the volume of a cylinder whose diameter is 2 feet, and whose length is 10 feet. 2. In form the Winchester bushel is a cylinder, 18| inches in diameter and 8 inches deep. How many cubic inches does it contain ? 3. A well is 26 feet deep and 4 feet in diameter. How many gallons of water are in it when it is half full ? MENSURATION. 393 4. A cylindrical pail 8 inches in diameter will hold 2 gallons. What is its depth ? 5. Take an oblong paper 4 inches by 6 inches and roll it to form a cylinder. What is the length of the cylinder ? The circumference ? 6. If a hollow cylinder is cut and spread into a flat surface, wliat form has it ? 7. What dimension of the cylinder is equal to the length of the rectangle ? What to the width ? 8. Then how may the lateral surface of the cylinder be found ? 9. What is the lateral surface of a cylinder whose diameter is 2 feet and whose length is 6 feet ? 10. How many square feet of material in a piece of stove- pipe 6 inches in diameter and 2 feet 8 inches in length ? 11. How many square feet of tin will be required to make 100 feet of spouting 2-^ inches in diameter ? THE CONE AND THE PYRAMID. What is the form of the base of the solid ABC ? Notice how the solid tapers to a point. 550. A solid that tapers uniformly from a circular base to a point is called a Cone. It is described by revolving a right-angled triangle about one of its sides as an axis. 1. The point is called the vertex ; as C. 2. The distance from the vertex to the center of the base is the altitude ; as CO. 3. The shortest distance from the vertex to the circumference of the base is called the slant height; as CA. 4. Name several objects that have the form of a cone. Make a paper cylinder and a paper cone of equal base and 394 SCHOOL ARITHMETIC. altitude. Fill the cone with salt, and empty it into the cylinder. How many conefuls will fill the cylinder ? It is shown in geometry that 551. A cone has one third the volume of a cylinder of the same base and altitude. Then how may the volume of a cone be found ? 1. What is the volume of a cone whose base is 12 feet in diameter, and whose altitude is 18 feet ? 2. Find the solid contents of a cone, the diameter of whose base is 10 feet, and whose height is 1 5 feet. 3. A conical pile of grain is 3 feet high, and the diameter of its base is 6 feet. How many bushels in the pile ? 4. If a cubic yard of granite weighs 4700 lb., what is the weight of a granite cone 6 feet high, the diameter of the base being 4 feet ? The slant height is the hypotenuse of the right-angled triangle re- volved to describe the cone, and the radius of the cone's base is the base of the triangle. (See figure.) If the slant height and radius of base are known, how can the altitude be found ? 5. If the slant height of a cone is 10 feet, and the diam- eter of the base is 12 feet, what is the altitude ? The vol- ume ? What form has the base of the solid E ABCDE ? What form has each side ? Where do the sides meet ? 552. A solid whose base is a polygon and whose sides are triangles meeting in a point is called a Pyramid. 1. The point in which the sides meet ^ is the vertex ; as E. 2. The distance from the vertex to the centre of the base is the altitude : as EO. MENSURATION. 396 3. The distance from the vertex to the middle of a side of the base is the slant height. 4. Make a paper prism and a paper pyramid of the same base and altitude from pasteboard. If the latter is filled three times with salt, and the contents poured into the prism, will the latter be exactly full ? It is shown in geometry that 553. A pyramid has one third the volume of a prism of the same base and altitude. Then how may the volume of a pyramid be found ? 1. What is the volume of a pyramid whose height is 15 feet, and whose base is 8 feet square ? 2. Find the solid contents of a pyramid whose altitude is 20 feet, and whose base is a rectangle 12 feet by 8 feet. 3. Find the lateral surface of a pentagonal pyramid whose slant height is 20 feet, each side of the base being 8 feet. (a). The lateral surface is com- posed of five equal sides, each a tri- angle whose dimensions are given. Thus, in the triangle ACD, the base CD is 8 feet, and the slant height AH is 20 feet. (b). Since the triangles have the same altitude, they are together equal to one triangle whose altitude is 20 feet, and whose base is 40 feet, the perimeter of the base of the pyramid. Hence the lateral surface = perimeter of base x ^ slant height. (c). If the number of the sides of the pyramid be increased indefi- nitely, the bases of the triangles will become extremely small, and the perimeter may be regarded as the circumference of the base of a cone whose lateral surface is equal to that of the pyramid. Hence the lateral 396 SCHOOL ARITHMETIC. surface of a cone = circumference of base X i slant height. (d). If the lateral surface of a cone be imagined as unrolled from the solid itself, it will appear as a portion (sector) of a circle, as shown in the figure. Its area is equal to that of a triangle whose base equals the arc of the sector and whose altitude is the radius of the circle. The altitude VD is the slant height of the cone. What is the base g' ADE? 554. The lateral surface of a cone = circumference of base X I sla7it height. 1. Find the lateral surface of a pyramid whose base is 12 feet square, and whose slant height is 20 feet. 2. The slant height of a pentagonal pyramid is 9 feet, and each side of the base is 2 feet. What is the lateral surface ? 3. Find the lateral surface of a cone whose diameter at the base is 16 feet, and wliose slant height is 24 feet. 4. The circumference of the base of a cone is 40 feet, and the slant height is 20 feet. What is the lateral surface ? 5. The cupola of a building is 16 feet in diameter at the base, and measures 22 feet from tlie vertex to the circumfer- ence of the base. What will be the cost of painting it at $.35 a square yard ? 6. How many yards of canvas, 54 inches wide, must be bought to make a conical tent having a slant height of 11 feet, and a circumference at base of 27 feet ? 7. The base of the pyramid Cheops in Egypt is 763.4 feet square, and the slant height is 612 feet. How many acres of surface in its sides ? 8. Find the weight of each of the following, a cubic foot of marble weighing 170 pounds : (a). A marble cylinder — lengtli 3 feet, diameter 1 foot, (b). A marble prism — length 3 feet, base 1 square foot. MENSURATION. 397 (c). A marble cone — height 3 feet, radius of base 1 foot, (d). A marble pyramid — height 3 feet, base 1 foot square. THE SPHERE. 555. A solid bounded by a surface whose every point is equidistant from a point within, called the center, is a Sphere. (a). Bisect a sphere and observe the two surfaces exposed. These are called great circles. Is the diameter of these circles also the diameter of the sphere ? (b). The curved surface of a hemisphere is larger than its flat surface. Is it twice as large ? Take a wooden hemisphere and investi- gate by winding the surface of the hemisphere with a waxed cord, and then winding a great circle of the sphere with the cord. (c). It is proved in geometry that the curved surface of a hemisphere is twice the flat surface. If the radius of the great circle is r, the area is Trr'. Then what is the area of tlie curved surface ? (d). How many great circles in the curved surface of two hemispheres, or one sphere ? Then the surface of a sphere = how many times irr* ? The area of a circle = nr^. The surface of a sphere = 4;rr'. 1. How many square inches in the surface of an 8-inch globe ? 2. If the diameter of the moon is reckoned at 2000 miles, how many square miles in its surface ? 3. If the diameter of the earth is reckoned at 8000 miles, its area is how many times that of the moon ? 4. The surface of a sphere is 64 square feet. What is its diameter ? 5. The area of a great circle of a sphere is 100 square inches. Find the cost of gilding the sphere at 75^ a square foot. If we join three points on a sphere with the center, we mark out a 398 SCHOOL ARITHMETIC. solid which is nearly a pyramid. A sphere may be regarded as made up of a very great number of such pyramids whose common vertex is the center of the sphere, and whose bases are small portions of the sur- face. The altitude of each pyramid is the radius of the sphere, and the area of all the bases is equal to the surface of the sphere. Investigate this by cutting a sphere Can apple will do) into pyramids. Volume of pyramid = area of base x ^ altitude. Volume of sphere = surface x ^ radius. " = 4:7tr' X ir = i7rr\ 6. What is the volume of a sphere whose diameter is 12 inches ? 7. How many cubic inches in a rubber ball if its diameter is 2 inches ? 8. The diameter of a cannon ball is 6 inches. If the specific gravity of iron is 7.48, what is the balFs weight ? 9. What is the diameter of the largest sphere that can be cut out of a cube whose edge is 10 inches ? What is the volume ? What per cent of the cube is cut away ? 10. Find the number of cubic miles in the earth, consider- ing the distance from the surface to the center as 4000 miles. 11. A 12-inch shell has an inside diameter of 10 inches. How many cubic inches of iron were used in casting it ? (From the entire volume subtract the volume of the hollow portion.) SIMILAR FIGURES. 556. Figures that have the same shape are called similar figures. Thus, lines, squares, triangles whose angles are respectively equal, circles, cubes, or spheres are similar figures. May similar figures be regarded as enlarged or reduced copies of one another ? 