ILNE'S iACTICAlffEflMETIC [^ rVv Cincinnati, Philadelphia, Chicago. GIFT OF |V THE PRACTICAL ARITHMETIC ON THE INDUCTIVE PLAN, INCLUDING ORAL AND WRITTEN EXERCISES. BY WILLIAM J. MILNE, A. M. PRINCIPAL OF THE STATE NORMAL SCHOOL, GENESEO, N. Y. JONES BKOTHEES & COMPANY: CINCINNATI, PHILADELPHIA, CHICAGO, MEMPHIS. 1879, M-rf COPYRIGHT, 1877, BY JOHN T. JONES. ELECTROTYPED AT THE FRANKLIN TYPE FOUNDRY, CINCINNATI. THE design of the author in preparing this work has been to embrace within moderate compass all the essentials for A PRACTICAL COURSE IN ARITHMETIC, and to present every subject in such a manner as to secure the highest mental development of the learner. To accomplish these results the author has spent much time in investigation, and in consul- tation with eminent educators and successful business men, and he believes that he has included in this volume all the subjects necessary for the arithmetical part of a business education. The method of introducing each subject is such that the student is led to truth in the path of the original investi- gator certainly the most natural and delightful road to the acquisition of knowledge. It is because of this special feat- ure in connection with every subject that the series has been called THE INDUCTIVE SERIES. The work contains oral and written exercises sufficient in number to enable the student to master the principles un- derlying each subject and to give him facility in numerical processes. (iii) 438905 IV PREFACE. In the problems given for solution it has been the aim of the author to use the language of trade, when no error is conveyed thereby, thus accustoming the student to the forms of expression needed in after life ; and in general the author has striven after clearness of statement rather than technical accuracy of expression. It would be pedantry to specify the departments in which excellence or originality may be found, but it is hoped that a careful examination will exhibit the logical sequence of the steps in all the processes, the perspicuity and accuracy of the analyses, and the brevity and correctness of the definitions, principles, and rules. The author takes pleasure in acknowledging his indebted- ness to Prof. J. B. DE MOTTE, of Indiana Asbury University, and to several other teachers of ability and experience, for timely and valuable suggestions. Trusting that the book will, in some measure, supply the popular demand for a brief and comprehensive treatise upon Arithmetic, the author presents his work to the public. W. J. M. STATE NORMAL SCHOOL, GENESEO, N. Y., September, 1877. 4 GONTENTS f NTJMEBATION. Numbers 'may 'fee 'expressed 'l>y words or other characters, viz: figures and letters. 6. Notation is the method of expressing numbers by figures and letters. The Arabic Notation is the method of expressing numbers by means of figures. Its name is derived from the Arabs, by whom it was introduced into Europe. The Roman Notation is the method of expressing numbers by means of letters. It is so called because it was used by the ancient Romans. 7. Numeration is the method of reading numbers expressed by figures or letters. ARABIC SYSTEM. 8. In this system ten figures are employed to express numbers, viz: Figures. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Names. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. Each of these, except naught, is called a significant figure. Naught is also called zero and cipher. 9. By combining these figures according to certain prin- ciples, we can express any number. 10. PRINCIPLE. When figures are written side by side, the one at the right expresses units, the next tens, and the next hundreds. EXERCISES. 11. 1. In 79, what does the 7 express? What does the 9 express? Read the number, beginning at the left. 2. In 58, what does the 5 express? What does the 8 express? Read the number, beginning at the left. 3. In 740, what does the 7 express? What does the 4 express? What does the express? Read the number. 4. Begin at the left and read 76, 176, 106, 360, 203. NOTATION AND NUMEBATION. 9 12. Figures in units' place express units of the first order; those in tens' place, units of the second order; those in hundreds' place, units of the third order, etc. 13. Numbers between 1 ten and 2 tens are named thus: 1 ten and 1 unit or 11, eleven. 1 ten and 2 units or 12, twelve. 1 ten and 3 units or 13, thirteen. 1 ten and 4 units or 14, fourteen. 1 ten and 5 units or 15, fifteen. 1 ten and 6 units or 16, sixteen. 1 ten and 7 units or 17, seventeen. 1 ten and 8 units or 18, eighteen. 1 ten and 9 units or 19, nineteen. The words thirteen, fourteen, fifteen, etc., mean three and ten,/ow and ten, five and ten, etc. 14. The units of the second order are named as follows : 2 tens or 20, twenty. 3 tens or 30, thirty. 4 tens or 40, forty. 5 tens or 50, fifty. 6 tens or 60, sixty. 7 tens or 70, seventy. 8 tens or 80, eighty. 9 tens or 90, ninety. The suffix ty means ten. Thus, forty means four -tens, etc. The other numbers between 20 and 100 are read without the word and between the tens and units. Thus, 27 is read twenty-seven, instead of twenty and seven. EXERCISES. 15. Read the following: 28 99 43 73 67 41 64 83 75 86 45 31 39 78 60 51 32 92 47 32 21 55 82 25 10 NOTATION AND NUMERATION. 16. Express in figures the following: Three tens and eight units. Four tens and seven units. Two tens and two units. One ten and three units. Six tens and nine units. Five tens and seven units. Eight tens and one unit. Seven tens and three units. One ten and eight units. Four tens and two units. Three units of the second order, six of the first order. Two units of the second order, four of the first order. Write all the numbers between 10 and 20. Between 30 and 50. Between 70 and 90. 17. In reading a number expressed by three figures, the tens are read after the hundreds without the word and. Thus, 235 is read two hundred thirty-five instead of two hundred and thirty-five. 18. Read the following: 746 932 786 849 534 678 453 777 391 585 963 378 243 873 412 531 217 918 855 248 19. Express in figures the following: Two hundreds, three tens, five units. Six hundreds, two tens, nine units. Four hundreds, one ten, eight units. Three units of the third order, six of the second order. Three hundred eighteen. Eight hundred thirty. Four hundred four. Six hundred eighty-one. Seven hundred seventy. Seven hundred. Seventy. Seven. Seven hundred six. Six hundred forty. Two hundred six. One hundred eleven. Seven hundred seventy-seven. NOTATION AND NUMERATION. 11 From the previous examples we deduce the following gen- eral principles: 20. PRINCIPLES. 1. The representative value of a figure is increased tenfold by each removal one place to the left, and de- creased tenfold by each removal one place to the right. 2. The figure is used to give significant figures their positions. 21. In reading numbers a new name is given the order of units next higher than hundreds of any denomination. Thus, the order next higher than hundreds is called thousands, that next higher than hundreds of thousands, millions, etc. Therefore each denomination can have but three orders of units. 22. A Period is a group of figures containing the hun- dreds, tens, and units of any denomination. The present system of notation is illustrated by the following TABLE. PERIODS. 6th. 5th. 4th. 3d. 2d. 1st. CQ NAMES I* | 3 J PERIODS. 3^3 . 3 ~3 fi fi 02 in. in. 02 co cc Q} <3^ Q} Q} QP Q^ ORDERS. ^ 3 ^ ^ m w ^ * 30,230,160,700,401,690. This number is read thirty quadrillion, two hundred thirty trillion, one hundred sixty billion, seven hundred million, Jour hundred one thousand, six hundred ninety. 12 NOTATION AND NUMERATION. 1. In reading numbers, the name of units' period is omitted. 2. Each period, except the highest, must contain three figures. 3. The periods are separated from each other by commas. 23. The periods above Quadrillions, in their order, are Quintittions, Sextillions, Septillions, etc. 24. Give the number of each of the following periods : Millions. Thousands. Trillions. Units. 25. Give the names of the following : 5th period. 3d period. 2d period. 4th period. Billions. Quadrillions. 1st period. 6th period. 26. Repeat in order the names of the periods from : Units to billions. Units to quadrillions. Thousands to trillions. Billions to units. Quadrillions to units. Millions to thousands. 27. Copy and point off into periods: 1. 46825. 2. 239746. 3. 180040. 4. 14168843. 5. 38420058. 6. 33468204. 7. 8438206. 8. 436784. 9. 5284325684. 10. 7932468512. 11. 83749275867. 12. 1423789276586. 13. How many thousands are there in the first number? 14. How many thousands in the second number? 15. How many billions in the next to the last number? 16. How many trillions in the last number? How many billions? How many millions? How many thousands? How many units? 17. Point off into periods, and name in order, the billions, millions, thousands, and units of the next to the last number. 18. Point off into periods, and name in their order, the periods composing the 12th number. 19. In like manner point off and read each of the numbers. NOTATION AND NUMERATION. 13 28. Write in figures: 1. Thirty-four billion, eighteen thousand, forty. PROCESS. ANALYSIS. Since the highest period is bill- ions, which occupy the fourth period, we make ? co Si TH ^ our s P aces f r the periods. We write 34 in the fourth period, thus expressing the billions of 34 000 018 040 the given number; 18 in the second period, thus expressing the thousands; and 40 in the first Or, period, thus expressing the units. Since every 34 000 018 040 P er id except the highest must contain three figures, we fill the vacant places with ciphers. As soon as possible use commas instead of the lines, and cease to write both the number and name of the periods. Write in figures, and read the number: 2. Thirty-six in the 3d period, two hundred eighteen in the 2d, eight hundred forty-six in the 1st. 3. Eighty-four in the 4th period, five hundred forty in the 3d, six hundred in the 2d, forty in the 1st. 4. Two hundred one in the 5th period, seventy-five in the 4th, five hundred sixty-two in the 3d, twelve in the 2d, one in the 1st. 5. Sixty in the 5th period, four hundred two in the 4th, three hundred thirty-three in the 3d, two hundred in the 2d, one hundred eleven in the 1st. Write in figures: 6. Seventy-three million, two hundred fourteen thousand, seventy. 7. Eighty billion, forty million, six hundred twelve thou- sand, seven hundred eighty-eight. 8. Two hundred twenty-five million, six hundred forty- one thousand, three hundred fifty-one. 9. Three hundred fifty-four billion, six hundred four mill- ion, eight hundred ninety-two thousand, thirty-six. 14 NOTATION AND NUMERATION. 29. RULE FOR NOTATION. Begin at the left and write the hundreds, tens, and units of each period in their proper order, putting ciphers in all vacant places and periods. While writing, separate each period from the next by a comma. 30. RULE FOR NUMERATION. Begin at the right and sep- arate the numbers into periods of three figures each. Begin at the left hand and read each period as if it stood alone, adding its name. EXERCISES. 31. Copy, point off, and read: 1. 116234 8. 141120. 15. 7640. 2. 65231. 9. 101207. 16. 800900. 3. 20703. 10. 68978. 17. 2568242. 4. 71005. 11. 72020. 18. 1008003. 5. 3104. 12. 80001. 19. 212375647. 6. 48000. 13. 857000. 20. 609003588. 7. 60029. 14. 91029. 21. 897856846. 32. Write in figures, and read: 22. Two hundred in the 1st period. 23. Sixty in the 2d period, two in the 1st. 24. Seven hundred in the 3d period. 25. Two hundred thirty in the 3d period, sixty in the 1st. 26. Eighty-one in the 4th period, five hundred one in the 3d, seven in the 2d, twelve in the 1st. 27. Thirty in the 5th period, six hundred three in the 1st. 28. Seven hundred in the 5th period, eighty in the 4th. 29. Eight in the 4th period, seven in the 3d, fourteen in the 2d, and ten in the 1st. 30. Fifteen in the 6th period, eighteen in the 4th, two hundred seven in the 3d, and eighty-one in the 1st. NOTATION AND NUMERATION. 15 33. Copy, point off, and read: 1. 60701892. 2. 50607801. 3. 600000. 4. 49000000. 5. 593006070500. 6. 19019000190019019. 7. 163194568. 8. 3050050183. 9. 5000204. 10. 594900. 11. 12000012. 12. 200798013400019. 13. 2125'06067093012063067. 34. Write in figures: 14. Two in the 3d period, sixty in the 2d, one hundred fifty-three in the 1st. 15. Sixty in each of the 4th, 3d, 2d, and 1st periods. 16. 60 million, 200 thousand, 500. 17. 402 billion, 348 million, 213 thousand, 20. 18. 78 trillion, 640 billion, 9 million, 6 thousand, 16. 19. 6 billion, 542 million, 25. 20. Six billion, five hundred forty-two million, twenty-five. 21. Four hundred two billion, three hundred forty-eight million, two hundred thirteen thousand, twenty. 22. Five million,, two huiidred sixty-eight thousand, nine hundred forty-nine. 23. Two hundred million, three hundred thousand, eight hundred. 24. Twenty-nine billion, five hundred ninety-nine million, six hundred one. 25. Four trillion, five hundred fifty-eight million, two hundred forty-four thousand, seventy. 26. Thirty-two billion, sixty-one million, three hundred forty-three thousand, four hundred four. 27. Five hundred fifty-five million, seven hundred seventy- seven thousand, six hundred sixty-nine. 28. Eight hundred six billion, seventy million, three hun- dred eighty-five thousand, two hundred six. 16 NOTATION AND NUMERATION. 29. Nine hundred forty-one trillion, one hundred sixteen thousand, twenty-two. 30. Twenty-three billion, twenty-three million, twenty- three thousand, twenty-three. 31. Six hundred thousand, seventy-five. 32. Twelve billion, eight million, nine hundred eighty- eight thousand, thirteen. 33. Twenty-nine quadrillion, seven hundred fifty-seven trill- ion, four hundred eighty million, thirteen thousand, five hundred sixty-five. NOTATION AND NUMERATION OF U. S. MONEY. 35. The currency of the United States has a Decimal System of notation, thus: 10 mills make 1 cent. 10 cents make 1 dime. 10 dimes make 1 dollar. 36. The Sign of Dollars is $. It is written before the number. Thus, $16 is read, sixteen dollars. 37. In writing decimal currency a mark called the deci- mal point is placed before cents and mills. 38. Cents occupy the first two places at the right of the decimal point, and mills the third. Thus, $7.584 is read, seven dollars, fifty -eight cents, four mills; is read, sixty-nine cents, four mills. 39. If the number of cents is less than ten, write a cipher in the first place at the right of the decimal point. Thus, five dollars, eight cents, is written, $5.08; three dollars, seven cents, $3.07. NOTATION AND NUMERATION. 17 40. Read the following: $6.85 $7.843 $12.056 $31.095 $24.055 $20.20 $28.075 $40.04 $606.952 $500.50 $2103.094 $7000.16 $20000. $6001.101 $300.416 $212012.12 $695.955 $200.204 $613.495 $211.12 $69.69 $203.033 $216.16 $75.25 41. Write the following: 1. Two dollars, twenty-three cents, five mills. 2. Two hundred two dollars, two cents, five mills. 3. One hundred twelve dollars, twenty-five cents. 4. Six hundred two dollars, nine cents. 5. Twenty thousand dollars, thirty-two cents. 6. Twelve million, seven hundred thousand dollars. 7. Six million dollars, eighty-eight cents. 8. Twelve thousand three hundred dollars, fifteen cents. ROMAN SYSTEM. 42. In this system seven letters are used to express num- bers, viz: Letters. I, V, X, L, C, D, M. Values. 1, 5, 10, 50, 100, 500, 1000. By combining these letters according to certain principles any number can be expressed. PRINCIPLES. 1. When a letter is repeated its value is re- peated. Thus, I represents 1; II, two; III, three; X, ten; XX, twenty; XXX, thirty ; C, one hundred ; CCC, three hundred. 18 NOTATION AND NUMERATION. 2. When a letter is placed before another of greater value its value is to be taken from that of the greater. Thus, I represents one and V five, but IV represents four; IX, nine; XIX, nineteen; XL, forty; XC, ninety. 3. When a letter is placed after another of greater value their values are to be united. Thus, XV represents fifteen; LX, sixty; LXXX, eighty; DC, six hundred ; MD, fifteen hundred. 4. A bar placed over a number increases its value a thousand- fold. Thus, V represents five ; V, five thousand ; LX, sixty ; LX, sixty thousand ; M, one thousand ; M, one million. TABLE. I . . . . 1 XIV . . . ... 14 LX 60 II . . . . 2 XV . . 15 LXX 70 Ill . . . . . . 3 XVI . 16 LXXX 80 IV .... . . . . 4 XVII. . . . . . 17 XC 90 V .... . . . . 5 XVIII 18 c 100 VI .... . . . . 6 XIX . . . . . . 19 cc 200 VII .... . . . . 7 XX . . . ... 20 CCL 250 VIII . . . . . . . 8 XXI . . . ... 21 CCCC 400 IX .... . . . . 9 XXIX ... 29 D 500 X .... . ... 10 XXX . . ... 30 DCC 700 XI .... . ... 11 XXXIV . ... 34 M 1000 XII .... . ... 12 XL .... ... 40 MMM 3000 XIII . . 13 L . . 50 MDCCCLXXX 1880 Read the following numbers: XV; XXIV; XXXIX; XL; XLIX; JXCIX; LXXVII; CCCLXXXIX; DCCXXXVI; VDLV; DLDC; CCXDVI; LXXMMMDCCCXCIX; MDXCVDCCCLXIV. Express the following numbers by Roman Notation: 15, 18, 27, 81, 95, 86, 534, 684, 1050, 8004, 7000, 75869, IND UCTIVE EXERCISES. 43. 1. How many are 2 pears and 1 pear? 2 pears and 2 pears? 2. How many are 3 leaves and 2 leaves? 3 leaves and 3 leaves? How many are 3 and 1? 3 and 2? 3 and 3? 3. Jane has 3 apples and Mary has 4 apples. How many apples have both? How many are 3 and 4? 4 and 3? 4. George gave me 2 apples and Mary gave me 4. How many apples did both give me? How many are 4 and 2? 2 and 4? 5. A farmer had 2 horses and bought 6 more. How many horses had he then? How many are 2 and 6? 6 and 2? 6. Henry paid 5 cents for a pencil and 7 cents for a writing-book. How many cents did he pay for both? How many are 5 and 7? 7 and 5? 7. If a barrel of flour is worth $6, and a cord of wood $4, how much are both worth? How many are 6 and 4? 8. A man plowed 8 acres of land in one week and 6 acres the next week. How many acres did he plow in both weeks? 9. On the Fourth of July, Ned spent 10 cents for fire- crackers and 6 cents for torpedoes. How many cents did he spend for both? 10. Harry is 6 years old and his sister is four years older. How old is his sister? How many are 6 and 4? (19) 20 ADDITION. 11. At Christmas, Horace received 9 gifts from his par- ents, and 4 from other friends. How many gifts did he receive? 12. A certain house has 5 windows in one side and 7 in another. How many windows in the two sides? 13. How many are 5 oranges and 4 oranges? 6 boys and 3 boys? 5 horses and 6 cents? 14. Why can you not tell how many 5 horses and 6 cents are? 15. Why can you tell how many 5 oranges and 4 or- anges are? Numbers that express things of the same name are called Like Numbers. 16. What kind of numbers only can be united? DEFINITIONS. 44. Addition is the process of finding a number which shall be equal to two or more given numbers. 45. The Sum or Amount is the result obtained by adding. 46. The Sign of Addition is an upright cross: +. It is called plus, and is placed between numbers to be added. Thus, 3 + 4 is read 3 plus 4, and means that 3 and 4 are to be added. 47. The Sign of Equality is two short horizontal lines: =. It is read equals, or is equal to. Thus, 3 + 4 = 7, is read 3 plus 4 equals 7. The expression 3 + 4 = 7, or any other expression of equality, is called an Equation. ADDITION. 21 48. PRINCIPLES. 1. Only like numbers can be added. 2. The sum and numbers added must be like numbers. TABLE. 1 + 1= 2 1 + 2= 3 1 + 3= 4 1 + 4= 5 1+ 5= 6 2 + 1= 3 2 + 2= 4 2 + 3= 5 2 + 4= 6 2+ 5= 7 3 + 1= 4 3 + 2= 5 3 + 3= 6 3 + 4= 7 3+ 5= 8 4 + 1= 5 4 + 2= 6 4 + 3= 7 4 + 4= 8 4+ 5= 9 5 + 1= 6 5 + 2= 7 5 + 3= 8 5 + 4= 9 5+ 5 = 10 6 + 1= 7 6 + 2= 8 6 + 3= 9 6 + 4 = 10 6+ 5 = 11 7 + 1= 8 7 + 2= 9 7 + 3 = 10 7 + 4=11 7+ 5 = 12 8 + 1= 9 8 + 2 = 10 8 + 3 = 11 8 + 4=12 8+' 5 = 13 9 + 1 = 10 9 + 2 = 11 9 + 3 = 12 9 + 4=13 9+ 5 = 14 1 + 6= 7 1 + 7= 8 1 + 8= 9 1 + 9 = 10 1 + 10 = 11 2 + 6= 8 2 + 7= 9 2 + 8 = 10 2 + 9 = 11 2 + 10 = 12 3 + 6= 9 3 + 7 = 10 3 + 8 = 11 3 + 9 = 12 3 + 10=13 4 + 6 = 10 4 + 7 = 11 4 + 8 = 12 4 + 9 = 13 4 + 10=14 5 + 6 = 11 5 + 7 = 12 5 + 8 = 13 5 + 9 = 14 5 + 10=15 6 + 6 = 12 6 + 7 = 13 6 + 8 = 14 6 + 9 = 15 6+10 = 16 7 + 6 = 13 7 + 7 = 14 7 + 8 = 15 7 + 9 = 16 7 + 10=17 8 + 6 = 14 8 + 7 = 15 8 + 8 = 16 8 + 9 = 17 8 + 10=18 9 + 6 = 15 9 + 7 = 16 9 + 8 = 17 9 + 9=18 9 + 10 = 19 CASE I. 49* To add single columns. 1. A grocer sold 8 pounds of sugar to one man and 7 pounds to another. How many pounds did he sell to both? ANALYSIS. Since he sold 8 pounds to one man and 7 pounds to another, to both he sold the sum of 8 pounds and 7 pounds, which is 15 pounds. 2. A man rode 7 miles the first hour and 6 miles the sec- ond hour. How far did he ride in the two hours? 22 ADDITION. 3. On one tree are 8 birds, and on another 4 birds. How many birds are there on both? 4. Carl earned $2 in May, $5 in June, and $4 in July. How much did he earn in the three months? 5. I gave 6 nuts to one boy, 5 to another, and 3 to an- other. How many nuts did I give to all? 6. I paid 5 cents for paper, 3 cents for pens, and 5 cents for ink. How much did I pay for all? 7. A lemon cost 5 cents, an orange 6 cents, and a pine- apple 8 cents. What did they all cost? 8. Esther gave her teacher 5 pinks, 7 roses, and 4 pan- sies. How many flowers did she give her? 9. James shot 9 birds, Henry shot 6, and William 5. How many did they all shoot? 10. A woman picked 9 quarts of blackberries one morn- ing, while her son picked 3 quarts. How many quarts did both pick? 11. James solved 6 examples, John 5, William 8, and Henry 7. How many examples did they solve? 12. One boy picked 6 quarts of cherries, another 4 quarts, another 5 quarts. How many quarts did they all pick? 13. I gathered from one pear-tree, this year, 2 bushels of fruit, from another 4 bushels, from another 3 bushels, and from another 2 bushels. How many bushels did I gather from these four trees? 14. A merchant sold from a piece of cloth, 3 yards at one time, 6 yards at another, 8 yards at another, and 5 yards at another. How many yards did he sell in all? 15. A man picked 8 barrels of apples on Monday, 6 bar- rels on Tuesday, 4 barrels on Wednesday, and 5 barrels on Thursday. How many did he pick altogether? 16. Henry learned 7 verses of poetry on one day, 5 on another, 6 on another, and 8 on another. How many verses did he learn in the four days? ADDITION. 23 17. A man paid $9 for a coat, $4 for pants, and $2 for a hat. How much did he pay for all? 18. In a garden there are 8 apple-trees, 7 plum-trees, and 9 peach-trees. How many trees are there in the garden? 19. There are 4 boys and 7 girls in one class, and 6 boys and 8 girls in another. How many pupils in both classes? 20. Homer paid 8 dollars for a fur cap, and 5 dollars for a pair of skates. How much did both cost him ? 21. A boy gathered nuts for three days. The first day he brought home 8 quarts, the next day 7 quarts, the next day 9 quarts. How many quarts did he bring home ? 22. Repeat the addition table of ones. Of twos. Of threes. Of fours. Of fives. Of sixes. Of sevens. Of eights. Of nines. Of tens. 23. Count by 2's from to 20; thus: 0, 2, 4, 6, 8, 10, 12, etc. 24. Count by 3's from 2 to 26. From 26 to 41. 25. Count by 4's from to 36. From 5 to 53. 26. Count by 5's from 3 to 43. From 7 to 72. 27. Count by 6's from to 42. From 4 to 46. 28. Count by 7's from 4 to 39. From 11 to 60. 29. Count by 8's from 2 to 58. From 7 to 63. 30. Count by 9's from 7 to 70. From 8 to 71, WRITTEN EXERCISES. 50. 1. What is the sum of 5, 4, 7, and 6? PROCESS. ANALYSIS. We write the numbers to be added in a 5 column, and begin at the bottom to add; thus: 6, 13, 17, 22; and write the sum beneath. To prove the work we - may begin at the top and add downwards. If the result agrees with the one formerly obtained the work is proba- bly correct. In adding say, 6, 13, 17, etc., instead of 6 22 Sum. and 7 are 13. and 4 are 17, etc. 24 ADDITION. Copy, add, and prove: (2-) (3.) (4.) (5.) (6.) (7.) 5 6 5 6 7 8 3 . 7 6 4 3 9 4 8 2 3 8 8 2 1 3 4 5 7 (8.) (9.) (10.) (11.) (12.) (13.) 8 6 5 8 9 2 7 4 4 7 7 5 6 3 3 9 8 7 3 2 8 9 9 5 3 1 8 8 3 51. Kequired the sum of the following: 14. 6, 7, 5, 3, 2, 4, and 5. 15. 8, 2, 0, 3, 3, 2, and 4. 16. 5, 6, 7, 6, 4, 2, and 8. 17. 3, 2, 6, 5, 8, 7, and 9. 18. 8, 3, 0, 5, 3, 8, and 2. 19. 7, 6, 6, 4, 3, 6, and 3. 20. 7, 8, 8, 9, 0, 3, and 3. 21. 8, 9, 7, 8, 5, 8, and 2. 22. 7, 6, 5, 4, 3, 2, and 1. 23. 5, 4, 4, 3, 2, 6, and 7. 24. 4, 3, 4, 5, 6, 8, and 8. 25. 3, 6, 8, 6, 7, 0, and 5. 26. There are 8 chickens in one coop, 9 in another, 7 in another, and 5 in another. How many chickens are there in all the coops? 27. My father has 5 horses, 9 cows, 7 sheep, and 3 pigs. How many animals has he in all? '28. A man walked from A to B in four hours. He went 4 miles the first hour, 3 miles the second hour, 5 miles the third hour, and 6 miles the fourth hour. What was the distance between the two places? 29. A house had 8 windows on the east side, 7 on the west, and 9 on the south. How many were there in all? ADDITION. 25 CASE II. 52. To add several columns. 1. Count by 10's from 7 to 107; thus, 7, 17, 27, 37, 47, etc. 2. Count by 10's from 5 to 95. From 9 to 79. 3. Count by 20's from 5 to 85. From 9 to 89. 4. Add 2 to each of the series of numbers 6, 16, 26, etc., to 76. 5. Add 3 to each of the series of numbers from 8, 18, etc., to 88. 6. A gentleman paid $7 for a hat, $8 for a vest, and $13 for pantaloons. How much did he pay for all? ANALYSIS. Since he paid $7 for a hat, $8 for a vest, and $13 for pantaloons, for all he paid the sum of $7, $8, and $13, or $28. 7. James gave 25 cents to his brother and 20 cents to his sister. How much did he give to both? 25 and 20 are how many? 8. Horace earned 35 cents on Monday, 20 cents on Tues- day, and 9 cents on Wednesday. How much did he earn during the three days? How many are 35 and 29? 9. William saw two flocks of wild geese; the first of 27 geese, the second of 23 (20 + 3). How many geese did he see? How many are 27 and 23? 10. Paid 9 cents for raisins, 15 cents for plums, and 27 (20 + 7) cents for currants. How much did all cost? 11. During a certain recitation 29 questions were answered correctly and 16 incorrectly. How many questions were asked? How many are 29 and 10? 39 and 6? 12. Add 2 to each of the numbers 2, 12, 22, 32, 42, etc., to 72. 13. Add 3 to each of the numbers 4, 14, 24, etc., to 94. 14. Add 4 to each of the numbers 9, 19, 29, 39, to 99. 26 ADDITION. 15. Add each of the numbers 5, 6, 7, 8, and 9 to each of the numbers 6, 16, 26, etc., to 96. 16. A certain school had 40 girls and 30 boys in attend- ance. How many pupils were there in the school? 17. A music teacher paid $12 for a metronome and $15 for music. How much did she pay for both? 18. A boy bought a velocipede for $15 and a watch for $20. How much did both cost him? 19. Mary read 20 pages of history one day, 30 pages the next, and 25 the next. How many pages did she read in all? 20. In a certain book-case there were 18 books on the upper shelf, 20 on the next, 12 on the next, and 10 on the lowest. How many books in the case? 21. A merchant sold 15 yards of cloth to one woman, 25 to another, 30 to another, and 25 to another. How many yards did he sell to them all? 22. A postmaster sold on one day 50 three-cent stamps, 65 on another, and 55 on another. How many stamps did he sell in the three days? 23. James solved 31 oral problems and 24 written prob- lems. Harry solved 35 oral problems and 25 written prob- lems. How many problems did each solve? How many did both solve? 24. In an orchard there are 26 cherry-trees and 31 apple- trees. How many trees are there in the orchard? 25. Henry saw three flocks of wild ducks, the first con- taining 17 ducks, the second 25, and the third 30. How many ducks did he see? 26. James paid 28 cents for a slate, 20 cents for a writing- book, and 10 cents for ink. How much did he pay for all? 27. The month of January has 31 days, the month of Feb- ruary has 28 days, and the month of March has 31 days. How many days are there in these three months? ADDITION. 27 28. How many acres are there in three fields, containing respectively 22 acres, 33 acres, and 37 acres? WRITTEN EXERCISES. 53. 1. What is the sum of $535, $213, and $384? PROCESS. ANALYSIS. For convenience we arrange the numbers to $ 5 3 5 be added so that units of the same order shall stand in the same column. Beginning with the lowest order of units we add each column separately. Thus, 4+3 + 5 = 12, the sum of the units. 12 units are equal to 1 ten and 2 $ 1 1 3 2 units. We write 2 under the column of units and reserve the 1 to add with the tens. 1 reserved + 8 + 1 + 3 = 1 3, the sum of the tens. 13 tens are equal to 1 hundred and 3 tens. We write the 3 under the column of tens and reserve the 1 to add with the hundreds. 1 reserved + 3 + 2 + 5 = 11, the sum of the hundreds. 11 hun- dreds are equal to 1 thousand and 1 hundred, which we write in thousands' and hundreds' places in the sum. Hence the sum is $1132. 1. In adding, name results only. Thus, instead of saying 4 and 3 are 7 and 5 are 12, say 4 , 7, 12. 2. When the sum of any column is exactly 10, 20, or any number of tens, we write under the column added and reserve the 1, 2, 3, etc., to add with the next column. 54. RULE. Arrange the numbers so that units of the same order shall stand in the same column. Begin at the right, add each column separately, and unite the sum, if it is less than ten, under the column added. If the sum of any column be ten or more, write the unit figure only under that column and add the ten or tens with the next column. Write the entire sum of the last column. PROOF. Add each column in the reverse order. If the re- sults agree, the work is probably correct. 28 ADDITION. EXAMPLES. Copy, add, and prove: (2.) (3.) (4.) (5.) 310 512 $24.15 $12.25 114 415 10.21 9.08 523 371 8.34 7.15 (6.) (7.) (8.) (9.) POUNDS. HORSES. PLOWS. RODS. 4134 8104 3910 45 2460 3673 418 3061 3782 1856 1916 418 469 7206 39 6 10. Add 4834, 3910, 4826, 8404. 11. Add 3159, 7816, 5459, 3568. 12. Add $16.05, $10.38, $77.055. 13. Add $317.50, $600.10, $514.085, $6.16. 14. What is the sum of thirty-six thousand, three hundred five; eight hundred ninety-seven thousand, nineteen? 15. What is the sum of fifty-nine thousand ; seven thou- sand, three hundred twelve; sixty-eight thousand, four hun- dred twenty-seven? 16. What is the sum of three hundred forty-four million, eight hundred ninety-six thousand, four hundred thirty-five; five million, six thousand, three; forty-eight thousand, two hundred ? 17. What is the sum of eighteen dollars, five cents; fifty- one dollars, fifty-one cents ; ten dollars, ten cents ; eighteen dollars, twenty-four cents ; thirty-five dollars, four cents ? 18. A owns 345 sheep, B owns 295, C owns 436, D owns 524. How many sheep do all own? ADDITION. 29 19. A man sold his piano for $413, his collection of paint- ings for $536, his library for $719, his carpets for $728, other furniture for $1,736, his horses, carriage and two sets of har- ness for $1,324, and his house for $9,137. How much money did he obtain by the sale? 20. A fruit-dealer shipped for New York, 3,932 bushels of apples in one week, 2,436 in the next, 4,197 in the third, and, within the next month, 10,937 bushels. What was the entire number of bushels shipped by him during that time? 21. A man making his will, left $3,450 to his wife, $2,675 to his oldest son, $1,850 to his second son, and $1,290 to his youngest son. What amount of money was bequeathed in his will? 22. A man owns five horses. The first is worth $225, the second $325, the third $450, the fourth as much as the second and third, and the fifth as much as the first and fourth. What is the value of the five horses? 23. A and B were building a brick store. The first day A laid 2,136 bricks, and the second day he laid as many as the first day plus 207. B, on the first day, laid 1,936, and, on the second day> 341 more than on the first. How many bricks were laid by both in the two days? 24. The distance from Greening to Chatfield is 277 miles, from Chatfield to Glendale is 325 miles, from Glen- dale to Wyoming is 139 miles, from Wyoming to Dale is 193 miles. By this route what is the distance from Green- ing to Dale? 25. A man took 2,126 steps going from home to his place of business, 3,197 while in his store, 6,239 going from there to the park, 5,782 while in the park, 8,573 going from there home. What was the whole number of steps taken by him from the time he left until he re- entered his house? 30 ADDITION. 26. In the first story of a house, the hall contained 117 square feet, the parlor 327, the sitting-room 296, the dining- room 257. How many square feet of carpeting would be required to cover the floors of these rooms? 27. In 1870, the population of Buffalo was 117,714; that of Eochester, 62,386; that of Albany, 69,422; that of Brook- lyn, 396,099. How many inhabitants did these four cities contain? 28. The area of Spain is 195,773 square miles; that of France, 204,091; that of Switzerland, 15,922; that of Italy, 112,622. Over how many square miles do the four countries extend ? 29. A speculator bought five lots for $1,375 each. He sold the first for $125 more than cost, the second for $319 more than cost, the third for $291 more than cost, the fourth for $739 more than cost, and the fifth for $135 more than cost. How much money did he receive for all? 30. The State of Alabama contains 1,430 libraries and 576,882 volumes; Mississippi, 2,788 libraries and 488,482 volumes; Louisiana, 2,332 libraries and 847,406 volumes; Texas, 455 libraries and 87,111 volumes. How many libra- ries and how many volumes do the four States contain ? 31. The population of five of the principal cities of Ohio was in 1870 as follows: Cincinnati, 216,239; Cleveland, 92,829; Toledo, 31,584; Columbus, 31,274; Dayton, 30,473. What was the entire population of these cities in 1870? 32. The population of five of the principal cities of Illinois was in 1870 as follows: Chicago, 298,977; Quincy, 24,025; Peoria, 22,849; Springfield, 17,364; Bloomington, 14,590. What was the entire population of these cities at that time? 33. In 1870, the population of St. Louis, Mo., was 310,864; Memphis, Tenn., 40,226; Charleston, S. C., 48,956; Kich- mond, Va., 51,038; New Orleans, La., 191,418. What was the entire population of these cities at that time? ADDITION. 31 34. The Warsaw Manufacturing Company sawed 11,936 feet of pine on Monday, 12,117 feet of hemlock on Tues- day, 8,135 feet of maple on Wednesday, and 9,963 feet of ash on Thursday. How many feet of timber did they saw in the four days? 35. According to the census of 1870, the number of native Americans in Nebraska was 92,245; the number of Irish, 4,999; of Germans, 10,954; of English, 3,602; of Scotch, 792; of Canadians, 2,632; of French, 340; of Norwegians, 506; of Swedes, 2,352. What was the total population of the State in 1870? 36. In a certain State there were raised, last year, 7,771,009 bushels of potatoes, 278,798 bushels of wheat, 1,089,888 bushels of Indian corn, 2,351,354 bushels of oats, 658,816 bushels of barley. What was the entire number of bushels of farm products raised that year? 37. Mr. George Peabody gave to the poor of London $2,250,000, to the town of Danvers $60,000, to the Grin- nell Arctic Expedition $10,000, to the city of Baltimore $1,000,000, to Phillips' Academy $25,000, to the Massa- chusetts Historical Society $20,000, to Harvard College $150,000, to Yale College $150,000, to the Southwest $1,500,000. How much did this benevolent gentleman give away? 38. In 1870, there were, in the United States, 574 daily newspapers, with a circulation of 2,601,547; 107 tri-week- lies, with a circulation of 155,105; 115 semi- weeklies, with a circulation of 247,197; 4295 weeklies, with a circulation of 10,594,643; 96 semi-monthlies, with a circulation of 1,349,820; 622 monthlies, with a circulation of 5,650,843; 13 bi-monthlies, with a circulation of 31,650; 49 quarter- lies, with a circulation of 211,670. How many periodicals were there in the United States during that year, and what was their entire circulation? 32 ADDITION. 39. Mr. A. deposited in the First National Bank of Albany, N. Y., on July 3, 1877, $395.25; on July 5, $874.75; on July 8, $325.85. He also deposited in the National Park Bank of New York City, on July 12, 1877, $1,546.87; on July 16, $1,275; on July 20, $1,985.50. How much did he deposit in each of the banks? How much in both banks? (40.) (41.) (42.) (43.) 2134 6166 5873 46321 8060 5878 3858 69788 5032 9876 6430 76434 8797 7977 5082 68924 9888 6503 6353 96355 6432 4556 4202 88789 5421 6432 8792 93745 (44.) (45.) (46.) (47.) 813 760 3945 5063 976 500 9204 2050 432 750 8769 3254 397 694 9876 4200 788 942 8020 6131 643 293 5612 5945 564 978 3424 2763 '321 785 5861 4828 156 696 2188 7688 642 785 7654 3288 321 688 3210 5634 876 762 8765 6546 543 451 5849 3250 429 984 8574 7864 386 579 9836 9758 595 384 8759 8410 INDUCTIVE EXERCISES. 55. 1. If I have 6 peaches and give away 3 of them, how many will be left? 2. If James has 4 bunches of grapes and eats 2 of them, how many will be left? 3. If I have 7 bunches of grapes and give away 4 of them, how many will be left? 4. If you find 8 acorns and lose 4 of them, how many will be left? 5. How many are left when 4 things are taken from 8 things? How many are 8 less 4? 7 less 4? 5 less 4? 9 less 4? 6 less 4? 6. A farmer who had 7 horses, sold 3 of them. How many had he left? How many are 7 less 3? 9 less 3? 7. James earned, during the summer, $9. He spent $5 of the money for a coat, and the rest for a pair of boots. How much did the boots cost him? 8. Nine is how many more than 5? Than 6? Than 4? Than 3? 9. A boy who had 9 chickens, sold 3 of them. How many had he left? 10. Lawrence had 10 pictures in his room. He gave his sister 3 of them. How many were left in his room? 11. A man earned $11 per week and spent $7. How much did he save weekly? (33) 34 SUBTRACTION. 12. A hen had nine chickens, but 5 of them were killed. How many chickens were left? How many must be added to 5 to make 9? 13. When 5 is taken from 9, what number is left? 14. When 7 is taken from 10, what number remains? How many must be added to 7 to equal 10? 15. Howard is 10 years of age and Herbert is 8. What is the difference in their ages? What is the difference be- tween 10 and 8? 16. If the difference between 10 and 8 be added to 8, what will the result be? 17. If the difference between any two numbers be added to the smaller number, to what will the result be equal? 18. What is the difference between 6 horses and 4 horses? Between 6 horses and 5 cents? 19. Why can you not find the difference between 6 horses and 5 cents? 20. Why can you find the difference between 6 horses and 4 horses? 21. Between what kinds of numbers only can the differ- ence be found? DEFINITIONS. 56. Subtraction is the process of taking one number from another. 57. The Minuend is the number from which another is to be subtracted. 58. The Subtrahend is the number to be subtracted. 59. The Remainder, or Difference, is the result obtained by subtracting. 60. The Sign of Subtraction is a short horizontal line: . It is named minus. SUBTRACTION. 