557. It is proved in geometry that — 1. The corresponding lines of similar figures are propor- tional. MENSURATION. 399 2. The surfaces {areas) of similar figures are to each other as the squares of their corresponding dimensions. Conversely, their corresponding dimensions are to each other as the square roots of their surfaces. 3. The volumes of similar figures dre to each other as the cubes of their corresponding dimensions. Conversely, their corresponding dimensions are to each other as the cube roots of their volumes. 1. If the side of one square is twice that of another, is its area four times as great ? If the edge of one cube is twice that of another cube, is its volume eight times as great ? Il- lustrate by drawing figures. 2. Prove that the surfaces of two spheres are to each other as the squares of their radii. Let S and s represent the surfaces of two spheres, and li and r the radii. Then, S = ^vR\ and s = 4irr'. .-. -^ = 49ri2»-i-4irr» = :^, s r' or S : s= R"" : r\ 3. Show that the circumferences of two spheres are to each other as the radii, and the volumes as the cubes of the radii. 4. One rectangular field containing 12.15 acres is 36 rods long and 18 rods wide ; another field of the same shape is 27 rods wide. Find the length and area of the larger field. (a) 18 : 27 = 36 rd. : x rd. (b) IS'' : 27' = 12.15 A. : x A. 5. The weight of a ball whose diameter is 5 inches is 27 lb., and the weight of a similar ball is 64 lb. What is the diameter of the larger ball ? 5 in. : X in. = -^27 : ^6l =8:4. .*. the required diameter = 6f in. 6. How many circles 2 inches in diameter are equal to a circle whose diameter is 6 inches ? 400 SCHOOL ARITHMETIC. 7. An 8-inch square is equal to how many 2-inch squares ? 8. In a park are two circular flower beds, one three times as large as the other. Find the circumference of the larger, if the smaller is 25 feet ? 9. A rectangular lot* is 20 rods long and 4 rods wide. A similar lot contains 2^ acres, and is surrounded by a fence which cost $1.75 a rod. Find the cost of fencing it. 10. A cow is tied to a stake by a rope 9 yards long, and a horse is tied to another stake by a rope 6 yards in length. Upon how much more area can the cow graze than the horse ? 11. If 25 gallons of water flow through a pipe 2 inches in diameter in a minute, how many gallons an hour will flow through a pipe 6 inches in diameter ? 12. If a pipe 1 ft. 6 in. in diameter fills a cistern in 6 hours, what is the diameter of a pipe that will fill it in 1 hr. 30 min.? 13. How many square feet of zinc will be required to line the sides and bottom of a cubical box whose capacity is equal to that of a rectangular box 4 ft. 6 in. long, 3 ft. 3 in. wide, and 2 ft. 1^ in. deep ? 14. A cubical box is 2 ft. deep. What is the depth of an- other cubical box that holds three times as much ? 15. A pail 9 inches deep will hold 2 gallons. What is the depth of a similar pail that holds 2 quarts ? 16. The diameter of one cannon ball is 2J times that of another, which weighs 27 lb. What is the larger ball worth at 1^ a pound ? 17. What is the edge of a cube whose contents are equal to the contents of two cubes whose edges are respectively 3 feet and 5 feet ? 18. The Winchester bushel is 18J inches in diameter and 8 inches deep. What are the dimensions of a similar measure that holds ^ peck ? MENSURATION. 401 SUPPLEMENTARY PROBLEMS. 668. 1. The altitude of a pyramid is 12 feet, and its base is 18 feet square. What will be the cost of painting the lateral surface at $.45 a square yard ? 2. If a 3-inch pipe fills a cistern in 9J^ hours, how large a pipe will fill it in 12 hours ? 3. How many 2-inch balls can be made from a ball 6 inches in diameter ? 4. A box has a bottom 2 ft. 6 in. square, the top is 3 ft. 6 in. square, the height is 2 ft. 6 in. What will it cost to line with zinc at 20^ a square foot ? 5. What is the side of the largest cube that can be cut from a sphere 17 inches in diameter ? 6. If a pipe 1.5 inches in diameter fills a cistern in 5 hours, in what time will another whose diameter is 15 inches fill it ? 7. A conical candle is one inch thick at the bottom, and burns away the first inch in 15 minutes ; it continues to burn at the same rate, and is consumed in 54 hours. Find its length. 8. If a 4|-inch pipe fills a cistern in 5^ hours, how long will it take a 3-inch pipe to fill it ? 9. How many half-inch bullets can be made from a lead ball 5 inches in diameter ? 10. A cubic foot of brass is to be drawn into a wire ^ of an inch in diameter. What will be the length of the wire ? 11. How many inch-pipes will be required to empty a reservoir as fast as a foot-pipe fills it ? 12. The sides of 3 regular octagons are 3 ft., 4 ft., and 12 ft., respectively. Find the side of a fourth octagon whose area is equal to that of the first three. 13. How many square yards of cloth will be required to make a conical tent 10 ft. in diameter and 12^ ft. high ? 14. The diameter of the earth is about 4 times that of the 36 402 SCHOOL ARITHMETIC. moon. How many moons should weigh as much as the earth, assuming them to be composed of like material ? 15. A conical wine glass 2 inches in diameter and 3 inches deep is ^ full of water. What is the depth of the water ? 16. Three men bought a grindstone 3 feet in diameter. How much of the diameter must each grind off to use up his share of the stone, making no allowance for the eye, or aperture ? 17. Four women bought a ball of yarn 6 inches in diam- eter, and agreed that each should take her share in turn by winding from the outer part of the ball. How much of the diameter did each wind off ? 18. The number of oscillations that pendulums make in a given time is inversely as the square root of the numbers representing their lengths. The length of a 1-second pendu- lum being .994 m., what is the length of a pendulum that beats half-seconds ? 19. A cylinder 12 feet in diameter is equivalent to a cone 18 feet in diameter and 8 feet high. What is the height of the cylinder ? 20. Two circular plates of the same thickness and material have diameters, the one 7 inches, the other x inches. If the weight of the latter is 40^ of the former, find the value of x. 21. If a ball of yarn 4 inches in diameter makes one pair of gloves, how many similar pairs will a ball 8 inches in diameter make ? 22. Find the amount of tin necessary to make a tin pail 6 inches in diameter and 8 inches deep. 23. A hollow sphere 8 inches in diameter is filled with water. How many hollow cones, each 8 inches in altitude, and 8 inches in diameter at the base, can be filled with the water in the sphere ? 24. A cylindrical tank is 1.2 m. in diameter and 3 m. long. If it is full of petroleum, which is .7 as heavy as water, what is the weight of the petroleum ? MENSURATION. 408 26. Find the dimensions of a cylinder, having its diameter equal to its height, that will hold 1 liter. 26. The volume of a cone is 1 cu m. What are its dimen- sions if its height is equal to the radius of its base ? 27. If the air around the earth is 40 miles deep, and the diameter of the earth is taken as 7920 miles, how many cubic miles of air are there ? 28. Find the radius of that sphere the number of square centimeters of whose surface is three times the number of cubic centimeters of its volume. 29. A hollow sphere is 32 cm. in diameter, and the shell 38 mm. thick. If the weight of the metal is 7.2 as heavy as water, what is the weight of the sphere ? How much will it hold ? 30. A trapezoidal board 12 feet long is 16 inches wide at one end, and 8 inches at the other. How far from either end must it be cut so that each part may contain one half of it ? f Since the board, represented in the figure by A BOD, decreases 8 in. in 12 ft., its non-parallel sides will meet in ; a point in 24 ft., if imagined to be produced as indicated / by the dotted lines. / .-. area of ABG = i x 24 x 1^ x 1 sq. ft. = 16 sq. ft. Area of board ABCD = 12 sq. ft. Why ? .-. area of EFG = 16 sq. ft. - 6 sq. ft. = 10 sq. ft. From similar triangles, 16 : 10 = 24'' : GN'. Hence GN = V'SQO =: 18.97 + . .-. 18.97 ft. - 13 ft. = 6.97 ft., MN, the distance from the narrow end. 31. A pole 120 ft. long breaks so that the top touches the ground 40 ft. from the foot. What is the height of the stump ? Let AT represent the pole, C the place it breaks, and B the point where T touches the ground. 404 SCHOOL ARITHMETIC. AT = 120 ft., AB = 40 ft. Construct the squares on BC and CE, CE being taken equal to AC. Since P is equal to the square on AC, it is evident that the square on the hypotenuse ex- ceeds that on the perpendicular by the two rectangles^ and h, whose combined length is 120 ft. , and whose area is equal to the area of the square on AB, or 1600 sq. ft. 1600 -4- 120 = VM the common width i (130 ft. - 13^ ft.), or of the rectangles j9 and h is 13^ ft. But this width is the difference in the lengths of AC and BC. Their sum = 120 ft. Hence, i (120 ft. + 13^ ft.), or 66f ft., = BC ; and 53i ft, = AC. 32. A limekiln measured at the bottom 50 ft. long and 20 ft. wide ; at the top 40 ft. long and 16 ft. wide. The height was 6 ft. How many cubic feet of lime in the kiln ? The limekiln may be regarded as the frustum of a pyramid. The volume of the frustum of a pyramid (or cone) of bases B, b, and altitude h, as shown in geometry, is expressed by the formula V^'liB + h ^ VBh). The area of the lower base = 20 x 50 sq. ft. = 1000 sq. ft. The area of the upper base = 16 x 40 sq. ft. = 640 sq. ft. Hence the volume = f (1000 + 640 + VlOOO x 640) cu. ft. = 4880 cu. ft. GREATEST COMMON DIVISOR. 559. The Greatest Coniinoii Divisor of two or more numbers is the greatest numher that exactly divides each of them. Thus, 7 is the greatest common divisor of 21, 35, and 42. Why ? 1. Find the G. C. D. of 56, 84, and 140 by the method of factoring. 56 =: 2" X 2 X 7. Since the greatest common divisor is the 84 = 2^* X 3 X 7. greatest factor common to the three numbers, 140 = 2' X 5 X 7. it is 2" X 7, or 28. GREATEST COMMON DIVISOR. 405 2. Find, by factoring, the G. C. D. of : (a). 32, 48, 128. (b). 84, 12G, 128. (c). 187, 253, 341. 560. To find the greatest common divisor when the num- bers cannot be easily factored, tlie long division process (also called the ** Euclidean method," from Euclid, who first used it) is usually employed. This method depends upon two principles : 1. A factor of a number is a factor of any of its multiples. 2. Every common factor of ttvo numbers is also a factor of their sum and of their difference. Thus, 7, which is a factor of 14, is also a factor of 28, 35, etc. ; and, being a common factor of 35 and 126, it is also a factor of 161, their «Mm, and of 91, their difference. 1. Find the G. C. D. of 63 and 231 by the method of continued division. 63 )231( 3 Since 21 is a factor of itself and of 42, it is 189 a factor of 63, their sum. ~~7k \cn/ 1 Since 21 is a factor of 6), it is a factor of '^2 ^ ^ 63, or 189, a multiple of 03; and being a — factor of 42 and of 189, it is a factor of 42 + 189, 21)42(2 or 231, their sww. ^ .-. 21 is a common factor of 63 and 231. Again, every common factor of 63 and 231 is a factor of 3 x 63, or 189, a multiple of 63 ; and also a factor of 231 — 189, or 42, their dif- ference. Since every such factor is now a common factor of 63 and 42, it is a factor of 63 — 42, or 21, their difference. Since the greatest common factor of 63 and 231 is contained in 21, it cannot be greater than 21. .-. 21 is the G. C. D. of 63 and 231. Find by the last method the G. C. D. of : {a). 9801 and 33759. {b). 3864, 3404, and 3657. {c). 4656 and 5926. To find the G. C. D. of several numbers by this method, find the G. C. D. of two of them ; then of that result and a third number, and so on. 406 SCHOOL ARITHMETIC. COMPOUND PROPORTION. 661. The product of two or more ratios is called a Com- pound Ratio. Thus, 3 X 5 : 4 X 7, or 15 : 28, is the compound ratio of 3 : 4 and 5:7.' 562. Any equation each member of which is composed of two factors may be written as a proportion. Take the simple proportion 5 : 8 = 15 : 24. Solving, we have, 5 X 24 — 8 X 15. It will be observed that one member contains the extremes of the proportion, the other the means. The equation ah = cd may be written, a : c = d : b, ov, a : d = c : b. The compound ratio of a : c and a : d is a x a : c x d ; that of d : b and c : b is d X c : b X b. And since the simple ratios of each proportion are equal, their products are equal. Hence axa: cxd = dxc: bxb. That is, one compound ratio is equal to the other. 563. An expression of the equality of two compound ratios, or of a compound ratio and a simple one, is called a Compound Proportion. Thus, the equation f x f = ^^ may be written in the form of the compound proportion ^ • ^ = 10 : 28 3 :7 By taking the product the proportion is reduced to the simple one 15:42=10:28. Note. — In a compound proportion the product of the extremes is equal to the product of the means, as in the case of a simple proportion; hence the required number is found in the same way. In problems of this class it is convenient to make the number which is of the same kind as the answer the third one, and then to consider each of the remaining pairs of numbers separately, forming a first couplet from each, as in simple proportion. COMPOUND PROPORTION. 407 WRITTEN EXERCISES. 564. 1. If 8 horses eat 48 busliels of oats in 24 days, in how many days will 4 horses eat 38 bushels ? ^^^;|3 = 24days:( ,. .-. the required time is 8 x 38 x 24 da. ^ gg ^^^^ 4 X 48 The problem is to determine the ratio resulting from each compari- son, and how they affect the number of days which we are required to find. For convenience we make the 24 days the third number, as the answer is to be in days. It will require more daya for 4 horses to eat a given quantity than for 8 horses to eat the same amount. Therefore we make 4 the first number and 8 the second. It will require less time for the same number of horses to eat 38 bu. than to eat 48 bu. Therefore, we make 48 the first numl)er and 38 the second. In finding the fourth number, how may the work be simplified? 2. If 24 men in 5 days can build a wall 72 rd. long, how many rods of wall can 15 men build in 6 days ? 3. If 10 men can cut 46 cords of wood in 18 days, working 10 hours a day, how many cords can 40 men cut in 24 days, working 9 hours a day ? 4. If a railroad charges $15 for carrying 3 tons of goods 180 miles, what will it cost at the same rate to transport 15000 lb. of goods 140 miles ? 5. If 36 men earn $1296 in 18 days, how much will 42 men earn in 87 days ? 6. What is the weight of a block of granite 8 feet long, 4 feet wide, and 10 inches thick, if another block 10 feet long, 5 feet wide, and 16 inches thick weighs 5200 lb.? 7. A miller has a bin 9 ft. long, 4 ft. wide, and 2 ft. deep, holding 72 bushels of wheat. How long must he make an- other bin which is to be 5 ft. wide and 4 ft. deep in order that it may hold 192 bu. of wheat ? 8. If the cost of digging a cellar 36 ft. long, 25 ft. wide, 408 • SCHOOL ARITHMETIC. and' 6 ft. deep is $90, what is tlie cost of digging a cellar 45 ft. long, 28 ft. wide, and 8 ft. deep ? 9. If 120 men in 15 days can do J of a certain piece of work, how many men in 30 days can do f^ of the same work ? 10. If 58 men working 9 hours a day require 6 days to dig a trench 100 yd. long, 2 yd. wide, and 3 yd. deep, how many men working 10 hours a day for 9 days will be required to dig a trench 50 yd. long, 6 yd. wide, and 5 yd. deep, in ground twice as hard to dig ? INSURANCE. 065. 1. A house valued at $1000 was insured against fire for one year at 1^. What was the premium, or cost of insurance ? 2. A building worth $10000 is insured for ^ of its value at Ifo. What is the premium ? 3. If I pay $10 for having my property insured at 1^, for what amount do I get it insured ? 566. Insurance is a contract by which one party agrees to indemnify another for loss sustained in the event of cer- tain misfortunes. The three most common forms -Are Fire insura.nce, Accide fit insurance, and Life insurance. 567. The written contract of insurance is called the Policy. It contains a. promise to pay a specified sum in the event of certain contingencies. This sum is the face of the policy. Fire insurance companies do not usually insure more than f of the value of a property. 568. The sum paid for insurance is called the Premium. It is usually computed at a given sum for each $100 or $1000 of insurance ; but sometimes at a certain per cent of the face of the policy. INSURANCE. 409 WRITTEN EXERCISES. 569. 1. The owner of a store insures for $16750, at 75^ per 1100. How nuicli is the annujil premium ? 2. Wliat will it cost to insure a house for 14800, at If^ ? 3. AVIiat will it cost to insure a mill worth ILSOOO for ^ of its value, at 11.50 per $100 ? 4. What is the premium for insuring property against loss by fire for 1 yr. for $3500 at $1.20 per $100, and the con- tents for $7500 at $1.30 per $100 ? 5. A house worth $8000 was insured by 3 companies for -j^*^ of its value. The first took ^ of the risk at 2^%, the second ^ of the risk at 2^, and the tliird the remainder at 2^^. What was the total premium ? 6. For how much must property worth $21825 be insured at $3 per $100, to cover both property and premium ? 7. I paid $72 for insuring my house at 2^. What was the face of tlie policy ? 8. I paid $60.75 for insuring property worth $2700. What was the rate of insurance ? 9. A grain shipper paid $525 for the insurance of a cargo of wheat at $1.50 per $100. For how much was the wheat insured ? 10. If SSfo of the value of a ship is insured at a cost of $271.33^, at f^ premium, what is the value of the ship ? 11. A man 30 years old insures his life for $2500, at the rate of $22.50 for every $1000 of insurance. What is the annual premium ? How much d es he pay in 20 years ? 12. A man had his life insured for $10000 at $32.40 per $1000. Should he die after having paid 18 premiums, how much more would his heirs receive than he had paid in premiums ? 13. A man paid an insurance company for 30 years an an- nual premium on a life policy for $5000 at the rate of $2*^.85 per $1000. If 15^ of this premium was returned in divi- dends, how much did he pay for his insurance ? 410 SCHOOL ARITHMETIC. EXCHANGE. 570. The system of making payment of debts at distant places without the transmission of money is called Ex- change. 571. If John Doe, of Baltimore, owes Richard Roe, of Rochester, $500, he can cancel the debt in several ways with- out actually sending the money. He can send a chech, a draft, Q, postal money order, or an express money order. If Doe draws a check for the |500 payable to the order of Richard Roe, he sends it to the latter, who indorses it and has it cashed. The bank receiving it collects the sum named from the bank upon which it is drawn. 