35 When the sign minus is placed between two numbers it shows that the one after it Is to be subtracted from the one before it. Thus, 9 5 is read 9 minus 5, and means that 5 is to be sub- tracted from 9. 61. PRINCIPLE. 1. Only like numbers can be subtracted. 2. The sum of the subtrahend and remainder must be equal to the minuend. TABLE. 11= 22= OJ 33= 44= 0| 5 5= 21= 1 32= 1 43= 1 54= 1 6 5= 1 31= 2 42= 2 53= 2 64= 2 7 5= 2 41= 3 52= 3 63= 3 74= 3 8 5= 3 51= 4 62= 4 73= 4 84= 4 9 5= 4 61= 5 72= 5 83= 5 94= 5 10 5= 5 71= 6 82= 6 93= 6 104= 6 11 5= 6 81= 7 92= 7 103= 7 114= 7 12 5= 7 91= 8 102= 8 113= 8 124= 8 13 5= 8 101= 9 112= 9 123= 9 134= 9 14 .5= 9 111=10 122=10 13-3=10 144=10 155=10 66= 77= 8-8= 99= 1010= 76= 1 87= 1 98= 1 109= 1 1110= 1 8-6= 2 97= 2 108= 2 119= 2 1210= 2 96= 3 107= 3 118= 3 129= 3 1310= 3 10-6= 4 117= 4 12-8= 4 139= 4 1410= 4 116= 5 127= 5 138= 5 149= 5 1510= 5 126= 6 137= 6 148= 6 159= 6 1610= 6 136= 7 147= 7 158= 7 169= 7 1710= 7 146= 8 157= 8 168= 8 179= 8 1810= 8 156= 9 167= 9 178= 9 189= 9 1910= 9 166=10 177=10 188=10 199=10 2010=10 36 SUBTRACTION. CASE I. 62. When no figure of the subtrahend has a greater value than the corresponding figure of the minuend. 1. A merchant had 15 barrels of flour, and sold 4 of them. How many had he left? ANALYSIS. Since he had 15 barrels of flour and sold 4 of them, he had left the difference between 15 barrels and 4 barrels, which is 11 barrels. 2. Alice bought 18 cakes, and ate 6 of them. How many had she left? 3. James saw 17 birds on a tree, but 7 soon flew away. How many remained? 4. If a man earns $19 a week, and spends $9, how much will he save each week? 5. Lewis owed his brother $7, and paid him S3. How much did he still owe him? 6. Eliza had 16 plums, but gave 5 to her father. How many had she left? 7. James had $12, and lost $2. How many had he left? 8. If John is 19 years old, and Maggie 13, how much younger than John is Maggie? 9. In the same shop 6 boys and 17 men work. How many more men than boys are there in the shop? 10. There were 18 girls and 7 boys in a class. How many more girls than boys were there? 11. Laura had 14 cents, and lost 3 cents. How many had she then? 12. Henry solved 19 examples, and George solved 8. How many more did Henry solve than George? 13 William wrote 16 lines in his copy-book, and Peter wrote 5 lines less. How many did Peter write? SUBTRACTION. 37 14. One piece of cloth contained 20 yards and another 10 yards. How many yards more were there in the larger piece ? 15. A boy had 24 chickens, and 10 of them died. How many had he left? 16. Julia gave me 11 cents." If she had 16 cents at first, how many had she left? 17. A girl bought 18 eggs, and, on her way home, fell and broke 5 of them. How many had she left? 18. Subtract by 2's from 22 to 0; thus: 22, 20, 18, 16, 14, 12, etc. 19. Subtract by 3's from 35 to 2. From 45 to 0. 20. Subtract by 4's from 48 to 0. From 45 to 1. 21. Count back by 5's from 35 to 0. From 59 to 4. WRITTEN EXERCISES. 63. 1. From 547 subtract 235. PROCESS. ANALYSIS. For convenience we write the less Minuend 547 number under the greater, units under units, tens Subtrahend 235 un ^ er tens > etc -> an( * subtract each order of units separately from the same order of the minuend. Remainder 312 Thus, 7 units 5 units = 2 units, which we write under the units. 4 tens 3 tens = 1 ten, which we write under the tens. 5 hundreds 2 hundreds = 3 hundreds, which we write under the hundreds. Hence the remainder is 312. PROOF. 312, the remainder, plus 235, the subtrahend, equals 547, the minuend. Hence the result is correct. Copy, subtract, and prove: (2.) (3.) (4.) (5.) (6.) (7.) 713 458 986 854 795 7842 302 134 732 641 433 2310 38 SUBTRACTION. (8.) (9.) (10.) (11.) (12.) $48.25 $64.29 $45.78 $38.94 $41.89 23.13 30.29 34.65 27.83 20.45 13. A drover, having 1583 sheep, sold 1441 of them. How many had he left? 14. A speculator bought some land for $5849.75, and sold it for $6959.95. How much did he gain? 15. A cotton factory made 9875 yards of cloth in one week, and sold, during the same time, 7652 yards. How much more was made than sold? 16. A money-lender received for interest, during 1875, $1685.49, and during 1876, $2796.59. In which year did he receive the greater sum, and how much? 17. A man bought 7467 bricks, and carted away 3136. How many remained to be moved? 18. A has 736 sheep, and B has 213 less than A. How many sheep has B? 19. A man gave his note for $6792, without interest. In two years he had paid $3401. How much did he still owe on the note? 20. A man bought a house for $1765, and afterward sold it, thereby losing $504. For how much did he sell it? 21. A man bought a span of horses for $364, and a yoke of oxen for $120. How much more did he give for the horses than for the oxen? 22. A merchant having 6755 yards of cloth, sold 2532 yards. How many yards had he remaining? 23. A father having 3652 acres of land, gave his son 1230 acres. How many acres had he left? 24. A vintner had 38756 gallons of wine, and sold during the year, 34243 gallons. How much remained unsold? 25. The circulation of a newspaper in 1875 was 38293 copies, and in 1876, 37180. What was the decrease? SUBTRACTION. 39 CASE II. 64. When any figure of the subtrahend has a greater value than the corresponding figure of the minuend. 1. A gentleman bought a coat at $40, and a vest at $9; he gave the merchant a hundred-dollar bill. How much change ought he to receive? ANALYSIS. Since he paid $40 for a coat and $9 for a vest, for both he paid the sum of $40 and $9, or $49. And since he gave the mer- chant $100, he should receive the difference between $100 and $49. 4100 $40=$GO; $60 $9 = $51. Therefore he should receive $51. 2. A boy saw 15 birds on a tree, and 9 of them flew away. How many remained? 3. John is 16 years old, and James is 8. How much older than James is John? 4. A jeweler bought a watch for $75. and sold it for $100. How much did he gain by the operation? 5. A grocer bought a quantity of sugar for $36, and re- tailed the same for $50. How much did he gain by the sale? 6. A boy had 34 marbles, and gave away 9 of them. How many had he left? 7. A lady bought a chair for $3, and a table for $5; she gave a twenty-dollar bill to the cabinet-maker., How much change ought she to receive? 8. A man set out to walk 50 miles; he walked 20 miles the first day, and 19 the second day. How many miles were left for him to walk? 9. A man bought a cow for $35, and sold her for $43, after keeping her 4 weeks at an expense of $2 per week. How much did he gain? 10. A man who earned $60 a month, paid $25 a month for his board, and $15 a month for other expenses. How much did he save per month? 40 SUBTRACTION. 11. Count back by 10's from 107 to 7; thus: 107, 97, etc. 12. Count back by 10's from 95 to 5. From 79 to 9. 13. Count back by 10's from 83 to 13. From 98 to 18. 14. Subtract by 20's from 106 to 26; thus: 106, 86, etc. WRITTEN EXERCISES. 65. 1. From 643 subtract 456. PROCESS. ANALYSIS. We write the numbers as in the previous g 4 g case and begin at the right to subtract. Since 6 units can not be subtracted from 3 units, we unite with the 3 units a unit of the next higher order, 187 which is equal to 10 units, making 13 units: 6 units from 13 units leave 7 units, which we write under the units. Since one of tens was united with the units, there can be but 3 tens left. Because 5 tens can not be subtracted from 3 tens, we unite with the 3 tens as before, a unit of the next higher order, which is equal to 10 tens, making 13 tens: 5 tens from 13 tens leave 8 tens, which we write under the tens. Since one of the hundreds was united with the tens, there are but 5 hundreds left: 4 hundreds from 5 hundreds leave 1 hundred, which we write under the hundreds. Hence the result is 187. PROOF. 187, the remainder, plus 456, the subtrahend, equals 643 ? the minuend. Hence the result is correct. 66. RULE. Write the subtrahend under the minuend, units under units, tens under tens, ete. Begin at the right and subtract each figure of the subtrahend from the corresponding figure of the minuend, writing the result beneath. If a figure in the minuend has a less value than the corre- sponding figure in the subtrahend, increase the former by ten, and subtract ; then diminish by one, the units of the next higher order in the minuend, and subtract as before. PROOF. Add together the remainder and subtrahend. If the result be equal to the minuend the work is correct. SUBTRACTION. 41 EXAMPLES. Copy, subtract, and prove: (2.) (3.) (4.) (5.) (6.) 753 984 826 754 1426 448 756 534 482 547 (7.) (8.) (9.) (10.) (11.) 843 1846 1683 2897 3001 782 927 1395 1598 2851 (12.) (13.) (14.) (15.) (16.) $24.45 $39.18 $63.25 $71.89 $42.34 21.38 27.92 47.18 47.93 18.67 Find the difference between 17. 583 and 294. 18. 690 and 508. 19. 763 and 574. 20. 966 and 599. 21. 982 and 796. 22. 891 and 798. 23. 5833 and 4968. 24. 7521 and 3635. 25. 26. 27. 28. 29. 30. 31. 7812 and 8003 and 63004 and 65432 and 69721 and 78303 and 865932 and 32. 9050308 and 1984. 5872. 54872 54862. 49653. 49424. 785841. 563420. 33. A man set out on a journey of 861 miles. During the first day he traveled 297 miles, and during the second day 308 miles. How many miles had he yet to travel? 34. A merchant deposited in a bank on Monday $584; on Tuesday, $759; on Wednesday, $463. He drew out $1298 during that time. How much did his deposits exceed what he drew out? 42 SUBTRACTION. 35. A grocer had 3715 pounds of sugar on hand. On one day he sold 1235 pounds, on the next 1317; the third day he sold to C all the sugar that remained. How many pounds did C buy? 36. I bought a horse for $637, and a cow for $317. I sold the horse for $729, and the cow for $356. How much did I gain by the sale? 37. In the first of three pavements there are 1317 bricks, in the second there are 2357, in the third there are 1719 less than in both the others. How many bricks in the third pavement? 38. In 1869 there were 264,146,900 bushels of wheat raised in the United States, and 874,120,005 bushels of corn. How much more corn than wheat was produced? 39. A bought 351 acres of land, and B bought 27 acres more than A; B sold his land to C, who then had 537 acres. How many acres did C have at first? 40. A grocer retailed a quantity of sugar for $308. 40, and so gained $106.28. How much had he paid for it? 41. The year 1870 was just 378 years after the discovery of America by Columbus. In what year did that event take place ? 42. On Monday morning a bank had on hand $1826. Dur- ing the day $2191 were deposited and $3412 drawn out; on Tuesday $3256 were deposited and $2164 drawn out. How many dollars were on hand Wednesday morning? 43. E. S. Hill is worth $15795, of which $2895 is in- vested in bank stock, $3864 in mortgages and the rest in land. How much has he invested in land? 44. Of the two numbers 89346 and 56849, how much nearer is the one than the other to 68754? 45. The number of pupils who attended school in Boston in 1870 was 38944, and of this number 35442 attended the public schools. How many attended the other schools? MULTIPLICATION INDUCTIVE EXERCISES. 67. 1. How many books are there in 2 piles containing 3 books each? 2. If you place 4 apples in a group, how many apples are there in 3 such groups? In 4 groups? 3. When there are 3 roses in a cluster, how many are there in 3 clusters? In 4 clusters? In 5 clusters? 4. How many are 3 + 3 + 3 + 3, or four 3's? 5. How many are 4 + 4 + 4, or three 4's? 6. How many are four 4's? Four 5's? Four 6's? 7. James bought 5 pencils at 5 cents each. How much did they cost him? How many cents are 5 times 5 cents? How many are five 5's? 8. An orchard contains 5 rows of 6 trees each. How many trees are there in the orchard? How many trees are 5 times 6 trees? How many are 5 times 6? 9. James piled his blocks in 3 piles, each containing 5 blocks. How many blocks had he? How many are 3 times 5 blocks? How many are 3 times 5? 10. A boy earned $4 a week for 6 weeks. How much did he earn in all? How many dollars are 6 times $4? How many are 6 times 4? 11. Harry played 5 hours per day. How many hours did he play in 6 days? How many are 6 times 5 hours? How many are 6 times 5? (43) 44 MULTIPLICATION. 12. How does 5 times 4 compare with 4 times 5? 5 times 6 with 6 times 5? 13. When numbers are used without reference to any par- ticular thing, they are called Abstract Numbers. DEFINITIONS. 68. Multiplication is a short process of finding the sum of equal numbers; or, The process of repeating one number as many times as there are units in another. 69. The Multiplicand is the number to be repeated or multiplied. 70. The Multiplier is the number showing how many times the multiplicand is to be repeated. 71. The Product is the result obtained by multiplying. 72. The multiplicand and multiplier are called the factors of the product. 73. The Sign of Multiplication is an oblique' cross : X It is read, multiplied by, or times. When placed between two numbers it shows that they are to be multiplied together. Thus, 4 X ^ is read, 4 multiplied by 3, or 3 times 4. 74. PRINCIPLES. 1. The multiplier must be regarded as an abstract number. 2. The multiplicand and product must be like numbers. 3. Either of the factors may be used as multiplicand or multi- plier when both are abstract. In practice, for convenience, the smaller number is generally used as multiplier. MULTIPLICATION. 47 24. If 6 men can do a piece of work in 21 days, how long will it take one man to do the same work? 25. In a certain orchard there are 9 rows of trees and 27 trees in a row. How many trees are there in the orchard? 26. Count by 2's from to 24; thus: 2, 4, 6, 8, 10, etc. 27. Count by 3's from to 36. By 4's from to 48. 28. Repeat all the numbers of times 5 from once 5 to 10 times 5. Thus, once 5 is 5, 2 times 5 are 10, 3 times 5 are 15, etc. 29. Repeat from once 6 to 10 times 6, and back from 10 times 6 to once 6. 30. Repeat from once 7 to 10 times 7, and reverse. 31. Repeat from once 8 to 10 times 8, and reverse. 32. Repeat from once 9 to 10 times 9, and reverse. 33. Repeat from once 10 to 10 times 10, and reverse. 34. At 25 cents a pound, how much will 6 pounds of raisins cost? 35. If a man can dig 28 bushels of potatoes in one day, how many can he dig in 4 days? 36. If a person spend 25 cents a day for cigars, how much will he spend in 7 days? 37. If a boy earns 33 cents a day, how much will he earn in 9 days? 38. When butter is selling at 37 cents a pound, what will 7 pounds cost me? WRITTEN EXERCISES. 76. 1. How many are 3 times 275? IST PROCESS. ANALYSIS. Since multiplication is a short 275 process of adding equal numbers, it is evident 275 that we can determine by addition how many 3 2 rj p. times 275, or three 275's, are. The sum is 825. In practice, a shorter method is employed, Sum 825 which is given in the second process and analysis. 48 MULTIPLICATION. 2D PROCESS. Multiplicand 275 Multiplier 3 Product 825 ANALYSIS. For convenience we write the multiplier under the multiplicand, and begin at the right to multiply. Thus, 3 times 5 units are 15 units, or 1 ten and 5 units. We write the 5 units in units' place in the product and reserve the tens to add with the tens. 3 times 7 tens are 21 tens, plus 1 ten reserved are 22 tens, or 2 hundreds and 2 tens. We write the 2 tens in tens' place in the product and reserve the hundreds to add with the hundreds. 3 times 2 hundreds are 6 hundreds, plus 2 hundreds reserved are 8 hundreds, which we write in hundreds' place in the product. Hence the product is 825, the same as the sum in the first process. PROOF. If the results obtained by both processes agree, the work is probably correct. In multiplying, pronounce the results only. Thus, in the example given above, instead of saying 3 times 5 are 15, 3 times 7 are 21, plus 1 reserved are 22; 3 times 2 are 6, plus 2 reserved are 8; say 15, 22, 8. Solve and prove: 2. 3 times 314. 3. 4 times 568. How many are 8. 5 times 314? 9. 4 times 655? 10. 7 times 764? 4. 4 times 987. 5. 5 times 345. 11. 3 times 830? 12. 6 times 734? 13. 9 times $48? 6. 5 times $819. 7. 3 times $769. 14. 8 times $42? 15. 6 times $32? 16. 7 times $57? 17. If a man earns $17.25 per week, how much can he earn in 8 weeks? 18. A benevolent man paid annually for the support of the poor $2365. How much did he pay in 7 years? 19. A shoe dealer sold 9 pairs of shoes at $3.75 a pair. How much did he receive for all? 20. A man bought 8 cows at an average price of $31.27. How much did they all cost him? MULTIPLICATION, 49 21. If a ship sail 425 miles in one week, how far will she sail in 9 weeks? 22. A barrel of flour weighs 196 pounds. How much will 8 barrels weigh? 23. When wheat is worth $1.78 per bushel, how much can be realized from the sale of 9 bushels? 24. At $6.25 a pair, what will be the cost of 7 pairs of boots? 25. There are 5280 feet in a mile. How many feet in 7 miles? 26. At $37.50 an acre, what will be the cost of 8 acres of land? 27. What will be the cost of 7 thousand feet of lumber at $18.25 per thousand? 28. When broom corn is selling at $83.50 a ton what is the value of 8 tons? CASE II. 77. When the multiplier is expressed by more than one figure. 1. There are 9 square feet in one square yard. How many are there in 10 square yards? 2. How many square feet in 6 square yards? 3. Since 10 square yards contain 90 square feet, and 6 square yards contain 54 square feet, how may the number of square yards in 10 -f 6, or 16, square feet, be found? How, then, may you multiply by 16? By 18? By 13? 4. Find the cost of 17 yards of cloth at 18 cents a yard? 5. When eggs are 21 cents a dozen, what will 15 dozen cost? 6. Since 12 inches make one foot in length, how many inches are there in 18 feet? 7. A pound of sugar is equal to 16 ounces. How many ounces are there in a quantity of sugar weighing 16 pounds? 50 MULTIPLICATION. 7. Find the cost of 17 yards of cloth at 8 cents a yard, by finding the cost of 9 yards, and then of 8 yards. Of 10 yards and 7 yards. 8. What will be the cost of 11 primers at 25 cents each? 9. Find the cost of 16 yards of cloth at 8 cents a yard, by finding the cost of 10 yards and 6 yards. 9 yards and 7 yards. 8 yards and 8 yards. 10. James is in school 5 hours a day. How many hours is he in school during three weeks, or 15 school-days? 11. A bought 4 sets of forks, each set containing 6 forks. How much did the forks cost him at $2 each ? 12. Mary bought 15 pounds of sugar at 11 cents a pound, and 3 pounds of raisins at 15 cents a pound. After paying her bill she had 10 cents left. How much money had she at first? 13. A cooper can make 12 barrels a day. How many can he make in 12 days? 14. John bought 12 lead pencils at 8 cents each, and 2 erasers at 4 cents each. How much did all cost him? 15. The railroad fare from Rochester to New York is $7. How much will the tickets for a party of 9 cost? 16. If a cow give 9 quarts of milk a day, how much milk will she give in 9 days? 17. If a man put $8 in a savings-bank each month, how much will he deposit in a year? 18. At $4 a yard, what will 17 yards of broadcloth cost? 19. If a laborer can earn $2 a day, how much can he earn in 12 days? 20. What will 15 pairs of skates cost at $4 a pair? 21. At 20 cents a dozen, how much will 18 dozen eggs cost? 22. A coal dealer sold an average of 18 tons of coal per day for 12 days. How many tons did he sell in that time? 23. At 22 cents a pound, how much will 11 pounds of butter cost? MULTIPLICATION. 51 24. How far will a man travel in 15 days, if he travel 10 hours a day and 3 miles an hour? 25. A man bought 25 cows and 12 times as many sheep. How many sheep did he buy? WRITTEN EXER CISES. 78. 1. Multiply 327 by 123. 1ST PROCESS. 327 123 ANALYSIS. For convenience we write the numbers as in the preceding case. Since in multiplying we must multiply by the parts of the multiplier and add the partial products, to multiply by 123 we multiply by 3 units, 2 tens, and 1 hundred as partial multipliers. 3 times 327 are 981, the first partial product; 2 times 327 are 654 and 2 tens times 327 are 654 tens, or 6540, which we write for a second partial product. 1 time 327 equals 327, and 1 hundred times 327 are 327 hundreds, or 32700, which we write for a third partial product. The sum of the partial products will be the entire product. 1st Partial Prod. 2d Partial Prod. 3d Partial Prod. 981 6540 32700 Entire Prod. 40221 1st Partial Prod. 2d Partial Prod. 3d Partial Prod. 2D PROCESS. 327 123 981 654 327 ANALYSIS. In the second process the ciphers at the right of the partial prod- ucts are omitted, the significant figures still occupying their proper places. Thus, in multiplying by 2 tens the product was 654 tens, or 6 thousand, 5 hundred, 4 tens, which we write in their places in the partial product. In multiplying by hundreds, the low- est order of the product is hundreds, hence we write the first figure of the product under hundreds. PROOF. Multiply the multiplier by the multiplicand. (Prin. 3.) If the result agrees with that formerly obtained, the work is probably correct. Entire Prod. 40221 52 MULTIPLICATION. RULE. Write the multiplier under the multiplicand with units under units, tens under tens, etc. Multiply each figure of the multiplicand by each significant fig- ure of the multiplier successively, beginning with units. Place the right hand figure of each product under the figure of the multiplier used to obtain it, and add the partial products. PROOF. Review the work, or multiply the multiplier by the multiplicand. If the results agree the work is probably correct. EXAMPLES. (2.) (3.) (4.) (5.) (6.) Multiply 325 219 384 $2.81 $3.18 By 42 54 46 23 36 Multiply : Multiply : 7. 456 by 12. 23. 73982 by 321. 8. 389 by 23. 24. 42586 by 604. 9. 493 by 25. 25. 89258 by 703. 10. 374 by 27. 26. 84206 by 569. 11. 3625 by 28, 27. 156783 by 423. 12. 2413 by 31. 28. 248164 by 372. 13. 3681 by 63. 29. 182642 by 419. 14. 67021 by 52. 30. 192573 by 429. 15. 63583 by 62. 31. 234567 by 612. 16. 84216 by 78. 32. 467105 by 623. 17. 38413 by 35. 33. 398120 by 706. 18. 29615 by 45. 34. 683912 by 1684. 19. 23423 by 25. 35. 312465 by 1827. 20. 24542 by 64. 36. 468975 by 2946. 21. 45684 by 73. 37. 416004 by 3009. 22. 41075 by 62. 38. 329706 by 3802. MULTIPLICATION. 53 39. $ 18.61 by 73. 40. $115.81 by 45. 41. $164.32 by 81. 42. $123.45 by 804. 43. $415.05 by 367. 44. $ 18.37 by 127. 45. $113.41 by 613. 46. $281.69 by 247. 47. $312.09 by 684. 48. $425.27 by 618. 49. In a reaper factory an average of 2346 reapers is con : structed annually. At this rate how many would be made in 25 years? 50. A farmer counted the trees in his orchard and found that he had 104 rows, each row containing 106 trees. How many trees were there in the orchard? 51. In a croquet factory a man makes 835 balls daily. How many balls can he make in 312 days? 52. The distance between Rochester and Syracuse is 81 miles. How many miles per month of 31 days, will a loco- motive travel that goes from Rochester to Syracuse daily and returns? 53. Mr. Davis built 8 houses at a cost of $1925 each 6 at $2275 each, and 5 at $3897 each. What did they all cost him? 54. Sold my farm of 413 acres at $85 per acre. How much did I get for it? 55. How much will it cost to build 89 miles of railroad at an estimated expense of $57394 per mile? CASE III. 79. Where there are ciphers on the right of either or hoth factors. 1. How many are are 10 times 2? 3? 4? 5? 6? 7? 8? 9? 2. Write the above multiplicands and products side by side and compare them. 54 MULTIPLICATION. 3. How may the product be found from the multiplicand when the multiplier is 10 ? 4. How many are 100 times 2? 3? 4? 5? 6? 7? 8? 9? 5. How may any number be multiplied by 100? by 1000? 6. How may a number be multiplied by 1 with any num- ber of ciphers affixed ? 80. PRINCIPLE. In multiplying by 10, 100, 1000, etc., as many ciphers must be annexed to the right of the multiplicand as there are ciphers in the multiplier. 1. Multiply 36 by 1000. PROCESS. ANALYSIS. Since in multiplying by 1 with any 3 6 number of ciphers annexed, we annex as many 1000 ciphers to the multiplicand as there are in the mul- tiplier, to multiply by 1000 we annex three ciphers to the multiplicand, which gives the product 36000. 2. Multiply 2360 by 400. PROCESS. ANALYSIS. Since 2360 is equal to 236 X 10> an d 2360 40 is e< l ual to 4 X 100 > the P roduct of 236 X 400 may J. ^ e obtained by multiplying 236 by 4, and this product i-LjL by 10 times 100, or 1000. The product of 236 X 4 is 944000 944^ and this may be multiplied by 1000 by annexing three ciphers (Prin.), giving as a result 944000. RULE. Multiply without regard to the ciphers on the right, and to the product annex as many ciphers as there are on the right of both multiplier and multiplicand. 3. Multiply 375 by 10. By 100. By 40. By 300. 4. Multiply 845 by 30. By 70. By 600. By 900. 5. Multiply 176 by 500. By 700. By 400. By 1000. 6. Multiply 1385 by 200. By 2000. By 2200. By 3300. 7. Multiply 4860 by 250. By 3200. By 4200. By 6500. 8 Multiply 3120 by 210. By 3800. By 2700. By 4600. MULTIPLICATION. 55 9. In a mile there are 5280 feet. How many feet are there in 500 miles? 10. In an acre there are 160 square rods. How many square rods are there in a farm of 300 acres? 11. A farmer sold a flock of 260 sheep at $3.20 per head. How much did he get for them ? 12. A drover sold 1120 hogs at an average price of 816.30 per head. How much did he receive for them? EXAMPLES. 81. 1. What will be the cost of 896 chests of tea, each chest containing 58 pounds, at 63 cents a pound? 2. An agent sold 3923 Lyman's Historical Charts at $3.50 each. How much did he receive for them? 3. I have 6 bins that hold 119 bushels each. They are full of grain and I have already sold 515 bushels. It was all raised on my farm this year. How much grain was raised? 4. A. J. Newton & Co. bought 113 cases of calico, each case containing 64 pieces, and each piece 47 yards. How many yards did they buy? 5. A drover bought 25 oxen at $85 a head, 316 sheep at $4.50 a head, and 94 calves at $8 a head. What was the whole amount paid? 6. A man insured 2 houses valued at $3750 and $4650, respectively, at the rate of $2 per hundred dollars. How much did the insurance cost him? 7. If I have 219 acres of land, and each acre produces 47 bushels of corn, how many bushels do I receive? 8. How many quills can be obtained from 398 geese, if each wing furnishes 6 quills? 9. A grocer sold in one month 81 dozen eggs at 26 cents per dozen ; in the next, 53 dozen at 28 cents per dozen. How much money did he receive for the eggs? 56 MULTIPLICATION. 10. It requires 1716 pickets to fence one side of a square lot. How many pickets will be required to fence 13 lots of the same size and shape? 11. A sold 13 firkins of butter, each firkin containing 56 pounds, at $ .34 a pound. How much did he receive for it? 12. A coal dealer bought 13 car loads of coal, each load containing 10 tons, at $6.85 a ton. He retailed 48 tons of this at $7 per ton, 28 tons at $8.25 per ton, 27 tons at $8.75 per ton, and the remainder at $9.50 per ton. How much did he make by the transaction? 13. An army lost in battle 315 killed, 417 wounded; the enemy lost in killed and wounded, together, 13 times as many. How many soldiers were killed and wounded in this battle? 14. If two steamers should leave New York at the same time, and should sail in the same direction, the first at the rate of 18 miles an hour, the second at the rate of 15 miles an hour, how far apart would they be in 36 hours? 15. Mr. Hudson bought 350 bushels of corn at 65 cents a bushel, 215 bushels of wheat at $1.35 per bushel, and 273 bushels of oats at 43 cents a bushel. What did the whole cost him? 16. Mr. Henderson sold a farm of 325 acres at $65.50 per acre, and received in payment 345 sheep at $3.25 per head, a note for $2684.95 and the rest in cash. How much cash did he receive? 17. A cloth merchant sold two lots of cassimeres, the first containing 17 pieces of 28 yards each, at $1.75 per yard, the second containing 23 pieces averaging 29 yards each, at $1.85 per yard. What was the value of the whole ? 18. An excursion train composed of 13 passenger coaches, each containing 37 persons, went from Syracuse to Niagara Falls and back. If the fare to Niagara Falls and return to Syracuse, was $3.25 per ticket, how much did the railroad company receive? I S I O tsl O\1 !;VlH9L ^"i *'' ^^JSs*' mFwidrVr > iSBBB 1 / 'I llr^ K-7 A \| pTnnru n n n n ii i M ii n ii M ii n i ii ii 'i i7!rriii|^jiji n ii ii BT ii ii ii H ii 11 n .. * -, , ..MUMMI ||y # C-T > =; ~~ ~"^ INDUCTIVE EXERCISES. 82. 1. How many groups of 2 birds each can be formed from 6 birds? How many 2's are there in 6? 2. How many groups of 3 sheep each can be formed from 9 sheep? How many 3's are there in 9? 3. How many groups of 2 chickens each can be formed from 10 chickens? How many 2's are there in 10? 4. At 5 cents apiece, how many pencils can be bought for 10 cents? How many 5's are there in 10? 5. When milk is worth 7 cents a quart, how many quarts can be bought for 28 cents? How many 7's are there in 28? 6. There are 20 panes of glass in the front of a block of stores. If each window contains 4 panes, how many win- dows are there? How many 4's are there in 20? 7. At 8 cents a dozen, how many peaches can be bought for 24 cents? How many times 8 cents are 24 cents? How many 8's are there in 24? 8. How many groups of 4 things each can be formed from 16 things? How many 4's are there in 16? 9. A merchant had 30 yards of calico which he cut into pieces 5 yards long. How many pieces did it make? How many 5's are there in 30? How many times is 5 contained in 30? 10. How many 9's are there in 18? How many times is 9 contained in 18? (57) 58 DIVISION. 11. How many 5's are there in 10? In 15? In 20? In 25? In 30? 12. If 15 cents are divided equally among 3 boys, how many cents will each receive? When 15 cents are divided into 3 equal parts, how many cents will each part contain ? 13. If 12 peaches are arranged in 3 rows, how many will there be in each row? 14. What is one of the 4 equal parts of 8? Of 12? Of 16? 15. How many 3's are there in 30? How many are 10 threes, or 10 times 3? 16. How many 4's are there in 40? How many times is 4 contained in 40? How many are 10 fours? DEFINITIONS. 83. Division is the process of finding how many times one number is contained in another; or, The process of separating a number into equal parts. 84k The Dividend is the number to be divided. 85. The Divisor is the number by which we divide. It shows into how many equal parts the dividend is to be divided. 86. The Quotient is the result obtained by division. It shows how many times the divisor is contained in the dividend. 87. The part of the dividend remaining when the division is not exact is called the Remainder. 88. The Sign of Division is -r- . It is read divided by. When placed between two numbers it shows that the one at the left is to be divided by the one at the right. Thus, 154 -r- 7, is read 154 divided by 7. DIVISION. 59 Division is also indicated by placing the dividend above the divisor with a line between them, and by writing the divisor at the left of the dividend with a curved line between them. Thus, 154 divided by 7, may also be written -!-p, and 7)154. 89. PRINCIPLES. 1. The dividend and divisor must be like numbers. 2. The quotient must be an abstract number. 3. The product of the divisor by the quotient, plus the remain- der, is equal to the dividend. 9 1. In problems where it is required to separate a number 3 into equal parts, it is customary to regard the dividend and quotient as like numbers, and the divisor as an abstract number. 2. The example "How many 3's are there in 9" may be 3 solved, as in the margin, by subtraction. All examples in divis- 3 ion may be solved in the same manner. Hence, division may be regarded as a shwt method of subtracting equal numbers. 3. In multiplication two numbers are given to find their product, In division the product is given and one of the factors to find the other. Hence, division is the converse of multiplication. CASE I. 90, When the divisor is expressed hy one figure. 1. At $8 each, how many plows can be bought for $24? ANALYSIS. Since each plow costs $8, as many plows can be bought for $24 as $8 is contained times in $24, which is 3 times. Therefore 3 plows can be bought for $24. 2. If a man can earn $7 in a day, how long will it take him to earn $28? 3. At $4 each, how many hats can be bought for $24? 4. When flour is selling at $6 a hundred-weight, how many hundred-weight can be bought for $36? 60 DIVISION. 5. If a mason built 3 rods of walk per day, how long did it take him to build 21 rods? 6. B paid 96 cents for glass at 8 cents a pane. How many panes did he buy? 7. At $9 a cord, how many cords of wood can be bought for $45? 8. If a man earns $11 a week, how many weeks will he require to earn $66? 9. How many lots of 11 acres each can be made from a farm containing 132 acres? 10. If a farmer exchanges 6 firkins of butter worth $20 a firkin for cloth at $4 a yard, how many yards will he receive? 11. My coal cost me $35 at the rate of $7 a ton. How many tons did I purchase? 12. How many engravings must an artist sell for $12 apiece to realize $84? 13. When sugar is worth 9 cents a pound, how many pounds can be bought for 45 cents? 14. At the rate of $7 a rod, how many rods of fence can be built for $63? 15. I hired a man for $45 to do a piece of work at the rate of $5 a day. How many days did it take him ? 16. A lady bought some silk worth $3 a yard, paying $36 for it. How many yards did she buy? 17. How many barrels of flour at $8 a barrel can be bought for $48? 18. How many pounds of nails can be bought for 75 cents at the rate of 4 pounds for 20 cents? 19. I bought 6 sheep for $30. How much did I pay per head? 20. At $5 per head, how many head of sheep can be bought for $37? Am. 7 sheep and $2 left. 21. A man whose wages were $4 a day earned in a certain time $33. How many days did he work? Am. 8^ days. DIVISION. 61 91. From examples 20 and 21 it is apparent that the remainder may be written either after the quotient, as in the answer to the 20th, or as a part of it, as in the answer to the 21st, When written as a part of the quotient, the remainder is expressed by placing the divisor under it with a line between them. Such an expression shows that each unit of the re- mainder is to be divided into as many equal parts as there are units in the divisor. When any thing is divided into two equal parts, each of the parts is called one half. When into three equal parts, each part is called one third. When into four equal parts, each part is called one fourth. When into five, six, seven, etc., equal parts, the parts are called fifths, sixths, sevenths, etc. ^ expresses one half, or one of two equal parts of any thing. ^ expresses one fourth, or one of four equal parts of any thing. f expresses two fifths, or two of five equal parts of any thing. 2^ expresses five twenty-sevenths, or five of twenty-seven equal parts of any thing. 92. One or more of the equal parts of any thing is called a Fraction. 93. Read the following fractional expressions: A ft 1* It VV W A ft T 22. If James should divide 25 apples equally among 5 boys, what part of the whole would each receive? How many apples would each receive? 62 DIVISION. ANALYSIS. If he should divide 25 apples equally among 5 boys, each boy would receive (me-fifth of 25 apples, which is 5 apples. 23. If flour is worth $8 a barrel, what will one-half barrel cost? 24. Mr Smith bought 8 bushels of chestnuts for $24. How much did he pay per bushel? How much is one-eighth of 24? 25. What is one-sixth of 36? Of 42? 26. What is one-tenth of 50? Of 60? 27. What is one-seventh of 14? Of 28? Of 48? Of 60? Of 70? Of 80? Of 42? Of 49? WRITTEN EXERCISES. 94. 1. Divide 1396 by 1ST PROCESS. Divisor. Dividend. Quotient. 4)1396(300 times. 1200 40 times. 196 160 36 36 9 times. 349 times. 2D PROCESS. 1 1 1 4)1396(349 12 19 16 36 the partial dividend, there is 4. ANALYSIS. For convenience we write the divisor at the left, and the quotient at the right of the dividend, with curved lines between them, and begin at the left to divide. 4 is not contained in 1 thousand any thousand times, therefore the quotient can not contain units of any order higher than hundreds. Hence we find how many times 4 is con- tained in all the hundreds of the dividend. 1 thousand plus 3 hun- dreds equals 13 hundreds. 4 is con- tained in 13 hundreds 3 hundred times and a remainder. We write the 3 hundreds in the quotient and multiply the divisor by it, obtaining for a product 12 hundreds, or 1 thou- sand 2 hundred, which we write under units of the same order in the divi- dend. Subtracting this product from a remainder of 1 hundred. DIVISION. 63 1 hundred plus 9 tens equals 19 tens. 4 is contained in 19 tens 4 tens times and a remainder. We write the 4 tens in the quotient and multiply the divisor by it, obtaining for a product 16 tens, or 1 hundred and 6 tens, which we write under units of the same order in the partial dividend. Subtracting, there is a remainder of 3 tens and 6 units. 3 tens plus 6 units equals 36 units. 4 is contained in 36 units 9 times. We write the 9 units in the quotient and multiply the divisor by it, obtaining for a product 36 units, or 3 tens and 6 units, which we write under units of the same order in the partial dividend. Subtract- ing, there is no remainder. Hence the quotient is 349. In the second process all ciphers are omitted from the right of the products and the significant figures are written under units of the same order. The quotient also is expressed by writing the different orders of units in proper succession. PROOF. 349 the quotient, multiplied by 4 the divisor, is equal to 1396 the dividend. Hence the work is correct. (Prin. 3.) Solve in like manner and prove: 2. 738 -r- 3. 3. 845 -- 5. 4. 385-*- 7. 5. 4821 3. 6. 3462 6. 7. 3864 8. 8. 7848 -f-9. 9. 8432 --4. 10. 8308 -r-7. 95. The solution of the preceding examples may be short- ened by performing the multiplications and subtractions with- out writing the results. This process is called Short Division. The solution of Example 1 by Short Division is as follows: PROCESS. . ANALYSIS. 4 is contained in 13 hundred 3 4)1396 hundred times and 1 hundred remainder. We write 3 hundreds in the quotient under units of the same order in the dividend. 1 hundred remainder united with 9 tens makes 19 tens. 4 is contained in 19 tens 4 tens times and 3 tens remainder. We write the 4 tens in the quotient under tens of the dividend. 3 tens remain- der united with 6 units make 36 units. 4 is contained in 36 units 9 times. We write the 9 in the quotient. Hence the quotient is 349. 64 DIVISION. Solve by short division . 11. 4872- -4. 17. 12. 6830- -5. 18. 13. 2976- -6. 19. 14. 2985- -5. 20. 15. 4635- -3. 21. 16. 3936- -4. 22. 23. 4567- -5. 24. 8932- -6. 25. 8174- -9. 26. 9185- -4. 27. 8436- -7. 28. 3885- -8. CASE II. 96. When the divisor is expressed by more than one figure. 1. How many barrels of flour at $10 a barrel can be bought for $80? ANALYSIS. Since 1 barrel costs $10, as many barrels can be bought for $80 as $10 are contained times in $80 which is 8 times. Therefore 8 barrels can be bought for $80. 2. How many pounds of mutton at 10 cents a pound can be bought for 50 cents? How many 10's in 50? In 60^ In 70? In 80? 