572. A Bank Draft is a written order from one bank directing another bank to pay a specified sum of money to the order of the person named in the draft. The following is a common form of draft : flDercbante^ matlonal Banh. Baltimore, ^une f5, /poo. Pay to the order of B^lchaxd Sfboe $5oo.oo. cfive BSundzed^ ^^DollarS. ^. c^. "Walkct, Cashier, To the (S^dtoz €Bank, %e^ york 6iiy. John Doe may discharge his indebtedness to Roe by paying the money to his Baltimore bank, which in that case delivers to him a draft for the $500 on a New York bank, payable to his own order, or to the order of Richard Roe. This New York draft Doe sends to Roe, who indorses it and has it cashed as he did the check. For its service the Baltimore bank may charge Doe a small sum (cost EXCHANGE. 411 of exchange), probably 25^ or 50^, or it may be tV of 1% on the face of the draft. Many banks charge their patrons nothing for New York ex- change. 573. The cost of the exchange is either a merely nomi- nal sum to cover expenses, or a certain per cent of the face of the draft, usually r^^. 574. If the cost of the draft is greater than the face, exchange is said to be at a premium ; if less than the face, it is said to be at a discount. If the banks of any city, say New Orleans, have not sufficient funds on deposit in New York to meet tlie drafts they are making on that city, they must incur the expense of sending the money to meet these drafts. This raises the cost of drafts in New Orleans, and exclmnge on New York is at a premium. But if the New Orleans bankers have an abundance of funds standing to their credit in New York, they sell drafts on that city at a discount in order to get money for use at home without incurring the expense and risk of having it forwarded by express. 575. A Commercial Draft is a written order from one person to anotlier, directing him to pay a stated sum of money to the order of the bank named in the draft. Commercial drafts are extensively used by creditors to demand payment and collect debts through banks. The following is a common form : Rochester^ N, V., July f5, /poo. t Alght ^ — Pay to the %ke cflxAt National cBank of cBjocheAtex cfive BSundzed j% DollaVS. Shickaxd Oooe, To Jokn 2)oe, p8 ^ood St., malUmotc, SJBd. 412 SCHOOL ARITHMETIC. 576. If a draft is made payable on its presentation, it is called a sight draft ; if payable at a specified time after sight or after date, it is called a time draft. In many states 3 days of grace are allowed on time drafts, and in some states grace is also allowed on sight drafts. 1. If John Doe owes Richard Roe the $500 and is slow in paying it, the latter may make out a draft as above and deposit it for collection. The Rochester bank will then send it to some Baltimore bank, with a re- quest to collect and remit. This is called " drawing on " a debtor. 3. The Baltimore bank will present it to John Doe and demand pay- ment. If a sight draft, he may pay it on presentation, if a time draft (or a sight draft where grace is allowed on same), he may write the word "accepted," the date, and his name across the face. This is called " accepting the draft." He is then responsible for its payment, but is not liable unless and until he "accepts." At the proper time it will again be presented by the bank and payment demanded. 3. If he declines to accept it, or to pay it on presentation, it is returned to the Rochester bank and Roe is notified. If Doe pays the draft, the Baltimore bank remits to the Rochester bank, deducting a small sum (cost of exchange) for making the collection. 4. An "accepted" draft is in effect a note whose date is the date of acceptance if payable so many days after date ; otherwise the date of the draft is the date of the note. 577. The Postal Money Order is an order drawn by one postmaster on another, directing him to pay a specified sum of money to the person named therein, or to his order. The fees charged (cost of exchange) vary from 3^ to 30^*, ac- cording to the amount. 578. The Express Money Order is substantially like the postal money order. The fees are the same, except that on all orders not over $5 the fee is 5^. 579. Foreign Excliang-e is subject to the same general laws as exchange between different cities of this country — domestic exchange — differing chiefly as to currency and the manner of making quotations. 580. Foreign drafts are usually called Bills of Exchange, EXCHANGE. 413 and are now generally drawn in duplicate, formerly in sets of three. These are sent by different mails to avoid loss or de- lay. When one is accepted or paid, the others are void. The following is the usual form : £5oo. New York, o^ay i, tpct. At sight of this First of Exchange {Second of same tenor and date unpaid^ pay to the order of cRyichatd ffuoe, ffive &6undzcd £Sounda, ValuC rC- ceived, and charge same to account of To cBaxincf mtotliexd, '^ 3 i^ >on()on, onc/i 581. Exchange for sight drafts was quoted in New York, on June 30, 1900, as follows : On London, at $4.8G5 for 1 pound sterling, meaning that £1 in gold was worth $4,865 in gold, the exchange being quoted as so many dollars to the pound. On Paris, at 5.18 francs for $1, meaning that $1 would buy a draft for 5.18 francs. It is sometimes quoted as so many cents to the franc. On German cities, at 4 reichsmarks for $.945, meaning that $.945 will buy a draft for 4 marks. It is sometimes quoted as so many cents to the mark. WRITTEN EXERCISES. 582. 1. If exchange is at a premium of -hfo, and the bank's charge is j^^^, find the cost of a New York draft for $300. The exchange = i% of $300 = $1.50. The charge . = -,^0^ of $300 = .30. Total cost := $300 + $1.50 + $.30 = $301.80. 414 SCHOOL ARITHMETIC. 2. "When exchange was at a discount of f^ I bought a draft for $40. the bank's charge being 10^*. How much did the draft cost me ? 3. Find the cost in New York of a sight draft on London for £25 8s. £25 8s. = £25.4. 25.4 X 14.865 = $123.57. 4. When New York exchange is at ^^ premium, what is the cost of a draft for $600 ? 5. Find the cost of a $200 draft at 30^ a $1000 discount, the charge for issuing it being 15^. 6. At 50^ a $1000 premium, what is the cost in St. Louis of a draft on Boston for $540, if the western bank charges ■^^ for issuing ? 7. When exchange was ^^ discount I bought a New York draft for $1200, paying a local charge of -^^. Find the cost of the draft. 8. When a New York draft for $10000 can be bought in Chicago for $9800, is exchange at a premium or at a dis- count ? What is the rate of exchange ? The bank of which city, then, has a large balance to its credit in the other city ? 9. AVhat is the cost of a sight draft, or bill of exchange, on London for £300, exchange $4.90 ? Is exchange selling at a premium or^t a discount in this case ? 10. What is the cost of a sight draft on Paris for 1000 francs, exchange 5.18^ ? 11. What will be the cost of a bill of exchange on Berlin for 1200 marks, the rate of exchange being $.945 for 4 marks ? 12. If a New York merchant owes $2500 to a dealer in London, and remits by draft, what is the face of the draft, if the rate of exchange is $4.87 ? 13. Harry B. Naylor, of Pittsburg, draws on W. A. Saunders, of Toledo, at 30 days after sight for $320. The latter accepts July 3. Write the draft and acceptance. AVERAGE OF PAYMENTS. 415 AVERAGE OF PAYMENTS. 583. 1. For what time is the use of $1 worth us much as the use of $2 for 1 month ? 2. For what time is the use of 110 worth as much as the use of $5 for 4 months ? , 3. If one half of a debt is paid 1 month before maturity, when may the other half be paid without loss to either party ? 4. If A uses $100 of B's money for 2 months, how long may B use $200 of A's money to balance the favor ? 6. A owes B $100 due in 2 months, and $200 due in 3 months. If the first debt were paid in 1 month, who would gain ? How much ? If the payment of the second debt were deferred 1 month after maturity, who would lose ? How much ? Then at what time may both debts be paid by a single payment without gain or loss to either party ? WRITTEN EXERCISES. 584. 1. A owes B $300 due in 3 months, $400 due in 4 months, and $500 due in 7 months. In how many months can he pay the whole indebtedness at one time so that neither party shall lose ? The use of $300 for 3 mo. = the use for 1 mo. of $900. The use of $400 for 4 mo. = the use for 1 mo. ctf $1600. The use of $500 for 7 mo. = the use for 1 mo. of $3500. The use of $1200 for x mo. = the use for 1 mo. of $6000. .-.x = $6000 -I- $1200, or 5, the number of months. 2. $800 of a debt is due in 6 months, and $500 of it is due in 8 months. What is the average term of credit ? 3. Find the average (or equated) time for the payment of $2000 due in 3 mo., $1500 due in 4 mo., and $2500 due in 8 mo. 4. Find the average time for the payment of $300 due in 30 days, $500 due in 60 days, and $200 due in 90 days. 416 SCHOOL ARITHMETIC. 5. On Dec. 1, 1900, a merchant bought goods as follows : 1350 on 2 mo., 1500 on 3 mo., 1700 on 6 mo. He gave one note in payment. At what date should the note be made payable ? 6. Find the average time for the payment of $1000 due May 31, $1500 due June 18, and $2000 due July 9, reckoning the time from May 31. 7. $3000 is due in 8 months. If $1200 is paid in 5 months, and $900 in 6 months, how long after maturity should the balance be paid ? 8. Find the average time of payment on the following debts : Mar. 12, 1901, $1500 due in 3 months ; Apr. 16, 1901, $1000 due in 2 months ; May 19, 1901, $1250 due in 4 months. The earliest date at which any debt is due is June 12. The $1000 is due 4 days after, and the $1250 is due 99 days later. The use of $1500 for da. = The use of $1000 for 4 da. = the use for 1 da. of $4000. The use of $1250 for 99 da. = the use for 1 da. of |123750. .