3. A man measured a stick and found it to be 60 inches long. There are 12 inches in a foot. How many feet long was it? How many 12's in 60? In 72? In 84? In 96? 4. At $13 a ton how much hay can be bought for $26? 5. At 15 cents each how many toys can be bought for 30 cents? For 45 cents? For 60 cents? 6. Mr. Henderson sold 20 lambs for $80. How much did he get apiece for them? 7. 25 cents make a quarter of a dollar. How many quarters of a dollar has a boy who has 50 cents? 8. Henry's father gave him a dollar. How many pine- apples at 20 cents each can he buy with the money? 9. The railroad fare to a certain place is 35 cents. How many tickets can be bought with 70 cents? DIVISION. 65 10. If a boy earns 11 cents an hour, how long will it take him to earn 55 cents? 66 cents? 88 cents? 11. There are 20 hundred-weight in a ton. How many tons are there in 45 hundred-weight? How many in 55? 12. In one day there are 24 hours. How many days are there in 50 hours? In 60 hours? In 72 hours? 13. 12 articles make a dozen. How many dozen are there in 39 articles? In 48? In 51? In 60? ' In 65? 14. A farm of 60 acres was divided into 15 equal lots. How many acres were there in each lot? 15. At 18 cents a dozen, how many dozen of eggs can be bought for 36 cents? For 40 cents? For 54 cents? 16. When butter is 30 cents a pound, how many pounds can be bought for 90 cents? For $1:20? For $1.50? 17. How many 20's are there in 40? In 50? In 60? 18. How many 25's are there in 50? In 60? In 75? 19. By selling brooms at 25 cents each, I received $1.25. How many brooms did I sell? 20. In 100 how many 10's are there? How many ITs? 12's? 13's? 15's? 16's? 20's? 23's? 25's? WRITTEN EXERCISES. 97. 1. Divide 7975 by 26. PROCESS. ANALYSIS. 26 is not contained in 7 Divisor. Dividend. Quotient. thousands any thousands times; hence 26)7975(306- 1 ^- we un i te the thousands with the hun- rj g dreds, making 79 hundreds. 26 is con- tained in 79 hundreds 3 hundred times 1 5 an( j a remainder. We write the 3 hun- 156 dreds in the quotient and multiply the ^ 9 divisor by it, obtaining for a product 78 hundreds, or 7 thousands and 8 hun- dreds, which we write under units of the same order in the dividend. Subtracting, there is a remainder of 1 hundred. 5 66 DIVISION. We unite the 1 hundred with the 7 tens, making 17 tens. 26 is not contained in 17 tens any tens times ; therefore there are no tens in the quotient, and we write a cipher there. We unite the 17 tens with the 5 units, making 175 units. 26 is contained in 175 units 6 times and a remainder. We write the 6 in units 7 place in the quotient and multiply the divisor by it, obtain- ing for a product 156 units, or 1 hundred, 5 tens, and 6 units, which we write under units of same order in the partial dividend. Sub- tracting, there is a remainder of 19. We write the remainder over the divisor as a part of the quotient. Hence the quotient is 306 |f. PROOF. 306X26 + 19 = 7975. Hence the work is correct. (Prin. 3.) 98. When the steps in the solution of an example in divis- ion are written, the process is called Long Division. RULE. Write the divisor at the left of the dividend with a curved line between them. Find how many times the divisor is contained in the fewest figures on the left hand of the dividend that will contain it, and write the quotient on the right. Multiply the divisor by this quotient and place the product under the figures divided. Subtract the result from the partial dividend used, and to the remainder annex the next figure of the dividend. Divide as before until all the figures of the dividend have been annexed to the remainder. If any partial dividend will not contain the divisor, write a cipher in the quotient, then annex the next figure of the dividend and proceed as before. If there is a remainder after the last division write it after the quotient, or with the divisor under it as part of the quotient. PROOF. Multiply the divisor by the quotient, and to the prod- uct add the remainder, if any. If the work is correct, the result , ivill equal the dividend. 1. To find the quotient figure, see how many times the first figure of the divisor is contained in first figure* of the partial dividend that will con- DIVISION. 67 tain it, making allowance for the addition of the tens from the prod- uct of the second figure of the divisor. 2. If the product of the divisor by the quotient figure be greater than the partial dividend from which it is to be subtracted, the quo- tient figure is too large. 3. Each remainder must be less than the divisor; otherwise the quotient figure is too small. 4. When there is no remainder the divisor is said to be exact. EXAMPLES. 99. Divide: Divide : 2. 1240 by 10. 25. 12456 by 24. 3. 3443 by 11. 26. 28350 by 54. 4. 2592 by 12. 27. 50854 by 94. 5. 3978 by 13. 28. 58176 by 96. 6. 5684 by 14. 29. 56394 by 78. 7. 6480 by 15. 30. 54944 by 101. 8. 8736 by 21. 31. 90992 by 121. 9. 1472 by 32. 32. 199864 by 301. 10. 9672 by 31. 33. 475524 by 612. 11. 9724 by 22. 34. 1445204 by 802. 12. 2952 by 72. 35. 1760225 by 905. 13. 1188 by 54. 36. 3156584 by 722. 14. 4235 by 55. 37. 5173302 by 834. 15. 5356 by 52. 38. 5926431 by 643. 16. 8733 by 41. 39. 3214664 by 566. 17. 9639 by 81. 40. 6923471 by 555. 18. 7991 by 61. 41. 14293624 by 675. 19. 2508 by 22. 42. 56243121 by 686. 20. 7332 by 52. 43. 692348726 by 897. 21. 4824 by 72. 44. 496839715 by 1047. 22. 16665 by 33. 45. 786935846 by 3118. 23. 13545 by 43. 46. 1234589640 by 96813. 24. 25578 by 63. 47. 31964875932 by 37425. 68 DIVISION. 48. Into how many lots of 39 acres each, can a tract of land containing 6318 acres be divided? 49. Wm. Wallace has 17 horses, the aggregate value of which is $4386. What is the average worth of each horse? 50. A surveyor traveled 41600 rods in one week. How many miles did he travel, there being 320 rods in a mile? 51. How many eggs at $.38 per dozen can be bought for $6.84? 52. In 24 hours the earth moves 1575000 miles. How far does it move in one minute, 60 minutes making an hour? 53. Mount Everest in Asia is 29100 feet high. There are 5280 feet in a mile. How many miles high is it? 54. It required 4375480 bricks to build an orphan asylum. How many days did it require 5 teams to draw the bricks, if they drew 5 loads per day and 1250 bricks at a load? 55. The earth is 91500000 miles from the sun. How many seconds does it take light to come from the sun to the earth, if it travels 185000 miles per second? 56. A man bought a farm of 278 acres at $63 an acre. He paid $1275. down and agreed to pay the rest in 8 equal annual payments. How much was he required to pay yearly? 57. The earnings of a certain railroad were $3681452 during the year. The number of days in a year is 365. What was the average income per day? 58. How many feet are there in a mile, if 42 miles contain 221760 feet? 59. If the average wages of a laboring man are $500 per year, how many men will it require to earn $50000 per year, Jhe salary of the President? 60. The area of the State of North Carolina is 50704 square miles, and the population, according to the census of 1870, was 1071400. How many persons, on an average, were there living on a square mile? DIVISION. 69 CASE III. 100. When the divisor has ciphers on the right. 1. How many 10's and what remainder in 46? 84? 97? 2. When a number is divided by 10, what part is re- mainder? What part is quotient? 3. How many hundreds, and what remainder in 434? 516? 639? 758? 4. When a number is divided by 100 what part of it is remainder? What part is quotient? *5. When a number is divided by 1000, what part is re- mainder? What part is quotient? 101. PRINCIPLE. In dividing by 10, 100, 1000, etc., the remainder will be as many of the figures at the right of the divi- dend as there are ciphers on the right of the divisor. The rest of the number is quotient. 102. 1. Divide 6374 by 1000. PROCESS. ANALYSIS. Since the divisor contains no 11000^61374 or der of units lower than thousands, in divid- ing we may omit or cut off from the dividend 6 iVcrb" for a remainder, all orders of units lower than thousands. (Prin.) Dividing 6 thousands by 1 thousand we obtain 6 for the quotient and 374 for the remainder or 2. Divide 39321 by 6000. ANALYSIS. Since the divisor con- i tains no order of units lower than i thousands, in dividing we may omit 6 Rem. 3000 or cut off from the dividend all or- 321 ders of units lower than thousands. 6 thousands are contained in 39 Entire remainder 6621,, , ^. jo^ j thousands 6 times and 3 thousand remainder. 3 thousand plus the other partial remainder equals the entire remainder. Hence the quotient is 70 DIVISION. RULE. Cut off the ciphers from the right of the divisor, and as many figures from the right of the dividend. Divide the rest of the dividend by the rest of the divisor. Annex to the remainder the figures cut off: the result will be the true remainder. Divide the following: 3. Divide 1869 by 100. 4. Divide 12345 by 200. 5. Divide 89325 by 700. 6. Divide 35968 by 900. 7. Divide 2465 by 1000. 8. Divide 13692 by 4000. 9. Divide 83005 by 1100. 10. Divide 75684 by 1500. 11. How many miles of railroad at $50000 a mile can be constructed for $38968457? 12. How many schooners carrying 8300 bushels of wheat will it require to carry 984364 bushels? 13. The area of the State of New York is 47000 square miles, or 30080000 acres. How many acres in a square mile? 103. RELATION OF DIVIDEND, DIVISOR, AND QUOTIENT. The value of the quotient depends upon that of the divi- dend and divisor. If one of these is changed, while the other remains the same, the quotient will be changed. If both are changed, the quotient may not be changed. The changes may be illustrated as follows : FUNDAMENTAL EQUATION. 64 -=-8 = 8. CHANGED EQUATIONS. 1. 128 -5- 8 = 16 "I 1. Multiplying the dividend changed. 1 D* V7 rl ' by 2 multiplies the quotient \ by 2. 2. 32 -H 8 = 4 | 2. Dividing the dividend by 2 divides the quotient by 2. DIVISION. 71 2. Divisor changed. 3. Both changed. 1. 64 --16:= 4 2. 64 -*- 4 = 16 1. 128-*- 16 = 8 2. 32-- 4 = 8 1. Multiplying the divisor by 2 divides the quotient by 2. 2. Dividing the divisor by 2 multiplies the quotient by 2. Multiplying or dividing both dividend and divisor by 2 does not change the quo- tient. From these illustrations the following principles are de- duced : 104. PRINCIPLES. 1. Multiplying the dividend or dividing the divisor, multiplies the quotient. 2. Dividing the dividend or multiplying the divisor, divides tlie quotient. 3. Multiplying or dividing both dividend and divisor by the same number, does not change the quotient. ANALYSIS AND KEVIEW. 105. Analysis is the process of solving problems by tracing the relation of the parts. In analyzing we commonly reason from the given number to one, and then from one to the required number. 1. If 8 yards of cloth cost $16, what will 12 yards cost? PROCESS. ANALYSIS. Since 8 yards cost $16> 1 yard 8 yards=$16. will cost one-eighth of $16, or $2; and since 1 " $ 2. 1 yard costs $2, 12 yards will cost 12 times 12 " =$24. $2, or $24. 2. If 8 horses cost $2400, what will 6 horses cost? 3. If 8 lemons cost 40 cents, what will 11 lemons cost? 4. How much will 12 hats cost, if 8 hats cost $16? 72 DIVISION. 5. If 25 pounds of sugar cost $2.50, what will 36 pounds cost? 6. If 12 men can build a school-house in 25 days, how long will it take 25 men to build it ? 7. If 12 barrels of flour are worth $132, what are 22 bar- rels worth? 8. If it requires 576 feet of boards to build 18 rods of fence, how many feet will be required to build 13 rods? 9. If 6 men can do a piece of work in 10 days, how long will it take 5 men to do it? 10. If I exchanged 18 barrels of flour for 61 yards of cloth at $4 a yard, how much did I get per barrel for the flour? 106. The Parenthesis, ( ), shows that the numbers included within it are to be subjected to the same operation. Thus, ( 5 + 6 2 ) X 3 shows that 5 + 6 2, or 9, is to be multi- plied by 3. 107. The Vinculum, " ~~, may be used instead of the parenthesis. Thus, instead of ( 5 + 6 2 ) X 3, we may write 5 + 6 2X 3. Find the value of the following : 11. (12 + 7 9) X 5. 12. (13 6 + 8)x6. 13. (11 2 + 5) X8. 14. (3 + 4)x9 (3 + 6)-3. 15. (5 + 7 3)x3 + (3 + 5 4)-v-4. 16. (36 7)x5 + (102 + 6)~ 9. 17. ( 99 3 ) -i- 8 ( 86 + 10 ) 12 + ( 3 + 6 ) -^ 3. 18. -(7 2) + 6. 19. A man dying, left the following tracts of land to be divided equally among his five children. The first tract con- tained 1118 acres; the second, 3 times as much lacking 193 DIVISION. 73 acres; the third, twice as much as the other two lacking 105 acres. What was each one's share? 20. A gentleman bought 1516 head of cattle at $39 per head. During the summer 97 died of disease, but he sold the remainder so as to gain on the whole number $1819. How much did he get for his cattle per head? 21. If a young man who has a salary of $30 per week, pays $7.25 for his board and $4.25 for other expenses, how long will it take him to save $1500? 22. A man bought a horse for $115 and after keeping him three months, sold him for $155. If lie paid $30 for his keep- ing and received $50 for the use of him during that time, how much did he gain? 23. A speculator purchased a certain number of bushels of wheat for $8735. He sold it for $9215 and in so doing gained $ .25 per bushel. How many bushels did he buy? 24. I bought 25 barrels of flour for $200. For what must it be sold per barrel to gain $50? What will be the gain per barrel ? 25. A tailor having $585 wished to purchase with this an equal number of yards of two kinds of broadcloth. One kind was worth $6 a yard, the other $7 a yard. How many yards of each kind could he buy ? 26. Two men leave the same place at the same time and travel in opposite directions, one at the rate of 48 miles per day, the other at the rate of 52 miles per day. How far apart will they be at the end of 5 days? 27. If 20 men can do a piece of work in 31 days, how many days will be required to do an equal amount of work if 11 additional men are employed? 28. A farmer wished to obtain $120. He sold 16 barrels of apples at $3.50 per barrel, and enough barley at $ .80 a bushel to make up the sum required. How many bushels of barley did he sell? 74 DIVISION. 29. Mr. B. bought 140 acres of land for $17500, and sold enough at $120 per acre to amount to $9600. The rest of the land he sold at cost. How many acres did he sell at cost, and what was the entire loss? 30. A man pays $628 a year for groceries, $350 for house rent, $262 for clothes, twice as much for traveling expenses as for house rent, $175 for annual premium for life insurance, and saves in 4 years enough money to purchase 130 acres of land at $53 an acre. What is his yearly income? 31. In October, 1871, the great fire in Chicago burned ovei an area of 2124 acres. The estimated loss occasioned by the fire was $196000000. What was the average loss per acre? 32. A boy has a velocipede which he can run at the rate of 140 rods in 4 minutes. How many minutes will it take him to run it 630 rods? 33. A farmer has 1000 head of cattle in 5 fields. In the first he has 315 head, in the second 175 head, in the third 300 head, and in the fourth the same number as in the fifth. How many has he in the fifth ? 34. A man gave away $45000 in three equal amounts. One share he gave to his son, one share to his daughter, and the rest to his grandchildren, giving them $1500 apiece. How many grandchildren had he ? 35. In the Centennial Exhibition, at Philadelphia, a section of a cable in process of construction for the new suspension bridge at New York was shown. It was composed of 6000 galvanized steel wires, and its ultimate strength was 22,300,000 pounds. What weight was each wire capable of sustaining ? 36. The main building of the Centennial Exhibition at Philadelphia, the largest building in the world, contained on the ground floor an area of 872320 square feet, on the upper floors in projections 37344 square feet, in towers 26344 square feet. If there are 43560 square feet in an acre, how many acres did the floors of the building contain ? PROPERTIES OF NUMBERS 108. 1. What is the product of 4 times 5? What are 4 and 5 of their product? 2. What is 4 of 16? Of 24? What is 7 of 14? Of 28? 3. What numbers will exactly divide 18? 24? 36? 72? 4. Give the exact divisors of 42. 96. 108. 48. 32?. 5. What are the factors of 30? 24? 40? 56? 64? 6. What numbers between and 10 can not be divided by any number except themselves and 1? Between 10 and 20? 7. What numbers between and 10 can be divided by other numbers than themselves and 1? Between 10 and 20? DEFINITIONS. 109. An Integer or Integral Number is one that expresses whole units. Thus, 281, 36 houses, 46 men, are integral numbers. 110. An Exact Divisor of a number is an integer that will divide it without a remainder. Thus, 2, 4, 6 and 12 are exact divisors of 24. 111. The Factors of a number are the integers which being multiplied together will produce the number. Thus, 6 and 8 are factors of 48. The exact divisors of a number are factors of it. (75) 76 PROPERTIES OF NUMBERS. 112. A Prime Number is one that has no exact divisors except itself and 1. Thus, 1, 3, 5 and 7 are prime numbers. L3. A Composite Number is one that has exact divisors besides itself and 1. Thus, 18 and 24 are composite numbers, for 18 is divisible by 6, and 24 by 8. 111. An Even Number is one that is exactly divisible by 2. Thus, 2, 4, 6, 8, etc., are even numbers. 115. An Odd Number is one that is not exactly divis- ible by 2. Thus, 1, 3, 5, 7, 9, etc., are odd numbers. DIVISIBILITY OF NUMBERS. 116. In determining by inspection the divisibility of num- bers, the following facts will be found valuable. 1. Two is an exact divisor of any even number. Thus, 2 is an exact divisor of 12, 16, 30 and 44. 2. Three is an exact divisor of any number, the sum of whose digits is divisible by 3. Thus, 3 is an exact divisor of 312, 135, 423, and 3816. 3. Four is an exact divisor of a number, if the number expressed by its two right hand figures is divisible by 4. Thus, 4 is an exact divisor of 264, 1284, 1368, and 7932. 4. Five is an exact divisor of any number whose right hand figure is or 5. Thus, 5 is an exact divisor of 360, 1795, 3810, and 7895. DIVISIBILITY OF NUMBERS. 77 5. Six is an exact divisor of any even number, the sum of whose digits is divisible by 3. Thus, 6 is an exact divisor of 732, 534, 798, and 8226. 6. Eight is an exact divisor of a number, if the number expressed by its three right hand figures is divisible by 8. Thus, 8 is an exact divisor of 4328, 3856, 61360, and 5920. 7. Nine is an exact divisor of any number, the sum of whose digits is divisible by 9. Thus, 9 is an exact divisor of 513, 1314, 252, 1341, and 312462. 8. 10, 100, 1000, etc., are exact divisors of any numbers that end respectively with one, two, three, etc., ciphers. Thus, 10, 100, 1000, etc., are exact divisors respectively of 80, 800, 8000, etc. 9. If an even number is divisible by an odd number it is divisible by twice that number. Thus, 72 is divisible by 9 and by twice 9 or 18. 312 by 3 and 6. 10. An exact divisor of a number is an exact divisor of any number of times that number. Thus, 3 is an exact divisor of 12, and of any number of times 12, as 36. 11. An exact divisor of each of two numbers is an exact divisor of their sum and of their difference. Thus, 3 is an exact divisor of 9 and 12 respectively, and therefore of 9 + 12, or 21 ; of 12 9, or 3. 117. Find by inspection some of the exact divisors of the following numbers: 1. 1524. 5. 2556. 9. 42840. 13. 376250. 2. 3432. 6. 7236. 10. 92475. 14. 428328. 3. 4264. 7. 27360. 11. 362088. 15. 4183200. 4. 9360. 8. 23661. 12. 438408. 16. 6853744. 78 PROPERTIES OF NUMBERS. FACTORING. 118. 1. What are the factors of 6? 8? 12? 16? 2. What factors of 18 are prime numbers or prime factors? 3. What are the prime factors of 30? 4. What are all the exact divisors of 30? 5. What numbers besides the prime factors of 30 are its exact divisors? How are they obtained from the prime factors ? 6. Of what number are 2, 3, and 5, the prime factors? 7. How can a number be obtained from its prime factors? 8. The prime factors of a number are 2, 2, and 5. What is the number? Give all the exact divisors of this number. 9. What are the exact divisors of 60? 72? 96? 144? DEFINITIONS. 119. Factoring is the process of separating a number into its factors. 120. Prime Factors are factors that are prime numbers. 121. The number of times a number is used as a factor is indicated by a small figure called an exponent. It is written above and at the right of the number. Thus, 4 X 4 X 4 = 4 3 , and the 3 indicates that 4 is used as a factor three times. 122. PRINCIPLES. 1. Every prime factor of a number is an exact divisor of that number. 2. The only exact divisors of a number are its prime factors or the product of two or more of them. 3- Every number is equal to the product of its prime factors. FACTORING. 79 1. What are the prime factors of 756? PROCESS. 2)756 2)378 3)189 3)63 3)21 7 ANALYSIS. Since every prime factor of a number is an exact divisor of the number, we may find the prime factors of 756 by finding all the prime numbers that are exact divisors of 756. Since the number is even, we divide by 2. Since the quotient obtained is an even number, we divide again by 2. Then we di- vide by the prime numbers 3, 3, 3, successively, and the last quotient is 7, which is a prime number. Hence the prime factors are 2, 2, 3, 3, 3, 7, or 2 2 , 3 3 , 7. RULE. Divide the given number by any prime number that will exactly divide it. Divide this quotient by another prime num- ber, and so continue until the quotient is a prime number. The several divisors and last quotient will be the prime factors. What are the 2. Of 35? 3. Of 64? 4. Of 336? 5. Of 168? 6. Of 144? 7. Of 315? 8. Of 198? 9. Of 224? 10. Of 786? 11. Of 316? prime factors 12. Of 484? 13. Of 1280? 14. Of 1008? 15. Of 1140? 16. Of 1184? 17. Of 1872? 18. Of 7644? 19. Of 2310? 20. Of 3204? 21. Of 4725? 22. Of 23. Of 24. Of 25. Of 26. Of 27. Of 28. Of 29. Of 30. Of 31. Of 3913? 3812? 7007 ? 3980? 26840? 38148? 11340? 24024? 18500? 124416? MULTIPLICATION BY FACTOES. 123. 1. What are the factors of 12? 16? 18? 20? 2. What are the factors of 24? 42? 36? 30? 27? 3. What are the factors of 45? 48? 56? 63? 72? 4. When a number is multiplied by 4 and the product by 6, by what is the number multiplied? 80 PROPERTIES OF NUMBERS. 5. What will 20 carriages cost at $346 each? PROCESS. ANALYSIS. Since 20 is 5 times 4, 20 $ 3 4 6 cost of 1 carriage. carriages will cost 5 times as much as 4 4 carriages. 4 carriages will cost 4 times $346, or $1384, and 20 carriages >st of 4 carriages. wiR ^ g times ^ mu ^ ^ 4 carriageR> or 5 times $1384, which is $6920. $ 6 9 2 cost of 20carriages. Hence, 20 carriages will cost $6920. RULE. Multiply the multiplicand by one factor of the plier, the product thus obtained by another factor, and so continue until all the factors have been used successively as multipliers. The last product will be the product sought. Multiply in same manner, using the factors of the multiplier : 6. 425 by 32; by 36; by 48; by 72. 7. 1824 by 56; by 27; by 45; by 108. 8. What will be the cost of 35 cows at $64 each ? 9/ What will 21 cords of wood cost at $5.35 a cord? 10. What will 72 yoke of oxen cost at $168 per yoke? 11. What will 36 boxes of lemons cost at $6.25 per box? 12. What will 48 acres of land cost at $46 per acre? 13. What will 24 paintings cost at $55 each? 14. What will 45 cases of boots cost at $36 a case ? 15. What will 56 barrels of salt cost at $2.35 a barrel? DIVISION BY FACTORS. 124. 1. What are the factors of 32? 25? 64? 96? 2. If a number is divided by 8, by what must the quotient be divided that the number may be divided by 16? 3. If a number is divided by 8 and the quotient by 6, by what is the number divided? 4. What factors may be used to divide a number by 36 ? 5. What factors may be used to divide a number by 48 ? FACTORING. 81 6. A miller put tip 500 pounds of hominy in packages con- taining 4 pounds each, and packed them in boxes containing 10 packages each. How many packages and how many boxes did he have ? 7, Divide 888 by 24, using factors. ANALYSIS. 24 is equal to 6 times 4. Hence to divide by 24 we may divide by 6 times 4. 888 -f- 4 = 222. But since we were to divide by 6 times 4, this quotient is 6 times too great, hence we must divide it by 6. 222 ~ 6 = 37 the true quotient. PROCESS. 4)888 6)222 37 8. Divide 5863 by 32, using factors. PROCESS. 4)5683 3 2)1420. 4)710 177, 3 + (2 X 8) = 19 trueRem. 1 7 7 if Quotient. mainder is 3 units, and the second, 2 eights, or 16 ; hence the entire remainder is 3 + 16, or 19, and the quotient is 177-g-f. ANALYSIS. 32 is equal to 4 X 2 X 4. Dividing 5683 by 4 gives a quotient of 1420 fours and 3 units remaining. Dividing 1420 fours by 2 gives a quotient ot 710 eights. Dividing 710 eights by 4 gives a quotient of 177 thirty-twos and 2 eights remainder. The first partial re- EULE. Divide the dividend* by one factor of the divisor, the quotient thus obtained by another factor, and so continue until all the factors have been used successively as divisors. If there be remainders, multiply each remainder by all the preced- ing divisors except the one that produced it. The sum of these prod- ucts ivill be the true remainder. 17. 3275 by 56. 18. 3276 by 27. 19. 4104 by 45. 20. 7304 by 24. Divide, using fact 9. 1704 by 24. 10. 4725 by 15. 11. 5740 by 28. 12. 1428 by 42. 6 ors: 13. 1288 by 56. 14. 3528 by 72. 15. 3824 by 32. 16. 2184 by 49. 82 PROPERTIES OF NUMBERS. 21. A wholesale grocer put up 1120 pounds of tea in 35-pound packages, containing 5-pound canisters. How many packages and canisters were there? 22. A paper manufacturer put up his paper so that each quire contained 4 packages of 6 sheets each. How many packages and quires were made up from 912 sheets? CANCELLATION. 125. 1. How many times is 2 times 5 contained in 4 times 5 ? 2 times 7 in 4 times 7 ? 2 times 9 in 4 times 9 ? 2 times 24 in 4 times 24? 2 times any number in 4 times that number? 2. How many times is 4 times 8 contained in 12 times 8? 4 times 25 in 12 times 25? 4 times 75 in 12 times 75? 4 times any number in 12 times the same number? 3. How many times is 6 X 48 contained in 24 X 48 ? 6X 144 in 24 X 144? 4. In determining the quotient, what numbers may be omitted from both dividend and divisor? 126. Cancellation is the process of shortening compu- tations by rejecting equal factors from the dividend and divisor. 127. PRINCIPLE. Rejecting equal factors from both dividend and divisor does not alter the value of the quotient. 1. Divide 66 times 36 by 24 times 11. PROCESS. ANALYSIS. We write 66X36 ^X^X^X3X3 the numbers as in divis- = - 9 ion, the dividend above. 24X11 0X^X;j the divisor below a line. Instead of multiplying 66 by 36 we resolve 66 into its factors 11 and 6, and 36 into its factors 4, 3 and 3, and in the divisor resolve 24 into the factors 6 and 4. CANCELLATION. 83 Cancelling equal factors from both dividend and divisor, which is the same as dividing both by the same number, and does not alter the value of the quotient, we have remaining in the dividend the factors 3 and 3, or 9, which is the quotient. 2. Divide 72 X 66 X 49 by 63 X 40 X 21. PROCESS. ANALYSIS. We write the num- g 22 y kers as before. Since 9 is a factor !72V$$V4$ 22 ^ k tn 72 and 63 it may be rejected * = = 4f . from both, leaving 8 instead of 72 in jjTX ' j* X $? 5 the dividend, and 7 instead of 63 in /f 5 fl the divisor. We next cancel 8 from 8 and 40, leaving 5 instead of 40 in the divisor. We next cancel 7 from 7 and 49, leaving 7 instead of 49 in the dividend, and 7 again from 7 and 21, leaving 3 instead of 21. Rejecting the factor 3 from both 66 and 3, there is left for a dividend 22, and for a divisor 5, which gives a quotient of 4f . RULE. Reject from the dividend and divisor all factors common to both 9 and then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor. When all the factors of both dividend and divisor are cancelled, the quotient is 1, for the dividend will then exactly contain the divisor once. EXAMPLES. Divide, using -cancellation : 3. 7 X 5 X 3 X 11 by 5 X 11 X 3. 4. 12 X 14 by 6 X 7 X 2. 5. 6 X 3 X 5 X 2 by 3 X 5 X 2 X 2. 6. 4 X 2 X 8 X 24 by 36 X 8 X 2. 7. 24 X 32 by 8 X 6 X 4. 8. 45 X 60 X 7 by 49 X 12 X 9. 9. 2X3X5X8X7 by 6X5X2X7. 10. 5 X 8 X 12 X 6 by 20 X 16 X 2. 11. 12 X 60 X 36 X 35 by 7 X 30 X 18 X 24. 12. 30 X 49 X 64 X 25 by 35 X 15 X 24. 84 PKOPERTIES OF NUMBERS. 13. Divide the product of 26 times 18 times 35, by 78 times 30. 14. Find the quotient of 99 times 360 times 365, divided by 11 times 72. 15. Find the quotient of 175 X 28 X 72 times 363, divided by 12 X 11 X 9. 16. Four farms containing 80 acres each, worth $65 per acre, were exchanged for 5 farms containing 95 acres each. What was the value per acre of the farms received in exchange? 17. A farmer buys 3 pieces of muslin each containing 44 yards at 11 cents a yard, and pays for it in wheat at $2 per bushel. How many bushels are required? 18. A merchant bought 13 tubs of butter, each containing 39 pounds, at 32 cents a pound, paying for it in 4 patterns of silk of 13 yards each. How much was the silk a yard? COMMON DIVISORS. 128. 1. What numbers will exactly divide 12? 15? 20? 2. What numbers will exactly divide both 12 and 15? 15 and 20? 24 and 48? 63 and 72? 3. What numbers will exactly divide both 12 and 24? What is the largest number that will exactly 'divide them? 4. What is the largest number that will exactly divide both 15 and 30? 16 and 32? 16 and 24? 24 and 32? 5. Name all the divisors common to 15 and 30. 6. Name all the prime divisors or factors common to 15 and 30? 7. How is the greatest divisor common to 15 and 30 found from the prime factors of those numbers? 8. What is the greatest divisor common to 24 and 30? 9. How is the greatest divisor common to 24 and 30 obtained from the prime factors of those numbers ? COMMON DIVISORS. 85 DEFINITIONS. 129. A Common Divisor of two or more numbers is an exact divisor of each of them. Thus, 6 is a common divisor of 12, 24, 48 ; 8 of 16, 24 and 64. 130. The Greatest Common Divisor of two or more numbers is the greatest number that is an exact divisor of each of them. Thus, 24 is the greatest common divisor of 24 and 48. 131. When numbers have no common divisor they are said to be Prime to each other. Thus, 7, 8 and 9 are prime to each other. A common divisor is sometimes called a common measure and the greatest common divisor the greatest common measure. 132. PRINCIPLE. The greatest common divisor of two or more numbers is the product of all their common prime factors. 1. What is the greatest common divisor of 45, 60, and 75? IST PROCESS. ANALYSIS. Since the greatest coni- 45 = 3v3v5 mon divisor is equal to the product of n f\ _ c> \/ c> \/ > ^/ K a ll the prime factors common to the O U L X -A X d X i ^i ^ given numbers, we separate the num- ' ** = ^ ' bers into their prime factors. The only 3 X 5 = 15 prime factors common to all these num- bers are 3 and 5. Hence their product, 15, is the greatest common divisor of the given numbers. 2o PROCESS. ANALYSIS. 3 will divide each of the 3145 60 75 given numbers, and is therefore a factor -^ - - of the greatest common divisor. 5 will divide each of the resulting quotients 3 4 .5 and is therefore a factor of the greatest 3 X 5 = 15 common divisor. The quotients 3, 4, and 5, have no common divisor; there- fore 3 and 5 are the only factors of the greatest common divisor, 15. 86 PROPERTIES OF NUMBERS. RULE. Separate the numbers into their prime factors and find the product of all the common factors. Or, Divide the numbers by any common divisor, the resulti7ig quo- tients by another common divisor, and so continue to divide until quotients are obtained that have no common divisor. The product of the divisors will be the greatest common divisor. EXAMPLES. What is the greatest common divisor of 2. 12, 16, 20, 24? 3. 18, 27, 36, 45? 4. 24, 48, 60, 72? 5. 36, 60, 72, 66? 6. 48, 72, 96, 84? 7. 18, 81, 72, 54? 8. 32, 48, 80, 96? 9. 45, 63, 99, 81 ? 10. 35, 56, 84, 63? 133. When the numbers can not be factored readily, the following method is employed: 1. What is the greatest common divisor of 35 and 168 ? PROCESS. ANALYSIS. The greatest common 11. 12. 13. 14. 15. 16. 17. 18. 19. 16, 36, 30, 30, 14, 24, 33, 24, 42, 40, 60, 55, 54, 42, 28, 77, 72, 84, 72; 84, 85, 66, 63, 120, 143,- 120, 252, 88? 96? 90? 78? 91? 144? 154? 168? 294? 35)168(4 140 28)35(1 28 divisor can not be greater than the smaller number; therefore 35 will be the greatest common divisor if it is exactly contained in 168. By trial it is found that it is not an exact divisor 7)28(4 ^ 168, since there is a remainder of 2 Q 28. Therefore 35 is not the greatest common divisor. Since 168 and 140, which is 4 times 35, are each divisible by the greatest common divisor, their difference, 28, must contain the greatest common divisor; therefore the greatest common divisor can not be COMMON BIVISOES. 87 greater than 28. 28 will be the greatest common divisor if it is exactly contained in 35; since if it be contained in 35, it will be contained in 140, and in 28 plus 140, or 168. By trial we find that it is not an exact divisor of 35, for there is a remainder of 7. Therefore 28 is not the greatest common divisor. Since 28 and 35 are each divisible by the greatest common divisor, their difference, 7, must contain the greatest common divisor; therefore the greatest common divisor can not be greater than 7. 7 will be the greatest common divisor if it is exactly contained in 28 ; since if it be contained in itself and 28, it will be contained in their sum, 35, and also in 168, which is the sum of 28 and 4 times 35, or 140. By trial we find that it is an exact divisor of 28. Hence 7 is the greatest common divisor. RULE. Divide the greater number by the less and if there be a remainder divide the less number by it, then the preceding divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor. If more than two numbers are given, find the greatest common divisor of any two, then of this divisor and another of the given num- bers, and so on. The last divisor will be the greatest common divisor. Find the greatest common divisor of 2. 169 and 195. 3. 187 and 209. 4. 372 and 492. 5. 119 and 187. 6. 243 and 297. 7. 322 and 391. 8. 252 and 294. 9. 156 and 208. 10. 702 and 945. 11. 1029 and 1197. 12. 1666 and 1938. 13. 3596 and 3768. What is the greatest common divisor of 14. 672, 352, 992? 15. 714, 867, 1088? 16. 462, 759, 1155? 17. 630, 1134, 1386? 18. 462, 1764, 2562? 19. 7955, 8769, 6401 ? 20. In a village some of the walks are 56 inches wide, some 70 inches, and others 84 inches. What is the width of the widest flagging that will suit all the walks? 88 PROPERTIES OF NUMBERS. 21. A merchant has 60 pounds of tea of one kind, 75 pounds of another, and 100 pounds of another, which he wishes to put up in the largest possible equal packages with- out mixing the different kinds. How many pounds should be put in each package? 22. Mr. A. has 324 acres of land in one farm and 78 acres in another. He wishes to divide these into the largest possible fields of equal size. How many fields will there be, and how many acres in each field? MULTIPLES. 134. 1. What numbers less than 25 will exactly contain 4? 5? 6? 2. What numbers less than 25 will exactly contain both 4 and 6? 3. Name some numbers that are exactly divisible by 5. By 4. By both 5 and 4. 4. Name some numbers that are exactly divisible by 2. By 3. By 4. By 2 and 4. 5. What is the smallest number that is exactly divisible by each of the numbers 2, 3, and 4? 6. What is the least number that will contain 10 and 15? 7. What common prime factors have 10 and 15? What factor occurs in 10 that does not in 15? What factor is found in 15 that is not found in 10? 8. AVhat are all the different prime factors of 10 and 15? 9. How may the least number that will contain 10 and 15 be formed from their prime factors? What is the least number that will exactly contain 10. 3, 6 and 9? 11. 3, 5 and 6? 12. 4, 8 and 12? 13. 2, 3, 5 and 6? 14. 3, 4, 5 and 6? 15. 3, 6, 8 and 12? MULTIPLES. 89 DEFINITIONS. 135. A Multiple of a number is a number that will exactly contain it. A multiple of a number is obtained by multiplying the given number by some integer. 136. A Common Multiple of two or more numbers, is a number that will exactly contain each of them. 137. The Least Common Multiple of two or more numbers, is the least number that will exactly contain each of them. 138. PRINCIPLE. The least common multiple of two or more numbers is equal to the product of all the prime factors of the numbers, and no other factors. WRITTEN EXERCISES. 139. 1. Find the least common multiple of 30, 28 and 60? IST PROCESS. ANALYSIS. Since the least com- OQ 2 Y 3 "X 5 mon mu l t ipl e i s equal to the product ~ Q 9 \/ 9 v 7 of all the different prime factors of the numbers and no other factors, (Prin.) the numbers must be sepa- 2 X2X3X5X7 420 rated into their prime factors, and the product of all the different prime factors found. The prime factors of 60, the largest number, are 2, 2, 3 and 5. 28 contains a factor, 7, which is not found in 60. 60 con- tains all the factors of the other number, 30. Therefore all the different prime factors of the given numbers are 2, 2, 3, 5 and 7, and their product, 420, is the least common multiple. RULE. Separate the given numbers into their prime factors. Find the product of all the different prime factors, using each factor the greatest number of times it occurs in any of the given numbers. 90 PROPERTIES OF NUMBERS. Find the least common multiple of 2. 28, 32 and 64. 3. 36, 72 and 144. 4. 45, 70 and 90. 5. 12, 16, 18 and 24. 6. 15, 20, 25 and 30. 7. 18, 54, 90 and 180. What is the least common multiple of 8. 22, 55 and 77? 9. 48, 60 and 180? 10. 10, 64 and 96? 11. 81, 63 and 135? 12. 25, 70 and 95? 13. 12, 18, 24 and 96? 14. 32, 56, 64 and 80? 15. 14, 35, 50 and 28? 16. 33, 99, 84 and 135? 17. 17, 51, 65 and 121? PROCESS. 2 2 5 16 20 ' 30 8 10 15 4 5 15 4 1 3 2X2X5X4X3 = 18. Find the least common multiple of 16, 20, and 30. ANALYSIS. Since 2 is a prime fac- tor of each of the numbers, it is also a factor of the least common multiple. (Prin.) Dividing, there remain as the other factors of the numbers, 8, 10, and 15. 2 is a prime factor of 8 and 10, and is therefore a factor of the least common multiple. Divid- ing, there remain 4, 5, and 15. 5 is a prime factor of 5 and 15, and is therefore another factor of the least common multiple. Dividing, there remain 4, 1, and 3, which are prime to each other. Therefore the product of the factors 2, 2, 5, 4, and 3, will be the least common multiple. RULE. Write the given numbers in a horizontal line. Divide by any prime number that is an exact divisor of two or more of the given numbers, and write the quotients and undivided numbers in a line beneath. Thus continue to divide until the quotients and undivided num- bers are prime to each other. The product of the divisors, and the numbers in the last horizontal line, will be the least common multiple. MULTIPLES. 91 In finding the least common multiple, all numbers that are factors of other given numbers may be disregarded. Thus, the multiples of 8, 16, 32, 64, 80, and 128 are the same as the multiples of 80 and 128. EXAMPLES. Find the least common multiple of 19. 60, 40, 120 and 72. 20. 81, 45, 108 and 135. 21. 40, 60, 80 and 120. 22. 32, 36, 72 and 80. 23. 30, 75, 60 and 90. 24. 24, 44, 65 and 100. What is the least common multiple of 25. 8, 12, 16, 24 and 48? 26. 16, 20, 24, 32 and 40? 27. 25, 40, 75, 80 and 120? 28. 32, 45, 70, 64 and 90? 29. What is the smallest number that will exactly contain 16, 24, and 30? 30. How long must a box be that no room may be lost in packing in it books 6 inches, 8 inches, or 12 inches long? 31. A lady desires to purchase a piece of cloth that can be cut without waste, into parts 4, 5, or 6 yards long. How many yards must the piece contain? 32. I have a certain number of pennies which I can ar- range in either 4, 6, 8, 10, or 12 equal piles. What num- ber of pennies have I, if it is the least number that admits of such arrangement? 33. How many bushels will the smallest bin contain that can be emptied by taking out either 7 bushels, 10 bushels, or 30 bushels at a time ? 34. Four agents start from New York at the same time. The first makes his trip in 8 weeks, the second in 9 weeks, the third in 15 weeks, and the fourth in 20 weeks. How many weeks will pass by before they will again start out from New York together ? 92 PROPERTIES OF NUMBERS. 35. Three men walk around a circular island, the circum- ference of which is 360 miles. A \valks 15 miles a day, B 18 miles a day, and C 24 miles a day. If they start together and walk in the same direction, how many days will elapse before they will be together again ? 36. Divide 5 X 15 X 80 X 56 X 81 by 10 X 5 X 16 X 78. 37. If a man buys a lot whose sides measure respectively 48 feet, 60 feet, 96 feet and 108 feet, what will be the length of the longest boards which he can use to fence all the sides without cutting? 38. Find the greatest common divisor of 1744, 9564 and 8524. 39. What is the smallest number which can be divided by 250, 350, and 525 respectively, and leave a remainder of 25 ? 40. What is the greatest common divisor of 1728, 6912, and 8640? 41. A stock buyer wishes to invest the same amount of money in sheep at $3 each, hogs at 814 each, and cows at $21 each, as he does in beef cattle at $48 each. What is the smallest possible amount which he can invest in each ? 42. Jones Brothers & Co., of Cincinnati, O., received an order for a number of Lyman's Historical Charts. It was found that if the charts were packed in boxes containing either 24, 28, 32, or 36 charts each, there was a remainder of 9 each time, but if packed in boxes containing 25 each, there was no remainder. How many charts were ordered? 43. A dealer in real estate purchased 3 lots of land whose width on the street were respectively 152 rods, 288 rods, and 184 rods. What is the width of the largest lots of equal size which can be formed from them ? 44. Divide 3 X 5 X 20 X 10 X 3X 13 by 26 X 9 X 3 X 4. 45. Find the greatest common divisor of 2219, 4501, and 5964. 46. Divide 5x8x3x7X 28 X 99 by 11x4x7x5x4. 140. 1. When a line is divided into two equal parts, what is each part called? 2. When a line is divided into three equal parts, what is each part called? What are two of the parts called? 3. When a line is divided into four equals parts, what is each part called? What are two of the parts called ? called? one- fifth What are three of the parts 4. When any thing is divided into five equal parts, what is each part called? What are three parts called? What are four parts called? 5. When things are divided into 6, 7, 8, 9, 10, 15 equal parts respectively, what is each of the parts called? What are four of them called ? 6. How many halves are there in any thing? How many thirds? Fourths? Fifths? Sixths? Tenths? 7. If 10 marbles are separated into 5 equal groups, what part of the marbles will be in each group ? 8. How many are one-fifth of 10? Two-fifths? Three- fifths? 9. How many are one-sixth of 12? Two-sixths? (93) 94 COMMON FBACTIONS. DEFINITIONS. 141. A Fraction is one or more of the equal parts of a unit. 142. The Unit of a Fraction is the unit which is divided into equal parts. A fraction in which the unit has been divided into any number of equal parts is called a Common Fraction. A fraction in which the unit has been divided into tenths, hun- dredths, thousandths, etc., is called a Decimal Fraction. 143. A Fractional Unit is one of the equal parts into which a unit is divided. 144. Since a fraction is one or more of the equal parts of any thing, to express a fraction two numbers are necessary, one to express the number of equal parts into which the unit has been divided, the other to express how many make the fraction. These numbers are written one above the other with a horizontal line between them. 145. The Denominator is the number which shows into how many equal parts the unit is divided. It is written below the line. Thus, in the fraction f , 7 is the denominator. It shows that the unit of the fraction has been divided into 7 equal parts. 146. The Numerator is the number which shows how many fractional units form the fraction. It is written above the line. Thus, in the fraction \ , 5 is the numerator and shows how many fractional units form the fraction. 147. The numerator and denominator are called the Terms of a Fraction. COMMON FRACTIONS. 95 148. Fractional units are named from the number of parts into which the unit is divided. Thus, ^ is read one-sixth; ^, one-seventh. Fractions are read by naming the number and kind of frac- tional units. Thus, -f is read five-sixths ; ^, five twenty-firsts ; g-f , thirteen thirty-fifths. 149. Read the following: TB~ TT TT5" t 325 829 469, 385 639 "3~T5" "83T 986 Express by figures: 1. Three elevenths. Five thirteenths. Eight twenty- firsts. 2. Forty-eight fiftieths. Twenty-seven eighty-fifths. 3. Sixty forty-eighths. Fifty-seven ninety-ninths. 4. Forty-two eighty-sevenths. Thirty-nine ninety-thirds. 5. Seventy-four one-hundredths. Ninety-seven one-hun- dred-fifths. 6. Fifty-two seventy-eighths. Thirty-six eighty-fourths. 7. Two hundred three-hundred-ninetieths. 8. Seven hundred seventy-one eight-hundred-sixtieths. 9. Two hundred forty-nine three-hundredths. 10. Five hundred sixty-six seven-hundred-fiftieths. 11. One hundred eleven two-hundredths. 12. Four thousand six hundred thirty five-thousandths. Fractions are classified with reference to the relation of numerator and denominator thus: 150. A Proper Fraction is one in which the numer- ator is less than the denominator. Thus, }, f, -|f , etc., are proper fractions. The value of a proper fraction is therefore less than 1. 96 COMMON FRACTIONS. 151. An Improper Fraction is a fraction in which the numerator equals or exceeds the denominator. Thus, f, |, |f, are improper fractions. The value of an improper fraction is therefore 1 or more than 1. 152. A Mixed Number is a number expressed by an integer and a fraction. Thus, 2 1, 5J, are mixed numbers. Mixed numbers are read by naming the fraction after the whole number. Thus, 2f is read two and three-fourths. Fractions may be regarded as expressing unexecuted divis- ion. Thus, -ig 6 : is equal to 16 -f- 8 ; \ 5 - is read 15-7-3. 153. 1. Interpret the expression -f-. ANALYSIS. f represents 5 of 7 equal parts into which any thing is divided. It also represents one-seventh of five, and 5 divided by 7. It is read five-sevenths. In like manner interpret: 2. f 5. if 8. 196' 11. 3. -jSp 6. If 9. AV 12. 4- A- 7. T^- 10. iff- 13. REDUCTION. CASE I. 154. To reduce fractions to larger, or higher terms. 1. In -|- of an apple how many fourths are there? How many eighths ? 2. How many sixths are there in ? How many ninths? How are the terms of the fraction f obtained from those of l? ffromi? 3. How many eighths are there in J? How many twelfths? REDUCTION. 97 4. How do the terms of the fraction -f compare with the terms of the fraction ^? 5. In what equivalent fraction can -J- be expressed ? 6. How do the terms of the fraction -J- compare with those of -j^? 7. How are the terms of the fraction -^ obtained from those of ? 8. How are the terms of the fraction f obtained from ^? 9. How are the terms of the fraction f obtained from |-? 10. What then may be done to the terms of a fraction without changing "the value of the fraction ? 11. Change | to 24ths. 12. Change f to 16ths. 13. Change f to 24ths. 14. Change f to 12ths. 15. Change -& to 36ths. 16. Change f to 20ths. 17. Change % to 14ths. 18. Change f to 18ths. 155. Reduction of Fractions is the process of changing their form without changing their value. 156. A fraction is expressed in Larger or Higher Terms when its numerator and denominator are expressed by larger numbers. 157. PRINCIPLE. Multiplying both terms of afraetion by the same number, does not change the value of the fraction. WRITTEN EXERCISES. 1. Change ^ to forty-fifths. PROCESS. ANALYSIS. Since there are 45 forty-fifths in 4 5 -f- 1 5 ^=. 3 1, in T ^ there are 3 forty -fifths; and in T 7 ^ there 7 v q 91 Since the denominator of the required frac- 15 X 3 = 45 tion is 3 times that of the given fraction, we must multiply the terms of the fraction by 3. 98 COMMON FRACTIONS. RULE. Multiply the terms of the fraction by such a number as will change the given denominator to the required denominator. Reduce : 2. |f to 50ths. 3. U to 60ths. 4. |f to 70ths. 5. || to 84ths. 6. f to 40ths. 7. H to 54ths. 8. || to 66ths. 9. ff to 54ths. 10. - to 84ths. Reduce : 11. If to 120ths. 12. f| to 64ths. 13. || to 74ths. 14. ff to 210ths. 15. |f to 240ths. 16. ff to 225ths. 17. f| to 348ths. 18. f| to 558ths. 19. f f to 235ths. CASE II. 158. To reduce fractious to smaller, or lower terms. 1. How many fourths are there in |? How many in ^? 2. How many thirds are there in |? How many in ^? 3. How does the number of eighilis of any thing compare with the fourths? Thirds with sixths? Halves with' eights? 4. How do the terms of the fraction ^ compare with those of | ? How with those of ^? 5. How do the terms of the fraction | compare with those of | ? How with those of &? 6. How are the terms of the fraction ^ obtained from those of the fraction |? How from those of ^-? 7. How are the terms of the faction | obtained from f ? 8. What then may be done to the terms of a fraction without changing the value of the fraction? 9. Express T V; -^-, &, in smaller or lower terms. 10. Express -i-f , -|f , |4> ^ n smaller or lower terms. 11. Reduce ^-, f, f|, to smaller or lower terms. 12. Reduce ff , -j^j-, -j^, to smaller or lower terms. REDUCTION. 99 159. A fraction is expressed in Smaller 9 or Lower Terms \\hon its numerator and denominator are expressed in smaller numbers. 160. A fraction is expressed in the Smallest, Or JjOtrest Terms when its numerator and denominator have no common divisor. 161. PRINCIPLE. Dividing both terms of a fraction by the same member does not change the value of the fraction. WRITTEN EXERCISES. 162. 1. Change -|| to an equivalent fraction expressed in its smallest, or lowest terms. PROCESS. ANALYSIS. Since the denominator of 32 _4 48 ~4 8 2 the required fraction is to be smaller than 7^ = 17 that of the given fraction, we may obtain an equivalent fraction having smaller Or, terms, by dividing the terms of the given Q9 Q9 1fi 9 fraction by any exact divisor, as 4 (Prin.), '. and the terms of the resulting fraction by 4. 48 48 -r- 16 3 We thus obtain the fraction -f , whose terms have no common divisor. The fraction is therefore in its smallest terms. Or, Since fractions are in their smallest terms when their numerator and denominator have no common divisor, to reduce them to their smallest terms we may divide both terms by their greatest common divisor. RULE. Divide the numerator and denominator by any common divisor, and continue to divide thus until the terms have no common divisor, Or, Divide both terms by their greatest common divisor. 2. Reduce ^f , $$, |-f , $--0-, to their smallest terms. 3. Reduce f|, ffo, |, }-, to their smallest terms. 100 COMMON FRACTIONS. Keduce to their smallest, or lowest terms: 4 n 6. -ift. 7. 9. m- 10. B$. 11. &* 12. 13. 14. 15. 17. 18. ffff. 19. 20. 21 1710 22. ^ffv 23. CASE III. 163* To reduce integers or mixed numbers to improper fractions. 1. How many halves are there in 1 apple? In 4 apples? In 6 apples? 2. How many thirds are there in 1 orange? In 3 oranges? In 5 oranges? 3. How many fourths are there in 2? In 3? In 4? 4. How many fifths are there in 3? In 4? In 6? 5. How many fourths are there in 1^? In 2|? 6. How many thirds are there in 2|? In 3J ? In 6|? WRITTEN EXERCISES. 164. 1. Reduce 8f to sevenths. PROCESS. ANALYSIS. Since in 1 there are 7 g __ .56. sevenths, in 8 there are 8 times 7 sevenths, or - 5 7 6 -; and in 8 + f- there are RULE. Multiply the integers by the given denominator, to this product add the numerator of the fractional part, if there be any, and write the result over the given denominator. 2. Change 5^ to fourths. 3. Change 15 to fifths. 4. Reduce 13^- to sixths. 5. Reduce 18^ to elevenths. REDUCTION. 6. Change 5^- to ninths. To eighteenths. 7. Change 6^- to twelfths. To twenty-fourths. 8. Change 8 T 3 to fourteenths. To forty-seconds. 9. Eeduce 9^ to twentieths. To sixtieths. Reduce to improper fractions: 10. 13f. 14. 25f 18. 421ff. 22. 867^- 11. 12f 15. 29-fr. 19. 540i|. 23. 904Jf. 12. 18&. 16. 37||. 20. 763f 24. 314^. 13. 23f. 17. 56ff 21. 419jfr. OF: 791 1 2 ^U. i ^/J.-T-2^" CASE IV. 165. To reduce improper fractions to integers or mixed numbers. 1. How many days are there in 6 half-days? In 8 half- days? In 14 half-days? 2. How many yards are there in 9 thirds of a yard ? In 15 thirds? In 18 thirds? 3. If a boy pick 1 bushel of peaches per hour, how many bushels can he pick in 10 hours? How many are 10 halves ? 4. If a man can earn ^ of a dollar per hour, how much can he earn in 12 hours? How many are 12 fourths? 5. How many units are there in 6. How many dollars are there in WRITTEN EXERCISES. 166. 1. Reduce to a mixed number. ANALYSIS. Since 7 sevenths equal 1 unit, 123 sevenths are equal to as many units as 7 sevenths are contained , or 17f times. Therefore if3. = 17f PROCESS. 12_3 __ 123 _^_ 7 174 times in RULE. Divide the numerator by the denominator. 105 N FRACTIONS. 2. Change S- 1 ^/ to dollars. 3. Change f pounds to pounds. 4. Change -^- ounces to ounces. 5. Change if- 5 - to a mixed number. 6. Change ^- to a mixed number. 7. Change -^ to a mixed number. Keduce to integers or mixed numbers : 8. 9. 10. 11. 12. 13. 14. 15. TO- 16. 17. 18. 19. 20. 21. 22. 23- Wo 4 /- CASE V. 167. To reduce dissimilar fractions to similar frac- tions. 1. How many fourths are there in \ of an orange? 2. How many sixths of a field are there in ^ of a field? 3. How many eighths in f ? How many ninths in f ? 4. Express each of the fractions f , f , and f as twelfths. 5. Express each of the fractions ^ and f as twentieths. 6. If ^ is divided into 3 equal parts, how large is each part? 7. If i is divided into 2 equal parts, how large is each part? 8. When \ and \ are divided into equal parts, what parts are common to both ? 9. When \ and \ are divided into equal parts, what parts are common to both ? 10. What equal parts are common to both \ and -|-? 11. When i, ^ and ^ are divided into equal parts, what parts are common to all? / 12. Change , 1, 1, to equivalent fractions having the same fractional unit. Express the resulting fractions in equivalent fractions having their least common denominator. REDUCTION. 103 Keduce to fractious having the same fractional unit : 11. | and f. 14. f and T \. 17. -^ and 12. | and T V 15. and -A?. O \. A 18. y\ and 13. | and f. o t> 16. f and ^. 19. T 5 2 and 168. Similar Fractions are those that have the same fractional unit. 169. Dissimilar Fractions are those that have not the same fractional unit. 170. Similar fractions have a Common Denomi- nator. 171. When similar fractions are expressed in their small- est terms they have their Least Common Denomi- nator. 172. PRINCIPLES. 1. A common denominator of two or more fractions is a common multiple of their denominators. 2. The least common denominator of tivo or more fractions is the least common multiple of their denominators. WRITTEN EXERCISES. 173. 1. Reduce f and -| to similar fractions. PROCESS. ANALYSIS. Since similar fractions have a 3. _ - 3. x g - 14. common denominator, to make these fractions similar we must change them to equivalent IT ><4~32 fractions having a common denominator. Since a common denominator of two or more fractions is a common multiple of their denominators (Prin.), we find a common multiple of the denominators 4 and 8, which is 32. We then multiply the terms of each fraction by such a number as will change the fraction to thirty-seconds. 104 COMMON FE ACTIONS. 2. Reduce f , f and -f to similar fractions having their least common denominator. PROCESS. ANALYSIS. The least common denomina- __ 2. x 4 __ ^g, tor of several fractions is the least common multiple of their denominators (Prin. 2); 1" x 3 = T2 therefore we find the least common multiple I = * 2 _ ii of 3, 4, and 6, which is 12. We then multiply the terms of each frac- tion by such a number as will change it to twelfths, or to a fraction whose denominator is 12. Or, Since 1 is equal to f , | is equal to J of ^f , or T 4 ^, and f are equal to 2 times T 4 2, or T 8 ^, etc. RULE. Find the common, or least common multiple of the denominators for a common, or least common denominator. Divide this denominator by the denominator of each fraction and multiply both terms of the fraction by the quotient Keduce all mixed numbers to improper fractions and all fractions to their smallest terms. Change the following to similar fractions having their least common denominator : 3. * 5 20' "3TP " T> T8> "35"' 7. f , f , A- 8. |, A, A- " "5"' 2~0> 2T* 10. A. if. If 11. 3f, If, #. 19 13 14 36 JLZ;. o 5 j 5> 90* 13. H, if. if 14. M , A. ADDITION. 1. James has 2 fifths of a dollar, and his brother has 4 fifths of a dollar. How many fifths have both ? 2. George spent $f on Monday, and $f on Tuesday. How much did he spend in both days ? How many sevenths are f and ? ADDITION. 105 3. Mr. A. sold \ of his farm at one time, and f at another time. What part of it did he sell ? What is the sum of -J- and |? 4. James caught a fish in the morning that weighed f of a pound, and another in the afternoon that weighed f of a pound. What did both weigh? What is the sum of f and f ? 5. Marian gave $J for a book and $^ for some w r riting paper. How much did she pay for both? What is the sum of and i? 6. Ella gave f of her apple to a poor beggar and Julia gave him \ of hers. How many fourths did he receive? What is the sum of f and \ ? 7. I bought \ of an acre of ground for a site for a house, and \ of an acre for a site for a barn. How much land did I buy? What is the sum of % and ? Of \ and J? Of f 8. Mr. A. gave $^- to one man and $-| to another. How much did he give to both? 9. A merchant sold f of a bushel of clover seed to one farmer, and f of a bushel to another. How much did he sell to both ? 10. Sarah paid $f for eggs and $-J for butter. How much did both cost her ? 11. I paid $f for turnips and $f for squashes. How much did I pay for both ? 12. A merchant sold f of a yard of silk to one lady and -f of a yard to another. How much did he sell to both? 13. A boy earned $-| in the forenoon and 8-g- in the after- noon. How much did he earn that day ? 14. What is the sum of f and T y and T 7 ^ ? & and T 8 ? 15. What kind of fractions can be added without changing their form? 16. What must be done to dissimilar fractions before they can be added? How are dissimilar fractions made similar? 106 COMMON FRACTIONS. 175. PRINCIPLES. 1. Only similar fractions can be added. 2. Dissimilar fractions must be reduced to similar fractions before adding. WRITTEN EXERCISES. 176. 1. What is the sum off, f and f ? PROCESS. ANALYSIS. Since the frac- I + f + | = f + H + A = M ti0nS ^ "^ Simllar ' bef re adding we must change them to similar fractions, or equivalent fractions having a common denom- inator. The least common denominator of the given fractions is 36; and | 1^ 1 = 11, and % = -$. Hence the sum of the given fractions must be equal to the sum of f, f-J, and -f$, which is f f, or Iff. 2. What is the sum of 5}, 6| and 2f ? PROCESS. ANALYSIS. Since the numbers are composed of 51 5 6 both integers and fractions, we may add each sep- 2 g arately and unite the sums. Thus, the sum of the 1 2 fractional parts is f f , or \\\ ; the sum of the inte- 2| = 2^- gers is 13; and the sum of both, 14 JJ. RULE. Reduce the given fractions to similar fractions, add their numerators and write the sum over the common denomi- nator. When there are mixed numbers, or integers, add the fractions and integers separately and then add the results. If the sum be an improper fraction, reduce to an integral or mixed number. Find the sum 3. Of f, f , |, | and 4. Of i -I, \, i and 5. Of |, \ , |, | and 6. Of f , |, f $ and 7. Of f , H, |f and fi-. 8. Of 24-, 4, 3^ and 5f . 9. Of 27 T V 8f and 40f . 10. Of 13, 15f and 20|i. SUBTRACTION. 107 Add the following: 11 4. 15. 18. _5 7 7' 21' 14' 28' 35* 12. 4f , 5|, 8, 2f, 7|, 4f. 13. 9t, 7|, 8| 14. 7 T %, 8, 15- ft, tf , 2ft, 3A, 16. 3f , 4&, 6, 17. A farmer received $18f for hay, $65f for a cow, and $161f for a horse. How much did he receive for all? 18. A man earns $67f per month, and each of his two sons $23f per month. How much do all earn per month? 19. A pedestrian walked 45| miles on Monday, 47f on Tuesday, 50f on Wednesday. How far did he walk? 20. A has 5J acres of land, B has lOf acres more than A, C has as much as both A and B. How many acres have B and C together? SUBTRACTION. 177. 1. Mary earned 5 ninths of a dollar and spent 2 ninths. How many ninths of a dollar had she left? 2. Mr. A. owning ^ of a flouring mill, sold |- of it. How many sevenths did he then own? 3. From -f subtract f . From f subtract -|. From -^ subtract -f^. 4. From \\ subtract -f%. From T 8 -g- subtract -^. 5. Mr. B. owned a lot containing |- of an acre. How much had he left after selling \ of an acre? 6. A boy paid $- for a whip, but sold it after a time for %\. How much did he lose? 7. Find the difference between \ and \. \ and -|. 8. What kind of fractions can be subtracted without changing their form? 9. What must be done to dissimilar fractions before they can be subtracted? How are dissimilar fractions made similar? 108 COMMON FRACTIONS. 178. PRINCIPLES. 1. Only similar fractions can be subtracted. 2. Dissimilar fractions must be reduced to similar fractions before subtracting. WRITTEN EXERCISES. 179. 1. What is the difference between and -f ? PROCESS. ANALYSIS. Since the fractions are not 1 1 & 33 8__ similar, before subtracting we must change them to similar fractions. ~3lT~ 3~6 The common denominator of the given fractions is 36; and ij = f}, and J = -^-. Hence the difference between the given fractions is equal to the dif- ference between f f and ^ 8 B -, which is f f . 2. What is the difference between 23J and 4f . PROCESS. ANALYSIS. Since the numbers are 2 3 JL - 2 3 3 _ 2 2 J-- composed of both integers and frac- tions, we may subtract each sepa- rately. We first reduce the given fractions to similar fractions. Since we can not take |f from T 3 f , we unite with the T 3 2 1, or ||, taken from 23, making -J-f . Then 22^f 4}| = 18j 5 2, the remainder. RULE. Reduce the fractions to similar fractions. Find the difference of the numerators and write it over the com- mon denominator. When there are mixed numbers or integers, subtract the frac- tions and integers separately. Mixed numbers may be reduced to improper fractions and sub- tracted according to the first part of the rule. (3.) (4.) (5.) (6.) (7.) (8.) From |- -- T 6 3" M io" ir Take f ^ T V A A MULTIPLICATION. 1 09 9. From f take r V | 15. From 10 take ff. 10. From f take f. 11. From T 6 7 take T 3 -g-. 12. From / take T \. 13. From $ take ^. 14. From take |f 16. From 66f take 331 17. From. 210^ take 109f 18. From 112 take 75. 19. From 606| take 70J-. 20. From 589| take 67f . 21. If from a bin containing 506| tons of coal, 418|- tons are taken, how many tons still remain? 22. A lady having $25, paid $2| for a pair of gloves, $15f for a bonnet, and $3f for some lace. How much money had she left? 23. A man owned a farm of 412 acres. He sold three parcels of land from it, the first containing 60f acres, the second 45^ acres, and the third 116^- acres. How many acres did he sell, and how many had he remaining? 24. A clerk earned $50| per month. He paid $20f for board, $5f for washing, and $4f for other expenses. How much did he save per month? MULTIPLICATION. CASE I. 180. To multiply a fraction by an integer. 1. At $- a yard what will 3 yards of cambric cost? 2. If a man can earn $^ per hour, how much can he earn in 5 hours? How much can he earn in 8 hours? 3. James gave f of an apple to each of 5 children. How many apples did he give to all? How mtfch is 5 times f? 4. How many fifths are there in 6 times ? In 7 times f ? 5. If Mr. A. spends $2^- per day, how much will he spend in 5 days? How much in 10 days? 110 COMMON FRACTIONS. 6. How much is 2 times -f- ? How does the result com- pare with f ? How is it obtained from |? 7. In multiplying a fraction what part of the fraction do we multiply? 8. Multiply f by 2. f by 3. -- by 7. 9. Express 2 times f in smallest terms. How is this result obtained from the fraction f ? 10. In what other way then may we multiply a fraction? 11. How much is 3 times f ? 4 times f ? 6 times -f^. 12. How much is 5 times -|? 6 times -f ? 9 times f ? 13. How much is 4 times -|? 3 times T 5 ^? 5 times -J ? 14. How much is 4 times f ? 7 times ^-? 9 times 181. PRINCIPLE. Multiplying the numerator or dividing the denominator of a fraction by any number, multiplies the fraction by that number. WRITTEN EXERCISES. 1. Multiply |f by 6. PROCESS. Or, ANALYSIS. 6 times 13 twen- ty-fourths are 78 twenty -fourths, or 3. Or, Since dividing the denomi- nator multiplies the fraction (Prin.), 6 times Jf are ^, or 3. RULE. Multiply the numerator or divide the denominator by the integer. Multiply : Multiply : Multiply: 2. A by 5. 7. || by 7. 12. by 9. 3. A by 7. 8. A by 6. 13. if by 13. 4- A ^ 5. 9. Tfr by 8. 14. if by 14. 5. & by 3. 10. if by 3. 15. if by 18. 6. .tf by 17. 11. -Lf by 7. 16. M by 75. MULTIPLICATION. Ill 17. What is the value of a load of 17 bushels of apples at $1 a bushel? 18. If a boy earns $f per day, how much can he earn in 9 days? 19. At 87| a barrel what will 7 barrels of flour cost? PROCESS. ANALYSIS. In multiplying a mixed number, we mul- $ 7 f- tiply the fractional part and integer separately and add 7 the results. ~~^~I Thus, 7 times $ = $y = $5 J. 7 times $7 are $49, and 4 9 4 the sum of $49 and $5} is $54J. We may reduce the mixed number to an improper * fraction before multiplying. 20. If a man travel 21f miles per day, how far can he- travel in 4 days? 21. What will 13 yards of cloth cost at $6| a yard? 22. If a steamship sails 17^- miles an hour, how far can she sail in 9 hours? CASE II. 182. To multiply an integer by a fraction. 1. Henry had 6 rabbits and sold ^ of them to James. How many did he sell? 2. Jane had 16 cherries and gave \ of them to her sister. How many did she give to her sister? 3. How much is of $18? f of $18? 4. How much is | of 7 apples? 1 of 9 bushels? of 5 ounces? \ of 3 lemons? 5. How much is of 5? f of 5? f of 5? 6. How much is f of 7? f of 8? f of 12? 7. What is f of 36? f of 32? of 54? 8. How much is f of 35 tons ? f- of 49 horses ? f of 80? 183. PRINCIPLE. Multiplying by a fraction is taking such a part of a number as is imlicated by the fraction. 112 COMMON FRACTIONS. WRITTEN EXEJtCI SES. 1. Multiply 75 by f? PROCESS. ANALYSIS. To multiply 75 by f is 1 5 to find f of 75. | = 3 times \. of 7 5 X f fr'* $ x3 ' = 45 75 is 15 > and f = 3 times 15, or 45. Or, Since f = of 3, f of 75 = of 3 times 75, or -*Ap = 45. RULE. Multiply the integer by the numerator of the multiplier, and divide the product by the denominator. When possible use cancellation. 2. Multiply 9 by T V 3. Multiply 17 by &. 4. Multiply 12 by f. 5. Multiply 18 by ff. 6. Multiply 100 by ^. 7. Multiply 144 by ft. 8. Multiply 51 by T 5 T . 9. Multiply 79 by T or 3 times i Of f i of 7 = 7 Jy, and f are 3 times ^ or ^ = H- RULE. Reduce all integers and mixed numbers to improper fractions. Multiply the numerators together for the numerator of the product, and the denominators together for its denominator. 1. When possible use cancellation. 2. The word of between fractions is equivalent to the sign of multipli- cation. Such expressions are sometimes called compound fractions. Thus, J of | is equal to J X 3. Integers may be expressed in the form of fractions by writing 1 as a denominator. Thus, 4 may be written as J. 8 114 COMMON FKACTIOJSS. Multiply : 2. -h by f 3. | byf 4 ^ bv . Multiply : 5. | by f 6. T V by ^. 7. Jt by |. Multiply: 9- 10. Mby*. Find the value of 11. xx 14. is. 16. 17. || X U X 18. 19. 20. 21. 22. I X I x f x X x x X f x T 5 8- x if ft 23. Multiply f of 1| by f o PROCESS. f X | X | X T 9 2- X f = 4 , or 14 of 4. ANALYSIS. All the mixed Mmbera must be changed to improper fractions, and all whole numbers expressed in the fractional form. Multiplying and cancelling we have f, or 1 J. 24. Multiply | of f of 5 by -^ of | of 3. 25. Multiply f of -^ of 8 by f of T 9 <> of 15. 26. Multiply 3 times f by 4 times f of 7. 27. Multiply 5 times of 18 by f of 3 times 4- of 4. 28. What will be the cost of T 9 of a yard of cloth at $f a yard? 29. 16^- feet make a rod. How many feet are there in 5| rods? 30. A man who Owned f of a mill, sold -f- of his share. What part. of the mill did he sell? 31. How many yards of cloth are there in 12^ pieces of cloth, each piece containing 42f yards? 32. At $16f per ton how much can be realized from the sale of 4f tons of hay ? DIVISION. 115 33. What is the value of f of -fr of | of -^ of 15? 34. What is the value of ^ of 3f- of ff of 29? DIVISION. CASE I. 185. To divide a fraction by an integer. 1. Mr. Allen divided 3 fourths of a dollar equally between 3 boys. How much did each receive? 2. A man divided f of an acre into 3 equal lots. How large were they? 3. If 7 yards of cloth were bought for $$, what was the -cost per yard? 4. If 4 books cost $if , what is the cost of each? 5. Mr. Hurd put f of his crop of wheat on 3 wagons. What part of his crop was on each wagon? How much isf-3? _ 6. In dividing a fraction, what part of the fraction is divided? 7. A gentleman diyided ^ a barrel of flour equally between 2 people. What part of a barrel did each receive? How much is \ -r- 2 ? 8. 3 boys spent altogether $J. If each spent the same amount, what part of a dollar did each spend? How much is i-4-3? 9. Mr, Smith divided ^ of his farm into 3 equal fields. What part of his farm did each field contain ? How much is -J- -=- 3? How is the result obtained from the fraction ^? 10. In what other way, then, besides dividing the numera- tor may a fraction be divided ? 11. When a number is divided by 3 what part of it is found? 12. When a fraction is divided by 7 what part of it is found? 116 COMMON FRACTIONS. 13. What is 1 off? f--3? i 14. What is the value of -r- 2? Of f +- 3? Of -- ~- 9? 186. PRINCIPLE. Dividing the numerator or multiplying the denominator of a fraction by any number, divides the fraction by that number. WRITTEN EXERCISES. 1. Divide ff- by 6. PROCESS. ANALYSIS. Since dividing the nu- merator of a fraction divides the frac- tion, the fraction |f may be divided by 6 by dividing the numerator by 6. Or, Since multiplying the denominator divides the fraction, the fraction may be divided by 6 by multiplying the denominator by 6. The result by both processes is T 2 7 . RULE. Divide the numerator or multiply the denominator by the given integer. Or, TT ~^~ == TTx 6" I=r TT Divide : Divide : Divide : 2- f by 4. 6 ? 4 by 8. 10. |f by 21. 3. | by 8. 7. 45 by 15. 11. ft by 18. 4 12 2 9 by 3. 8- |I by 7. 12. 1! by 38. 5. || by 6. 981 T3 by 18. 13. It by 35. 14. Divide 16f by 5. PROCESS. Or, 5)16f ANALYSIS. We may reduce 16| to an improper fraction, and divide as before. Or, We may divide without reducing to an improper fraction. Thus, 5 is con- tained in 16 1, 3 times and a remainder of If, or |, and J divided by 5, equals -fa. Therefore the quotient is 3 2 V DIVISION. 117 Divide : 15. 17f by 6. 16. 25f by 4. Divide : 17. 38f by 8. 18. 24fV by 9. Divide : 19. 361 by 10. 20. 25f by 7. 21. A man gave each of his 5 sons an equal share of -f of his estate. What part of the whole did each receive? 22. 8 men built | of a mile of wall in 10 days. What part of a mile did each man build daily? 23. Mr. B. earned $35 by working 8 days. How much did he earn per day? 24. I paid $10| for 27 pounds of butter. What did I pay a pound? 25. A farmer realized $233^ for 21 bushels of clover seed. How much did he get per bushel? 26. Three men who have been partners in business gain $3216^. If they share equally, what will be each one's part of the gain ? CASE II. 187. To divide an integer by a fraction. 1. How many fourths are there in an apple? In 2 apples? 2. How many apples, at ^ cent each, can be bought for 2 cents? For 3 cents? For 10 cents? 3. At $-|- a yard, how many yards of cloth can be bought for $1? For $2? For $3? For $10? 4. At $-J- an ounce, how many ounces of nutmegs can be bought for $2? How many at $f an ounce can be bought for $2? 5. If a man can mow -f of a field per day, how long will it take him to mow the entire field? How long if he can mow |- per day? 6. At %\ a bushel, how many bushels of lime can be bought for $5? At $f a bushel how many bushels can be bought for5? 118 COMMON FRACTIONS. 7. When apples are worth f of a cent apiece, how many can be bought for 12 cents? How many times is f contained in 12? 8. How many pieces of cloth f of a yard long can be cut from a piece 9 yards long? How many times is f contained in 9? 9. How many times is f contained in 8? f in 9? 10. What is the quotient when 8 is divided by f? 10 by f ? 11. What is the value of 10--|? 7--|? 9---f? WRITTEN EXERCISES. 1. Divide 12 by f PROCESS. Or, ANALYSIS. \ is contained in 12 7 times 12, or 84 times; and f, one- sixth of 84 times, or 14 times. Or, Reducing 12 to sevenths, we have 84 sevenths. 6 sevenths are contained in 84 sevenths 14 times. RULE. Multiply the integer by the denominator of the fraction and divide the product by the numerator. Or, Reduce the dividend and the divisor to similar fractions, and divide the numerator of the dividend by the numerator of the divisor. Divide : Divide : Divide: 2. 18 by f. 10. 72 by f . 18. 51 by if 3. 64 by f . 11. 34 by f 19. 69 by |f. 4. 26 by \. 12. 15 by ft. 20. 70byf 5. 48 by f . 13. 49 by fi 21. 65 by |f. 6. 75 by i-f . 14. 91 by V 3 - 22. 90 by ff . 7. 31 by A- 15. 64 by if 23. 36 by i-f. 8. 45 by |f 16. 54 by ff. 24. 39 by ff . 9. 39 by T V 17. 39 by if 25. 24 by #. DIVISION. 119 26. Divide 24 by 3f PROCESS. ANALYSIS. When the divisor is a oj^ 2. mixed number we reduce it to an im- proper fraction and proceed according = 6 f to the rule. Thus, 3J = J; and 24 di- vided by J = 48, and by } = \ of 48, Or, 31 2_ 7 24 or 6f. Or, We may reduce 3J- to halves, and 48 24 to halves, and divide the numera- ~7T^ tor of the dividend by the numerator of the divisor. Divide the following by both processes : 27. 15 by 3$. 28. 23 by 6f. 29. 26 by 7f . 30. 35 by 6f 31. 39 by 8f. 32. 46 by 33. At $f a yard, how many yards of cloth can be bought for $15? 34. When corn is worth $ a bushel, how many bushels must a man sell to get money enough to pay $18 taxes? 35. A man invested $32 in peaches at $lf per basket. How many baskets did he purchase? 36. What is the price of hay, when 34 tons sell for $37? 37. Mr. Shaw paid $6 for 14| pounds of Java coffee. What did it cost him per pound? 38. It requires 656 pounds of meat to supply 34 soldiers 3-| weeks. How much does each soldier eat daily? CASE III. 188. To divide a fraction by a fraction. 1. How many fourths are there in 1? How many fifths? eighths ? tenths ? fifteenths ? twentieths ? 2. How many pieces of cloth -i- of a yard long, can be cut from a yard? 120 COMMON FRACTIONS. 3. If the pieces were J- of a yard long, how many would there be ? How does the number compare with the number when the pieces are -|- of a yard long? 4. If the pieces were f of a yard long, how many would there be? How does the number of pieces compare with the number when the pieces are ^ of a yard long? 5. How many times is ^ contained inl? f ? f ? |? f? 6. Since -g- is contained in 1, eight times, how many times will it be contained in |? What part of 8 times will it be contained in ^-? 7. Since f is contained in 1 -^ of 8 times, or f times, how many times will it be contained in ^? How many times in ^? 8. What is the value of 1--|? Of 1-f ? f 9. Into how many parts of 3 eighths of a dollar each, can 6 eighths of a dollar be divided ? 10. How many sacks containing ^ of a barrel each, can be filled from -f$ of a barrel of flour? How many times is ^ contained in T %? f in f ? ^ in ^ ? ^ in ^? 11. How many pine-apples at $J each, can be bought for $1 ? 12. How many times is ^ contained in ^? ^ in ^? f in f ? 13. How many times is ^ contained in 1? In ? In ^? 14. How many times is | contained in 1? In i? In f ? WRITTEN EXERCISES. 1. Divide ^ by f. PROCESS. ANALYSIS. ^ is contained in 1, 5 4 _._ & _ \/ 5. 20 times; and f is contained in 1, one- third of 5 times, or f times. Or, And since f is contained in 1, f 4 _._ _ 2o_ i _.2JL _ 2_o times, in f it will be contained f- of ^ f = if times. Or, \ is equal to f, and f is equal to f J. 21 thirty-fifths are contained in 20 thirty-fifths ff times. DIVISION. 121 RULE. Multiply the dividend by the divisor inverted. Or, Reduce the dividend and divisor to similar fractions and divide the numerator of the dividend by the numerator of the divisor. When possible use cancellation. Divide: Divide : Divide : 2. k- by f 3. MbyM- 4- H by &. 5- If by A. 6- if by A. 7. if by f . 8. ff by A- 9- M by A- 10. ifby T V H- Mbyif. 12- ft by A. 13. fi by &. 14. What is the quotient of f of f of 5| divided by f of ^ off? PROCESS. ANALYSIS. In the so- | of f of - 1 / _^_ I of -f- of |- = huion of examples like this, all mixed numbers i X 4 X "^ X 4 X -^ X -^ = 3 -^ = 6-i should be changed to im- proper fractions, and all fractions that are factors of the divisor, inverted, and the product found as in previous examples. 15. Divide f of f of 16 by f of f of 5J-. 16. Divide f of || of 5J by 4J times of 16. 17. Divide of f of f by f of f. 18. Divide J- of f of T 9 T by 5 times f of f . 19. Divide | of f of 15 by f of of 6. 20. Divide f of T \ of 22 by T \ of f of 16. 21. Divide -& of 3| of 6 by f of 6 times If. 22. Divide 8}- times i of 7^ by f of f of 5. 23. How many pieces of ribbon T 2 ^ of a yard in length, can be made from ^ of T 9 ^ of a yard ? 24. If a man spends $| per day for cigars, in how many days will he spend $17? 25. How many yards of cloth at $3f per yard can be bought for $317f ? 122 COMMON FBACTIONS. 26. At $f per bushel, how many bushels of potatoes can be bought for 61 7J? 27. If a family uses f of a barrel of flour a week, how long will 5} barrels last? 28. If a boy earns $f daily, how long will it take him to earn $3f ? 29. A certain number multiplied by f is equal* to f^-. AVhat is that number? 