-. the use of $3750 for x da. = the use for 1 da. of $127750. .'.x = $127750 -^ $3750, or 34 + . June 12, 1901, + 34 days = July 16, 1901. 9. Find the average time of payment of the following debts : May 5, 1902, $1250 due in 30 days ; May 15, 1902, $900 due in 90 days ; May 25, 1902, $1150 due in 60 days. 10. A man, Feb. 11, 1900, gave a note for $850 payable in 4 mo.; but he paid Mar. 22, $200; Apr. 20, $110; May 10, $150. When was the balance due ? CASTING OUT NINES. 585. Every power of 10 is one more than some multiple of 9. Thus, 10 = 9 + 1 ; 10^ =-- 11 x 9 + 1 ; 10=^ = 111 x 9 + 1, etc. 586. Every product of a power of 10 by a number of one CASTING OUT NINES. 417 digit is therefore some multiple of 9, plus the number repre- sented by that digit. Thus, 40 = 4 X 9 4- 4 ; 500 = 55 x 9 + 5 ; 7000 = 777 x 9 + 7, etc. 587. As every number greater tlian consists of the sum of such products, it follows that every such number is a multiple of 9, plus the sum of the numbers represented by its digits. 7000 = 777 X 9 + 7 GOO = G6 X 9 + 6 50 = 5x9 + 5 __4= 4 7654 = 848 X 9 + (7 + 6 + 5 + 4) -7654 = 850 nines 4-4. (7 + 6 + 5 + 4) = 2 nines + 4. It is thus seen that the excess of nines iy, any number equals the excess of nines in the sum of the numbers represented by its digits. This prin- ciple may be applied to test the accuracy of the work in the simple pro- cesses of arithmetic. 1. Multiply 857 by (5S. 857 ...8 + 5 + 7 = 2 nines + 2 68 6 + 8 = 1 nine -f_5 6856. 10 = 1 nine + l-i ^^42 equal excesses. 58276. ...5 + 8 + 2 + 7-f6 =3 nines + 1 J The excess of nines in the product of the numbers equals the excess in the product of the excesses in the factors. Therefore, the work is correct, unless it contains errors that balance, which is quite improbable. 2. Divide 46718 by 263. 46718 -J- 263 = 177, with remainder 167. The excess of nines In the dividend (46718) is 8 In the divisor (263) is 2 In the quotient (177) is . 6 In the product (12) is 3 In the remainder (167) is 5 In the sum (8) is . . 8 27 418 SCHOOL ARITHMETIC. Since the dividend equals the product of the quotient and divisor, plus the remainder, the excess of nines in the dividend = the excess in the sum of the excess in the product of the excesses of divisor and quo- tient, and the excess in the remainder. Therefore, the work may be assumed to be correct. 3. Find, by casting out the nines, which of the following products are incorrect : (a). 7777 X 864 = 6,712,328. (b). 67853 X 2976 = 201,930,028. (c). 3769 X 235 = 885,715. 4. Find, by casting out the nines, which of the following quotients are correct : (a). 1,348,708 -^ 498 = 2708, with remainder 129. (b). 87614 -f- 563 = 155, with remainder 349. (c). 4000 -T- 23 = 173, with remainder 18. MEASURES OF TEMPERATURE. Centigrade. Fahrenheit. Boiling point of water. Freezing point of water. 588. Temperature is measured by an instru- ment called a Ther- mometer. There are three scales for measuring tempera- ture by means of the thermometer. The Fahrenheit, used in this country in ordi- nary business, has the freezing point of water marked 32°, and the boiling point 212°. The Centigrade^ generally used in science, has the freezing point 0°, and the boiling point 100°. The Reaumur, which is also frequently used, has the freez- ing point 0°, and the boiling point 80°. Degrees below 0° are indicated by the sign — . Thus, — 20 means 20° below zero. SPECIFIC GRAVITY. 419 WRITTEN EXERCISES. 689. 1. 80° Fahrenheit corresponds to what temperature Centigrade ? 212° F. - 32° F., or 180° F. = 100° C. 1° F. = \U° C. = §° C. 80° F. = 80° F. - 32° F., or 48° F. above freezing. 48 X i° = 26.67°. That is, 80° F. = 26.67° C. 2. 60° C. corresponds to what temperature F. ? 100° C. = 180° F. 60° C. = 1%% of 180° F., or 108° F. above freezing. That is, 60° C. = 108° F. + 32° F., or 140° F. 3. 80° C. corresponds to what temperature R. ? 100° C. = 80° R. 80° C. = AH) of 80° R., or 64° R. 4. Express 30° F. in Centigrade scale ; in Reaumur's scale. 6. Express —35° F. in Centigrade scale; in Reaumur's scale. 6. Express — 40° C. in Fahrenheit's scale ; in Reaumur's scale. 7. Express — 33° C. in Fahrenheit's scale ; in Reaumur's scale. 8. The temperature of a room is 63° F. Find the tem- perature in C. In R. 9. Express in Centigrade scale the following melting points : (a) lead, 630° F.; (b) ice, 32° F.; (c) silver, 873° F.; (d) tin, 455° F. 10. Express in Fahrenheit scale the following boiling points : (a) alcohol, 78° C; (b) ether, 35° C; (c) mercury, 357° C. SPECIFIC GRAVITY. 690. The Specific Gravity of any substance is the ratio of its weight to the weight of an equal bulk of water. Thus, a cubic foot of zinc weighs 7000 oz., and a cubic foot of water 1000 oz. The ratio of 7000 oz. to 1000 oz. is 7 ; hence the specific gravity of zinc is 7. That is, zinc is 7 times as heavy as water. 420 SCHOOL ARITHMETIC. Table of Specific Gravity. Copper 8.9 Nickel 8.9 Cork....... .24 Gold 19.3 Silver 10.5 Granite.. .. 2.7 Lead 11.3 Sulphur 2.0 Steel 7.8- Alcohol 79 Petroleum .7 Mercury. . .13.596 591. If a substance is in water, the water buoys it up by just the weight of the water displaced by it. That is, it loses a portion of its Aveight just equal to the weight of the water displaced. WRITTEN EXERCISES. 592. 1. If a cubic foot of iron Aveighs 487.5 lb., and an equal volume of water weighs 62.5 lb., what is the specific gravity of the iron ? 487.5 lb. -i- 62.5 lb. = 7.8, the specific gravity. . 2. What is the weight of a cubic inch of silver ? 1 cubic foot of water weighs 1000 oz. 1 cubic inch of water weighs |9f ^ oz. ,*. 1 cubic inch of silver weighs 10,5 x |^|*^ oz. =6+ oz, 3. If a body weighs 3.71 Kg. in air and 2.38 Kg, in water, what is its specific gravity ? 3.71 Kg. - 2.38 Kg. = 1.33 Kg. Since the body weighs 1.33 Kg. less in water than in air, 1.33 Kg. is the weight of the water displaced by it. 3.71 Kg. -^ 1.33 Kg, = 2.8 nearly, the specific gravity of the body. 4. What does a bar of aluminum 113 mm. long, 17 mm. wide, and 13 mm. tliick weigh if its specific gravity is 2.57 ? 5. If a bar of iron 18 in. long, 2^ in. wide, If in. thick, weighs 18 lb. 9 oz., what is the specific gravity of the iron ? 6. How many pounds does a man lift in raising a cubic foot of stone under water if its specific gravity is 2.5 ? 7. If the specific gravity of gold is 19.3, find the number of cubic inches of gold to the pound. 8. How many cubic feet of sea water weigh a ton, if its specific gravity is 1.026 ? 9. The specific gravity of ice is .92, of sea water J.. 025. To what depth will a cubic foot of ice sink in sea water ? INTRODUCTION TO ALGEBRA. 593. In passing from arithmetic to algebra the meaning of number and the method of representing it are extended, but there is nothing contradictory to what has been already learned in arithmetic. Algebra may be regarded as but a more comprehensive arithmetic. The symbols, 1, 2, 3, etc., are retained in algebra with their arithmetic- al meanings, and the same symbols, +, — , x, -*-, ( ), =, are used in each. Fractions, powers, and roots have the same meaning and. are written in the same form. Literal or General Number. 594. An important difference between arithmetic and algebra comes from the frequent and extended use in the latter of letters to represent numbers. Just as, in interest problems, p may stand for principal, r for rate per cent, t for the time, i for interest, and a for amount, so in any case such symbols as a, b, x, y may be used to represent any numbers whatever. In arithmetic we speakof 5 books, meaning a certain number of books, or of $10, meaning a certain number of dollars ; in algebra we speak of 71 books, meaning any number or an unknoivn number of books, of x dol- lars, meaning any number or an unknown number of dollars, 595. Numbers represented by letters are called Literal or General Numbers. The reasoning is the same whether num- bers are represented by letters or by figures. Thus, if a stands for a certain number, say the number of pupils in a room, then 2a stands for twice this number, 3a for three times this num- ber, etc. 422 SCHOOL ARITHMETIC. 1. If n stands for the number of books in my library, what is the meaning of 3w ? Of bn ? OiQn? Of ^n ? 2. If X, y, and z stand for the cost of a horse, a cow, and a sheep respectively, for what does x + y stand ? x + y + z? 2x + ^ + z? 3. If a: = 5, ?/ = 9, and 2; — 7, what is the value of x -\- y? Otx + y — z? 4. If in a number of two digits, the digit in the ones' place is 5, and the digit in the tens' place is 2, the number is 10 X 2 + 5. Write a number containing a ones and b tens. Positive and Negative Numbers. 596. Sometimes quantities that are measured by the same unit are of opposite qualities. Thus, assets and liabilities are both measured by the unit dollar, the readings of a ther- mometer above and below zero are given in degrees, and dates A.D. and dates B.C. are both given in years. In the case of assets and liabilities, the unit dollar may be taken either as a dollar of assets or as a dollar of liabilities ; if as a dollar of assets, then assets are regarded as positive ; and liabilities, for the sake of dis- tinction, as negative. In order to represent quantities that have opposite qualities we need to extend the idea of number as given in arithmetic so as to include numbers that count negative units. 697. The numbers arithmetic deals with are greater than zero, and are called positive numbers ; but algebra treats also of numbers that in relation to positive numbers are regarded as less than zero, and these are known as negative numbers. 