30. If a man can saw 1^ cords of wood in one day, how long will he require to saw 17^- cords? 31. If a horse eats 12^ bushels of oats in 5 weeks, how much does he eat in a day? 32. When wheat is selling at 81 per bushel, how many bushels can be bought for $3168? FRACTIONAL FORMS. 189. Expressions of unexecuted division are often written in the form of a fraction. 190. A fractional form having an integral denominator and a fractional numerator is called a Complex Frac- tion. 3 A 1 Thus, ~ and ~ are complex fractions, o y Expressions which have a fraction in the "denominator can not properly be regarded as Complex Fractions, though they are commonly classified as such. 1. Find the value of the fractional form 1C. PROCESS. ANALYSIS. -?- is an expression A 4 i JL = -=- . =4- X -f~ -fy f division, and is the same as f -f- 1, "8 3 which is equal to |f. FRACTIONAL RELATION OF NUMBERS. 123 Reduce to simple fractions: 6. A. 7.1. 3, 4. 5. A 9. 5* 10. 11. Li. A 12. H. 13. H 14. 16. 17. | of 9 of 3 T~ 3 FRACTIONAL RELATION OF NUMBERS. CASE I. 191. To find the relation of one number to another. 1. What part of 5 cents is 1 cent? 2 cents? 3 cents? 4 cents? 2. What part of 9 acres is 5 acres? 7 acres? 3 acres? 4 acres ? 3. What part of 4 apples is 1 apple? ^ of 1 apple? f of 1 apple? | of 1 apple? 4. What part of $5 is $2? $| ? *i ? $i ? $f ? 5. What part of $6 is $1? $? $|? $f ? 6. Henry had $5 and gave his brother $f . What part of his money did he give his brother? 7. James earned 17, and his brother $2. What part of the whole did each earn? PRINCIPLE. Only like numbers can have relation to each other. 8. What is the relation of 5 to 9? ANALYSIS. 1 is $ of 9, and 5 is 5 times of 9, or, f . Hence 5 is 4 of 9. 124 COMMON FRACTIONS. What is the relation 9. Of 7 to 21? 10. Of 12 to 16? 11. Of 10 to 28? 12. Of 9 to 18? 13. Of 8 to 32? 14. Of 32 to 48? 15. Of 15 to 24? 16. Of 14 to 35? 17. Of 18 to 81? 18. What is the relation of 4 to 2 ? ANALYSIS. 1 is J of 2, and f of 1 is f of \ of 2, or Hence f is T % of 2. of 2. What is the relation 19. Of | to 4? 20. Of % to 9? 21. Of | to 6? 22. Of | to 8? 23. Of | to 15? 24. Of to 25? 25. Of | to 6? 26. Of f to 18? 27. Of | to 12? 28. What is the relation of f to -f ? ANALYSIS. \ is \ of f, and 1 is 7 times \ of f, or J of f ; and since 1 is | of f, | of 1 is f of | of f , or Jf of f . Hence f is || of f- . What is the relation 29. Of | to |? 30. Of f to |? 31. Of to 4? 32. Of f to 33. Of | to | ? 34. Of f to f ? 35. Of 36. Of 37. Of to to to CASE II. 192. A number and its relation to another number given, to find the other number. 1. 2 cents are ^ of how many cents? ^ of how many cents? 2. 3 is ^ of what number? \ of what number? number? 3. 8 is ^ of what number? -| of what number? |- of what number ? of how many cents? of what REVIEW EXERCISES. 125 4 12 is -J- of what number? what number? 5. \ is ^ of what number? 6. f is ^ of what number? f of what number? of what number? of what number? f of 7. 24 is f of what number? ANALYSIS. Since 24 is f of a certain number, 1 fifth of the num- ber is J of 24, or 6; and since 6 is ^ of the number, the number must be 5 times 6, or 30. Hence 24 is f of 30. 8. 24 is | of what number ? 9. 18 is f of what number? 10. 24 is ^ of what number? 11. 45 is f of what number? 56 is f- of what number? 49 is -J of what number? 42 is f of what number ? 96 is of what number? 12. ^ is | of what number? ANALYSIS. Since f is f of a certain number, J of the number is J of f , or f ; and since f , or f. Hence y is of the number is f , the number must be 3 times of f . 13. -f^- is 4 of what number? -|f is - 14. J-f is f of what number? f| is 15. ff is T 7 3- of what number ? f| is 16. - is 9 - of what number ? is of what number? of what number ? of what number? of what number ? REVIEW EXERCISES. 193. 1. Mr. B. bought a barrel of flour for $7|, a cord of wood for S5-|, and gave the clerk a twenty-dollar bill. How much change should Mr. B. receive ? 2. A merchant bought 360 pounds of sugar at 11| cents a pound, 50 pounds of tea at 621 cents a pound. How much did he pay for both ? 3. If a man can cut in one day ^ of a field containing 7 acres of wheat, how many acres can he cut in ^ of a day ? 126 COMMON FRACTIONS. 4. What will be the cost of 3^- dozen eggs at 18| cents a dozen ? 5. If a man can hoe a field in 7^ days, how long will it take 3 men to hoe a field 2-| times as large ? 6. From a barrel of kerosene containing 41^ gallons it was estimated that -f leaked out. If I paid $6.15 for it, at what price per gallon must I sell the remainder to balance the loss sustained by leakage ? 7. James Henderson sold f of his farm of 155 acres to Mr. Paine, and Mr. Paine soon sold f of what he had bought, to Mr. Banker. How many acres did Mr. Banker buy ? 8. Mr. A. built a block of stores which cost him $3122 for brick, $1368^ for lumber,. $3258$ for labor, and $1325^ for other expenses. He sold the block for $10000. Did he gain or lose, and how much? 9. There are 272^ square feet in a square rod. How many square feet are there in f of a square rod ? 10. I sold a house and lot for $3215, which was If times what it cost me. How much did it cost? 11. How much will 8 carpenters earn in 6| days at $2-| per day? 12. If a man walks 3J miles per hour, in how many hours can he walk 30^ miles? 13. Mr. Jones left an estate valued at $19000. f of it was divided equally among 4 sons and the rest equally among 3 daughters. What was the share of each? 14. The price of maple sugar this year is only -J of what it was last year. How much more would I have received last year for 3140 pounds which I sold this year at $ .20 a pound? 15. Custom millers take \ of the quantity of grain as pay for grinding it. How many bushels must a man carry to the mill so that he may bring back 14 bushels of ground provender? 16. If $45 is f of my money, what part of it will that sum plus$4be? REVIEW EXERCISES. 127 17. A farmer had two fields in which he kept his sheep. In one there was -| of the whole number of sheep, and in the other there were 148 sheep. How many sheep had he? 18. A merchant exchanged 21 barrels of flour worth $7-f a barrel, for 24f cords of wood. What did the wood cost him per cord? 19. If I give A of my money, B of it, and C of it, what part of my money have I left? 20. Mr. A. owns f of a vessel valued at $18326. If he sells f of his share to Mr. B., what part of the whole will he have left? What part will Mr. B. have? What is the value of Mr. A.'s share? Of Mr. B.'s share? 21. After buying a suit of clothes for $60 I found I had f of my money left. How much had I at first? 22. A man sold \ of 3 cords of wood for f of $8|. How much did he receive for it per cord? 23. How many tons of hay will be required to keep 7 horses for 6 months, if 9 horses eat 16^ tons in that time? 24. If f of a farm is sold for $8516, what would be the worth of the whole at the same rate ? 25. A gentleman spent \ of his annual income traveling, and \ of the remainder in the purchase of books. The rest, which was $8526, he expended upon paintings and other works of art. What was his annual income? 26. Two men dug a ditch for $53; one man worked 3|- days and dug 14|- rods; the other worked as many days as the first dug rods per day. How much did each receive, if they shared in proportion to the time they worked? 27. Two brothers together own \ of a flouring mill valued at $13000. One owns -f as much as the other. What is the value of each one's share? 28. The loss caused by a fire was $3865. The sum was paid by an insurance company which insured the stock for -f of its value. What was the entire value of the stock? 128 COMMON FRACTIONS. 29. A can do a piece of work in 10 days. What part of it can he do in 1 day ? If B can do the same piece of work in 8 days, what part of it can he do in 1 day ? What part can both together do in 1 day? How many days would be re- quired for both to do the work? 30. If A can do a piece of work in 5 days and B in 8 days, how long will it take both to do it? 31. If A and B can do a piece of work in 10 days, and A can do it alone in 15 days, how long will it take B to do it? 32. A tree 124 feet high was broken in two pieces by falling, f of the length of the shorter piece, equaled y of the length of the longer piece. What was the length of each piece? 33. A man who had spent % his money and $-|- more, found that he had $21 left. How much money had he at first? 34. A, B and C can do a piece of work in 9 days. A can do it in 25 days, and B in 30 days. In what time can C do it? 35. Two ladders will together just reach the top of a build- ing 75 feet high. If the shorter ladder is |- the length of the longer one, what is the length of each? 36. There are two numbers whose sum is 140, one of which is f the other. What are the numbers? 37. In 1875 a merchant's profits were ^ of his receipts; in 1876 they decreased -|, which diminished his profits ^ of $2756i. What were his receipts in 1875? 38. A and B together had $5700. | of A's money was equal to f of B's. How much had each? 39. A man engaged to work a year for $240 and a suit of clothes. At the end of 9 months an equitable settlement was made by giving him $168 and the suit of clothes. What was the value of the clothes? 40. A and B can do a piece of work in 12 days. Assum- ing that A can do f- as much as B, how long will it take each to do it? DECIMAL FRACTIONS 194. 1. If an apple be divided into ten equal parts, what part of the apple will one of these parts be? Two parts? Five parts? 2. If one-tenth of an apple be divided into ten equal parts, what part of the apple will one of these parts be? Two parts? Whatpartoflis T Vof T y -&of T V? T 8 oOf T V 3. If one-hundredth of a dollar be divided into ten equal parts, what part of a dollar will one of these parts be ? Eight parts? What part of 1 is -^ of y-J^? 4. What part of one-tenth is one-hundredth ? Of 2 tenths is 2 hundredths? Of 3 tenths is 3 hundredths ? Of 9 tenths is 9 hundredths? 5. What part of one-hundredth is one-thousandth ? Of 2 hundredths is 2 thousandths? Of 8 hundredths is 8 thou- sandths ? 6. What are the divisons of any thing into tenths, hun- dredths, thousandths, ten-thousandths, etc., called? Atis. Decimal divisions. DEFINITIONS 195. A Decimal Fraction is one or more of the decimal divisions of a unit. The word decimal is derived from the Latin word decent, which signifies ten. 9 (129) 130 DECIMAL FRACTIONS. Decimal fractions, for the sake of brevity, are usually called decimals. 196. Since tenths are equal to ten times as many hun- dredths, and hundredths are equal to ten times as many thousandths, thousandths to ten times as many ten-thou- sandths, etc., it is evident that decimals have the same law of increase and decrease as integers, and that the denominator may therefore be indicated by the position of the figures. According to the decimal system of notation, figures de- crease in tenfold ratio in passing from left to right; therefore a figure at the right of units will express tentlis, at the right of tenths, hundredths, at the right of hundredtlis, tJiousandths, etc. , as is exhibited by the following expressions : a g *j 3 o s 8 a 2 a jg W ti P H B H 234 147 313 6 05 005 -234ft = 147y% 164 034 = 164yf^- 1 64 03400 164 T -|f 01 64 _LJL inn From this mode of expressing decimal fractions the follow- ing principles are deduced : 197. PRINCIPLES. 1. Decimals conform to the same prin- ciples of notation as integers. 2. Each decimal cipher prefixed to a decimal diminishes its value tenfold, since it removes each figure one place to the right. 3. Annexing ciphers to a decimal does not alter its value, since it does not change the place of any figure of the decimal. 4. The denominator of a decimal, when expressed, is 1 with as many ciphers annexed as tliere are orders in the decimal. DECIMAL FRACTIONS. 131 The Decimal Point is a period placed before the decimal. Thus, .6 represents T 6 ^; .54 represents -ffa* The decimal point is also called the Separatrix, since it is also used to separate integers from decimals. 198. A Pure Decimal Number is one which con- sists of decimals only; as .387. 199. A Mixed Decimal Number is one which con- sists of an integer and a decimal; as 46.3, which is equal to 46^. 200. A Complex Decimal is one which has a com- mon fraction at the right of the decimal; as .3-f, which is . equal to 3. 10 02. NUMEKATION TABLE. 03 CO ^ rf ^ H cj n Sc 93 02 i-M O2 O 02 O 'd 2 T5 O 02 T5 C c5 02 -j ^ 'd . _ 02 dredths. isandths. thousand O 3 T3 T5 02 rg J millionth 1 13 S T3 02 rg J IW rg 1 IB 1 g w fl O S s w 1 1 "S p M i- O H 1 C W 1 i O S M W H i 1 2 7 5 4 8 3 4 . 6 8 4 7 3 2 4 2 5 / INTEGERS. DECIMALS. By examining this table it will be seen that tenths occupy the first decimal place, hundredth the second, thousandths the third, ten-thousandths the fourth, hundred-thousandths the fifth, millionths the sixth, etc. Hence, The place occupied by any order of decimals is one less than that occupied by the corresponding order of integers. 132 DECIMAL FRACTIONS. 201. What order of decimals occupies 1st place? 4th place? 3d place? 7th place? 5th place? 2d place? 6th place? 10th place? 9th place? 8th place? 2d place? 3d place? What decimal place is occupied by hundredths? Tenths? Hundred-million ths? Thousandths? Ten-thousandths? Ten- millionths ? Millionths ? Billionths ? Hundred-thousandths ? EXAMPLES IN NUMERATION. 202. 1. Read the decimal 4.246. ANALYSIS. The figures of the decimal express 2 tenths, 4 hun- dredths and 6 thousandths, which, reduced to equivalent fractions having a common denominator, become 200 thousandths, 40 thou- sandths and 6 thousandths, or 246 thousandths. The whole expression is read 4 and 246 thousandths. RULE. Read the decimal as an integral number and give it the denomination of the right-hand figure. Read the following: 2. .684. 14. .6231. 26. 4.16. 3. .084. 15. .4896. 27. 5.8406. 4. .004. 16. .3893. 28. .60000. 5. 6.839. 17. 18.468. 29. .00006. 6. 68.39. 18. 23.8009. 30. .40508. 7. 683.9. 19. 649.3804. 31. 40.0004. 8. .00450. 20. .0020064. . 32. 4000.004. 9. 3,02304. 21. .4120465. 33. 518.6800. 10. .050600. 22. 6.932474. 34. 4000.129. 11. 4.00008. 23. 2.234006. 35. 80000.86. 12. .000000856. 24. 3.000600. 36. 8000.086. 13. 1.000003894. 25. 4.006006. 37. 800.0086. NOTATION. 133 EXAMPLES IN NOTATION. 203. 1. Express decimally forty-three thousandths. ANALYSIS. Since 43 thousandths are equal to 4 hundreclths and 3 thousandths, we write 3 in thousandths' place and 4 in hundredths' place, and as there are no tenths, in tenths' place. Hence, forty-three thousandths .043. RULE. Write the numerator of the decimal, prefix cipher* if necessary to indicate the denominator, and place the decimal point be/ore tenths. Express decimally: 2. Eight tenths. Nine tenths. Five tenths. 3. Two hundreclths. Eight hundreclths. Six hundredths. 4. Six thousandths. Four-hundrecl-two thousandths. 5. Nine ten-thousandths. Eight hundred ten-thousandths. 6. Seventeen hundreclths. Fifteen hundred-thousandths. 7. Forty-eight thousandths. Five hundred ten-millionths. 8. Ninety-three ten-thousandths. Ninety-three billionths. 9. Fifty-one hundred-thousandths. Fifty-one thousandths. 10. 107 millionths. 306 ten-millionths. 11. 3259 hundred-thousandths. 4268 hunclrecl-millionths. 12. 429 ten-millionths. 3842 hundred-thousandths. 13. 4300 billionths. 38496 hundrecl-billionths. 14. 85 hundred-millionths. 85 hundrecl-billionths. 15. Six thousand ten-thousandths. Five ten-billion ths. 16. Five and six-tenths. Eight and nine ten-thousandths. 17. Eighty ten-thousandths. Forty hundred-thousandths. Express decimally : 18. 20- 21. 4ft. 22. i o o o 25. 26. 27. . 10 0* 60- 29. T 134 DECIMAL FRACTIONS. In reading expressions of United States currency, the cents, mills, etc., may be read as decimals of a dollar. Thus, $4.7235 may be read 4 dollars 72 T 3 o 5 Q- cents, or REDUCTION. CASE I. 204. To reduce dissimilar decimals to similar deci- mals. 1. How many tenths of an apple are there in 1 apple ? How many hundredths in 10 apples? How many thousandths? 2. How many hundredths are there in 6 tenths ? How many thousandths? How many ten-thousandths? 3. Express 6 hundredths as thousandths. As ten-thou- sandths. As hundred-thousandths. As millionths. 4. Express 8 thousandths as ten-thousandths. As hun- dred-thousandths. As millionths. 205. PRINCIPLE. Annexing ciphers to a decimal does not alter its value. WRITTEN EXERCISES. 1. Eeduce .5, .36, .406 and 3.3109 to similar fractions. PROCESS. ANALYSIS. The lowest order of deci- 5 __. 5000 nials in the given numbers is ten-thou- sandths, and to reduce the decimals to similar decimals, we must change them .406 = .4060 all to ten-thousandths, or to other deci- 3 3109 = 3 3109 nials having an equal number of places. Since annexing ciphers to a decimal does not alter its value, we give to each number four decimal places by annexing ciphers, and this renders them similar. RULE. Give to all the given decimals the same number of decimal places by annexing ciphers. DEDUCTION. 135 2. Reduce .6, .75, .089, to similar decimals. 3. Keduce .15, .0406, .0035, .051, to similar decimals. 4. Keduce .0045, .3846, .51, .51040, to similar decimals. 5. Reduce 3.35, .345, to similar decimals. Reduce the following dissimilar decimals to similar decimals : 6. .0436, .04506, .82. 7. .05, 4.825, 3.6046. 8. .3854, .729, 8.053. 9. .8104, .0008, 8000.4. 10. *5, .5, .005, 50. 11. 3.5, .416, .34, 14. 12. .214, 8.3, .8, 4.6. 13. 8.1, 43, .68, 3.90. CASE II. 206. To reduce a decimal to a common fraction* 1. If 5 tenths be written as a common fraction what will be the numerator? What will be the denominator? 2. AVhat is the numerator and what the denominator of the decimal 18 hundredths, when expressed as a common fraction ? 3. Express the value of the decimal 50 hundredths, by a common fraction in its smallest terms. 4. Express by a common fraction in its smallest terms, the following decimals : 20 hundredths. 30 hundredths. 50 hun- dredths. 250 thousandths. 375 thousandths. WRITTEN EXERCISES. 1. Reduce .75 to its equivalent common fraction. PROCESS. ANALYSIS. .75 expressed as a common 7 5 _ _7_5 __ 3 fraction is T ^\, which, being reduced to its smallest terms, equals f. RULE. Omit the decimal point, supply the denominator, and reduce the fraction to its lowest terms. 136 DECIMAL FRACTIONS. Reduce the following decimals to equivalent common frac- tions in their smallest terms: 2. .054. 6. 4.0125. 10. .354. 14. .5675. 3. .03875. 7. .4355. 11. .00625. 15. 3.216. 4. .05625. 8. .0005. 12. .05375. 16. .4824. 5. .4375. 9. .5000. 13. .06506. 17. .005396. 18. Reduce .15^ to an equivalent common fraction. PROCESS. ANALYSIS. The expres- 1 C I Too i. Keducingthedenom- inator also to sevenths, the expression becomes J$, or / 5 3 Tr . Change the following to equivalent common fractions, or to mixed numbers : 19. .1% - 20. .33$. 21. .16|. 22. .87$. 23. .04f. 24. .037$. 25. .562$. 26. .003|. 27. .078^. 28. .0003$. 29. 2.756$. 30. 13.8i|. f ? f ? CASE III. 207. To reduce a common fraction to a decimal. 1. One half of an apple is equal to how many tenths of an apple ? 2. How many tenths are there in -J-? -f ? f ? 3. How many hundredths are there in 4. How many hundredths are there in 5. How many hundredths are there in divided by 2? How many in J? 6. How many hundredths are there in f-, or 400 hundredths divided by 5? How many in f ? 7. How many thousandths are there in -f , or 5000 thou- sandths divided by 8? How many in f ? How many in ? f ? or 100 hundredths DEDUCTION. 137 .625 Or, WRITTEN EXERCISES. 1. Reduce -f to an equivalent decimal. PROCESS. ANALYSIS. f is \ of 5, or 50 8)5 000 tenths ; and J of 50 tenths is 6 tenths anc^ 2 tenths remaining. 2 tenths are equal to 20 hun- dredths, and | of 20 hundredths is 2 hundredths and 4 hundredths remaining. 4 hundredths are equal to 40 thousandths, and J of 40 thousandths is 5 thousandths. Hence f is equal to 6 tenths + 2 hun- dredths + 5 thousandths, or .625. Or we may multiply both terms of the fraction by 1000 and divide the resulting terms by 8, and obtain the decimal '-fffat or -625. RULE. Annex ciphers to the numerator and divide by the de- nominator. Point off as many decimal places in the quotient as there are ciphers annexed. In many cases the division is not exact. In such instances the re- mainder may be expressed as a common fraction, or the sign -f- may be employed after the decimal to show that the result is not complete; thus: $ = ,166f, or .166 +. Reduce the following to equivalent decimals : 2. i. 8- 'A- 14. f. 20. A. 3. f. 9. A- 15. A- 21- if*- 4. f. 10. tf. 16. &. 22. f. 5. f. 11. . 17. &. 23. ^. 6. f. 12. . 18. ff. 24. T V 7. f. 13. ff 19- if*- 25. A- Change the following to the decimal form: 26. 15f. 27. 24|. 28. .821 29. 3.4. 30. .23f. 31. .62^. 32. .871 33. .43f. 34. 4.21|. 35. 37.5 T V 36. 20. Of. 37. .OOQtf. 138 DECIMAL FKACTIONS. ADDITION. 208. 1. What is the sum of & and &? & and &? .3 and .7? 2. What is the sum of T \ O Q- and yfo? ^ and -$y .12 and .20? 3. What is the sum of T ^ and T1I Vo ? TWIT and ^jfo? .005 and .043? 4. Find the sum of T 2 ^ and T |^. Of .5 and .06. .7 and .19. 5. Find the sum of .6, .31, .004. Of .5, .08 and .006. 209. PRINCIPLES. The principles are the same as for ad- dition of integers. WRITTEN EXERCISES. 1. What is the sum of .36, 2.136 and 4.5004? PROCESS. ANALYSIS. We write the numbers g g ___ 3600 so that units of the same order shall o 1 Q _o 1 Q ft stand in the same column, and add as L . 1 o Z.loOU ' . A . ,11- i A K(\(\ A A ^f\(\ A in mte g ers > separating the decimal part 4 ' 5UU4 == 4 ' 5004 of the sum from the integral part by 6 9964 6 9964 ^ e Decimal point. The decimals are made similar by annexing ciphers until all the decimals have the same number of places. It is not usual to make the decimals similar, for if they are written so that decimals of the same order stand in the same column it is unnecessary to supply the ciphers. RULE. The rule is the same as for addition of integers. What is the sum of 2. 4.15, 3.86 and .487? 3. 3.9, 4.84 and .0507? 4. .004, 5. 86 and 3. 05? 5. 6.843, 48.25 and 17.286? 6. .35, .046 and .00435? 7. 106, .106, 1.06 and 10. 6? SUBTRACTION. 139 8. What is the sum of $5.18, $3.09, $46. and $54.185? 9. Find the sum of $18.23, $12.08, $31.255 and $6.625. 10. Add $34.73, $206.357, $1200.18, $3816 and $137. 11. Express as decimals and add 6J, 3f, 5f, 6i and 9f. 12. A laborer earned $7.25 in one week, $7.12^ in another, $9.18f in another, and $8f in another. How much did he earn during that time? 13. What is the sum of 18 thousandths, 15 millipnths, 81 hundredths, 146 ten-thousandths, 834 hundred- thou sandths ? 14. What is the sum of 8 dollars 5 cents, 13 dollars 19 cents, 18 dollars 3 cents 8 mills, 25 dollars 37 cents 5 mills, 12f dollars, and -^ of a dollar? 15. Mr. A. paid the following bills for repairs upon his premises, viz : carpenter-w r ork, $381.45; plastering, $215.385; plumbing, $323.94; and other expenses, $181.57. How much did he pay for repairs ? 16. A farmer purchased cloth for $13f-, boots for $8^-, crockery for $10{^, and groceries for $15.49. How much did he pay for all his purchases ? SUBTRACTION. 210. 1. From T 5 o take T V From .9 take .5. 2. Find the difference between T -|-Q and yf^; -fo and .19 and .08. 3. Find the difference between y^Vfr an( ^ ToVo>' TTOTF Y^-; .007 and .005. 4. What is the difference between -$ and y|~o? .5 and .06? .7 and .09? 5. What is the difference between .16 and .03? .15 and .08? .45 and .3? 211. PRINCIPLES. The principles are the same as for the subtraction of integers. 140 DECIMAL FRACTIONS. WRITTEN EXERCISES. I. From 34.634 take 5.6857. PROCESS. ANALYSIS. We write the numbers so that units 34.6340 f tne same order shall stand in the same column, 5.6857 an d subtract as in integers, separating the decimal 28 9 4 8 3 P art ^ ^ ie rema ^ n( ^ er fr m tne integral part by a decimal point. Or, . In the first process the decimals are made simi- ~ P o < l ar ^7 annexing a cipher to the minuend. _ * ~ r In the second process, which is the one com- - monly employed, the cipher is not written, but we 28.9483 suppose it to be annexed. RULE. The rule is the same as for subtraction of integers. (2.) (3.) (4.) (5.) From 48.356 39.82 $43.25 $118.375 Subtract 23.453 13.856 $18.375 $ 43.50 6. What is the difference between .7134 and .50645? 7. What is the difference between 8.34 and 6.3168? 8. What is the difference between 100 and .03846? 9. From 84 millionths take 84 ten-millionths. 10. From 80 thousand take 80 thousandths. II. From 29 dollars 3 cents take 17 dollars 9 cents. 12. From 27 dollars 8 cents take 9 dollars 37 cents 5 mills. 13. If I spend $45. 89^- for merchandise, how much will I have left after paying for it with a fifty-dollar bill ? 14. A gentleman's income w r as $12384.16, and his expenses the same year were $9864.18. How much of his income was left? 15. The receipts of a reaper manufactory for the year 1876 were $1374837.64, and the expenditures $1298369.58. What was the surplus ? MULTIPLICATION. 141 MULTIPLICATION. 212. 1. What is the product of & X 2 ? -& X 3 ? .4X2? 2. How many decimal figures are there in the product of tenths by units? 3. What is the product of -fa X 4 ? yf ^ X 4 ? .04 X 2 ? 4. How many decimal figures are there in the product of hundredths by units? 5. What is the product of T 2 X T 3 o ? fVX T V .4 X .2? 6. How many decimal places are required to express the product of tenths multiplied by tenths ? 7. What is the product of ^ X T fo ? T B O X Tth) ? -4 X .02? 8. How many decimal places are required to express the product of tenths multiplied by hundredths ? Tenths by thou- sandths ? Hundredths by thousandths ? 9. If the multiplicand has two decimal places, and the mul- tiplier three, how many will there be in the product ? 10. How does the number of places required to express the product of two decimals compare with the number of decimal places in the factors? 213. PRINCIPLE. The product of two decimals contains as many decimal places as there are decimal places in both factors. WRITTEN EXERCISES. 1. Multiply .312 by .24. PROCESS. ANALYSIS. .31 2 X -24 = J$fo X T 2 o 4 <3 = iU o = .312 -07488. Hence .31 2 X .24 = .07488. Or, 9 4 We may multiply as if the numbers were integers; and since the multiplicand has three decimal places, 1248 an( ] the multiplier two places, the product must have 624 five places. (Prin.) Or, thousandths multiplied by 07488 hundredths give hundred-thousandths, the denomi- nation of the product. 142 DECIMAL FRACTIONS. RULE. Multiply as if the numbers were integers, and from the right of the product point off as many figures for decimals as there are decimal places in both factors. If the product does not contain as many figures as there are decimals in both factors the deficiency must be supplied by prefixing ciphers. Multiply: Multiply : 2. .65 by .34. 14. $2.75 by 8|. 3. .45 by 4.5. 15. $31.16 by 5|. 4. .436 by .46. 16. 34.165 by 3f. 5. 348 by .44. 17. 3.845 by 7.3. 6. 3.48 by .64. 18. $15.18 by .666f. 7. 34.8 by .74. 19. 500.15 by 5.36. 8. 8.75 by 8.5. 20. 37.856 by 30.04. 9. .579 by .035. 21. 70.05 by .0405. . 10. 486 by 3.75. 22. 63.18 by 2.308. 11. 2.48 by 2.37. 23. -64.032 by .0634. 12. 3.94 by 3.84. 24. 51.27 by 5.321. 13. 5384 by .0064. 25. | of .55 by of 6.5. 26. Multiply 4.639 by 100. PROCESS. 4.639 100 463.900 ANALYSIS. Since each removal of a figure one place to the left increases its value tenfold, the removal of the decimal point one place to the right multiplies by 10, and the removal of the point two places to the right multiplies by 100. Hence, RULE. To multiply by 1 with any number of ciphers annexed, remove the decimal point as many places to the right as there are ciphers annexed. 27. Multiply 384.64 by 100. By 10. By 1000. 28. Multiply 1.8465 by 100. By 1000. By 10000. 29. What will be the cost of 34.5 yards of cloth at per yard? 3.15 DIVISION. 143 30. When land is worth $137.18 per acre, how much must be paid for a farm of 38 acres? 31. Since 16.5 feet make a rod, how many feet are there in 23.7 rods? 32. What will be the cost of 9 houses at $3847.93 each? 33. When wheat is worth $1.62^ per bushel, what will 37.3 bushels cost? 34. What is the value of 57 barrels of flour at $8.37 a barrel ? 35. Mr. Orr sold 8 horses at an average price of $213.27 each. How much did he receive for them? 36. A lady made the following purchases, viz: 37 yards of bleached sheeting at $ .13^ per yard, 8 yards of velvet ribbon at $ .37| per yard, 27 yards of silk at $2.35 per yard. What was the entire cost of her purchases ? DIVISION. 214. 1. What is the product of .6 by 8? 2. 4.8 is the product of two factors, one of which is 8: what is the other factor? 3. What is the product of .6 by .8? 4. .48 is the product of two factors, one of which is .6: what is the other factor? 5. What is the product of .06 by .8? 6. .048 is the product of two factors, one of which is .06* what is the other factor? 7. What is the product of .06 by .08? 8. .0048 is the product of two factors, one of which is .06: what is the other factor? 9. How many decimal places are there in the quotient when tenths are divided by units ? Hundredths by tenths ? Thou- sandths by hundredths ? Ten-thousandths by hundredths ? 144 DECIMAL FRACTIONS. 10. How many decimal places are there in the product of any two factors? 11. If the product and one of two factors are given, how may the number of decimals in the other factor be found ? 12. How may the factor be found? 13. Since the factor sought will be the quotient, how many decimal places will there be in the quotient? 215. PRINCIPLE. The quotient will contain as many deci- mal places as the number of decimal places in the dividend ex- ceeds those in the divisor. WRITTEN EXERCISES. 1. Divide 8.88 by 2.4. PROCESS. ANALYSIS. 8.88 -~ 2.4 f J J -*-}$ = f ft 2.4)8.88(3.7 XJJ = tt = 3.7^ Or, 7 2 We divide as if the numbers were inte- gers ; and since the dividend has two deci- mal places, and the divisor one, the quotient will have one. (Prin.) RULE. Divide as if the numbers were integers, and from the right of the quotient point off as many figures for decimals as the number of decimal places in the dividend exceeds the number of those in the divisor. 1. If the quotient does not contain a sufficient number of decimal places the deficiency must be supplied by prefixing ciphers. 2. Before commencing the division, the number of decimal places in the dividend should be made at least equal to the number of deci- mal places in the divisor. 3. When there is a remainder after using all the figures of the divi- dend, annex decimal ciphers and continue the division. 4. For the ordinary purposes of business it is not necessary to carry the division further than to obtain four or five decimal figures in the quotient. DIVISION. 145 Divide : Divide : 2. 2.450 by 9.8. 9. .04905 by .327. 3. .00335 by .67. 10. 135.05 by .037. 4. 6.2512 by 3.7. 11. 687.50 by .025. 5. .05475 by 15. 12. 34.368 by .013. 6. 18.312 by .24. 13. .014582 by .0692. 7. 105.70 by 3.5. 14. 71.142 by .0071. 8. .11928 by .056. 15. .027538 by .0326. 16. Divide 423.68 by 100. PKOCESS. 100)423.68 ANALYSIS. Since each removal of a figure one place to the right decreases its value ten- fold, the removal of the decimal point one place to the left divides by 10, and the removal of 4.2368 the decimal point two places to the left divides by 100. Hence, RULE. To divide by 1 with any number of ciphers annexed, remove the decimal point as many places to the left as there are ciphers annexed. Divide : 17. 48.26 by 100. 18. 382.457 by 1000. 19. 13.8542 by 1000. Divide : 20. 4.897 by 100. 21. .06045 by 1000. 22. 3845.63 by 10000. 23. At $8.25 per ton, how much hay can be bought for $29.35? 24. At $.18 per dozen, how many dozen eggs can be bought for $32.40? 25. If I pay $106.40 for 35 hats, how much do they each cost me? 26. How many hogsheads of molasses, at $57.38 each, can be purchased for $1319.74? 27. How many stoves, at $21.35 each, can be bought for $789.95? 10 146 DECIMAL FRACTIONS. SHORT PROCESSES. 216. Many methods have been devised for abbreviating the processes of computation, among which the following are of much practical value : CASE I. 217. To multiply by a number a little less than a unit or the next higher order. 1. How much less than 10 is 9? Than 100 is 99? Than 1000 is 999? Than 100 is 98? Than 100 is 97? Than 100 is 96? 2. How much less than 10 times a number is 9 times the number? Than 100 times is 99 times? Than 1000 times is 999 times? Than 100 times is 97 times? Than 1000 times is 997 times? WRITTEN EXEMCISES. 1. Multiply 4685 by 97. PKOCESS. ANALYSIS. Ninety-seven 468500 = 100 times 4685 times a n " mb er is 100 times 14055 = 3 times 4685 the number minus 3 times the number. We annex two 454445= 97 times 4685 ciphers to the multiplicand, thus multiplying by 100, and then subtract from the product three times the multiplicand, leaving 97 times the multiplicand. Multiply : 2. 3856 by 99. 3. 4832 by 998. 4. 48567 by 999. 5. 89736 by 98. Multiply : 6. 346725 by 997. . 7. 486965 by 97. 8. 843256 by 96. 9. 586436 by 9996. I SHOUT PROCESSES. 147 10. What will be the cost of 385 bushels of corn at $ .97 per bushel? 11. How much must be paid for 1373 pounds of tea at $.96 per pound? 12. At $97 per acre, what will a farm of 139 acres cost? CASE II. 218. To multiply when one part of the multiplier is a factor of another part. 1. Multiply 4 by 8; 4 by 8 tens; 4 by 16 tens. 2. Multiply 7 by 6; 7 by 6 tens; 7 by 18 tens. 3. When 7 times a number is known, how may 21 times a number be found? 21 teris times? 21 hundreds times? WRITTEN EXERCISES. 1. Multiply 3684 by 124. PROCESS. ANALYSIS. The multiplicand may be regarded 3684 as com P ose d of 12 tens and 4 units, or 3 times as -^24 many tens as units. We therefore first multiply by : 4 units; and since there are 3 times as many tens 1 A 7 Q A as units, we multiply this product by 3, and write the result as tens by placing it one place to the right 456816 of units - Multiply : 2. 3825 by 63. 3.. 5973 by 93. 4. 8126 by 123. 5. 6924 by 213. 6. 14273 by 246. 7. 28653 by 328. 8. 68435 by 217. Multiply : 9. 3692 by 357. 10. 6384 by 248. 11. 4239 by 369. 12. 12783 by 189. 13. 36412 by 279. 14. 36485 by 2408. 15. 29753 by 3609! 148 DECIMAL, FKACTIONS. an aliquot Is 121? CASE III. 219. To multiply by a number which part of some higher unit. 1. What part of 10 is 2? Is 5? Is 2J? Is 3? 2. What part of 100 is 10? Is 20? Is 25? Is 50? Is 331? 3. What part of a dollar are 50 cents? 25 cents? 20 cents? 10 cents? 12| cents? 4. What part of a dollar are 33 cents? 16| cents? 220. An Aliquot Part of a number is such a part as will exactly divide the number. Thus, 5, 10, 12J, etc., are aliquot parts of 100. The aliquot parts of 10, commonly used, are : 5 = 1 of 10. | 3^1 of 10. | 21= \ of 10. The aliquot parts of 100, commonly used, are: 50 = | of 100. 25 = \ of 100. 20 = | of 100. 33 = of 100. 16f = | of 100. 12J = i of 100. 10 = i^ of 100. 8^ = t L f 100. 6^ = ^ of 100. Other parts of 100 are : 40 = | of 100. 37| = | of 100. 60 f of 100. 621 = | of 100. 80 = | of 100. 87f = | of 100. 66| = | of 100. 75 = f of 100. 431 = ^ of 100. WRITTEN EXERCISES. 1. Multiply 434 by 25. PROCESS. ANALYSIS. Since 25 is \ of 100, we may mul- 4)43400 tiply by 25 by first multiplying by 100, and then 10850 taking \ of the product. SHORT PROCESSES. 149 2. What will 85 yards of cloth cost at $ .33| per yard? ANALYSIS. At $1 per yard, the cloth would 3)85 cost $85; and at $.33J, or | of a dollar, it will 2 8 3 3 l cost i of $ 85 > or Multiply : 3. 688 by 12|. 4. 402 by 16|. 5. 5056 by 25. 6. 75630 by 33|. Multiply : 7. 8404 by 50. 8. 2160 by 37|. 9. 4236 by 66f . 10. 7288 by 75. 11. What will be the cost of 27 yards of cloth at $ .25 per yard? 12. When butter is worth 33^ cents a pound, what will 824 pounds be worth? 13. What will be the cost of 216 pounds of tea at $ .75 per pound ? 14. What will 287 bushels of oats cost at 37^- cents per bushel? 15. What will 394 bushels of potatoes cost at 62|- cents per bushel? 16. What is the value of 319 bushels of wheat at $1.37| per bushel? CASE IV. 221. To find the cost when the quantity and the price of 1OO or 1OOO are given. 1. When the cost of 100 articles is known, how can the cost of 500 be found? 600? 800? 900? 2. When the cost of 100 articles is given, how can the cost of 250 be found? 350? 750? 850? 950? 3. How many times 100 is 250? 275? 280? 285? 4. How many times 1000 is 3000? 3500? 3750? 4585? 150 DECIMAL FRACTIONS. WRITTEN EXERCISES. I. What will be the cost of 375 pounds of fish at $6.75 per 100 pounds? PROCESS. ANALYSIS. Since 100 pounds cost $6.75, 375 $ 6 7 5 pounds, or 3.75 times 100 pounds, will cost 3.75 g 75 times $6.75, or $25.31 {. Or, the price may be multiplied by the quantity, and the decimal point $25.3125 removed two places to the left in the product. The letters C and M are used instead of the words hundred and thousand, respectively. 2. How much will 6075 pounds of coal cost at $ .35 per hundred-weight ? 3. When shingles cost $4.75 per M, how much will 8609 shingles cost? 4. What is the price of a load of hay weighing 1925 pounds, at $9.50 per ton (2000 pounds)? 5. What is the cost of 16795 pounds of plaster at $4.50 per 100 pounds? 6. How much will 129765 laths cost at $2.75 per M? 7. What is the cost of 6975 bricks @ $3.25 per M? 8. What is the cost of 1825 pounds of iron @ $45 per ton? 9. How much must be paid for 6780 envelopes @ $2.75 per M? 10. What would be the cost of 550 pine-apples at $13.25 per C? II. What will be the cost of 1592 pounds of beef at $4.50 per hundred pounds? 12. What will 15000 pounds of coal cost at $7.50 a ton? 13. What will be the cost of 2294 pounds of broom-corn at $55 per ton? 14. What will be the cost of 1964 pounds of maple-sugar at $13.45 per hundred- weight? ACCOUNTS AND BILLS. 151 ACCOUNTS AND BILLS. 222. A Debt is an amount which one person owes to another. 223. A Credit is an amount which is due to a person, or a sum paid towards discharging a debt. 224. A Debtor is a party owing a debt. 225. A Creditor is a party to whom a debt is due. 226. An Account is a record of debts and credits be- tween parties doing business with each other. 227. The Balance of an Account is the difference between the amount of the debts and credits. 228. A Bill is a written statement given by the seller to the buyer, of the quantity and price of each article sold, and the amount of the whole. 229. The Footing of a Bill is the total cost of all the articles. 230. A bill is Receipted when the words Received Pay- ment are written at the bottom, and the creditor's name is signed either by himself or some authorized person. 231. The following abbreviations are in common use: @, At. Do, The same. Mdse., Merchandise. %> Account. Doz, Dozen. No, Number. Acc't, Account. Dr, Debtor. Pay't, Payment. Bal, Balance. Fr't, Freight, Pd, Paid. Bbl., Barrel. Hhd, Hogshead. Per, By. Bo't, Bought. Inst, This month. Rec'd, Received. Co, Company. Int., Interest. Yd, Yard. Or, Creditor. Lb, Pound. Yr, Year. 152 ^5) DECIMAL, FRACTIONS. v N ! 1 \ X ^ ^' ^ ^ X Xk ^ x \ ^ y V- 1 k SfeSi H ACCOUNTS AND BILLS. 153 Copy, fill out and find the footings of each of the following : (2.) KOCHESTER, March 1, 1877. MR. J. B. ADAMS, Bought of HOWE & ROGERS : 75 J yards of Carpeting . 