698. The primary notion of a negative number is that of one which, when taken with a positive number of the same kind, goes to diminish it, cancel it, or reverse it. Thus, liabilities neutralize so much assets, thereby diminishing the net assets, canceling them, or leaving a net liability. 699. Negative numbers may be regarded as arising through the extension of the operation of subtraction to the INTRODUCTION TO ALGEBRA. 423 case ill which the minuend is less than the subtrahend, which, from an arithmetical point of view, is impossible. Note the following : 6-4 = 2. 5-4 = 1. 4-4 = 0. 3 — 4 = - 1 ; that is, a numher one unit less than 0. 2 — 4 = -2 ; that is, a number two units less than 0. Observe that, as the minuend decreases by 1, 3, or more units, the subtrahend remaining the same, the remainder decreases by an equal number of units, becoming when the minuend is equal to the subtra- hend. If, then, the minuend becomes less than the subtrahend by 1, 2, or more units^the remainder must decrease by an equal number of units, and therefore become less than by 1, 2, or more units. But the opera- tion of subtracting a greater number from a less is possible only when numbers less than zero are introduced. 600. The negative remainder, -1, does not mean that more units were taken from the minuend 3 than it con- tained ; it merely shows that the subtrahend is 1 unit greater than the minuend. 601. The absolute value of a number is the number of units contained in it without regard to their quality. The numbers of arithmetic are the absolute values of the positive and negative numbers of algebra. Since letters are used to represent numbers which may have any values whatever, they can represent either positive or negative numbers. 602. In the expression 6 — 4, the minus sign indicates that 4 is to be taken from 6. It is a symbol of operation, and does not show 4 to be a negative number. Both 6 and 4 are positive numbers. But the same sign has another use, namely, to denote negative numbers ; and the sign + is used to denote positive numbers. When so used these signs are symbols of quality, and do not indicate any operation whatever. Thus, -4 means a number four units less than 0, and +4 a number four units greater than 0. 424 SCHOOL ARITHMETIC. 603. Positive and negative numbers are called opposite numbers, and may represent any quantities that are opposite in their relation to each other. Thus, degrees ahoi^e zero on a thermometer may be called positive ; degrees below zero, negative ; distance east, positive ; distance west, negative ; assets, positive ; liabilities, negative. 604. Zero is neither a positive nor a negative number ; it is the starting point from wliich positive and negative num- bers are counted. Thus, opposite temperatures are counted from zero on the ther- mometer, —5° meaning 5 degrees helow zero, and +5° meaning 5 degrees above zero. 605. Cash received and cash spent are opposite quantities, and may be represented by positive and negative numbers. Thus, $50 received may be represented by + $50. by -$50. A man's cash account might be kept as in the left-hand column, or as in the right. Friday he looks over his fig- ures to see how much cash he ought to have on hand. He adds the sums received, which amount to $44, and the sums expended, which amount to $44. These cancel each other. That is, $44 received united with $44 spent is equal to neither cash on hand nor debt ; or $44 received + $44 spent -— 0. Or, he may add the positive numbers in the column to the right, get- ting + $44, and the negative numbers, getting -$44. This may be ex- pressed algebraically thus : + $44 + -$44 = 0. 606. We have already seen that zero is the difference be- tween two equal numbers. From the preceding article we learn that it is also the sum of two equal and opposite num- bers. $50 spent <( (( Monday, received $25, or +125 (( spent $ 8, or -$ 8 Tuesday, spent $13, or -$13 Wednesday, received $ 9, or +$ 9 <( spent 116, or -$1G Thursday $ 7, or -$ 7 (' received $10, or +$10 Friday, Cash on hand, INTRODUCTION TO ALGEBRA. 425 Thus, gain $5 + loss $5 = 0. Or, in algebraic language, +$5 + -|5 = 0. Carefully examine the following statements : 20 dollars gain + 20 dollars loss = 0. +^0 + -20 = 0. This result means neither ffain nor loss. 20 dollars gain + 15 dollars loss = 5 dollars gain. +20 + -15 = +5. This result means a net gain of 5 dollars. 20 dollars gain + 30 dollars loss =^ 10 dollars loss. + 20 + -30 = -10. This result means a net loss of 10 dollars. 5 miles east + 5 miles west = the starting point. + 5 + -5 = 0. This result means that the traveler has returned to the point from which he started. 10 miles noftJi + 15 miles south = 5 miles south. + 10 + -15 = -5. This result means that the traveler stopped 5 miles south of his starting point. Make similar statements for each of the following : 1. 180 gain, |50 loss. 2. 175 gain, $100 loss. 3. 40 miles east, 30 miles west. 4. A rise of 20° in tem'perature, then a fall of 18°. 6. An army balloon ascended 3000 feet, then fell 1800 feet. ADDITION. 607. In algebra the process of adding two or more pos- itive or negative numbers is the same as that of adding in arithmetic, except that the sigti of quality is to be prefixed to the sum. Thus, +7 added to +3 = +10 ; -7 added to -3 = -10. 426 SCHOOL ARITHMETIC. Add the following : 1. +9to+l(). 4. +8.4 to +9.9. 7. ^8{a + b) to +3{a -^ b). 2. +84 to +48. 5. +8yto+e)y. 8. -7{m — n) to-6{m ■- n). 3. -72 to -28. 6. -Via to -oa. 9. +5a; and +x to +9a;. 608. Terms containing the same letters with the same exponents are called Similar Terms. Thus, dy^ and -5y^ are similar terms, as are 7 {x + y) and 2 (x + y). 600. Principle. — As in arithmetic only like numbers can be added, so in algebra only similar algebraic 7iumbers can be united by addition into one term. Although unlike numbers can not be united by addition into one term, an indicated operation is regarded as their algebraic sum. Thus, m + n is called the sum of m and n. Add the following : 1. 2. 3. 4. 5. +^x -X 5ab "^my (a + b) ^x -dx eab -'Hmy b(a + b) +9a: -9x ab -7 my 4(« -\- b) When numbers are positive, the symbol of quality ( + ) is usually omitted, as in examples 3 and 5 above. When no symbol is written, + is understood. The sign — is never omitted. 610. The following equations were considered in article 606: +20 + -20 = • (1). +20 + -15 = +5 (2). +20 + -30 ^ -10 (3). In (1) we see that +20 and -20 cancel each other, that is, that their sum is 0. In (2) we see that -15 cancels +15, leaving 5 positive units (+5), which is the sum. In (3) we observe that +20 cancels -20, leaving 10 nega- tive units (-10), which is the sum. Queries.— How is the 5 in (2) obtained ? Why has it the + sign ? In INTRODUCTION TO ALGEBRA. 427 (3) how is the 10 obtained ? Why has it the — sign ? Is -2x the sum of 5x and -7x ? Why ? Add the following : 1. +8 to -20. 4. +3x to -12a;. 7. Sx'y to -llx'y. 2. -T to +15. 6. ISab to -5«^. 8. 8(m + ?i) to -7{m + n). 3. 25 to -16. 6. -2% to +30^. 9. -{x - y) to 10(a; — y). 10. Find the algebraic sum of bx, -7a:, +9a;, -4a;, and a;. 1. The sum of the positive numbers is 15a: ; the sum of the negative numbers is -11a;. These two sums united = Ax. 2. In this example the 5, 7, 9, and 4 are coefficients. When no coef- ficient is expressed, 1 is understood. Add the following : 11. 12. 13. 14. ^ax -97nn SUcd 72 {b - a) -5ax -mn -2obcd -48 (b — a) + 7ax +4mw -9bcd -50 (b - a) Express in the simplest form : 15. 8a; 4- 3a: — 5a: + a; - 4a; + 12a; — 7a;. 16. 15Z»a: — 6bx -{- bx — 9bx — bx + 18bx — lObx. 17. 5(m — 7i) + 13(m — 7i) — ll(m - n) + Q(m-n) — 20(m — ii). 18. Add 3a: + « — 2y, 5a; — 4« + 6z/, 7« — 8y, and y — 4a; + 6a. 3a: + a — 2y bx — 4ca + 6y For convenience we write similar terms in the ^a — 8y same column. The sum of the first column is -4a: + 6a -^ y +4x, of the second +10a, and of the third -3y. 4a: -f- 10a — 3y 19. Add 6y — 4c, 3?/ + 8c, 5c — 4?/, y — c, y — 10c. 20. Add 6m + 2?^ — bb, 7?i — 3& — 4???, b + 8m — 9n, m + bb. 21. If a man has a sons, ^ daughters, and 1 wife, how many persons are in the family ? 428 SCHOOL ARITHMETIC. 22. A boy who had 15 cents found in cents and earned 4m cents. How much had he then ? 23. My house is d feet long and c feet wide. What is the distance around it ? 24. Tom walked due east m hours, then due west n hours. If his rate was 3 miles an hour, at what distance from his starting point did he stop ? 25. If m in the above problem is equal to 7i, where did Tom stop ? 26. If m = 5 and n = d, how far, and in what direction, from his starting point did he stop ? 27. Locate his stopping place if m = 4 and n = Q. 28. D earns a dollars each week and spends b dollars. How much will he have at the end of 8 weeks ? 29. What will be his financial condition it a = $15 and d = UO? What will it be if « = $12 and ^> = $16 ? SUBTRACTION. 611. In algebra, as in arithmetic, the minuend is the sum of the subtrahend and the remainder. Thus, 10 + -4 = 0, the sum, which we may regard as a minuend. Taking 10 as the subtrahend, we have 6 — 10 = -4, the remainder. Taking -4 as the subtrahend, we have 6 — -4 = 10, the remainder. In either case 6 is the sum of 10 and -4. Carefully examine the following: 8-2 = 6. -11 - 2 = -13. 1 - 2 = -1. 8 + -2 = 6. -11 + -2 = -13. 1 + -2 r^ -1. 8=6 + 2. -11 = -13 + +2. 1 = -1 + 2. Observe l}\&t subtracting +2 is equivalent to adding -2 ; also, that the minuend is the sum of the remainder and the subtrahend. INTRODUCTION TO ALGEBRA. 429 Examine these equations : 7 - -4 = 11. -12 - -4 = -8. 2 - -4 = 6. 7 + 4 = 11.. -12 + 4 = -8. 2 + 4 = 6. 7 = 11 + -4. -12 = -8 + -4. 2=6 + -4. Observe that subtracting -4 is equivalent to adding +4 ;.also, that the minuend = remainder + subtrahend. 012. In general, to subtract n positive number is equivalent to adding an equal negative number ; to subtract a negative number is equivalent to adding an equal positive number. Illustration I. A man wliose income is $100 a month spends $00, and saves $40. If his income is reduced $10 a month, he will save $30. $90 - $60 = $30. Or, if his expenses are increased $10 a month, he will save $30. $100 - $70 = $30. Hence, to tahe away $10 income is equivalent to adding $10 expenses. Either reduces his savings to $30. Calling income and ^ixsmg^ positive, and expenses ^e^a^eve, we have the following algebraic expression of this relation : $40 - +$10 = $30. $40 + -$10 = $30. Illustration II. If his income is increased $10 a month, he will save $50. $110 - $60 = $50. Or, if his expenses are reduced $10 a month, he will save $50. $100 - $50 = $50. Hence, to tahe away $10' expenses is equivalent to adding $10 income. Either increases his savings to $50. The rela- tion is algebraically expressed thus : $40 - -$10 = $50. $40 + +$10 = $50. 430 SCHOOL ARITHMETIC. 1. A has $10 ; B has no money and is $5 in debt. How much more is A worth than B ? 2. Tom has $50 in bank. Harry has $20 in cash, but owes John 140. Tom is worth how much more than Harry ? 3. Albert has 24 marbles and Ed has none, but owes Albert 16. How many more marbles than Ed has Albert ? 4. Mary has 8 jacks and Alice has -5 (i.e., owes 5). Alice has how many fewer than Mary ? 5. In the schoolroom the temperature is 70° above zero, while outside it is 5° below zero. How many degrees warmer is it inside tlian outside ? 75° — -5° = ( ). 6. A is $35 in debt and B is $50 in debt. How much bet- ter off is A than B ? -$35 - -$50 = ( ). 613. In algebra when the subtrahend is greater than the minuend, the remainder is negative, and shows liotv nmchiYiQ subtrahend exceeds the minuend. Thus, 8 - 11 = -3 ; 5a: - 9a: = --4a; ; -\2y - -5y = -ly. Subtract the following : 1. 8 from 5. 5.-1 from -11. 9. 15« — 22a. 2. 9 from 1. 6. -8 from -5. 10. -4y from -lly. 3. 25 from 7. 7. -27 from -2. 11. -9a: from -20:r. 4. -6 from -10. 8. 7a: - 15.r. 12. -2U from -18^. 614. In arithmetic the remainder is never greater than the minuend ; but in algebra it is often greater. Note the following : (a) (b) (c) (d) Minuend +31 -19 +12 -35 Subtrahend . . +16 +13 ^ -24 Remainder +15 -32 +20 -1 In (a) and (b) the remainders are less than the minuends ; in (c) and (d) they are greater. Which subtrahends are positive ? These examples illustrate the following : 1. When the subtrahend is j^ositive the remainder is less INTRODUCTION TO ALGEBRA. 431 than the minuend. Subtracting a positive number is equiva- lent to adding an equal negative number. 2. When the subtrahend is negative the remainder is greater than the minuend. Subtracting a negative number is equivalent to adding an equal positive number. 1. From 12a; subtract bx. 12x 12a; — +bx = Ix (subtracting a positive number); hx or 12a; + ~5a; = 7a; (adding a negative number). ~x \. Subtract : I8y from Iby. 15y 15y- - +18y = -3y. 18y i5y ■ f -18y = -3y. -3y 1. From 7a take -5a ',. 7« 7« - -5a = 12a. -5a 7a + 5a = 12a. 12a \, From 4a; — 3y take x + ^y^ 4a; - Zy 4a; - - X = Zx, and -%y- + 2^ = -oy, x + 2y or - -3y 4- -%y = -5y. 3x — by Notice that in each of these examples we have changed the 8ign of the subtrahend and then added. Subtract : 6. 8a; from 3a;. 9. 7a; — 2«/ from 8a; — 5y. 6. 13?/ from -17^. 10. 9«J + '6d from 12«^' - d. 7. -5a from -12a. 11. a — h from a + ^. 8. -Qh from 4^. 12. a; — «/ + from x + y -\- z. 13. 3aa; — 7^ + 55 from 6aa; — 4i/ — 3J. 14. 4c(5 — ^) + 6am from 10c(J — and we have 5a + 7b — Za — 2b, or 2a + 56. 16. 12.C + Sy - (dx - y) = what ? I2x + ^y - ^x + ij = ^x + 9y. Complete the following equations : 17. 3^ + 17:?; - {y + 16a;) = 18. 5(m -\- n) — Ixy — (m 4- w) + 4:xy = 19. 12(a -{- b + c) - 8 {a + b -h c) ~ U -h 20g = 20. By selling a cow for a dollars I gained b dollars. What did the cow cost ? 21. A and B together have 10 children, of whom A has n. How many has B ? 22. Smith has b dollars and Jones is c dollars in debt. How much more is Smith worth than Jones ? 23. H' I have 8 children and m of them are girls, how many boys have I ? 24. A man who earns d dollars a month spends p dollars for rent and q dollars for other purposes. How much does he save in a month ? 25. A boy wishing to ride n miles on his wheel rode a miles the first day and b miles on each of the next two days. How many miles had he yet to ride ? 26. A man engaged to fence a field c rods long and d rods wide. If he built (c — d) rods a day for 3 days, how many rods had he yet to build ? 27. A drover paid |m for b pigs, and $wfor c sheep. He sold the former at $8 and the latter at $5 a head. Find his gain. MULTIPLICATION. 615. When the multiplier ispositive, the process of multi- plication in algebra is the same as that in arithmetic, except that the sign of quality of the multiplicand is to be written before the product. Thus, INTRODUCTION TO ALGEBRA. . 433 (a). +3 X +5 = +15. (b). +3 X -5 = -15. Observe that three times 5 positive units gives 15 positive units as a iroduct, and three times 5 negative units gives 15 negative units as a iroduct. 616. When the multiplier is negative^ it gives to the )rodiict a sign opposite to that given by a positive multiplier; hat is, the quality of the multiplicand is reversed in the )roduct. Note the following :• (c). -3 X +5 = -15. (d). -3 X -5 = +15. Since +3 + -3 = 0, -3 = - +3, or -(+3) ; hence in (c), -3 x +5 lay be regarded as signifying that +5 is to be taken three times, and hen tlie result reversed (subtracted), giving the product the sign opposite that of the multiplicand. Similarly in (d), -3 x -5 may be regarded as signifying that -5 is to e taken three times, and the result reversed, that is, -15 is to be sub- racted. Now, subtracting -15 is equivalent to adding +15. .'. -3 x 5 = +15. Queries. — 1. In (a) and (b) the product has the sign of the raultipli- and. Is this true in (c) and (d) where the multiplier is negative? 2. When both factors are positive, as in (a), what is the sign of the iroduct ? When both are negative, as in (d) ? When they have unlike igns, as in (b) and (c) ? 617. If a and h stand for any two numbers, we have +a X +h = -^ah, +a X -b = -ah, -ax +b = -ab, ^ -a X -b = +ab. That is, when two factors have like signs the product is wsitivej when they have unlike signs the product is negative. Illustrations. A train whose speed is 20 miles an hour runs north and outh past a point P, passing at 12 m. Locate the train at 5 '.M. and at 7 a.m. 28 iBi SCHOOL ARITHMETIC. Consider the following as positive : (1) distances north, (2) the train's rate northward, (3) time after 12 m. Consider the opposites negative. Then, if the train is running north- ward, (a) in 5 hours after 12 M. it will be 100 miles north of P, which is expressed algebraically by + 5 X +20 = +100. (b) 5 hours before 12 M. it will be 100 miles south of P, expressed algebraically by -5 X +20 = -100. If the train is running southward, (c) in 5 hours after 12 M. it will be 100 miles south of P, expressed algebraically by + 5 X -20 = -100. (d) 5 hours before 12 m. it will be 100 miles north of P, expressed algebraically by -5 X -20 = +100. Multiply the following : 1. 4a by 3. 6. 9xy by -a. 2. -b by 7. 7. -ab by 10. 3. -6x by 2. 8. -c'd by -5. 4. -6y by -d. 9. -12 by ax. 5. -8m by -1. 10. -1 by 15/?. 11. What is the product of a — b + 2x multiplied by Sa ? ^ ^ ' '^'^ Each terra of the multiplicand is multiplied ^a by 3a. The algebraic sum of these products is 3«' — 'dab + 6ax "^^^ required product. Multiplication is usually indicated by writing letters, or a figure and one or more letters, side by side. INTRODUCTION TO ALGEBRA. 485 12. Find the product of x^ + ^x^y — Zxy^ — y" multiplied by ic - y. x^ + '^x^y - 2>xy'' — y^ X — y X* + ^x^y — Sx^y^— xy^ = product of multiplicand by x. — x^y — 3a;y + 3xy* + y* = product of ^^ by -y. X* + 2x^y— 6a;y + 2xy^ + y* = complete product. It is a convenient arrangement to write the multiplier under the multiplicand, and place like terms of the partial products in columns. Observe that in multiplying we take the product of the coeflBcients and the sum of the exponents of the same letters. 19. 8«m + Qbn by bah — h. 20. a;' + 2xy + y^ hy x + y. 21. {i?i — ny by (m — n). 22. a'-h ab + hc- V by «*+ l\ 23. r' - r 4- 1 by r' + r + 1. 24. m^ + Sm'^i + wi/^' by W2 — w. w Multiply the following : 13. a ^l\>y a ^ b. 14. flj — J by a — ^. 16. a -\- b\iy a — b. 16. a; + 9 by a; - 9. 17. m + 5 by m — 2. 18. Zy + 7;2 by 4;? + by. 25. How many square rods in a field m ■\- n rods square ? 26. How many acres in a field whose length is 20/» rods, and whose width is 16m rods ? 27. A has hb acres, B has 76' acres, and C has Via acres. They offer to sell out to D at 3w dollars an acre. Find what D would have to pay. 28. A man having "la horses sold b of them at %r each, and the remainder at %%r each. If they cost him %2>ar, how much did he gain ? Find the value of the following : 29. {a + b)\ 31. {c 4- 3)'. 30. {a - b)\ 32. (y + 2) (?/ + 2). m« mn mn n^ 436 SCHOOL ARlfHMEf id. 33. (x' + 7) {x' - 7). 35. (m - 7i) (m - n), 34. {c + 4.d) {4d + c). 36. {x + 4:) {x + 5). 