37 yards of Drugget . . (fy $2.12J " 1 20 $ 8 Edge 5 Mats . . . . " 4.16 " 2 37 ^ 18 yards Oil-cloth. . . . . . " 1.08 9 yards Carpet Lining 3 Carpet-sweepers . . . 2 doz. Stair-rods .... . . .12J . . " 2.00 . . " 8.25 * Received Payment, HOWE & ROGERS. (3.) MEMPHIS, May 20, 1877. MR. GEORGE B. SHERMAN, To SAMUEL B. SMALLWOOD, Dr. To 37 bbl. Pork ....... $24.35 s " 127 bbl. Flour. ...... " 8.15 " 3 hhd. Molasses 169 gal. . . " 29 firkins Butter 2120 Ib. . . " 3 boxes Raisins .43 .31 " 4.65 " 5 bbl. Kerosene 207 gal. . . " 25 doz. cans Fruit " 3 packages Tobacco 318 Ib. . " 13 doz. Spices .18} 2.40 .45 " 1.10 $ Received Payment by note at 60 days, SAM'L B. SMALLWOOD. 154 DECIMAL FKACTIONS. (4.) MR. ERASTUS P. GATES, NEW YORK, April 1, 1877. To STUKDEVANT & Co., Dr. 1877. Jan. 9 To 3 Gold Watches $124.50, $61.24, $57.18 $ u 13 " 437 pwt. Gold Chains . . @ $1.15 Feb. 3 " 35 sets Plated Tea-service. " 43.10 " 15 17 a a a " 51. Mar. 8 11 5 Silver Pie-knives . . " 12. u 12 " 12 Plated Ice-pitchers . . " 12.50 Or. $ 1877. Jan. 24 By Cash $21 20 Feb. 10 Draft 327 50 18 " Mdse. returned 78 67 $ How much is still due Sturdevant & Co.? Make out in proper form and receipt the following: 5. Mrs. M. T. Dana bought of G. C. Smith & Co., 25 yd. of calico @ 10 cents, 37 yd. of sheeting @ 18^ cents, 2 pairs of gloves @ $1.50, 1 sun-umbrella @ $6.75, 5 yd. of Ham- burg edging @ 25 cents, 7 pairs of hose @ $ .85. 6. Mr. C. C. Lovell bought of R. P. Lawton 7568 feet of hemlock @ $12.75 per M, 8539 feet of pine flooring @ $23.50 per M, 5608 feet of clear pine @ $45 per M, 3815 feet of oak joists @ $32 per M, 7346 feet of ash flooring @ $34 per M. 7. Mr. George M. Line bought of Steele & Avery 15 reams of commercial note paper @ $1.25, 7500 envelopes @ $3.65 per M, 18 gross steel pens @ $ .75 per gross, 24 Ridpath's Histories @ $1.25, 9 Webster's Dictionaries @ $10.25. DECIMAL FRACTIONS. 155 REVIEW EXERCISES. 1. A farmer sold his butter at 34 cents a pound, and received for it $123.59. How many pounds did he sell? 2. A gallon of distilled water weighs 8.339 pounds. How much will 15^ gallons weigh? 3. A square rod contains 272^ square feet. How many square feet are there in 7|- square rods? 4. The best anthracite coal is said to weigh 55.32 pounds per cubic foot. How many cubic feet will weigh a ton of 2000 pounds? 5. The number of cubic inches in a bushel is 2150.42. How many cubic inches are there in 1000 bushels? 6. What is the quotient when 3 is divided by 3 thou- sandths? 7. What is the quotient when 300 is divided by 3000 hundred-millionths ? 8. A lumber merchant had 2182565 ft. of lumber. After selling .20, or 20 per cent., of it, he lost 15 per cent, of the remainder by fire. How many feet of lumber were burned? 9. What will 385 pounds of flour cost at $4.25 per hun- dred-weight? 10. At $ .111 p er pound, how many pounds of sugar can be bought for 131.25? 11. Bought 26 yards of broadcloth at $4.37| per yard, and paid for it in pork at $7.25 per hundred-weight. How much pork will it take to pay for the cloth? 12. If 15 tons of hay cost $125.25, what will 35 tons cost? 13. If Ridpath's histories retail at $1.25 each, what will be received for 350 sold at that rate? 14. When pork is selling at $6.25 per hundred-weight, how much can be bought for $325? 15. When 8000 is divided by .004, what is the quotient? 156 DECIMAL FRACTIONS. 16. When .0008 is divided by 40000, what is the quotient? 17. How many days must a laborer work at $1.37-^ per day, to pay for 8 cords of wood at $4.43f per cord? 18. A lady bought the following articles: 27 yards of silk at $2.75 per yard, 11 yards of lace at $6.37| per yard, 9 pairs of gloves at $2.15 per pair, 10 pairs of hose at $1.10 per pair. What was the amount of the purchase? 19. If a man earns $12^- per week, and spends $7f per week, in how many weeks can he save $500? 20. What is the value of 95150 bricks at $7.25 per M? 21. What is the value of a farm of 195 acres if 91 acres are worth $6688.50, and the remainder $1.12| per acre more? 22. A drover bought 375 sheep at $4.50 per head. He sold 200 of them at a loss of $ .20 per head, and gained enough on the rest to balance the loss. What did he get per head for the rest? 23. The expenses of conducting a business enterprise were .40 of the entire profits. If the profits were .15 of the value of the goods sold, how much was received from the sale of goods if the profits were $9000 more than the expenses? . ,-, (|-iV)X(3 + f) 24. Express as a decimal (1 , + , } + (3 _ lf } x 5 ' 25. A speculator bought 5000 bushels of corn at $ .65 per bushel. He sold .25 of it for $ .70 per bushel, and the re- mainder for such price that he realized a profit on the whole of $447.50. How much did he get per bushel for the re- mainder? . 26. The estimated value of Mr. A.'s farm was $6500. If he sold a portion of it, at its estimated value per acre, for $2275, what decimal part of the farm did he sell? 27. A, B and C divide 645^- bushels of wheat among them- selves. A takes .37^, B ^-, and C the remainder. How many bushels had each? DEFINITIONS. 232. A Concrete Number is a number used in con- nection with some specified thing. Thus, 5 books, 7 trees, 8 horses, are concrete numbers. 233. An Abstract Number is a number that is not used in connection with any specified thing. Thus, 5, 7, 8, are abstract numbers. 234. A Denominate Number is a concrete number in which the unit of measure is established by law or custom. Thus, 5 yards, 3 feet, 7 pounds, 3 ounces, are denominate numbers. 235. A Simple Denominate Number is a denom- inate number composed of units of the same denominations. Thus, 5 feet, 9 pounds, 3 miles, are simple denominate numbers. 236. A Compound Denominate Number is a denominate number composed of units of two or more denom- inations which are related to each other. Thus, 6 feet and 4 inches, 8 hours and 32 minutes, are compound denominate numbers. 237. A Standard Unit is a unit of measure from which the other units of the same kind may be derived. Thus, the yard is the standard unit from which all measures of length are formed; the Troy pound the standard unit of weight. (157) 158 DENOMINATE NUMBERS. 238. A Scale is the ratio by which numbers increase or decrease. Scales are either uniform or varying. MEASURES OF VALUE. 239. Money is the measure of value. It is also called Currency, and is of two kinds, viz: coin and paper money. 240. Coin or Specie is stamped pieces of metal having a value fixed by law. 241. Paper Money is notes and bills issued by the Government and banks, and authorized to be used as money. UNITED STATES MONEY. 242. The unit of United States or Federal money is the Dollar. TABLE. 10 Mills (m.) = 1 Cent . . . ct. 10 Cents = 1 Dime . . . d. 10 Dimes = 1 Dollar. . . $ 10 Dollars = 1 Eagle . . . E. $ d. ct. m. 1 = 10 = 100 = 1000 Scale Decimal. The coins of the United States are : The double-eagle, eagle, half-eagle, quarter-eagle, three- dollar piece, one-dollar piece. Silver: The dollar, half-dollar, quarter-dollar, the twenty-cent piece, the ten-cent piece. Nic kel : The five-cent piece and three-cent piece. Bronze.: The one-cent piece. There are various other coins of the United States in circulation, but they are not coined now. MEASUEES OF VALUE. 159 The denominations dimes and eagles are rarely used, the dimes being regarded as cents, and the eagles as dollars. No examples in Reduction of U. S. Money are given, because the pupil has been familiarized with the process from the beginning. CANADA MONEY. 243. The currency of Canada is decimal, and the table and denominations are the same as those of United States money. English money is still used to some extent. The coins of Canada, are, for the most part, of the same denomina- tions as those of the United States, except the gold coins, which are the sovereign and half-sovereign. ENGLISH OR STERLING MONEY. 244. English money is the currency of Great Britain. The unit is the Pound or Sovereign. TABLE. 4 Farthings (far.) = 1 Penny. . . d. 12 Pence = 1 Shilling . . s. OA 01 iv f 1 Pound, or) 20 Shillings = 1 . ! > ( 1 Sovereign J s. d. far. 1 = 20 = 240 = 960 Scale 4, 12, 20. 1. Farthings are commonly written as fractions of a penny. Thus, 7 pence 3 farthings is written 7|d.; 5 pence 1 farthing, 5^d. 2. The value of 1 or sovereign is $4.8665 in American gold. The coins of Great Britain in general use are Gold: Sovereign, half-sovereign, and guinea, which is equal to 21 shillings. Silver : The crown (equal to 5 shillings), half-crown, florin (equal to 2 shillings), shilling, six-penny and three-penny pieces. Copper: Penny, half-penny, and farthing. 160 DENOMINATE NUMBERS. REDUCTION DESCENDING. 245. 1. How many farthings are there in 2 pence? In 5 pence? In 7 pence? In 8 pence? In 6 pence? 2. How many pence are there in 2 shillings? In 5 shil- lings? In 7 shillings? In 8 shillings? In 6 shillings? 3. How many pence are there in 5s. ? In 5s. and 3d. ? In 7s. 4d.? In 4s. 5d.? In 6s. 8d.? 4. How many farthings are there in 5d. ? In 6d. 3 far. ? In 5id.? In6id.? InSfd.? InG^d.? InlOfd.? 5. How many shillings are there in 2 5s. ? In 3 5s. ? 246. Reduction of a denominate number is the process of changing it from one denomination to another without altering its value. 247. Reduction Descending is the process of chang- ing a denominate number to an equivalent number of a lower denomination. WRITTEN EXERCISES. 1. How many farthings are there in 3 5s. 6f d. ? PROCESS. ANALYSIS. Since in 1 pound -S ^s 6^-d there are 20 shillings, in 3 pounds ^ ^ there are 3 times 20 shillings, or 60 shillings; and 60 shillings + 5 shil- 65s. = 3 5 S. lings = 65 shillings. 1 2 Since in 1 shilling there are 12 786d 35s 6d pence, in 65 shillings there are 65 ^ times 12 pence, or 780 pence; and 780 pence + 6 pence = 786 pence. 3 1 47 far. = 3 5s. 6|d. Since in 1 penny there are 4 far- things, in 786 pence there are 3144 farthings; and 3144 farthings + 3 farthings = 3147 farthings. Hence in 3 5s. 6|d. there are 3147 farthings. REDUCTION DESCENDING. 161 2. How many pence are there in 2 10s. 6d. ? 3. How many shillings are there in 13 5s. ? 4. How many farthings are there in 4 6s. 5d. ? 5. How many pence are there in |-? PROCESS. ANALYSIS. Since in 1 pound 3 _ - 3. of 20 s = - 6 --S there are 20 shillings, in f of a pound there are f of 20 shil- 6/s. = Sf. of 12 d. = -^-pd. i ingSj or _6 7 o of a shilling. 720 j __ 102^d. Since in 1 shilling there are 12 pence, in sf- of a shilling there are - 6 ^ of 12 pence, or -S-f 9 - pence; and -^f 2 - pence = 102fd. Therefore in f there are 102fd. EULE. Multiply the number of the highest denomination given, by the number of units of the next lower denomination which is equal to one of the next higher, and to the product add the num- ber given of this lower denomination. Proceed in like manner with this and each successive result thus obtained, until the number is reduced to the denomination required. 6. How many pence are there in f ? 7. How many pence are there in ? 8. How many farthings are there in -f s. ? 9. How many pence are there in T 5 T ? 10. How many shillings are there in 5 6s.? How many farthings ? 11. Keduce 12s. 5d. 2 far. to farthings. 12. How many pence are there in 7 9s. 5d. ? 13. Eeduce 17s. 6fd. to farthings. 14. What is the value of f jn units of lower denomina- tions? 15. Find the number of farthings in 5 13s. 3d. 16. Reduce 35 6s. 8d. to pence. 17. Reduce 45 3s. 9|d. to farthings. 18. Reduce 29 18s. 5d. to farthings. 11 162 DENOMINATE NUMBERS. REDUCTION ASCENDING. 248. 1. How many pence are there in 12 farthings? In 16 farthings? In 20 farthings? 2. How many shillings are there in 24 pence? In 60 pence? 84 pence? 96 pence? 3. How many pounds are there in 40 shillings? In 60 shillings? In 120 shillings? 4. How many pounds sterling must be paid for 10 pairs of boots at 6 shillings a pair? 5. At 5 shillings each how many pounds sterling must be paid for 16 hats? For 20 hats? 6. Sold 8 pairs of skates at 5 shillings a pair. How many pounds sterling did I receive for them? Reduction Ascending is the process of changing a denominate number to an equivalent number of a higher denomination. WRITTEN EXERCISES. 1. How many pounds sterling are there in 7254 pence? PROCESS. ANALYSIS. Since 12 pence are 12)7254 equal to 1 shilling, there must be as many shillin S s in pence as 12 pence are contained 30 .... 4 times in that number. 12 pence n r~ A i ^ o A n i are contained in 7254 pence 604 7254 d. =30 4s. 6d. A . . Al . , *7- times with a remainder ot 6 pence, therefore 7254 pence are equal to 604s. 6d. Since 20 shillings are equal to 1 pound, there must be as many pounds in 604 shillings as 20 shillings are contained times in that number. 20 shillings are contained in 604 shillings 30 times and a remainder of 4 shillings. Therefore 7254 pence are equal to 30 4s. 6d. REDUCTION ASCENDING. 163 2. How many shillings are there in 345 farthings? 3. How many pounds are there in 456 shillings? 4. How many pounds are there in 1586 pence? 5. Reduce 3864 farthings to pounds. 6. Reduce f d. to a fraction of a pound. PROCESS. ANALYSIS. Since 1 penny is 3 rl - 3 rvf i Q 3 A of a shilling, f of a penny is 1 T2 s - - -8* s - equa i to f of ^ of a shilling, or / S. = & Of 3^ == T ^ A of a shilling. _ Since 1 shilling is -fa of a pound, -fx of a shilling is equal to y \ of Y V f a pound, or T g 3 ^ of a pound. RULE. Divide the given number by the number of that de- nomination which is equal to a unit of the next higher denomi- nation. Divide the quotient in like manner, and thus proceed until the required denomination is reached. The last quotient and the several remainders ivill be the result sought. 7. Change f of a shilling to a fraction of a pound. 8. Change -f- of a farthing to a fraction of a shilling. 9. Change 384 pence to units of higher denominations. 10. Change 3146 shillings to pounds. Reduce : 11. 3596d. to pounds. Reduce : 20. 15 8s. to farthings. 12. 3846 far. to shillings. 21. 15 to dollars. 13. 4856s. to pounds. 22. $456 to pounds. 14. 5968 far. to pounds. 15. 3984d. to pounds. 16. 4685 far. to shillings. 17. 48567 far. to pounds. 18. 3 14s. 5d. to far. 19. 48596 far. to pounds. 23. $394.45 to pounds. 24. $37.50 to pounds. 25. 25 to dollars. 26. 15 10s. to farthings. 27. $973.30 to pounds. 28. $1216.625 to pounds. 164 DENOMINATE NUMBEKS. FRENCH MONEY. 249. In France the currency is decimal. The unit is the Franc. TABLE. 10 Centimes (ct.) [pronounced son-teems] = 1 Decime . . dc. 10 Decimes [pronounced des-seems] = 1 Franc . . . fr. Scale Decimal. The value of the franc, as determined by the Secretary of the Treas- ury, is $ .193 in United States money. 1. How many centimes are there in 1 franc? In 5 francs? 2. How many decimes are there in 1 franc ? In 7 francs ? 3. How many centimes are there in 4 decimes? 4. How many dollars are there in 10 francs? In 20 francs? 5. In 3684 centimes how many francs are there? 6. How many francs are there in $19.30? In $9.65? In $3.86? MEASURES OF SPACE. 250. Space is extension in any direction. It has three dimensions or measurements length, breadth and thickness. 251. A Line is that which has only length. Thus, the edge of any thing, or the distance between any two objects or places, is a line. 252. A Surface is that which has only length and breadth. Thus, the floor, this page, or the outside of any thing, is a surface. 253. A Solid is that which has length,, breadth and thickness. Thus, a stone, an apple, a block, a book, etc., are solids. MEASURES OF SPACE. 165 LINEAR MEASURES. 254, Linear Measures are used in measuring lengths and distances. SUEVEYOES' LINEAE MEAS. LINEAE MEASUEE. 12 Inches (in.) = 1 Foot . ft. 3 Feet = 1 Yard . yd. 5J Yards| 16 Feet J = 1 Eod . rd. 320 Eods = 1 Mile . mi. mi. rd. yd. 7.92 Inches 1 Link . 25 Links = 1 Eod . 4 Eods 100 Links 80 Chains = 1 Mile ft. in. 1 Chain 1. rd. ch. mi. Scale - 1 = 320 = 1760 = 5280 = 63360 -12, 3, 51 and 320. The following are also used: 3 Barleycorns 4 Inches 6 Feet 3 Feet 5 Paces 8 Furlongs 1.15 Statute Miles 3 Geographical Miles 60 Geographic Miles ^) 69.16 Statute Miles f Used in pacing distances. : 1 Inch. Used by shoemakers. : 1 Hand. Used to measure the height of horses. : 1 Fathom. Used to measure depths at sea. = 1 Pace.) = 1 Eod. J :1 Mile. = 1 Geographical, or Nautical Mile. League. fof Latitude on a Meridian, or ' Longitude on the Equator. = lDegree{j 1. For the purpose of measuring cloth and other goods sold by the yard, the yard is divided into halves, fourths, eighths, and sixteenths. 2. The length of a degree of latitude varies. 69.16 is the average length, and is that adopted by the United States Coast Survey. 166 DENOMINATE NUMBEKS. 1. How many inches are there in 4 feet? 6 feet? ^8 feet? 10 feet? 12 feet? 2. How many feet are there in 2 rods? 3 rods? 4 rods? 3. How many inches are there in 2 yards? 4 yards? 5 yards ? 4. How many inches are there in 2 yards and 2 inches? 3 yards and 4 inches? 5. How many rods are there in 2 miles? 3 miles? 6. How many feet are there in 1 rod and 2 yards? 2 rods and 3 yards? 7. How many feet are there in 45 inches? In 63 inches? 8. How many yards are in 22 feet? In 47 feet? In 34 feet? 9. How many miles in 640 rods? In 480 rods? 10. How many inches in 10 links? In 100 links? 11. How many links in 5 rods? In 3 rods? In 6 rods? 12. The length of a road was 400 links. What was its length in rods? 13. In 160 chains how many miles? WRITTEN EXERCISES. 14. Reduce 5 mi. 18 rd. 4 yd. to yards. 15. Reduce 7 rd. 5 ft. 6 in. to inches. 16. How many inches are there in 7 miles? In 9 miles? 17. A building was 327 ft. long. How many rods was it in length? 18. A man sold a piece of wire 36828 in. long. How many rods was it in length? 19. In 3960 rods how many miles are there? 20. Reduce 15 mi. 8 rd. 5 yd. 3 ft. 4 in. to inches, 21. Reduce 8 mi. 14 rd. 5 ft. 4 in. to inches. 22. Reduce 66454 inches to miles, etc. 23. Reduce 158964 inches to miles, etc. MEASUKES OF SPACE. 167 24. The diameter of the earth is 7912 miles. How many feet is it? 25. How high is a horse that measures 15 hands? 26. My farm is 67 ch. 83 1. long. How many rods long is it? 27. Keduce 59 ch. 75 1. to inches. : SURFACE MEASURES. 255. An Angle is the difference in the direction of two lines that meet. 256. A Square is a figure that has four equal sides, and four equal angles. A square inch is a square whose side is one inch. A square foot, a square whose side is one foot. The angles of a square are called right angles. 257. A Rectangle is a figure that has four straight sides and four equal angles. The angles of a rectangle are all right angles. 258. The Area or extent of any surface is the number of square units it contains. Thus, if a rectangle is 4 inches long and 3 inches wide the area will be 12 square inches. For it may be divided into 4 rows, each containing 3 square inches or units, and the entire area will be 12 square inches. The method of computing the area of fig- ures that are not rectangular is given in MENSURATION. ANGLE. SQUARE. KECTAJSGLE. 168 DENOMINATE NUMBERS. 259. The area of a rectangle is equal to the product of the numbers that express its length and breadth. The length and breadth must be expressed in units of the same de- nomination. 1. How many square inches are there in p, rectangle 6 inches long and 5 inches wide? 8 inches long and 3 inches wide? 7 inches long and 5 inches wide? 2. How many square feet are there in a rectangle 4 feet long and 3 feet wide? 7 feet long and 5 feet wide? 3. How many square inches are there in a square whose side is 2 inches? 5 inches? 8 inches? 12 inches? 4. How many square yards are there in a square whose side is 2 yards? 5 yards? 7 yards? 10 yards? 5. How many square feet are there in a square whose side is 1 yard long? 3 yards? 5 yards? 7 yards? 10 yards? 6. How many square rods are there in a lot 5 rods long and 4 rods broad? In a square whose side is 6 rods? 7. How many square feet in a square whose side is 3 yards? In a rectangle whose length is 4 yards and breadth 3 yards? 8. How many square inches are there in a square foot? Square feet in a square yard? Square yards in a square rod? SQUARE MEASURE. TABLE. 144 Square Inches (sq. in.) = 1 Square Foot . . . sq. ft. 9 Square Feet = 1 Square Yard . . . sq. yd. 30| Square Yards = 1 Square Hod . . . sq. rd. 160 Square Rods = 1 Acre A. 640 Acres = 1 Square Mile . . . sq. mi. sq.mi. A. sq.rd. sg. yd. sq.ft. sq.in. 1 = 640 = 102400 = 3097600 = 27878400 = 4014489600 Scale 144, 9, 30J, 160, 640. MEASUKES OF SPACE. 169 1. Plastering, ceiling, etc., are commonly estimated by the square yard; paving, glazing, and stone-cutting, by the square foot. 2. Hoofing, flooring and. slating are commonly estimated by the square of 100 feet. SURVEYORS' SQUARE MEASURE. TABLE. 625 Square Links = 1 sq. rd. 16 Square Kods 1 sq. chain. 10 Square Chains =1 acre. 640 Acres 1 sq. mi. In some parts of the country a Township contains 36 square miles, or is 6 miles square. 1. How many square feet are there in 4 square yards? 7 square yards? 9 square yards? 2. How many square inches are there in 2 square feet? 3 square feet? 5 square feet? 3. How many square yards are there in 27 square feet? 36 square feet? 81 square feet? 4. How many square yards are there in 10 square rods? 5. How many square chains are there in 48 square rods ? 64 square rods? 96 square rods? 6. How many square rods are there in 3 acres? In 5 acres? 7. How many acres are there in 480 square rods? 8. How many square feet are there in 288 square inches? 9. How many acres are there in 30 square chains? In 50? WRITTEN EXERCISES. 10. Reduce 9 sq. yd. 3 sq. ft. 15 sq. in. to square inches. 11. Reduce 3 sq. mi. 15 sq. rd. to square inches. 12. Reduce 262685 sq. ft. to acres, etc. 13. Reduce 2 A. 37 sq. rd. 5 sq. yd. 7 sq. ft. to sq. in. 14. Reduce 184265 sq. in. to units of higher denominations. 170 DENOMINATE NUMBERS. 15. Reduce -f- of an acre to units of lower denominations. PROCESS. f A. X 1 60 r= -00 gq. r( J. H42 gq> r( J. f sq. rd. X 30 1 = f sq. rd. X H 1 = W s q 7 d - &sq.yd.X 9 = H sq.ft. = 5^ sq.ft. iJ sq.ft. X144 = iff4sq.m. = H3^Bq.in. Therefore f A. 114 sq. rd. 8 sq. yd. 5 sq. ft. 113f sq. in. ANALYSIS. We multiply by that number in the scale which will reduce the number to the next lower denomination, and so continue to multiply each fraction until the lowest denomination is reached. 16. Express f of an acre in lower denominations. 17. What part of an acre are 100 sq. rd.? 80 sq. rd.? 120 sq. rd.? 18. Change f of a sq. rd. to lower denominations. 19. How many sq. in. are there in a rectangle 7 inches wide by 11 inches long? 20. How many square feet are there in a floor 8 feet long by 15 feet wide? 21. How many square yards are there in a ceiling that is 18 feet wide by 21 feet long? 22. What is the area of a square whose side is 5 feet? 23. How many square yards are there in a floor 18 feet wide by 24 feet long? How much would it cost to carpet it at $1.15 per square yard? 24. How many yards of carpeting 1 yard wide will be required to cover a room 18 ft. long by 17 ft. wide? 25. What will it cost to carpet a room 18 ft. long by 15f ft. wide, with carpet f of a yard wide, at $1.90 per yard? 26. If the width of a lot is 66 feet, how long must it be to contain \ of an acre? What will be the cost of it at $3.25 per square foot? 27. A pasture containing 10 acres had a width of 20 rods? How long was it? MEASURES OF VOLUME. 171 28. Mr. A. sold a lot of land whose width was 20 rd. and whose length was 80 rd. at $47.25 per acre. How much did he get for it? 29. What is the difference between 10 square feet and 10 feet square? Illustrate this by drawings. 30. What will be the expense of painting a roof 48 feet long and 22 feet wide at $.30 a square yard? 31. What will be the cost of cementing the bottom of a cellar 45 feet by 32 feet at $.30 per square yard? 32. How many yards of plastering are there in the sides of a room 18 ft. long, 17 ft. wide, and 11 ft. high? How many in the ceiling ? What will be the cost of plastering at $.37 a square yard? 33. What will be the cost of papering the side walls of the above room at $.25 per square yard? MEASURES OF VOLUME. 260. A Solid has length, breadth, and thickness. 261. A Cube is a solid having six equal square sides called faces. 262. A Cubic Inch is a solid whose sides or faces are each a square inch. 263. A Cubic Foot is a solid whose sides are each a square foot. 264. The Volume, or Solid Contents, of any body is the number of solid units it contains. Thus, if a solid is 4 ft. long, 3 ft. wide, and 3 ft. thick, its volume will 172 DENOMINATE NUMBERS. be 36 cubic feet. For it may be divided into 3 blocks, each contain- ing 12 cubic feet, making in all 36 cubic feet. That is, the number of cubic feet in each block will be equal to the product of the num- bers expressing its length and breadth, and the number of blocks is equal to the number expressing the thickness. Therefore, 265. The volume of any rectangular solid is equal to the prod- uct of the numbers expressing its length, breadth, and thickness. The length, breadth and thickness must be expressed in units of the same denomination. 1. How many cubic feet are there in a rectangular solid whose length is 3 ft. , its breadth 2 ft. , and its thickness 2 ft. ? 2. How many cubic feet are there in a cube whose dimen- sions are each 3 feet; or, how many cubic feet are there in a cubic yard ? In a cube whose sides are 5 ft. long ? 3. How many cubic inches are there in a cube whose di- mensions are each 12 inches; or, how many cubic inches are there in a cubic foot? In a cube whose sides are 10 in. long? 4. What is the volume of a cube whose sides are each 4 inches square? 9 inches square? 16 inches square? CUBIC MEASURE. TABLE. 1728 Cubic Inches (cu. in.) = 1 Cubic Foot 27 Cubic Feet = 1 Cubic Yard cu. ft. cu. yd. A cord of wood or stone is a pile 8 feet long, 4 feet wide and 4 feet high. A pile that is 1 foot long, 4 feet wide and 4 feet high, is a cord foot. MEASURES OF VOLUME. 173 following are the denominations: 16 Cubic Feet = 1 Cord Foot . . . cd. ft. 8 Cord Feet) 128 Cubic Feetf" 1. A perch of stone or masonry is 16J ft. long, 1| ft. thick, and 1 foot high, and contains 24} cu. ft. 2. A cubic yard of earth is considered a load. 3. Brick-work is commonly estimated by the thousand bricks. 4. Brick-layers, masons and joiners commonly make a deduction of one-half the space occupied by windows and doors in the walls of buildings. 5. In computing the contents of walls, masons and brick-layers mul- tiply the entire distance around on the outside of the wall by the height and thickness. The corners are thus measured twice. WRITTEN EXERCISES. 1. How many cubic inches are there in 2 cubic feet? In 3 cu. ft. ? In 15 cu. ft. ? In 32 cu. ft. ? 2. How many cubic feet are there in 2 cubic yards? In 3 cu. yd. ? In 13 cu. yd. ? In 25 cu. yd. ? 3. How many cubic feet are there in 5 cords? In 8 cords? 4. How many perch of masonry are there in 418 cubic feet? What will be the cost of laying it at $1.75 per perch? 5. How many perch of masonry are there in a wall 38 feet long, 4 feet high, and 1^ feet thick? 6. How many yards or loads, of earth, must be removed in digging a cellar 35 feet by 20, 8 feet deep? 7. Keduce 32 cu. ft. 114 cu. in. to cubic inches. 8. Keduce 13 cu. yd. 18 cu. ft. to cubic feet. 9. Reduce 15 perch 13^- cu. ft. to cubic feet. 10. How many cubic blocks of one foot on a side can be cut from a cube that is 8 yards long on each edge? 11. How many cubic feet in a block of marble 9 feet long, 5 feet wide, and 3 feet thick ? 174 DENOMINATE NUMBEKS. 12. A man sawed a pile of wood 40 ft. long, 4 ft. wide, and 5| ft. high, for $1.50 per cord. How much did he earn? 13. A bin is 8 ft. long, 7 ft. wide, and 5 ft. high. How many cubic feet are there in it? How many cubic inches? How many bushels will it hold if a bushel contains 2150.4 cubic inches? 14. What will it cost to excavate a cellar 80 by 35 ft. , and 8 ft. deep, at $ .42 per yd. ? What will be the expense of build- ing a stone wall around it 1^ ft. thick, at $3.75 a perch? 15. How many bricks will it require to build a wall 35^- ft. long, 19 ft. high, and 3 ft. thick, allowing 22 bricks to the cubic foot when laid ? BOARD MEASURE. 266. In measuring lumber, when a board is one inch thick, the number of feet board measure is obtained by multiplying the length in feet by the breadth expressed in feet. When the lumber is more than one inch thick, the number of feet, board measure, may be obtained by multiplying the length in feet by the breadth in feet, and this product by the number expressing the inches in thickness. When a board tapers uniformly, the average or mean width is equal to half the sum of the two ends. Board measure may also be computed by multiplying the number of feet in length by the number of inches in width, and then dividing the product by 12. EXERCISES. How many feet are there in the following boards: 1. 18 ft. by 16 in.? 2. 15 ft. by 11 in.? 3. 10 ft. by 13 in.? 4. 13 ft. by 15 in. ? MEASURES OF CAPACITY. 175 5. How many feet of timber are there in a stick 40 feet tig, 9 inches wide, and 6 inches thick? 6. Mr. B. bought 318 fence boards 16 feet long and 8 inches vide. What did they cost at $11 per thousand feet? 7. A lumber dealer bought 35 three-inch planks, 22 feet long and 16 inches wide, at $17.50 per M. How much did they cost? 8. What will it cost to floor a room 35 feet by 18, with 1^ inch flooring, at $30 per M, allowing ^ for matching? 9. What will be the expense of flooring a room 20 feet by 25 with 1| inch flooring, at $25 per M, allowing -|- for matching ? MEASURES OF CAPACITY. LIQUID MEASURE. 267. Liquid Measure is used in measuring liquids. TABLE. 4 Gills (gi.) = 1 Pint . . . pt. 2 Pints =1 Quart . . . qt. 4 Quarts =1 Gallon . . . gal. gal. qt. pt. gL I = 4 = 8 = 32 Scale 4, 2, 4. 1. In determining the capacity of cisterns, reservoirs, etc., 31 \ gallons are considered a barrel (bbl.), and 2 barrels, or 63 gallons a hogshead (hhd.). In commerce, however, the barrel and hogshead are not fixed measures. 2. Cash of large size do not hold any fixed quantity. Their ca- pacity is usually marked upon them. 3. The standard gallon of the United States contains 231 cubic inches. 4. The beer gallon is not now in use. It contained 282 cubic inches. 176 DENOMINATE NUMBERS. EX EXCISES. 1. How many gills are there in 3 pints? 5 pints? 7 pints? 2. How many gills are there in 2 quarts? 3 quarts? 3. How many pints are there in 3 quarts? 8 quarts? 4. How many pints are there in a cask which contains 37 gallons? 5. A man sold 684 pints of milk at 20 cents a gallon. How much did he get for it? How many gallons were there? 6. Eeduce 3846 gi. to gal. 4869 pt. to gal. 7. Reduce 3 gal. 4 qt. 1 pt. 3 gi. to gi. 8. Reduce 4 bbl. 6 gal. to gi. 484 pt. to gal. 9. Reduce 24 gal. to pt. 8459 gi. to bbl. 10. How many cubic inches are there in 7 gal. ? 11. How many gallons will a vessel hold that contains 3846 cubic inches? 12. How many barrels of water will a cistern hold that is 15 feet long, 10 feet wide, and 8 feet deep? APOTHECARIES' LIQUID MEASURE. 268. Apothecaries 9 Liquid Measure is used in compounding and measuring liquid medicines. TABLE. 60 Drops (gtt.) or minims (Til) 1 Fluid drachm . /. 8 Fluid drachms = 1 Fluid ounce . /. 16 Fluid ounces =lPint ... 0. 8 Pints = 1 Gallon . . . Cong. 1. The abbreviation Cong, is from the Latin congim, a gallon. A pint being one-eighth of a gallon the abbreviation is 0., from the Latin octavus, one-eighth. 2. In writing prescriptions, physicians write the number after the symbol; thus: 0. 5, / 2, etc. MEASURES OF CAPACITY. 177 DRY MEASURE. 269. Dry Measure is used in measuring grain, roots, fruit, etc. TABLE. 2 Pints (pt.) 1 Quart . . . qt. 8 Quarts = 1 Peck . . . pk. 4 Pecks = 1 Bushel . . . bu. bu. pk. qt. pt. 1 = 4 = 32 = 64 Scale 2, 8, 4. 1. In measuring grain, seeds, or small fruits, the measure must be even full or stricken. In measuring large fruits, coarse vegetables, corn in the ear, etc., the measure should be heaped at least six inches. 2. Five stricken bushels are considered equal to 4 heaped bushels. 3. A standard bushel contains 2150.4 cubic inches. 4. A pint, quart, or gallon, dry measure, is more than the same quantity liquid measure, for a quart is ^ of a bushel, or ^ of 2150.4 cubic inches, which is about 67 J cubic inches, while a quart liquid measure is \ of 231 cubic inches, or 57f cubic inches. Cn. In. in Cn. In. in Cii.In.in Cii.In.in One Gal. OneQt. One Pt. OneGi. Liquid Meets. 231 57| 28J 7& DryMeas. 268f 67J 33f Sf EXERCISES. 1. How many pints are there in 3 quarts? 7 quarts? 2. How many quarts are there in 2 pecks? 3 pecks? 5 pecks? 7 pecks? 3. How many pints are there in 1 bushel? 3 bushels? 5 bushels? 8 bushels? 4. How many pints are there in 3 bu. 3 pk. 5 qt. 1 pt. ? 5. How many pints are there in 8 bu. 5 qt. 3 pt. ? 6. Change 16845 qt. to units of higher denominations. 12 178 DENOMINATE NUMBERS. 7. Change 13965 pt. to units of higher denominations. 8. Change 57364 qt. to units of higher denominations. 9. Change 35 bu. 3 pk. 6 qt. 1 pt. to pints. 10. How many cubic inches are there in 7 bu. ? 8 bu. ? 10 bu. ? 20 bu. ? 11. How many bushels are there in 13846 cu. in.? 35769 cu. in. ? 48695 cu. in. 12. How many cubic inches are there in a bin 8 ft. long, 7 ft. wide, and 5 ft. high ? How many bushels will it hold ? 13. How many bushels will a bill hold that is 9 ft. long, 6ft. wide, and 6 ft. high? MEASURES OF WEIGHT. 270. Weight is the measure of the force that attracts bodies to the earth. AVOIRDUPOIS WEIGHT. 271. Avoirdupois Weight is used in measuring all coarse and heavy articles, as hay, grain, groceries, coal, etc., and the metals, except gold and silver. TABLE. * 16 Ounces (oz.) = 1 Pound Ib. 100 Pounds = 1 Hundred- weight . cvvt. 20 Hundred-weight == 1 Ton T. T. cwt. Ib. oz. 1 = 20 = 2000 = 32000 Scale 16, 100,20. 1. In weighing coal at the mines and in levying duties at the United States Custom House, the lony ton of 2240 Ib. is sometimes used. 2. The ounce is considered as 16 drams. MEASURES OF WEIGHT. 179 The following denominations are also used: 56 Ib. Butter 100 Ib. Grain or Flour 100 Ib. Dried Fish 100 Ib. Nails 196 Ib. Flour 200 Ib. Pork or Beef = 1 Firkin. = 1 Cental. = 1 Quintal. = 1 Keg. = 1 Barrel. = 1 Barrel. 280 Ib. Salt at N. Y. Works = I Barrel. The following are the pounds in a bushel in the States named : a * 5 ^ ^' ^ ti 3 * CO V: rC: .v) ^ I 1 J % S fej & '' O e Penna. \ ^- 1 C5 * Wheat GO 52 32 50 40 54 50 5C> 28 45 56 60 56 GO 52 32 48 40 54 60 45 00 56 32 4S 50 56 60 45 GO 5(5 32 48 52 56 60 45 60 56 33% 48 52 56 60 45 60 56 32 32 32 30 60 56 30 46 46 56 60 56 32 48 42 56 (SO GO 56 32 48 42 56 60 GO 52 35 48 52 56 (JO 45 30 60 56 30 48 50 56 64 60 56 32 48 48 56 GO 44 GO 56 32 48 56 60 60 56 34 46 42 56 60 60 56 32 47 48 56 60 56 32 46 4G> 56 60 56 36 45 42 5G, 60 GO 56 32 48 42 56 60 46 60 54 48 50 Indian Corn Oats Barley Buckwheat Rye Clover Seed. Timothy Seed....... EXERCISES. 1. How many ounces are there in 5 Ib. ? In 3 Ib. 5 oz. ? 2. How many pounds are there in 5 cwt. ? In 6 cwt. ? 3. How many pounds are there in 1 ton ? In 3 T. ? 4. How many pounds are there in 3 T. 2 cwt. 5 Ib. ? 5. How many pounds are there in 5 T. 216 Ib. ? 6-. How much will 5 Ib. 7 oz. of indigo cost at $.12|- per oz. ? 7. What will 3-J- Ib. of confectionery cost at $ .Q4 per oz. ? 8. At 8 cents a pound, what must be paid for 5 cwt. 28 Ib. of sugar? 180 DENOMINATE NUMBERS. 9. How many pounds are there in \ barrel of pork? In \ barrel of salt? In \ barrel of flour? In \ keg of nails? 10. What will be the value of \ barrel of flour at $8.50 per cwt. ? 11. What will \ quintal of codfish cost at $ .06^ per lb.? 12. What will be the cost of 13 cwt. 18 lb. of hay at $15 per ton? 13. When flour is $10 a barrel, how many pounds can I buy for $2.80? . 14. A merchant sold 3 cwt. 19 lb. 9 oz. of cheese at $.17 per lb. How much did he receive for it? 15. If a merchant buys flour at $9 per barrel and sells it at $5 per cental, how much will be his profit on the sale of 15 barrels? 16. How many barrels of salt are there in 275000 lb. ? 17. If the weight of a bushel of wheat is 60 lb., how many bags that hold 2 bu. each will be required to carry away 3 T. 4 cwt. 20 lb. of wheat? TROY WEIGHT. 272. Troy Weight is used in weighing gold, silver, and jewels. TABLE. 24 Grains (gr.) 1 Pennyweight . . . pwt. 20 Pennyweights = 1 Ounce oz. 12 ounces = 1 Pound lb. lb. oz. pwt. gr. 1 = 12 ==240 = 5760 Scale 24, 20, 12. 1. In weighing diamonds, pearls, and other jewels, the unit com- monly employed is the carat, which is equal to 4 grains. 2. The term carat is also used to express the fineness of gold, and means -fa part. Thus, gold that is 18 carats fine is |f gold and f alloy. MEASURES OF WEIGHT. 181 APOTHECARIES' WEIGHT. 273. Apothecaries 9 Weight is used by apothecaries id physicians in weighing medicines. TABLE. 20 Grains (gr.) = 1 Scruple . . . sc., or 9 3 Scruples = 1 Dram .... dr., or 3 8 Drams = 1 Ounce .... oz., or 12 Ounces = 1 Pound .... lb., or flb Ib. oz. dr. sc. gr. 1 = 12 = 9Q = 28S = 5760 Scale 20, 3, 8, 12. 1. In writing prescriptions, physicians express the number in Ko- man characters, using j instead of i final. They also write the symbol first; thus: v, gvj, ^ij. 2. Medicines are bought and sold in large quantities by Avoirdu- pois Weight. 1 lb. Avoirdupois = 7000 gr. 1 lb. JT**/ 1 * 1 . , j = 5760 gr. (Apothecaries') 1 oz. -= 437i gr. 1 oz. " == 480 gr. EXERCISES. 1. How many grains are there in 3 pwt. ? In 5 pwt. ? 2. How many pennyweights are there in 5 oz. ? In 7 oz. ? 3. How many grains are there in 7 oz. 5 pwt. 18 gr. ? 4. Express 3456 grains Troy in higher units. 5. What will be the value of an ornament weighing 2 oz. 15 pwt, at $1.35 per pwt.? 6. How many spoons, weighing 5 ounces each, can be made from 3 lb. 5 oz. of silver ? 7. How many powders, of 5 grains each, can be made from 5 oz. 7 dr. of quinine ? 182 DENOMINATE NUMBERS. MEASURES OF TIME. 274. The following are the ordinary divisions of time : 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 . . . , vr 366 days = 1 Leap Year . , vr 100 years = 1 Century . . , , cen. yr. mo. da. hr. min. sec. 1 = 12 = 365 = 8760 = 525600 = 31536000 Soak 60, 60, 24, 365, 100. 1. In most business computations 30 days are considered a month, and 12 months a year. For many purposes 4 weeks constitute a month. 2. The common year contains 52 weeks and 1 day, the leap year 52 weeks and 2 days. Hence, commonly, each year begins one day later in the week, but the year succeeding leap year begins tico days later. 3. The time required for the earth to revolve around the sun is one year, which is 365 da. 5 hr. 48 min. 49.7 sec., or very nearly 365J days. Instead of reckoning this part of a day each year, it is disregarded, and an addition made when this would amount to one day, which would be very nearly every fourth year. This addition of one day is made to the month of February. Since the part of a day that is disregarded when 365 days are considered as a year, is a little less than one-quarter of a day, the addition of one day every fourth year is a little too much, and, to correct this excess, addition is made to only every fourth cen- tennial year. With this correction the error does not amount to much more than a day in 4000 years. Therefore, Centennial years whose number is exactly divisible by 400, and other years whose number is exactly divisible by 4, are Leap Tears. MEASUKES OF TIME. 183 The year begins with the month of January, and ends with the month of December. The months, their names and the number of days in each, are as follows : January, 31 da. . . Jan. February, 28 or 29 da. Feb. March, 31 da. . . Mar. April, 30 da. . . Apr. May, 31 da. . . May. June, 30 da. . . June. July, 31 da. . . July. August, 31 da. . . Aug. September, 30 da. . . Sept. October, 31 da. . . Oct. November, 30 da. . . Nov. December, 31 da. . . Dec. EXERCISES. 