37. Show that the square of the sum of two numbers is equal to the square of the first number, plus twice the prod- uct of the two numbers, plus the square of the second number. 38. Square a -\- b and a — h, and compare the results. The square of the difference of two numlers is equal to what ? 39. Multiply x + yhy x — y, and note the product. The 2)roduct of the sum and difference of two numbers is equal to the difference of what ? Write the products of the following : 40. {c + d) {c + d). 43. {Zy - 5) (3y - 5). 41. (c -¥ d) (c- d). 44. (2a; + y) (2a; - y). 42. (m -n) (m- n). 45. (2« + b) {2a + b). 46. What two equal factors produce a"^ + 2ax + x^ ? 47. What two equal factors give the product ¥— 2by +^''? 48. What two factors produce m^ — n^ ? x"^ — y"^ "^ 49. How many square feet in a room a -\- b feet long and a — b feet wide ? How many square yards in the room, if a = 15 and Z* =r 12 ? 50. Find the volume of a cube whose edge \& m — n inches. DIVISION. 618. Division is the inverse of multiplication. In the latter two factors are given, the product required. In the former the product (dividend) and one factor (divisor) are given, the other factor (quotient) required. Hence the law of signs may be derived from that in multiplication, as follows : Since +3 x +5 = +15, Since +3 x -5 = -15, Since ~3 x +5 = -15, Since -3 x -5 = +15, 15 -^ + 3 = + 5. 15 -^ +3 = -5. 15 -^ -3 =: +5. 15 -r -3 = -5. INTRODUCTION TO ALGEBRA. 437 k Or, using a and h to denote anyjtwo numbers, we have +« X +J =: +flf.^, .-. ^ah -^ +« = +5. +a X -5 = -ah, . '. -ab -r- +a = -b. -ax +b = -ab, .*. -ab -^ -a = +^. -a X -b ~ ^ab, .*. ^ab -^ -a — -b. That is, like signs of dividend and divisor give a positive quotient ; unlike signs, a negative quotient. Divide the folio-wing : 1. Qab by 2. 6. a'^b by -ab. 2. %x by U. 7. -2;.y' by -rcy. 3. 16c by -4c. 8. -a^b'^c by rt^>c. 4. -1% by 2. 9. 20w' by -4m. 5. -12«m by -3w. 10. IGaVy by Saa:^. 11. Divide 4a'a; - 6a^/ + 2ac' by 2a. 2a [ 4a'a; — Ga^y' + 2ac' 2ax — Uy"" + c' 12. Divide x^ — 3x''y + 3xy'' — y' by x — y. x'-3x'y 4- 3xy'-y' X - -y x\x-y) =^x'- x'y x'- - 'Zxy + y^. - %x^y H- 3a;^* -'Ixyix — y) — — "Zx^y + 2a:?/ xf - f f{^ -y) = ^f - f Divide the followinor : 13. a' + 2ab + b^ hy a + b. 14. m^ — 2mn + /^^ hj m — n. 15. 15^>^ - 8Jc - 12c^ by 35 +2c. 16. x' + 3a:> + ^xy"" -f «/' by a: + ^y. 17. «/^ +3y*+2by«/ + l. 18. a' - a - 90 by a 4- 9. 19. x^ — y"^ hj X — y. 20. 9^^ - 4a^ bv U + 2a. 4:38 School arithmetic. 21. y" - \^yz - 242;' by ^ + %z. 22. 16a;' - %^^xy + 9«/' by 4a: - 3y. 23. There are 16Jc + 24c square feet in a hall 8c feet widd. What is the length ? 24. The area of a field « + c rods wide is a^ + «6 + 3«c + Jc + 2c' square rods. Find the length. Divide the following by the highest factor common to all the terms : 25. %a^b - l^ahy. 27. 2c«/' - Qc'x + ^bc. 26. 8Jaj' - 6aZ»'a;. 28. 18m' - ^7nn + 12m^i'. 29. Divide «' — Z>' first by a + ^ and then hy a — h, and compare results. The difference of the squares of two num- bers is divisible by what ? Then what are the two factors of a' - V ? Write from inspection the factors of : 30. x^ - y\ 33. 4 - h\ 31. x^ - 9. 34. x^ - a\j\ 32. a' - 1. 35. JV - d\ EQUATIONS. 619. An equation has been defined as a statement that two numbers or expressions are equal. The principles that apply to the transformation and solution of equa- tions, as given in Arts. 256-262, should here be reviewed. 020. In solving simple integral equations the following direction will be found useful : Transpose all the terms containing the unknown number to the first member, and all other terms to the second member. Unite like terms, arid divide both members by the coefficient of the unknown number, 621. If the value found for the unknown number is sub- stituted in tbe original equation, and the equation reduces to an identity, the value of the unknown number (called the root of the equation) is said to be verified. INTRODUCTION TO ALGEBRA. 43^ Find the value of x, and verify the answer : 1. 5a; = 28 - 2x. 4. 4a; - 14 = a; - 2. 2. _ 3a; - 7 = - 4.T + 7. 5. Hx — (ox + 5) = 7. 3. 4ic — 2(2 — x) = Q. 6. ax = mx — n. XX X 7. Solve the equation 4 + j=— — S+.y. X X ^ X Clearing of fractions, 48 + 3a; = 6a; — 3G + ^x. Uniting terms, — 7a; = —84. Dividing by — 7, x = 12. An equation may be cleared of fnictions by multiplying both mem- bers by the least common denominator of the fractions, which in this example is 12, Solve the following : S. x +^=15. 12. 2/ + I + f := 11. 4 -'2 3 9.. +1 = 18. I3.i + i + | = l8. 10.2« + i=U. 144^-1 + 1 = 8-^. 11. 0J+* j+f. 15. l±I-l^^ ^UzA = I. 4 4 4 o 2 16. Find the value of x in the equation a;" — 4 = 5. x' -4: = b. cc' = 5 + 4 = 9. X = ±3, by extracting the square root. (Art. 521.) The sign ± before the 3, read plus or tninus, shows that the root is either + or — . For +3 x +3 — 9, and -3 x -3 = 9. The negative value does not always have a meaning in particular problems. Solve the following : 17. 5x' = 80. 20. 3a;' + 1 = 2x'' + 10. 18. ^ - 5^5 = ^' - 4|. 21. (x + Q) {x-(j) = 28. 19. (3 - xy = 3(1 - xy. 22. ax' + i = bx' + a. 440 SCHOOL ARITHMETIC. PROBLEMS. 622. In stating problems, it is important to remember that the letter x should not be put for distance, time, weight, etc., but for the number of miles, of hours, of pounds, etc. In connection with the stating and solving of problems, Arts. 261 and 262 should be re-read. 1. The sum of the two digits of a number is 4. If the digits are interchanged, the resulting number is equal to the original one. What is the number ? Let X stand for the digit in the ones' place. Then 4 — a; is the digit in the tens' place. .'. 10(4 — ic) + a; = the original number. Why ? lOa^ + (4 — a;) = the second number. Hence, 10a; -|- (4 — a;) = 10(4 — :iS) -^^ x, the equation of the problem. Solving this equation, we obtain a; = 2, the digit in the ones' place ; whence 4 — a: = 2, the digit in the tens' place. .-. the original number is 10(4 — x) + x, or 22. 2. A son is one fourth as old as his father. Four years ago he was only one fifth as old as his father. What is the age of each ? Let X = the numher of years in the father's age. Then — = the number of years in the son's age. . 4 a: — 4 = the number of years in the father's age 4 years ago. ^ — 4 = the number of vears in the son's age 4 years ago. 4 .-. ^ — 4 = i(a: — 4), whence X = 64, and^ = 16. 4 Therefore, the father is 54 years old and the son is 16 years old. 3. A certain street contains 144 square rods, and the length is 16 times the width. Find the width. INTRODUCTION TO ALGEBRA. 441 Let X = the number of rods in the width of the street. Then 16a; = the number of rods in the length of the street. X X 16a; = tlie area of the street in square rods. 16a;' = 144 a;" = 9 X = ±3 Therefore, the width of the street is 3 rods. The negative root is not applicable to this particular problem. 4. A fulcrum is to be placed under a 3-foot lever so as to divide it into two parts such that 1.2 times the first shall equal 4.8 times the second. How far is it from either end ? 5. A man bought 10 yards of calico and 20 yards of silk for $30.60. The silk cost as many quarters a yard as the calico cost cents a yard. Find the price of each. 6. I bought a number of apples at the rate of 3 for a cent ; sold one third of them at 2 for a cent, and the remainder at 5 for 3 cents, gaining 7 cents. How many did I buy ? 7. Atmospheric air is a mixture of four parts of nitrogen with one of oxygen. How many cubic feet of oxygen are there in a room 10 yd. long, 5 yd. wide, and 12 ft. high ? 8. In a certain family each son has as many brothers as sisters, but each daughter has twice as many brothers as sis- ters. How many children are in the family ? 9. A number is composed of two digits whose sum is 8. If the digits are interchanged, the resulting number is greater by 18 than the original number. What is the number ? 10. One third of my sheep equals one ninth of them plus 8. How many have I ? 11. A man spends ^of his income for rent, ^ for groceries, and has $1140 left. What is his income ? 12. Divide 15 apples between A and B so that ^ of A's number shall equal -J of B's. 13. My wife's age plus mine equals 76 years, and f of her age minus two years equals ^ of my age plus 2 years. Find the age of each. 442 SCHOOL ARITHMETIC. 14. A has 8150 more than B, and has ^ as much as A and B. They all have $1000. How much has each ? 15. Sixty dollars was divided equally among a number of men. Had their number been 4 less, each would have re- ceived three times as much. How many men were there ? 16. Find two consecutive numbers such that J of the greater is 3 more than \ of the less. 17. What number added to the numerator and denomina- tor of f will give a fraction equal to | ? 18. Eleven sixteenths of a certain principal was at interest at 5 per cent, and the remainder at 4 per cent. The entire income was 11500. Find the principal. 19. Two numbers are to each other as 3 to 4. If 10 is sub- tracted from each, the smaller one will be f of the larger. What are the numbers ? 20. Two numbers are to each other as 2 to 3, and their product is 150. What are the numbers ? 21. A rectangular lot contains an acre, and its width is |- of its length. What is its width ? 22. A triangular field contains 5 acres, and its altitude is ■f of its base. What is the base ? 23. A circular pond contains 314.16 square yards. AVhat is its diameter ? 24. The area of one square field is twice that of another, and they together contain 867 square rods. 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