1. How many seconds are there in 5 minutes? In 6 min. ? 2. How many minutes are there in ^ hour? In ^ hr. ? 3. How many days are there in 4 weeks? In 5 wk. ? In 8 wk. ? In 10 wk. ? 4. How many hours are there in \ day ? In ^ da. ? 5. How many days are there in ^ year? In ^ month? 6. What part of an hour are 30 minutes? 15 min.? 7. How many hours are there in 90 minutes? In 120 min. ? In 240 min. ? 8. How many seconds are there in 5 hr. 15 min. 12 sec. ? 9. How many seconds are there in 6 hr. 27 min. 38 sec. ? 10. Express in units of higher orders 48695 sec. 11. Express in units of higher orders 38497 sec. 12. How many minutes are there in 5 yr. of 365 da. each ? 13. How many days are there from Jan. 1st to May 1st? 14. How many days are there from April 1st to Oct. 15th ? 15. Reduce 2 wk. 5 da. 13 hr. to hours. 16. Reduce 5 da. 10 hr. 15 min. to minutes. 17. Reduce 384600 sec. to higher denominations. 18. Reduce 15 hr. 12 min. 18 sec. to seconds. 19. Reduce 32965 min. to higher denominations. 184 DENOMINATE NUMBEKS. CIRCULAR OR ANGULAR MEASURE. 275. A Circle is a plane surface, bounded by a curved line every point of which is equally distant from a point within called the Center. 276. The Circumfer- ence is the line that bounds B the circle. 277. An Arc of a circle is any part of the circumference. 278. A Degree is ^ of the circumference of a circle. 279. The Measure of an Angle is that part of the circumference which is included between the lines which form the angle. Each of the arcs of the circumferences ab, cd, DE, is a measure of the same angle, and therefore contains the same number of degrees ; but since each degree is -3^ of the circumference, the length of a de- gree must vary. 280. Circular or Angular Measure is used to measure arcs of circles and angles, in determining latitude, longitude, direction, the position of vessels at sea, etc. TABLE. 60 Seconds (") = 1 Minute . . . x 60 Minutes 1 Degree . . . 360 Degrees = 1 Circumference . Cir. Cir. " 1 = 360 = 21600 = 1296000 Scale 60, 60, 360. MISCELLANEOUS. 185 1. A Quadrant is J of a circumference, or 90; a Semtant ^ of a circumference, or 60. 2. The length of a degree of longitude on the earth's surface at Equator is 69.16 miles. 3. In astronomical calculations 30 are called a Sign, and there therefore 12 signs in a circle. EXERCISES. 1. How many minutes are there in 5? 6? 2. In 35 degrees how many seconds are there? In 27? In 21 12' 18"? 3. How many seconds are there in 34 12' 43" ? 4. In 468560 seconds how many minutes, etc., are there? 5. In 384500 seconds how many minutes, etc., are there? 6. How many seconds are there in ^ Cir. ? In ^? In ^? 7. How many minutes are there in 2 quadrants? In 2 extants? COUNTING. 281. The following denominations are used in counting some classes of articles : 12 Things = 1 Dozen . . . doz. 12 Dozen = 1 Gross . . . gr. 12 Gross = 1 Great Gross . G. gr. Two things are often called a pair, six things a set, and twenty things a score; as a pair of birds, a set of spoons, a score of years. STATIONERS' TABLE. 282. The denominations used in the paper trade are : 24 Sheets = 1 Quire. 20 Quires = 1 Ream. 2 Reams 1 Bundle. 5 Bundles = 1 Bale. 186 DENOMINATE NUMBERS. The terms folio, quarto, octavo, applied to books, indicate the number of leaves into which a sheet of paper is folded. Thus, when a sheet of paper is folded into 2, 4, 8, 12, 16, 18, or 24 leaves, the forms are called respectively, folio, 4to, or quarto, 8vo, or octavo, 12nio, 16mo, 18mo, and 24mo. EXERCISES. 1. How many eggs are there in 5 dozen? 7 doz. ? 10 doz. ? 2. How many crayons are there in 2 gross ? 3 gr. 5 doz. ? 3. How many things are there in a great gross ? 4. What will be the cost of 3 dozen brushes at $.45 each ? 5. A man lived 3 score and 10 years. What was his age ? 6. What will 3 reams of paper sell for at $.15 per quire? REDUCTION OP DENOMINATE FRACTIONS. 283. The principles, processes and analyses are essentially the same as those of denominate integers. CASE I. 284. To reduce denominate fractions to equivalent numbers of lower denominations. EXERCISES. 1. How many hours are there in 1 day? In \ day? In | of a day? In f of a day? In of a day? 2. How many ounces are there in -J- pound avoirdupois? In^lb.? In|lb.? 3. How many pints are there in \ of a peck ? In f pk. ? 4. How many pecks and quarts are there in f of a bushel ? 5. How many pounds and ounces are there in f cwt. ? 6. How many inches are there in f of a foot ? f ft. ? f ft. ? REDUCTION OF DENOMINATE FKACTIONS. 187 7. Change -f- of a rd. to units of lower denominations. PROCESS. ANALYSIS. Since in 1 5 O f ILL yd. = |f yd. = 31f yd. y od th f e are 5 i ? ards > in f of a rod there will |f Of 3 ft. = : || ft. : = 2^ ft. be 5 of 5 1 yards , or 3|| |i of 12 in. = f in. = 9 T 6 in. yards. Since in 1 yard there are 3 feet, in of a yard there will be J of 3 ft,, or 2JJ ft. Since in 1 foot there are 12 inches, in -J-J of a foot there will be -JJ of 12 inches, or 9 T 6 in. Therefore f of a rod is equal to 3 yd. 2 ft. 9 T 6 ^ in. Change the following to lower denominations: 8. -| of a pound Troy. 9. f of a ton. 10. | of a furlong. 11. of an acre. 12. | of a peck. 13. T % of a day. 14. -| of a sq. rd. 15. ^ of a cu. yd. 16. Express y^ of a gallon as a fraction of a gill. ANALYSIS. Since in 1 gallon there are 32 gills, in T J-j of a gallon there are T ^ of 32 gi., or $ gi. Hence T ^ gal. = -gfo of a gill. 17. Express 18. Express 19. Express 20. Express 21. Express PROCESS. .685 12 8.220 oz. 20 4.400 pwt. 24 9.600 gr. O of a bushel as a fraction of a pint. T f a m ^ e as a fraction of a foot, of a pound as a fraction of a scruple. .006 of a bushel as a decimal of a pint, in lower denominations .685 of a pound Troy. ANALYSIS. Since in 1 pound there are 12 ounces, in .685 of a pound there are .685 of 12 ounces, or 8.220 ounces. Since there are 20 pennyweights in 1 ounce, in .220 of an ounce there are .220 of 20 penny- weights, or 4.400 pennyweights. Since in 1 pennyweight there are 24 grains, in .400 of a pennyweight there are .400 of 24 grains, or 9.600 grains. Therefore .685 Ib. is equal to 8 oz. 4 pwt. 9.6 gr. 188 DENOMINATE NUMBERS. Express in units of lower denominations: 22. .575. 23. . 1935 of a pound Troy. 24. .436 of a ream. 25. .1845 of a gallon. 26. .135 of a rod. 27. .455 of a mile. 28. .4832 of a bushel. 29. .684 of a league. CASE IT. 285. To change denominate fractions to equivalent fractions of higher denominations. EXERCISES. 1. What part of a pound Troy is 1 ounce? Is oz.? Is ioz.? 2. What part of a ton is 1 pound ? Is Ib. ? Is ^ Ib. ? 3. What part of a mile is 1 rod ? Is | rd. ? Is rd. ? 4. What part of a league is 1 mile ? Is 1 rd. ? Is -|- rd. ? 5. What part of an hour is of a minute ? Is |- of a min. ? 6. What part of a week is -f- of a day ? -| of a day ? 7. What part of a bushel is -f of a pint ? PROCESS. ANALYSIS. Since there are 64 pints in ^ y _i_ _ _s h n ^ bushel, 1 pint is ^ of a bushel, and f of a pint is f of & of a bushel, or 7 f Or, of a bushel. Or, g Since we are required to change pints to T2" 4 bushels we have an example in reduction T 2" ~^~ * """96" P^-' ascending, and hence we divide by 2, 8, and 9T -^ 4 = dhr bu - 4, respectively. 8. Reduce -^ of an inch to the fraction of a yard. 9. Change |- of a second to the fraction of an hour. 10. Express .375 of a week as a fraction of a year. 11. Express .35 of a pound as a fraction of a ton. 12. Express f of a cubic inch as a fraction of a cubic foot. REDUCTION OF DENOMINATE FRACTIONS. 189 13. Change f of a square yard to a fraction of an acre. 14. Reduce f of a pint to a fraction of a barrel. CASE III. 286. To express one denominate number as a frac- tion of another. 1. What part of a foot are 3 in. ? 6 in. ? 9 in. ? 2. What part of an hour are 30 min.? 15 min.? 45 min.? 3. What part of a gallon is 1 pint? 2 pints? 1 quart? 4. What part of a gallon are 2 quarts? 2 qt. 1 pt.? 3 qt. 1 pt. ? 5. What part of 3 ft. 6 in. are 2 ft. 3 in. ? ANALYSIS Since 3 ft. 6 in. = 42 in., and 2 ft. 3 in. = 27 in., 27 in. = f| of 42 in. 6. What part of 3 yd. 2 ft. are 2 yd. 2 ft. ? 7. What part of 5 gal. 3 qt. 1 pt. are 2 gal. 1 qt. 1 pt. ? 8. What part of 2 pounds Troy are 3 oz. 10 pwt. ? 9. What part of 3 pecks are 2 qt. 1 pt. ? 10. What part of 3 barrels are 13 gal. 3 qt. 2 pt. 2 gi. ? 11. Express 15s. 7d. in the decimal of a pound sterling: IST. PROCESS. ANALYSIS. In order to find what part .j - -, _ -\ 0.7 (] one number is of another, both must be reduced to the same denomination. 15s. 1 = 240d. 7d, = 187d. and .7791+ 15.5833 + s. = JL4,j$JLa O r .7791 + . 190 DENOMINATE NUMBERS. 12. Reduce 4 hr. 15 min. to the decimal of a day. 13. Reduce 3 pk. 2 qt. to the decimal of a bushel. 14. Reduce 3 ft. 6 in. to the decimal of a rod. 15. Reduce 18s. 5f d. to the fraction of a pound. 16. Reduce 18s. 5fd. to the decimal of a pound. 17. Reduce 16 Ib. 11 oz. to the fraction of a hundred-weight. 18. Reduce 37 rd. 14 ft. 3 in. to the decimal of a mile. 19. Reduce 3 da. 5 hr. 14 min. to the decimal of a week. 20. Reduce 8 quires, 15 sheets, to the decimal of a ream. 21. Change 3 cd. ft. 7 cu. ft. to the decimal of a cord. 22. Change 654 yd. 9 in. to the decimal of a mile. 23. Change 4 oz. 7 pwt. 13 gr. to the fraction of a pound Troy. 24. Write rules for each of the cases in denominate num- bers. REVIEW EXERCISES. 287. 1. What will be the cost of 15 Ib. 8 oz. of butter at 131 per pound? 2. What must be paid for 3 pk. 2 qt. of berries at 9 cents a quart? 3. Mr. A. sold 18 bu. 3 pk. of barley at $1.05 per bushel. How much did he get for it? 4. How much must be paid for making 42 rd. 7 ft. 8 in. of fence at $ .75 per foot? 5. How much butter at $ .30 a pound must be given for 12 gal. 3 qt. of molasses, at $ .50 per gallon? 6. Bought 15 bu. of oats at $ .37^ a bushel, and sold them at 15 cents a half-peck. How much did I gain? 7. How many cords of wood are there in a pile 4 ft. wide, 6 ft, high, 60 ft. long? What would it cost at 84.25 a cord? 8. A man built a cistern 10 ft. long and 6 ft. wide, that would hold 100 barrels. How high did he make it? REVIEW EXERCISES. 191 9. What is a druggist's profit if he buys opium at $.75 per ance Avoirdupois, and sells it at $1 per ounce Troy? 10. What are the contents of a field 15 rd. 8 ft. wide, 27 9 ft. long? What is its value at $150 per acre? 11. How many days of 10 hours each will it require to ake a million marks if I make 2 per second? 12. What is the value of a plank 18 ft. long, 16 in. wide, ad 4 in. thick, at $18 per M? 13. If at 10 cents a foot the Atlantic cable cost $1689600, vhat is its length? 14. A druggist put up 7 83 49 in two-grain pills. How any pills did he put up? 15. Bought paper at $2.55 per ream and sold it at 20 ents per quire. How much did I gain? 16. How much sugar at 12 cents a pound can be obtained br 13 Ib. 7 oz. butter at 27^- cents a pound? 17. A farmer sold 3 piles of wood at $4.60 per cord. The following are the dimensions of the piles: The first was 73 ft. 9 in. long, 6 ft. high, and 4 ft. wide; the second was 30 ft. long, 7 ft. 2 in. high, and 4 ft. wide; the third was 37 ft. long, 3 ft. 6 in. high, and 4 ft. wide. How much should he receive for his wood? 18. A printer used 4 reams 8 quires 12 sheets of paper for half-sheet posters. How many did he print? What did they cost at $6.50 per M? 19. Hay at $18 per ton is exchanged for flour at $6.85 per barrel. How many barrels are equal to a ton? 20. Two men who are equal partners, obtained from a field 327 bu. 3 pk. 5 qt. of oats. One of them claimed 167 bu. 3 pk. for his share. Did he claim too much or too little? How much? 21. A cubic foot of water weighs about 62 Ib. 8 oz. What will be the weight or pressure on a square yard where the sea is 20 fathoms deep? 192 DENOMINATE NUMBEES. ADDITION. 288. The processes of adding, subtracting, multiplying, and dividing compound numbers are based upon the same principles as those governing similar operations in simple numbers. The only difference between the processes is caused by com- pound numbers having a varying scale, while simple numbers have a uniform decimal scale. EXERCISES. 1. What is the sum of 130 rd. 5 yd. 1 ft. 6 in., 215 rd. 2 ft. 8 in., 304 rd. 4 yd. 11 in.? PROCESS. ANALYSIS. The numbers should -, -, , f . be written as in simple addition, so ra. yd. it. in. . OA IT -I n that units of the same denomma- tion stand in the same column, and for convenience we begin at the 304 4 11 r ight to add. 2 mi. 10 4|- 2 1 The sum of the inches is 25 in., /) i _ i a which i g equal to 2 ft. 1 in. We _> _ "2"~ write the 1 under the inches and 2 mi. 10 5 7 add the 2 ft. with the feet. The sum of the feet is 5 ft., or 1 yd. 2 ft. We write the 2 as feet in the sum and add the 1 yd. with the yards. The sum of the yards is 10 yd., or 1 rd. 4J yd. We write the 4J yd. as yards of the sum, and add the 1 rd. with the rods. The sum of the rods is 650 rd., or 2 mi. 10 rd., which we write as miles and rods of the sum. Therefore the sum is 2 mi. 10 rd. 4 yd. 2 ft. 1 in. Or, since \ yd. equals 1 ft. 6 in., the sum may be expressed as 2 mi. 10 rd. 5 yd. 7 in. RULE. Change the rule for the addition of simple numbers so that it may be applicable to denominate numbers. ADDITION. 193 2. What is the sum of 12 Ib. 5 oz. 13 pwt., 21 Ib. 8 oz. 15 pwt., 13 Ib. 7 oz. 10 pwt., 51 Ib. 3 oz. 17 pwt? 3. What is the sum of 71 6s. 5|d., 32 8s. 5d., 61 15s. Sid., 37 18s. 5fd., 115 lls. 7d.? 4. Find the sum of 10 mi. 217 rd. 2 yd. 3 ft. 4 in., 7 mi. 185 rd. 3 yd. 9 in., 19 mi. 37 rd. 6 yd. 5. Find the sum of 3 T. 7 cwt. 39 Ib. 8 oz., 8 T. 11 cwt. 48 Ib., 11 oz., 13 T. 33 Ib. 10 oz., 9 cwt. 18 Ib. 9 oz. 6. Find the sum of 18 gal. 3 qt. 1 pt. 3 gi., 15 gal. 2 qt. 1 pt. 2 gi., 11 gal. 2 qt. 2 gi., 3 qt. 1 pt. 1 gi. 7. A miller bought four loads of grain containing, respect- ively, 25 bu. 3 pk., 28 bu. 2 pk., 32 bu. 3 pk. 5 qt., 28 bu. 2 pk. 7 qt. How much grain did he buy ? 8. How much wood is there in 3 piles containing, respect- ively, 37 C. 21 cu. ft. 1140 cu. in., 29 C. 110 cu. ft. 708 cu. in., and 34 C. 121 cu. ft. 398 cu. in.? 9. Find the sum of f mi., .35 rd. and 2f rd. PROCESS. rd. ft. in. o . -, OP7 o A 9 ANALYSIS. Each of the f mi. 137 2 44 f ^ . -, . . 7 7 fractions is expressed in in- .00 rd. - "TO tegers of lower denominations, 2 1 rd. = 2 6 2^ and then they are added. 139 14 10. A merchant sold 12f yards of cloth to one person, 8f yards to another, 37^- yards to another, 39f yards to another. How many yards, feet and inches did he sell? 11. What is the amount of land in the following lots, the first containing -J- of an acre, the second f of an acre, the third 1291 sq. rd., and the fourth 118 sq. rd. ? 12. A merchant sold the following quantities of molasses, viz: On June 15, 24 gal. 2 qt. 3 pt. ; June 16, 45^ gal.; June 17, li bbl. (39f gal.) How much did he sell in that time? 13 194 DENOMINATE NUMBEKS. 13. James is 3 yr. 4 mo. 18 da. old, Henry is 2 yr. 8 mo. 6 da. older than James, William is 7 yr. 10 mo. 24 da. older than Henry, and Herbert is 20 mo. older than William. How old is Herbert? 14. Find the sum of 20 cwt., 16| T., 17 lb., 19 cwt. 18 lb. 7 oz., 15 lb. 8 oz., 2 T. 7 lb. 5 oz., f lb., f T., 2 T. 3 cwt. 57 lb. 4 oz. SUBTRACTION. 289. 1. Prom 127 rd. 3 yd. 1 ft. 7 in., subtract 100 rd. 4 yd. 2 ft. 9 in. PROCESS. ANALYSIS. The numbers should be rd. yd. ft. in written as in simple subtraction, so . ~ 7 o i 7 that units of the same order stand in the same column, and, for convenience, begin at the right to subtract. 26 31 1 1 Since we can not subtract 9 in. from Q l_-j a 7 in., we unite with 7 in. a unit of the ' next higher order, making 1 ft. 7 in., 26 4 4 or 19 in. Then 9 in. from 19 in. leaves 10 in., which we write as inches in the remainder. Inasmuch as 1 ft. was united with 7 in., there are no feet remaining in the minuend. Since we can not subtract 2 ft. from ft., we unite with ft, a unit of the next higher order, making 3 ft. Then 2 ft. from 3 ft. leaves 1 ft,, which we write as the feet of the remainder. Since 4 yd. can not be subtracted from 2 yd., we unite with 2 yd. a unit of the next higher order and proceed as before. The remainder will be 26 rd. 4 yd. ft. 4 in. RULE. Change the rule for subtraction of simple numbers so that it may be applicable to denominate numbers. 2. From 2 mi. 116 rd. 4 yd. ft. 4 in., take 1 mi. 120 rd. 2 yd. 1 ft. 8 in. SUBTRACTION. 195 3. From 15 cwt. 37 Ib. 10 oz., take 8 cwt. 42 Ib. 8 oz. 4. From 1 hhd. 38 gal. 3 qt. 2 pt., take 60 gal. 2 qt. 1 gi, 5. From 13 Ib. 8 oz. 13 pwt. 15 gr., take 8 Ib. 8 oz. 16 pwt. 15 gr. 6. From 18 33' 16", take 9 42' 28". 7. From 37 C. 7 cd. ft. 11 cu. ft., take 18 C. 7 cd. ft. 12 cu. ft. 8. From f bbl. take 7| gal. PROCESS. gal. qt. pt. gi. ANALYSIS. The fractions 3. bbl. =2321 are nrst expressed in integers 7 ffal =7 2 3 ^ l wer denominations and then subtracted. 16 9. From f of an acre of land a piece containing 72 sq. rd. 160 sq. ft. 39 sq. in. was sold. How much was left? 10. A merchant sold cloth for 384 6s. 5|d. which cost him 297 9s. 8|d. How much was his profit? 11. From a farm of 285 acres there were sold at one time 97f acres, and at another 38 A. 39 \ sq. rd. How' much was left? 12. A merchant bought 9 reams 18 quires 15 sheets of paper, from which he sold 3^ reams. How much remained unsold? 13. How long was it from Jan. 10, 1841, to May 7, 1853? PROCESS. ANALYSIS. Since the later date expresses 1853 5 7 * ne rea t er P er i O( l f time, we write it as the 1 & J. 1 1 10 minuend, and the earlier date as the subtra- hend, giving the month its number instead 12 3 27 of the name. We then subtract as in de- nominate numbers, considering 30 days one month, and 12 months one year. The remainder will be the time as correct as it can be expressed in months and days. 14. How long was it from Jan. 3, 1843, to Mar. 15, 1851? 196 DENOMINATE NUMBERS. 15. How old was a man who was born April 2, 1803, and who died Dec. 15, 1869? 16. A man bought a farm May 15, 1860, and paid for it Jan. 5, 1871. How long did it take to pay for it? 17. A legacy of $3000 was to be paid to a man 3 yr. 2 mo. 5 da. after Dec. 8, 1837. When was it to be paid? 18. How many years, months and days from the day of your birth? or, How old are you? 19. The American Civil War began April 11, 1861, and ended April 9, 1865. How long did it continue? 20. A note dated July 9, 1871, was paid October 10, 1876. How long did it run before it was paid? MULTIPLICATION. 290. 1. How much is 5 times 147 rd. 4 yd. 2 ft. 8 in.? PROCESS. ANALYSIS. We write the numbers as rd, yd. ft. in. * n s i m pl e numbers, and for convenience 147 4 2 8 begin at the right to multiply. P- 5 times 8 in. are 40 in., or 3 ft. 4 in. We write the 4 in. as inches in the prod- 2 mi. 9 9 2 1 4 uct, and reserve the 3 ft. to add with the product of feet. 5 times 2 ft. are 10 ft; 10 ft. + 3 ft. reserved equal 13 ft., or 4 yd. 1 ft. We write the 1 ft. in the product and reserve the 4 yd. to add to the product of yards. 5 times 4 yd. equal 20 yd. ; 20 yd. + 4 yd. reserved equal 24 yd., or 4 rd. 2 yd. We write the 2 yd. in the product and reserve the rods to add to the product of rods. 5 times 147 rd. are 735 rd.; 735 rd. + 4 rd. reserved equal 739 rd., or 2 mi. 99 rd., which we write in the product. Therefore the product is 2 mi. 99 rd. 2 yd. 1 ft. 4 in. RULE. Modify the rule for multiplication of simple numbers so that it may be applicable to denominate numbers. DIVISION. 197 2. Multiply 9 gal. 3 qt. 1 pt. 3 gi. by 7. 3. Multiply 17 Ib. 8 oz. 3 pwt. 15 gr. by 9. , 4. Multiply 1 T. 4 cwt. 35 Ib. 6 oz. by 10. 5. A farm consists of 7 fields each containing 18 A. 25 sq. rd. How much land does it comprise? 6. What is the length of a fence which encloses a square field each side of which is 28 rd. 5 yd. 2 ft. long? 7. How much wood is there in 7 piles, each containing 130. 7cd. ft. 24 cu. ft.? 8. What will 14^ yd. of lace cost at 2 5s. 6d. per yard? 9. What is the value of 4 loads of potatoes, each contain- ing 27 bu. 3 pk., at $.45 per bushel? DIVISION. 291. 1. Divide 27 bu. 3 pk. 5 qt. 1 pt. into 6 equal parts. PROCESS. ANALYSIS. Since the quan- 6 ) 2 7 bu. 3 pk. 5 qt. 1 pt. ^ is to be divided *> 6 e 1 u f l parts, each part will contain ^ 1~6" one-sixth of the quantity. One-sixth of 27 bu. is 4 bu. with a remainder of 3 bu. We write the 4 bu. in the quotient and unite the 3 bu. remaining with the number of the next lowest denomi- nation, making 15 pk. One-sixth of 15 pk. is 2 pk. and 3 pk. remaining. We write the 2 pk. in the quotient, and unite the 3 pk. remaining with the number of the next lower denomination, making 29 qt. One-sixth of 29 qt. is 4 qt. and 5 qt. remaining. We write the 4 qt. in the quotient, and unite the 5 qt. with the number of the next lowest denomination, making 11 pt. One-sixth of 11 pt. is If pt., which we write in the quotient. Therefore the quotient is 4 bu. 2 pk. 4 qt. If pt. RULE. Change the rule for the division of simple numbers so that it may be applicable to denominate numbers. 198 DENOMINATE NUMBERS. 2. In 8 bags there are 17 bu. 3 pk. 4 qt. How much does each bag contain ? 3. A gentleman divided his farm of 427 A. 131 sq. rd. equally among his 5 sons. What was the share of each ? 4. A brewer filled 4 casks of equal size from a vat con- taining 315 gal. 3 qt. How large was each cask? 5. 16 T. 1300 Ib. of hay was drawn at 9 loads. What was the average weight per load? 6. If a pile of wood containing 8 cords 100 cu. ft. be equally divided among 3 persons, how much will each receive ? 7. When 31 5s. 8d. is divided equally among 10 persons, how much does each receive? 8. If 31 cwt. 18 Ib. of tea is put up in packages, each containing 3 Ib. 8 oz. , how many packages will there be ? PROCESS. ANALYSIS. Since 31 cwt. 18 Ib. =49888 oz. the . divisor and . the dividend are similar 3 Ib. 8 oz. = 56 oz. denominate numbers, 4 9 8 8 8 oz. +- 5 6 oz. = 8 9 Of we may reduce them to the same denomi- nation, and then proceed to divide as in simple numbers. 9. How many times must a man dip with a dipper hold- ing 2 qt. 1 pt. so that he may empty a cask containing 31 gal. ? 10. If a man walks at an average rate of 23 mi. 160 rd. 4 yd. 2 ft. per day, how long will it take him to walk 100 miles? 11. If a man can travel 300 mi. in 13 days, how far can he travel daily? 12. How many barrels of sugar, each containing 2 cwt. 35 Ib., are there in 3 T. 4 cwt. 18 Ib.? 13. How many spoons, each weighing 2 oz. 10 pwt., can be made from 13 Ib. 7 oz. 15 pwt. of silver? 14. How many pickets 2 ft. 4 in. long and 2 in. wide, can be made out of 5 boards each 11 ft. 8 in. long arid 8 in. wide? LONGITUDE AND TIME. 199 LONGITUDE AND TIME. 292. 1. Where does the sun appear to rise? 2. How long will it be before it rises again? 3. Through how many degrees of space does it appear to pass in this daily motion? Ans. 360. 4. Since it seems to travel 360 in one day, or 24 hours, how great will be its apparent motion in 1 hour? 5. If the earth moves 15 in 1 hour, how far will it move in 1 minute? 6. If it moves 15' in one minute of time, how far will it move in 1 second? 7. How does the number of degrees passed over compare with the number of hours? The number of minutes of space with the number of minutes of time ? The number of seconds of space with the number of seconds of time? 8. When it is sunrise at New York, how long will it be before it is sunrise at a place 15 west of New York? 30 west? 45 west? 60 west? 9. When it is sunrise at New York, how long before was it sunrise at a place 15 east? 30 east? 45 east? 10. When it is sunrise at any place, how long will it be before it is sunrise at a place 15 west? 15 east? 30 west? 30 east? 11. When it is noon at any place, what time is it at a place 15 west? 15 east? 30 west? 30 east? 12. If I travel eastward will my watch become too slow or too fast? If I travel westward what change will take place? 13. What places have sunrise at the same time? Noon at the same time? Midnight at the same time? 293. A Meridian, is an imaginary line passing from the North Pole to the South Pole through any place. 200 DENOMINATE NUMBERS. 294:. Longitude is the distance east or west, from a given meridian. KELATION BETWEEN LONGITUDE AND TIME. 15 of Longitude make 1 Hour difference in time. 15 / of " make 1 Minute difference in time. 15 /x of " make 1 Second difference in time. 1 of " makes 4 Minutes difference in time. 1' of " makes 4 Seconds difference in time. WRITTEN EXERCISES. 1. The longitude of Boston is 71 3' 30" west; that of Ch> cinnati, 84 29' 31" west. What is the difference in time ? PROCESS. ANALYSIS. We first find the differ- o o Q' Q 1 " ence * n l n gi tu d e ^ ^ e two pl aces > an( l since there are 15 times as many de- 7 1 grees, minutes and seconds as there are 1 5 ^ 1 3 26 1 hours, minutes and seconds of time, we ~T~3 T-T- find ^ of 13 26' 1", which is 53 min. 44 sec., the difference in time. 2. When it is 12 o'clock M. at Philadelphia it is 5 o'clock, 10 min. P. M. at Paris. What is the longitude of Paris, the longitude of Philadelphia being 75 10' west? PROCESS. ANALYSTS. Since in 1 5 hr. 1 min. hour the earth moves 15 15 of distance, in 1 minute 15 7 - - o ^77T7 T^ . -r of distance, in 1 second 15 X/ 77 30' difference m Long. of distanc< ; the difference * ** in longitude will be 15 2 20' east, Long, of Paris. tim es as many degrees, min- utes and seconds of distance as there are hours, minutes and seconds of time. Since Philadelphia is 75 10 X west, and the difference is 77 3CK, the longitude of Paris is 2 20 7 east. LONGITUDE AND TIME. 201 To find the difference in time when the difference in longi- tude is given: Divide the difference in longitude, expressed in degrees, etc. , by 15; the several quotients will be the difference in time in hours, minutes, and seconds. To find the difference in longitude when the difference in time is given: Multiply the difference in time, expressed in hours, minutes and seconds, by 15; the several products will be the difference in longi- tude, in degrees, minutes, and seconds. 3. Two places are 32 IS' 24" apart. What is the differ- ence in time between them? 4. When it is noon at San Francisco it is 3 hr. 9 min. 7 sec. P. M. at Philadelphia. What is the longitude of San Francisco if that of Philadelphia is 75 10' west? 5. New York is 74 3' west longitude, and Paris, France, is 2 20' east. How much earlier is it sunrise in Paris than in New York? 6. Washington is 77 west of Greenwich, England. What is their difference in time? 7. When it is noon at Washington, which is 77 west, what time is it at New York, which is 74 3' west? 8. The difference in time between Halifax, Nova Scotia, and Charleston, S. C., is 1 hr. 5 min. 8 sec. What is their difference in longitude? 9. Pekin, China, is 116 27' 30" east longitude, and Wash- ington is 77 west longitude. When it is noon on January st at Washington, what time is it at Pekin? 10. A gentleman traveling found, on arriving at his des- nation, that his watch, which kept correct time, was 1 hr. min. slow. Which way was he traveling ? How far had traveled ? 295. The Metric System of weights and measures has been legalized by the United States, most of the coun- tries of Europe, and several countries of Central and South America. Although this system is extremely valuable on account of its sim- plicity, it is not in general use in this country, and hence is not treated as fully here as the other divisions of Denominate Numbers. 296. The Unit of Length, called the Metre (meeter), from which the system derives its name, is nearly one ten- millionth of a quadrant of the earth's circumference. 297. The Unit of Area, called the Are (air), is a square whose side is 10 metres. It contains 100 square metres. 298. The Unit of Solidity, called the Stere (stair), is a cube whose edge is one metre. 299. The Unit of Capacity, called the Litre (leeter), contains a volume equal to that of a cube whose edge is one- tenth of a metre. 300. The Unit of Weight, called the Gramme, is the weight of a cube of distilled water whose edge is one- hundredth of a metre. It must be weighed in a vacuum and at the period of its greatest density, 39.2 Fahrenheit. (202) METRIC SYSTEM. 203 301. From these standard units are derived the multiples and sub-multiples which are named to express units of higher or lower orders in the decimal scale. Thus, For multiples, Greek numerals are used: Deka, 10; Hecto, 100; Kilo, 1000; Myria, 10000. For sub-multiples the Latin ordinals are used: Deci, 10th; Centi, 100th; Milli, 1000th. Dekametre . . . . . means 10 Metres. Dekagramme .... " 10 Grammes. Hectometre " 100 Metres. Kilolitre " 1000 Litres. Myriagramme .... . 10000 Grammes. Centigramme .... " T J^ Gramme. Milligramme . . - . " j^ Gramme. MEASURES OF EXTENSION. 302. The Metre is the unit of length. TABLE. 10 Millimetres = 1 Centimetre = .3937079 in. 10 Centimetres = 1 Decimetre = 3.937079 in. 10 Decimetres = 1 Metre 39.37079 in. 10 Metres 1 Dekametre == 32.808992 ft. 10 Dekametres 1 Hectometre =c 19.927817 rd. 10 Hectometres = 1 Kilometre = .6213824 mi. 10 Kilometres = 1 Myriametre = 6.213824 mi. 303. The Are is the unit of land measure. TABLE. 1 Centiare = 1 Sq. Metre = 1.196034 sq. yd. 100 Centiares = 1 Are = 119.6034 sq. yd. 100 Ares =. 1 Hectare 2.47114 acres. 204 DENOMINATE NUMBERS. 304. The Square Metre is the unit for measuring ordinary surfaces, as flooring, ceiling, etc. TABLE. 100 Sq. Millimetres == 1 Sq. Centimetre .155 + sq. in. 100 Sq. Centimetres = 1 Sq. Decimetre = 15.5 + sq. in. 100 Sq. Decimetres = 1 Sq. Metre = 1.196 + sq. yd. 305. The Stere is the unit of wood and solid measure. TABLE. 1 Decistere = 3.531 + cu. ft. 10 Decisteres = 1 Stere = 35.316 + cu. ft, 10 Steres = 1 Dekastere = 13.079 + cu. yd. 306. The Cubic Metre is the unit for measuring many ordinary solids; as excavations, embankments, etc. TABLE. 1000 Cu. Millimetres = 1 Cu. Centimetre = .061 + cu. in. 1000 Cu. Centimetres = 1 Cu. Decimetre = 61.026 cu. in. 1000 Cu. Decimetres == 1 Cu. Metre. = 35.316 cu. ft. MEASURES OF CAPACITY. 307. The Litre is the unit of capacity, both of liquid and dry measure. It contains about a quart, liquid measure. TABLE. 10 Millilitres === 1 Centilitre = .6102 cu. in. = .338 fluid oz. 10 Centilitres = 1 Decilitre = 6.1022 cu. in. = .845 gills. 10 Decilitres *= 1 Litre = .908 quart = 1.0567 qt. 10 Litres = 1 Dekalitre = 9.08 quarts == 2.6417 gal. 10 Dekalitres === 1 Hectolitre = 2.8372 + bu. = 26.417 gal. 10 Hectolitres = 1 Kilolitre = 28.372 + bu. == 264.17 gal. 10 Kilolitres =* 1 Myrialitre *? 283.72 + bu. = 2641.7 + gal. METRIC SYSTEM. 205 MEASURES OF WEIGHT. 308. The Gramme is the unit of weight. TABLE. 10 Milligrammes 10 Centigrammes 10 Decigrammes 10 Grammes 10 Dekagrammes = 1 Centigramme = = 1 Decigramme = = 1 Gramme = = 1 Dekagramme = 1 Hectogramme = .15432 + gr. 1.54324+ " 15.43248+ " .35273 + oz. Av. 3.52739+ " " 10 Hectogrammes = 1 { K OTjSfeT} = 2.20462 + lb. " 10 Kilogrammes 10 Myriagrammes, 100 Kilogrammes 10 Quintals, or 1 000 Kilogrammes = 1 Myriagramme 1 == 1 Quintal ~| f Tonneau, or ") } = l { Ton r 22.04621+ " " 2204.62125+ " " The Kilogramme, or Kilo, is the unit of common weight in trade, and is a little less than 2^ Ib. Avoirdupois. The Tonneau is used for weighing very heavy articles, and is about 204^ Ib. more than a ton. EXERCISES. 1. What metric unit corresponds most nearly to our yard? How many metres are there in a rod? 2. What metric measure corresponds most nearly to our mile? 3. What unit expresses nearly one ton? 4. What unit is nearly equal to one quart? 5. Reduce 45 dekagrammes to grammes. 6. Express 7 dekametres, 25 centimetres, as metres. 7. How many litres are there in a kilolitre? In a deka- litre? 206 DENOMINATE NUMBERS. 8. What is the value of an acre in metric units? 9. What metric measure corresponds most nearly to one bushel? 10. Reduce 586.431 metres to decimetres. 11. What will be the cost of 324.16 hectogrammes of sugar at 22 cents per hectogramme? 12. A merchant bought 38 gal. of wine at $2.15 per gal. Did he gain or lose, and how much, by selling it at $5 per dekalitre? 13. Which is cheaper, to buy cloth at $3 per metre, or at $2.90 per yard? How much cheaper? 14. How much carpet a metre in width, is required to car- pet a room 4.2 metres long and 3.8 metres wide? 15. How long must a pile of wood be, so that it may con- tain 12 steres, if it is 3.5 metres wide and 3 metres high? 16. How much is gained by selling a piece of silk 100 metres in length, at $2.25 per yard, if it cost $2 per metre? 17. If a farm contains 1400 ares, what will be its value at $2.50 per are? 18. A barrel of flour contains 198 pounds. Express its weight in metric units. 19o How many hectares are there in a farm that is 1000 metres long and 180 metres broad? What is its value at $250 per hectare? 20. A bin is 5 metres square and 2.5 metres high. How many hectolitres of wheat will it hold ? 21. A room is 5.2 metres long, 4.5 metres wide, and 3.2 metres high. What will be the cost of plastering it at 35 cents per square metre? 22. Which is more profitable, and how much per ton, to sell sugar at 11 cents per Ib. or 23 cents per kilo? 23. A cask holding 2 hectolitres of molasses was sold at 18f cents per litre. How much more profitable would it be to sell the molasses at 90 cents per gallon? ii PERCENTAGE i jj 309. 1. In a quantity of sugar, 4 Ib. of every 100 Ib. were wasted. What part of it was wasted ? 2. A laborer digs potatoes for 10 bushels out of every 100. What part of the whole does he get ? 3. A merchant lost $3 out of every $100 worth of goods sold, on account of bad debts. What part of his sales did he lose? 4. Millers take 1 bushel out of every 10 bushels which they grind for customers, as pay for grinding. How many hundredths do they take ? 5. In a company of soldiers, 1 out of every 4 men w r as killed. How many was that per hundred, or per cent, f 6. In a school, 5 out of 20 pupils are more than 14 years old. How many is that per hundred ? How many per cent. ? 7. A man spent $3 out of every $4 earned. How many hundredths of his money did he spend? What per cent.? 8. A man whose income was $2500 annually, saved y 1 -^-, or 10 per cent, of it. How many dollars did he save ? 9. What is yfo-, or 5 per cent, of $800? 6 per cent, of $500? 10. What is 2 per cent, of $500? 4 per cent, of $900? 3 per cent, of $500? 11. What is 5 per cent, of $800? 6 per cent, of $900? 12. What is 8 per cent, of 500 bushels? 10 per cent, of 1000 pounds? (207) 208 PERCENTAGE. DEFINITIONS. 310. Per Cent, means by the hundred. It is a contraction of the Latin per centum, by the hundred. 311. The Sign offer Cent, is %. Thus, S% is read 8 per cent. 312. Percentage treats of computations which involve per cent. 313. Since per cent, is a number of hundredths, it is usu- ally expressed as a decimal. It may also be expressed as a common fraction. Thus, 2 per cent, is written 2^,, .02, or T J 7 . 5 per cent, is written 5^,, .05, or T f . 47 per cent, is written 47 ^>, .47, or T y 7 . 135 per cent, is written 135^, 1.35, or -}ff. 12J per cent, is written 12^, .12J, or f|$. f per cent, is written f ^>, .00 f, or T Q. 31J per cent, is written 31J^>, .31^, or fi$. 314. The expressions .12^, .31^, etc., may also be written .125, .3125, etc. ; and the complex fractional forms -J-g^, yf -Q, etc. , may be expressed as simple fractions : as, -^j-, -$$, etc. Express decimally, and in the smallest terms of their equiv- alent common fractions, the following: 1. 10%. 9. 6i%. 17. 18|% 2. 12%. 10. 125%. 18. 20|% 3. 20%. 11. 11%. 19. |% 4. 25%. 12. 33i%. 20. \% 5. 30%. 13. 16|%. 21. 4i% 6. 75%. 14. 2121%. 99 3 erf ~l"Tf /C 7. 87i%. 15. 31i%. 23. 7 T %% 8. 3%. 16. 66|%. 24. 37i% PERCENTAGE. 209 Express in hundredths or per cent., ^ of a number; \ of i> iV; "srV; 2T> t> T 8 Tr5 T^ A i5 f 5 i; i i> T> irJ r > "o T6 > 1 6 > "5 > 7T- Problems in Percentage involve the following elements: 315. The Base is the number of which the per cent, is aken. 316. The Hate is the number of hundredths taken. 317. The Percentage is the number which is a certain number of hundredths of the base. 318. The Amount is the sum of the base and per- centage. 319. The Difference is the base less the percentage. In the formulas, B. represents base ; R., rate ; P., percentage ; A., amount; and D., difference. CASE I. 320. To find the Percentage when the Base and Rate are given. EXERCISES. 1. What is 2. What is 3. What is 4. What is 5. What is 6. What is 7. What is 8. What is 9. What is 10. What is 11. What is 14 10 per cent. 5 per cent. 20 per cent. 2^ per cent. 15 per cent. per cent. per cent. 40 per cent. 4 per cent. 5 per cent. 25 per cent. , or ^o, of $150? , or -rfa, of $400? , or T 2 of any number is T 6 ^, or j 1 ^, of it, 6|^> of Or, $32.64 is T V of $32.64, which is $2.04. Since 6J^> of any number is .06^ of that number, 6J$, of $32.64 is FORMTILA.. f\i -C c^or nA 1*1 tno rk j .06J of $32.64, which is $2.04. RULE. Multiply the base by the rate. 13. Find 35% of $21.75. 14. Find 48% of $13.42. 15. Find 33|% of 465 gal. 16. Find 37|% of 816 mi. 17. A farmer who had a flock of 450 sheep, sold 33-|% of them. How many had he left? 18. A man whose salary was $2000 per year, spent 85% of it. How much did he save annually? 19. A farmer sold 37^% of his crop of 816 bu. of wheat at $1.56 per bu., and the rest at $1.60. How much did he realize from the sale of his wheat ? 20. If a merchant makes a deduction of 5% from a bill of $318.57, how much must be paid him? 21. A man bought a farm for $30000, and sold it for a gain of 25%. How much did he get for it? 22. Mr. Seymour sold $3000 worth of flour at a loss of 12|-%. How much did he realize from the sale? 23. A man having $40000, invested 15% in bank stock, 27% of it in bonds and mortgages, and the rest in a flouring mill. How much did the mill cost? 24. Two brothers each inherited $18500. The elder in- creased his inheritance 8% per year for 3 years. The younger lost 33-|% of his in the same time. What was then the value of the inheritance of each ? PERCENTAGE. 211 CASE II. 321. To find the Rate when the Base and Percentage given. 1. If a man earn $100 and spend $50 of it, what part of , does he spend ? How many hundredths ? How many per ent.? 2. In a piece of cloth containing 36 yards, 9 yards were imaged. What part of it was damaged? How many hun- Iredths of it? How many per cent.? 3. When I spend \ of my money, how many hundredths of it do I spend ? How many per cent. ? 4. If a farmer loses % of his crop by a flood, how many hundredths of it does he lose ? How many per cent. ? 5. If a merchant sells \ of his goods annually, how many hundredths does he sell ? How many per cent. ? 6. A farmer had 25 sheep, and 10 of them died. What part of his sheep died ? What per cent, of them ? 7. What part of $15 are $3? What per cent.? 8. What part of 12 bushels are 6 bushels? What per cent. ? 9. What per cent, of 24 cows are 8 cows? What per cent, are 12 cows? 10. What per cent, of 200 students are 40 students? Are 60 students? 11. What per cent, of 150 acres are 30 acres? Are 50 acres? Are 75 acres? 12. What per cent, of 80 hours are 16 hours? Are 20 hours? Are 40 hours? 13. What per cent, of 90 gallons are 30 gallons? Are 60 allons? Are 45 gallons? 14. If a man who earns $60 per month, expends $40 per aonth for necessary expenses, what per cent, of his earnings loes he save? 212 PERCENTAGE. 15. A merchant having 375 yards of cloth, sold 150 yards of it. What per cent, did he sell? PROCESS. ANALYSIS. 150 yards 1 5 yd. = m of 3 7 5 yd. , or are 2 "' or 1 of 375 yd., or 40^ of 375 yd. ^hu 40 hu Q r dredths; therefore 150 yards are .40, or 40%, 150 yd. -375yd. -.40, or 40% O f 375 yards. Or, Since the percentage is FORMULA. a product of the base by p _._ > __ jj the rate, if we divide the percentage by the base we shall obtain the rate. Therefore we divide 150 by 375, and obtain for a quotient .40, or 40^. RULE. Divide the percentage by the base. 16. What per cent, of 360 men are 60 men? 17. What per cent, of 840 men are 360 men? 18. What per cent, of 380 pages are 120 pages? 19. What per cent, of 45 hours are 25 hours? 20. What per cent, of 50 yards are 27 yards? 21. What per cent, of 36 pounds are 24 pounds? 22. A farmer who had a farm of 540 acres, sold 210 acres of it. What per cent, of it did he sell? 23. A man whose annual income is $1800, spends $1600 of it. What per cent, of it does he spend? What per cent, of it has he left? 24. A grocer sold tea for $1 that cost him $ .75. What per cent, of the cost did he gain? 25. What per cent, of 30000 bushels are 50 bushels? 26. What per cent, of the cost does a hatter gain by sell- ing hats at $7 each, that cost $5.50? 27. A real-estate agent gets $60 for selling my house for $4000. What % of the sale does he receive for his services? PERCENTAGE. 213 28. I paid SI 20 for insuring a boat-load of wheat valued at $10000. What % of the value of the cargo was received for insuring it? 29. A man who had 1000 acres of land, gave \ of it to his eldest son, \ of it to another, and the remainder he divided Dually between his 3 daughters. What % of the whole did ch receive? CASE III. 322. To find the Base when the Percentage and Rate Eire given. 1. A man spent $15, which was 10% or y 1 -^- of all the aoney he had. How much money had he ? 2. My net profit from an investment was $800, which was 25% or T 2 ^ 5 of the amount invested. How much had I in- vested? 3. Of what sum is 18 dollars 33^%, or fj$, or i? 4. Of how many days are 15 days 29% ? 30 days 37^% ? 5. Of what sum is 25 dollars 62^%, or f|f, or f ? 6. Of what number is 120 6% ? 150, 30% ? 180, 60% ? 7. Of what number is 40 80% ? 20, 60% ? 30, 150% ? 8. A drover lost 450 sheep, which was 75% of his flock. How many sheep had he? ANALYSIS. Since 75^ or -$ or f of the number is 450, J of the number is J of 450 or 150; and since 150 is \ "of the number, the whole number of sheep will be 4 times 150, or 600. Therefore he had 600 sheep. Or, Since the percentage is the product of the base by the rate, if the percentage is divided by the rate the quotient will be the base. Therefore we divide 450 by .75. PROCESS. or = 450 Whole = 600 Or, 450--. 75 = 600 FORMULA. RULE. Divide the percentage by tlie rate. 214 PERCENTAGE. Of what number is 9. 385 12$% ? , 10. 245 10%? 11. 125 15%? 12. 7.15 33^%? Of what number is 13. $53.25 14. 27.5bu. 8%? 15. 168 men 8%? 16. 231 oxen 7% ? 17. A farmer sold 275 barrels of apples, which were 75% of all he had. How many had he? 18. A man sold 25% of a mill for $3750. At this rate what was the mill worth ? 19. A man who owned 40% of a foundry sold 25% of his share for $10000. What was the value of the foundry ? 20. A farmer after selling 110 A. 43 sq. rd. of land had 90% of his land left. How much land had he at first? 21. A farm cost $3000. One-third of this sum was 62$% of what the house and barn on the farm cost. What was the cost of the house and barn? 22. A man indebted to me paid me $80, which was S$% of $ the amount due. How much did he still owe? 23. A merchant sold 4500 bushels of wheat at $1.60 per bushel. The amount received was 90% of the cost of the wheat. How much did it cost? 24. Mr. A. sold a lot for $8000, which was only 40% of the amount he paid for it. How much did he pay for it ? 25. A man pays $600 a year rent; 75% of this sum is just 33$% of $ his income. What is his income? 26. A man owning $ of a vessel sold 25% of his share for $3350.50. At that rate what was the value of the vessel? 27. The amount paid by insurance companies to the people of St. John, New Brunswick, for losses caused by the great fire in 1877, was about $7500000, which was 37$% of the estimated loss. What was the estimated loss ? 28. 25% of $ of 60 is 75% of $ of what number? 29. | of 40% of 100 is 5% of 10 times \ of what number? PEKCENTAGE. 215 CASE IV. 323. To find the Base when the Amount and Rate given. 1. A gentleman increased his collection of horses by an addition of ^ of the number, and then he had 15. How many had he before he made the addition ? 2. A coal dealer in selling coal at $6 a ton received 20% or -J- more than it cost him. What did it cost him ? 3. A grocer in selling sugar at $.11 a pound gains 10% or -^ of the cost. What was the cost? 4. A merchant sold cloth at an advance of 25% on the cost, receiving $1.25 per yard for it. What was the cost? 5. A man's monthly expenses were 33^-% more during 1876 than during 1875. During 1875 they were $120. What were they in 1876? 6. A certain number increased by 20% of itself is 36. What is the number? 7. After adding to a number 37-^% of it, the sum is 33. What is the number? 8. What number increased by 35% of itself equals 540? PROCESS. ANALYSIS. Since the number is in- 13. 5. _ 5 4 Q creased by 35^, or -f^ of itself, the amount ^1 will be l T 3 o 5 Q- or ^ times the number; T 7~ and since |J J of the number === 540, T fo The number = of u = _,_ of ^ which ig 4 . and gince Or, 4 is T ^ 7 of the number, the number will (1+35) be 100 times 4, which is 400. Or, A number increased by 35^ of itself equals 135/ or 1.35 of itself. And since 1.35 times the number equals 540, the FORMULA. number may be found by dividing 540 A~(1 + E)=B. by 1.35. RULE. Divide the amount by 1 -(- the rate. 216 PEKCENTAGE. 9. What number increased by 27% of itself equals 508? 10. What number increased by 33|-% of itself equals 492 ? 11. What number increased by 16f % of itself equals 329? 12. What number increased by 62^-% of itself equals 910? 13. A man owes $15400, which is 10% more than his property is worth. What is the value of his property ? 14. A man sold a horse for $345, which was 15% more than it cost him. How much did it cost ? 15. A clerk's salary was increased 30%, and now it is $1950. What was it before the increase? 16. A man expended $3750 in repairs upon his house. This sum was 25% more than -J- the cost of the house. How much did it cost? 17. The number of pupils in a certain school during 1876 was 872, which was 9% more than the number in attendance during 1875. What was the attendance during 1875 ? CASE V. 324. To find the Base when the Difference and Rate are given* 1. A gentleman sold 25% or \ of the number of his horses and had 15 left. How many had he ? 2. By selling coal at $6 per ton a coal dealer lost 20% or J- of the cost. What was the cost ? 3. A grocer sold sugar at 9c. per pound, and lost 10% or -^ of the cost. What did it cost? 4. A merchant sold cloth at $ .75 per yard, thereby losing 25% of the cost. What was the cost? 5. A man's monthly expenses are 33^% less this year than last year. This year they are $120 ; what were they last year ? 6. A certain number diminished by 20% of itself is 36. What is the number? 7. What number diminished by 10% of itself equals 45? PERCENTAGE. 217 8. After subtracting from a number 37|% of it, the re- minder is 25. What is the number? 9. What number diminished by 27% of itself equals 401.5? PROCESS. ANALYSIS. Since the number is de- 73 401 5 creased by 27^,, or T 2 ^ of itself, the 1 i f- remainder will be $ of the number, which equals 401.5; T ^ F of it equals The number 550 7 V of 401.5, which is 5.5; and since ^ 5.5 is T ^ of the number, the number ' will be 100 times 5.5, which is 550. (1 .27) Or, 401. 5-v-. 73 = 550 ^ number diminished by 27^ of itself, equals 73$,, or .73, of itself; and since .73 of the number equals FORMULA. AM , ,, , . , , 401.5, the number will be equal to D--(l R)==B. 401.5 -f- .73, which is 550. RULE. Divide the difference by 1 minus the rate. 10. What number diminished by 36% of itself equals 336? 11. What number diminished by 40% of itself equals 432? 12. What number diminished by 55% of itself equals 285? 13. What number diminished by 28% of itself equals 307? 14. A clerk, after paying out 75% of his salary, had $450 left. What- was his salary? 15. A farmer, after selling 30% of his wheat, found that he had 350 bushels left. How much had he at first? 16. A man sold some land for 30% less than he asked for it, getting $29.24 per acre. What was his asking price? 17. A regiment losing 15% of its men, had 527 left. How many had it at first? 18. A speculator lost 10% of his money during the year 1875 and 10% of the remainder during 1876. He then had $40500 left. How much had he at first? 19. A merchant's profit in 1876 was $10318, which was 23% less than in 1875. What was his profit in 1875? 74i --'-.,-.' .....---:: -^j^^^ xiltffr, ;mimiiimimninj JJMiMIA^ 325. 1. When a sum equal to 5% of the amount of money lent is paid for the use of it for one year, how much will be paid for the use of $100 for 1 year? For 2 years? 2. When the allowance for the use of money is 6% per year, what is the allowance for the use of $100 for 1 year? For 2 years? For 3 years? For 3^ years? 3. When the sum paid for the use of money is 8% per year, what must be paid per year for $50? For $500? 4. When the sum paid for the use of money is 12% yearly, what must be paid for the use of $100 for 1 year? 5. When the allowance for the use of money is 8% per year, what must be paid for the use of $100 for 6 months? For 1 month? For -J- month? For % month? For month? For 10 days? For 20 days? 6. When 6% is paid per year for the use of money, how much will $500 amount to in 2 years? In 3 years? 7. When $500 is loaned for 1^ years at 8% per year, what will be the amount? DEFINITIONS. 326. Interest is the sum paid for the use of money. 327. The Principal is the sum for the use of which interest is paid. (218) INTEREST. 219 328. The Amount is the sum of the principal and in- terest. 329. The Hate of Interest is the annual rate per cent. 330. Legal Interest is interest according to rate fixed by law. 331. Usury is interest computed at a higher rate than the law allows. 332. A Note, or Promissory Note, is a written promise to pay a sum of money at a given time. 333. PRINCIPLE. The interest is equal to the product of the principal, rate, and time expressed as years. 334. When the rate per cent, is not specified in notes, accounts, etc., the legal rate is always understood. On debts due the United States the rate is 6^ . The following table contains the rates of interest in the United States. The first column gives the legal rate, the second the rate that may be collected if agreed to in writing. NAME OF STATE. R; PER TE CENT. NAME OF STATE. RJ PER TE CENT. 8 Mississippi 6 10 Arkansas 6 10 6 10 10 Any. 10 Any. 10 Any 6 Con necticut 7 New Jersey 7 10 Any. New Mexico 6 Any. Dakota. ... ... . 7 12 New York. . . 7 Delaware 6 North Carolina 6 8 District of Columbia g 10 10 12 8 Any Nevada 10 7 12 Ohio 6 *Y- Idaho 10 24 10 12 Illinois 6 10 Pennsylvania 6 Indiana Indian Territory 6 10 Rhode Island South Carolina 6 7 Any. Any Iowa. g 10 10 Kansas 7 12 Texas. 8 Kentucky . . 6 10 Utah 10 Any Louisiana 5 g g Maine 6 Any Virginia g Maryland 6 West Virginia. g Massachusetts 6 Any Washington Territory 10 Any. Michigan 7 10 7 10 Minnesota 7 12 Wvominc 12 220 PEKCENTAGE. TO COMPUTE INTEREST. 335. 1. What is the interest of $200 for 1 year at 6^ ? 2. What is the interest of $200 for 2 years at 7^ ? 3. What is the interest of $300 for 3 years at 5^ ? 4. What is the interest of $400 for 1 years at 8 ft ? 5. What is the interest of $400 for 3 months at 6 ft ? 6. What is the interest of $600 for 1 month at 6^ ? 7. What is the interest of $600 for 10 days at 6^ ? 8. What is the interest of $500 for 15 days at 8^ ? 9. What will be the amount of $100 for 2 years at 6 ft ? 10. What will be the amount of $200 for 3 years at 4^ ? 11. What will be the amount of $300 for 2 years at 6^ ? 12. What will be the amount of $150 loaned for 1^ years at 5^? 13. Find the interest of $234.27 for 2 yr. 7 mo. 12 da. at 6^ ? PROCESS. ANALYSIS. Since the in- $ 2 3 4 27 terest for 1 year is 6^ of ^ ^ the principal, we find .06 of $234.27, which is $14.0562; $14.0562 Int. for 1 yr. and since $14.0562 is the in- 2 terest for 1 year, the interest $28.1124 Int. for 2 yr. for 2 7 ears wil1 be twice that Q a a n a T sum, which is $28.1124. The 8.6676 Int. for 7 2-5 mo. interest for 1 month is one- $36.78 Int. 2 yr. 7 mo. 12 da. twelfth of the interest for 1 ^ year, or $1.1713 ; and the in- ' terest for 7 mo. and 12 days, $ 2 3 4 2 7 or 7f nio., is 7| times $1.1713, or $8.6676. This added to the interest for 2 years, gives the 1 2 )$14.Q562 Int. for 1 yr. interest for 2 years, 7 months, 1.1713 Int. for 1 mo. 12 da ^ s - Or > o -. A We may find the interest - '. for 1 year as before, and then $36.78 Int. 2 yr. 7 mo. 12 da. for 1 month. We then mul- INTEREST. 221 tiply the interest for 1 month by the number of months and fractions of a month. Thus, in 2 years, 7 months, there are 31 months, and in 12 days there are Jf or T % of a month. Therefore, the entire interest may be found by multiplying $1.1713 by 31.4, which is $36.78. Since there are 30 days in a month, one-third of the number of days will be tenths of a month. EULE. I. Find the interest for 1 year and multiply this by the time expressed as years and fractions of a year. Or, II. Find the interest for 1 month and multiply this by the time expressed as months and fractions of a month. 14. What is the interest of $25.16 for 1 yr. 6 mo. at 6^ ? 15. What is the interest of $36.24 for 2 yr. 4 mo. at 1% ? 16. What is the interest of $48.20 for 2 yr. 4 mo. at 8^ ? 17. What is the interest of $2000 for 3 yr. 7 mo. at 9^ ? 18. Find the amount of $590.50 for 3 yr. 6 mo. at 1%. 19. Find the amount of $640.82 for 2 yr. 7 mo. at 8^. ( 20. Find the amount of $725.83 for 3 yr. 6 mo. at 21. Find the amount of $618.24 for 2 yr. 5 mo. at 22. Find the amount of $312.29 for 3 yr. 5 mo. at 23. Find the interest of $718.24 for 5 mo. 10 da. at 24. Find the interest of $127.46 for 3 mo. 15 da. at 25. Find the interest of $364.18 for 2 mo. 12 da. at 26. Find the interest of $318.29 for 9 mo. 10 da. at 27. Find the interest of $312.24 for 2 mo. 20 da. at 28. Find the interest of $1614.25 for 20 da. at 7 29. Find the interest of $1318.29 for 24 da. at 30. Find the interest of $4684.68 for 11 da. at 31. If you lend $500, how much will be due you in 3 yr. 6 mo. 21 da., interest at 7 ft ? 32. What is the interest on $784.25 from Aug. 7, 1874, to July 19, 1877, at 8^ ? What is the amount? 33. How much interest is due on $500, that has been loaned at interest since Jan. 1, 1876? 222 PEBCENTAGE. OTHER METHODS. ALIQUOT PARTS. 336. 1. What is the interest and amount of $520.32 for 2 yr. 5 mo. 15 da. at If ? PROCESS. ANALYSIS. Since the in- $520.32 terest is 7 ft of the principal, m 07 we find .07 of $520.32, which is $36.4224, the interest for 1 $36.4224lnt.forlyr. y ^ Twice ^^ ^ the interest for 2 years, which $72.8448 Int. for 2 yr. is $72.8448. One-third of the 12.1408 Int. for 4 mo. interest for 1 year is $12.1408, 3.0352lnt.forlmo. the interest f r 4 m nths ' ^ e . - n One-fourth of the interest for 1 . 5 1 7 O Int. for 15 da. A , . ^ no r ,, . 4 months is $3.0352, the in- $ 89.5384 Int. for 2 yr. 5 mo. 15 da, terest for 1 month. One-half $520.32 Principal. the interest for 1 month is ac AH Q $1.5176, the interest for 15 $609.86 Amount. ' ... days. The sum of these amounts is the interest for 2 years 5 months 15 days, which is $89.5384. This sum, added to the principal, gives the amount. Solve the following by aliquot parts: 2. What is the interest of $324.22 for 3 yr. 4 mo. at 6^ ? 3. What is the interest of $218.90 for 2 yr. 7 mo. at 7^ ? 4. What is the interest of $36.48 for 2 yr. 5 mo. 15 da. at 6^? 5. What is the interest of $40.28 for 1 yr. 7 mo. 20 da. at 6^? 6. What is the interest of $56.24 for 2 yr. 5 mo. 18 da. at 7^? 7. What is the interest of $24 96 for 3 yr. 1 mo. 6 da. at 8? INTEREST. 223 8. What is the interest of $48.72 for 2 yr. 2 mo. 16 da. 9. What is the interest of $36.18 for 2 yr. 5 mo. 21 da. 10. What is the interest of $20.25 for 3 yr. 1 mo. 16 da. 11. What is the interest of $30.24 for 2 yr. 8 mo. 15 da. t 8^? SIX PER CENT. METHOD. 337. The interest on $1, at 6^ per annum, For 12 months, is 06 For 2 months, is 01 For 1 month (30 days), is. . . .005 For 6 days (| month), is. . . .001 For 1 day, is 1. What is the interest of $125 for 2 yr. 3 mo. 16 da. at 6^ ? PROCESS. ANALYSIS. Int. of $1 for 2 yr. at 6^, is .12. a I 2 5 I n k ^ $1 f r 3 mo. at 6^> is .015. ., 7 4 Int. of $1 for 16 da. at 6f is .002$. The sum of these, .137f , is the interest of $1 for $17.208 the given time at the given rate, and since the interest of $1 is .137, the interest of $125 will be times that sum, which is $17.208. 1. When it is required to find the interest at any other rate than 6^>, first find it at 6^,, then increase or decrease this result by such a part of it as the given rate is greater or less than 6^. Thus, if the rate is 7^, increase the interest at 6^ by 1 of it; if the rate is 5^. decrease it by of it ; if the rate is 8^, increase it by -f , or | of it ; if the rate is 9^, increase it by J of it, etc. 2. Exact or Accurate interest requires that the year should be considered 365 days, for a common year, and 366 days for a leap year, instead of the ordinary method of considering 12 months of 30 days each, or 360 days a year. 224 PERCENTAGE. Find the interest on the following : 2. On a note for $185.26 for 1 yr. 4 mo. 13 da. at 3. On a note for $368.18 for 3 yr. 5 jno. 22 da. at 4. On a note for $284.25 for 2 yr. 7 mo. 18 da. at 5. On a note for $183.17 for 1 yr. 8 mo. 17 da. at 6. On a note for $215.25 for 3 yr. 2 mo. 18 da. at 7. On a note for $204.37 for 2 yr. 5 mo. 15 dar at 8. On a note for $186.15 for 3 yr. 7 mo. 23 da. at 9. On a note for $315.30 for 1 yr. 9 mo. 27 da. at 10. On a note for $379.15 for 1 yr. 8 mo. 11 da. at 11. On a note for $685.31 for 4 yr. 1 mo. 15 da. at 12. On a note for $516.28 for 3 yr. 6 mo. 28 da. at 13. On a note for $423.15 for 2 yr. 7 mo. 10 da. at 14. On a note for $304.27 for 1 yr. 3 mo. 21 da. at 15. On a note for $516.24 for 2 yr. 1 mo. 13 da. at 12% ? Find the amount of the following notes when due : 16. $150.15. CINCINNATI, 0., Jan. 31, 1877. Three months after date, for value received, I promise to pay John T. Jones, or order, One Hundred Fifty -fa Dollars, with interest at 6fi. CHARLES C. THOMSON. 17. $328.35. ST. PAUL, MINN., Oct. 1, 1877. On the 15th day of January, 1878, for value received, I promise ' to pay "to S. E. Hoyt, or order, Three Hundred Twenty-Eight T 3 ^ Dollars, with interest at 8%. J W KAY l&. : $31$.75*. BUFFALO, N. Y., May 3, 1876. For value received, on demand I promise to pay to J. C. Coe, 'of order, Three Hundred/ Fifteen -$fo Dollars, with interest. HENRY B. EOBESON. Paid, June 5th, 1877. How much was due? INTEKEST. 225 COMPOUND INTEREST. 338. Compound Interest is interest upon the prin- cipal and its unpaid interest, combined at regular intervals. It is usually compounded annually, semi-annually, or quarterly. Unless some other condition is mentioned in the written obligation, the interest is understood to be compounded annually. WRITTEN EXERCISES. 1. Find the compound interest of $250 for 2 yr. 3 mo. at ( $250 15 PROCESS. Prin. for 1st yr. Int. for 1st yr. Prin. for 2d yr. $265 15.90 Int. for2dyr. $280.90 Prin. for 3d yr. 4.21 Int. for 3 mo. $285.11 Amount for 2 yr. 3 mo. 250.00 First Principal. $35.11 Comp. Int. for 2 yr. 3 mo. inal principal, and obtain $35.11, the compound interest required. ANALYSIS. Since the in- terest is compounded an- nually, we first find the inter- est of $250 for 1 yr. We add this interest to the principal, and compute the interest on this amount for another year. We add this interest to the principal as before, and com- pute interest on this amount for 3 months, which we add to the principal. From this amount we subtract the orig- RULE. Find the interest of the principal for the first period of time at the end of which interest is due. Add this interest to the principal, and compute the interest upon this amount for the next period, and so continue. Subtract the given principal from the last amount, and the. re- mainder will be the compound interest. 1. If the interest is compounded semi-annually, the rate is consid- ered as one-half the annual rate; if quarterly, one-fourth, etc. 226 PERCENTAGE. 2. When the time consists of years, months, and days, find the compound interest for the greatest number of entire periods, and to this add the simple interest upon the amount for the rest of the time. 2. Compute the compound interest on $315 for 2 yr. 6 mo. at 6^. 3. Find the amount of $324.18 for 3 yr. 5 mo. at 7 compound interest. 4. What is the compound interest on $525.75 for 3 yr. 4 mo. at 6^ ? Computations in compound interest may be shortened very much by the use of the table on the following page. 5. What is the compound interest of $325.10 for 3 yr. 2 mo. at &% ? ANALYSIS. By referring to the table, the amount of $1 for 3 yr. is found to be $1.191016. Computing interest on this sum for the remain- ing 2 mo., the amount is $1.208881. Since the amount of $1 for the given time is $1.208881, the amount for $325.10 will be 325.10 times that sum. If from this product the principal is subtracted, the re- mainder is the compound interest. 6. Find the compound interest of $600.50 for 3 yr. 7 mo. at 6^. 7. Find the compound interest of $318.25 for 2 yr. 4 mo. at lf c . 8. Find the compound interest of $412.08 for 3 yr. 2 mo. 10 da. at 6^. 9. Find the compound interest of $310.24 for 2 yr. 5 mo. 15 da. at 8^. 10. What is the difference between the simple interest on $328 for 2 yr. 7 mo. at 7^, and the compound interest on same sum for the same time at 6^ ? 11. If I deposit $300 in a savings bank which compounds at 6^ semi-annually, how much will be due me in 3^ years ? INTEREST. 227 COMPOUND INTEREST TABLE, Showing the amount of $1, at various rates, compound int. from 1 to 20 years. Yrs. 1% per cent. 3 per cent. 3> per cent. 4 per cent. 5 per cent. 6 per cent. ~T~ 1.025000 1.030000 1.035000 1.040000 1.050000 1.060000 2 1.050625 1.060900 1.071225 1.081600 1.102500 1.123600 3 1.076891 1.092727 1.108718 1.124864 1.157625 1.191016 4 1.103813 1.125509 1.147523 1.169859 1.215506 1.262477 5 1.131408 1.159274 1.187686 1.216653 1.276282 1.338226 6 1.159693 1.194052 1.229255 1.265319 1.340096 1.418519 7 1.188686 1.229874 1.272279 1.315932 1.407100 1.503630 8 1.218403 1.266770 1.316809 1.368569 1.477455 1.593848 9 1.248863 1.304773 1.362897 1.423312 1.551328 1.689479 10 1.280085 1.343916 1.410599 1.480244 1.628895 1.790848 11 1.312087 1.384234 1.459970 1.539454 1.710339 1.898299 12 1.344889 1.425761 1.511069 1.601032 1.795856 2.012197 13 1.378511 1.468534 1.563956 1.665074 1.885649 2.132928 14 1.412974 1.512590 1.618695 1.731676 1.979932 2.260904 15 1.448298 1.557967 1.675349 1.800944 2.078928 2.396558 16 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352 17 1.521618 1.652848 1.794676 1.947901 2.292018 2.692773 18 1.559659 1.702433 1.857489 2.025817 2.406619 2.854339 19 1.598650 1.753506 1.922501 2.106849 2.526950 3.025600 20 1.638616 1.806111 1.989789 2.191123 2.653298 3.207136 Yrs. 7 per cent. 8 per cent. 9 per cent. 10 per cent 11 per cent. 12 per cent. 1 1.070000 1.080000 1.090000 1.100000 1.110000 1.120000 2 1.144900 1.166400 1.188100 1.210000 1.232100 1.254400 3 1.225043 1.269712 1.295029 1.331000 1.367631 1.404908 4 1.310796 1.360489 1.411582 1.464100 1.518070 1.573519 5 1.402552 1.469328 1.538624 1.610510 1.685058 1.762342 6 1.500730 1.586874 1.677100 1.771561 1.870414 1.973822 7 1.605781 1.713824 1.828039 1.948717 2.076160 2.210681 8 1.718186 1.850930 1.992563 2.143589 2.304537 2.475963 9 1.838459 1.999005 2.171893 2.357948 2.558036 2.773078 10 1.967151 2.158925 2.367364 2.593742 2.839420 3.105848 11 2.104852 2.331634 2.580426 2.853117 3.151757 3.478549 12 2.252192 2.518170 2.812665 3.138428 3.498450 3.895975 13 2.409845 2.719624 3.065805 3.452271 3.883279 4.363492 14 2.578534 2.937194 3.341727 3.797498 4.310440 4.887111 15 2.759031 3.172169 3.642482 4.177248 4.784588 5.473565 16 2.952164 3.425943 3.970306 4.594973 5.310893 6.130392 17 3.158815 3.700018 4.327633 5.054470 5.895091 6.866040 18 3.379932 3.996019 4.717120 5.559917 6.543551 7.689964 19 3.616527 4.315701 5.141661 6.115909 7.263342 8.612760 20 3.869684 4.660957 5.604411 6.727500 8.062309 9.646291 228 PERCENTAGE. ANNUAL INTEREST. 339. Anmial Interest is simple interest on the prin- cipal and upon any interest overdue, when the contract con- tains the words, "with annual interest," or, "with interest payable annually." Annual interest is not considered legal in some States. 1. Find the amount of $3500 for 4 yr. 6 mo., with inter- est payable annually at 6^5 < PROCESS. ANALYSIS. Since Int. of $3500 for 4J yr. == $945 annual interest is sim - Int. of $210 for 8 yr.== $100.80 pl ? ! nt f est n the principal and upon Annual Interest $1045.80 any over-due interest, 83500 + 81045.80 = 84545.80, Amt. we first find the inter ' est upon the principal, which is $945, and then upon the interest due. The interest for each year is $210. The interest for the first year remained unpaid for 3J years ; that for the second year, 2J years; that for the third year, 1| years; and that for the fourth year, for \ year; therefore the annual interest, $210, drew interest for 3J + 2+l+J years, or, 8 years, which is $100.80. This sum, added to the simple interest, $945 $1045.80, the annual interest. This sum added to the principal, $3500 = $4545.80, the amount due. RULE. Compute the interest on the principal for the entire time, and on each year's interest from the time it was due up to the end of the period. The sum of these interests will be the annual interest. 2. How much -is due upon a note of $350 which has run 4 years, interest at 8^, payable annually? 3. How much w r as due April 15, 1877, on a note for $750, dated Jan. 1, 1873, with interest at 6^, payable annually? INTEREST. 229 PARTIAL PAYMENTS. 340. A Partial Payment is a payment in part of a note or other obligation. 341. An Indorsement is the statement of the amount of a payment and the time when it was made. It is written on the back of the note or other written obligation. 34:2. Business men often settle notes and accounts running for one year or less by what is known as the Mercantile Rule. MERCANTILE RULE. Find the amount of the principal at the time of settlement. Find the amount of each payment from the time it was made until the time of settlement, and from the amount of the principal subtract the amounts of the payments. 1. A note for 8850, on demand with interest at Ifo dated Jan. 1, 1876, was indorsed as follows: April 10, 1876, $200; Sept. 15, 1876, $255. How much was due Nov. 15, 1876? 2. What is the balance due at the end of a year on a note for $1800, dated May 15, 1875, on which the following pay- ments had been made: Sept. 20, 1875, $300; Jan. 18, 1876, $200; April 20, 1876, $1000; when the rate is 1% ? 3. $585.25. BUFFALO, N. Y., March 3, 1876. Eight months after date, for value received I promise to pay to the order of E. 8. Farran, Five Hundred Eighty-five T 2 ^ Dollars, with interest at 1%. H. S. LAUPHIEE. This note was indorsed as follows: June 8, 1876, $325; Aug. 4, 1876, $84.30; Sept. 2, 1876, $100. What was due on the note at maturity? 230 PERCENTAGE. 343. Most of the States have adopted the United States Hule for computing the amount due upon any obligation where partial payments are made, based upon the following principle. PRINCIPLE. The indebtedness should be computed whenever a payment is made, but the principal must not be increased by the addition of interest. WRITTEN EXERCISES. 1. A note was given, Jan. 1, 1870, for $700. The fol- lowing payments were indorsed upon it: May 6, 1870, $85; July 1, 1871, $40; Aug. 20, 1871, $100; Jan. 10, 1873, $350. How much was due Sept. 30, 1874, interest at 6%? PROCESS. Principal $700.00 Int. to May 6, 1870, 4 mo. 5 da. . . . . . _ 1 4^>? Amount 714.58 First payment 85.00 New Principal 629,58 Int. from May 6, 1870, to July 1, 1871,! yr. 1 mo. 25 da 43.55 Second payment, less than interest due . . $40.00 Int. on $629.70 from July 1, 1871, to Aug. 20, 1871 1 mo. 19 da 5.14 Amount 678.27 Third payment to be added to second . . . $100 J40.00 New Principal . 538.27 Int. from Aug. 20, 1871, to Jan. 10, 1873,! yr. 4 mo. 20 da. ... 44.85 Amount . . . . . . . 583.12 Fourth payment 350.00 New Principal . 233.12 Int. from Jan. 10, 1873, to Sept. 30, 1874 1 yr. 8 mo. 20 da. 24.08 Amount due, Sept. 30, 1874 . . . $257.20 PARTIAL PAYMENTS. 231 UNITED STATES KULE. Find the amount of the principal tc a time when a payment, or the sum of the payments, equals or exceeds the interest due, and from this amount subtract such payment or payments. With the remainder as a new principal, proceed as before. 2. A note for $2500, dated July 10, 1871, bore the fol- lowing indorsements: Sept. 15, 1871, $150; Nov. 12, 1871, $300; Dec. 1, 1871, $100; April 3, 1872, $325; May 15, 1872, $275; Sept. 20, 1872, $1000. How much was due Jan. 1, 1873, the rate of interest being 6%? 3. How much was due at maturity on a note for $2150, dated Sept. 20, 1873, to run 2 years 6 months, on which the following payments were indorsed: Dec. 15, 1873, $75; Feb. 4, 1874, $200; April 3, 1874, $150; July 1, 1874, $500; Dec. 16, 1874, $1000, the rate of interest being 8%? 4. A note for $6725, dated Feb. 10, 1875, had the following indorsements: May 5, 1875, $275; Aug. 15, 1875, $50; Nov. 12, 1875, $1000; Jan. 3, 1876, $184.25; Sept. 13, 1876, $84.10; Dec. 23, 1876, $1000. How much was due Feb. 10, 1877, interest at 6% ? 5. A bond was given April 4, 1870, for $5825, with inter- est at 8%. The following payments were indorsed upon it: May 15, 1871, $728.50; April 8, 1872, $1000; Dec. 12, 1872, $125; July 9, 1873, $980; June 12, 1874, $1000. How much remained due upon the bond April 4, 1875? 6. Sept. 25, 1872, James Hanna gave his note for $895.75 with interest at 10%. He paid on it as follows: Jan. 10, 1873, $25; Oct. 12, 1873, $200; Jan. 18, 1874, $75; March 25, 1874, $187.50; Jan. 1, 1875, $375. How much was due when he paid the note, Nov. 15, 1875? 7. Required, the balance due on a note dated Jan. 1, 1875, for $580 at 5^, to run 2 years, on which a payment of was made every 3 months. 232 PERCENTAGE. 8. A note for $10000, with interest, dated Milwaukee, Wis., Dec. 12, 1875, was indorsed as follows: Feb. 23, 1876, $750; July 17, 1876, $108.25 ; Nov. 23, 1876, $3000 ; Jan. 18, 1877, $4000. How much was due May 12, 1877, interest at 8^ ? 9. Required, the balance due July 8, 1876, on a note for $3124.75, at 8^ interest, dated Feb. 15, 1874, on which a payment was made Dec. 23, 1874, of $985 ; another of $875.35, Feb. 15, 1875; another of $1025, Feb. 20, 1876. 10. $1885.75. SCHENECTADY, N. Y., Feb. 10, 1874. Two years after date, for value received I promise to pay W. W. Heilbronner or order. One Thousand Eight Hundred Eighty-five -fifa Dollars, with interest. c H VIDRARD. On this note were the following indorsements: June 30, 1874, $50; Nov. 8, 1874, $100; Feb. 5, 1875, $125; April 17, 1875, $500; Dec. 1, 1875, $500. How much remained un- paid Mar. 1, 1876? PROBLEMS IN INTEREST. PROBLEM I. 344. Principal, rate, and time given, to find the interest. This has already been solved. (See page 220.) RULE. Multiply the interest of $1, for the given rate and time, by the principal. PEOBLEM II. 345. Principal, rate, and interest given, to find the time. 1. How much is the interest of $100 for a year at 6^ ? For 2 years ? For 3 years ? PAETIAL PAYMENTS. 233 2. When $100, loaned at 6^, brings an income of for how long a time was it loaned ? How long when the in- terest was $18? $24? $3? $4? $2? $1.50? 3. When $50, loaned at 10 ft, brings an income of $10, for how long a time was it loaned? KULE. Divide the given interest by the interest of the prin- cipal for .one year. In what time will 4. $250 produce $30 interest at % ? 5. $600 produce $24 interest at 8^ ? 6. $115 produce $13.80 interest at 6^ ? 7. $12.60 produce $4.15 interest at 1% ? 8. $35.25 produce $13.25 interest at 1% ? 9. $25 produce $25 interest at 6ft> ? 10. $150 double itself at 8^ ? 11. Any sum double itself at 5^ ? 6^ ? 7^ ? 12. Any sum triple itself at 5^ ? 6^ ? If ? PROBLEM III. 346. Principal, time, and interest given, to find the rate. 1. What is the interest of $100 for 1 year at lf c ? At 2^ ? At 3^ ? 2. When the interest of $100 for 1 year was $8, what was the rate? 3. When the interest of $100 for 2 years was $14, what was the rate? 4. When the interest of $50 for 3 years was $15, what' was the rate? RULE. Divide the given interest, by the interest of the prin- cipal for the given time, at 1 per cent. 234 PERCENTAGE. What is the rate per cent, when the interest 5. Of $125 for 2 years is $15? 6. Of $250 for 6 months is $8.75? 7. Of $415 for 2 years 6 months is $56.025? 8. Of $317 for 1 year 5 months is $31.44? 9. Of $215 for 2 years 7 months 10 days is $39.30? 10. Of $325.18 for 5 months 26 days is $11.13? 11. Of $30.18 for 63 days is $ .32? 12. Of $24.36 for 93 days is $ .44? 13. Of $25.40 for 45 days is $ .397 ? PKOBLEM IV. 347. Rate, time, and interest given, to find the prin- cipal. 1. At 6^, what sum will produce $6 yearly? 2. At 6^g, what sum will produce $12 yearly? What $18 yearly? What $12 in two years? What $18 in 2 years ? RULE. Divide the given interest by the interest of $1, for the given time at the given rate. What sum of money will produce 3. $36.60 interest in 2 years at 6^ ? 4. $35.70 interest in 2 years 6 months at 8^ ? 5. $51.20 interest in 5 years 6 months at 5 ft ? 6. $50.84 interest in 6 months 27 days at 6^ ? 7. $39.18 interest in 5 months 18 days at 6^ ? 8. $41.25 interest in 3 months 15 days at 9^ ? 9. $87.50 interest in 1 month 12 days at l in Discounting, A -i rr: T> x e TV require the simple in- .0155, Kate of Discount. . , terest in advance, we $5.9675, Discount. find the interest of $385 $5. 9675 = $379. 0325, Proceeds. $385 for 3 mo - 3 da - at 6^, which is $5.9675. And since the present worth is the face of the debt minus the dis- count, we subtract $5.9675 from $385, which gives $379.0325, the pro- ceeds or avails. KULE. To find the bank discount: Compute the interest on the face of the note for 3 days more than the specified time at the given rate. To find the proceeds : Subtract the bank discount from the face of the note. If the note bears interest, find the discount on the amount of the note at its maturity. This cmiountj less the discount, will be the pro- ceeds. 242 PERCENTAGE Find the bank discount of: 2. $ 318.25 for 2 mo. 15 da. at 3. $3846.18 for 1 mo. 17 da. at 4. $2184.39 for 3 mo. 10 da. at 5. $ 824.17 for 3 mo. 28 da. at 6. $4484.15 for 3 mo. 18 da. at 7. $7318.69 for 2 mo. 15 da. at 8. $8984.15 for 30 da. at 9. $ 765.30 for 65 da. at 10. $375. CHICAGO, ILL., Dec. 20, 1876. Two months after date, for value received I promise to pay E. D. Bronson, or order, Three Hundred Seventy- Five Dollars, with interest at 10^, at the First National Bank, Chicago, III S. HOWARD BLACKWELL. This note was discounted Jan. 23, 1877, at 10%. What was the bank discount? What were the proceeds? 11. What is the bank discount on a note for $890.25 at 2 months, the rate being 1% ? What are the proceeds? 12. What is the difference between the bank discount and the true discount, on a note for 3 months for $3725.85, the rate being 8 % ? 13. $15,725.95. NEW YORK, May 15, 1876. For value received, two months after date " The North River Sugar Refining Co." promises to pay Messrs. Smith & Haughton, or order, Fifteen Thousand Seven Hundred Twenty-five T 9 ^- Dollars, at The Merchants National Bank, New Orleans, La. NORTH RIVER SUGAR REFINING Co., BY C. MOLLER, Sec'y. The above note was discounted May 25, 1876, at 6%. What was the discount? What were the proceeds? BANK DISCOUNT. 243 14. Mr. A sold goods amounting to $3782.75, and received in payment a note at 90 days, without interest, which he had discounted at a bank after holding it 30 days. How much did he realize from the- sale, the rate of discount being 8% ? 15. A merchant sold 32 yards of silk velvet at $8.75 per yard; 48 yards of silk at $2.80 per yard; 25 yards of English broadcloth at $6.25 per yard; receiving in payment a bank-note payable in 45 days, without interest, which he had discounted on the same day at 9%. How much did he realize in cash from the sale? CASE II. 374. To find the face of a note when the proceeds, time, and rate are given. 1. What is the bank discount of $1 for 2 mo. 27 da. at 6%? What the proceeds? 2. Since $.985 is the proceeds of $1 when discounted at 2 mo. 27 da. at 6%, of what sum is $1.97 the proceeds for the same time and rate? Of what sum is $2.955 the pro- ceeds for the same time and rate? 3. Of what sum is $394 the proceeds, when discounted at a bank for 1 mo. 27 da. at 6%? Of what sum is $197 the proceeds ? 4. Of what sum is $396 the proceeds, when discounted at a bank for 1 mo. 27 da. at 6% ? 5. The proceeds of a note discounted at a bank for 2 mo. 12 da. at 1% were $1182.50. What was the face of it? PROCESS. ANALYSIS. $1 $.0145f = $.98541 The proceeds of 1. Since the l 3 ' $1182.50--.9854|-=:$1200 The face of the note. Ceed l J** are $ .9854J, $1182.50 are the proceeds of as many dollars as $ .9854 \ is contained times in $1182.50, which is 1200 times. Therefore the face of the note was $1200. 244 PEKCENTAGE. RULE. Divide the proceeds by the proceeds of $1, at the given rate for 3 days more than the specified time. 6. The proceeds of a note discounted at a bank for 2 mo. at 6% are $989.50. What was its face? 7. What must be the face of a note so that when dis- counted at a bank for 3 mo. at 6%, the proceeds will be $1969? 8. A merchant wishes to use $975, which he can secure by giving a bank-note payable in 60 days, to be discounted at 1%. For what sum must the note be written? 9. For what sum must a note be drawn payable in 90 days, so that when discounted at 6% the proceeds may be $1000? 10. A man owes a debt of $1375.38, which he can meet by giving his note payable in 3 mo., discounted at 9^. For what sum must the note be written so that the avails may be just large enough to discharge the debt? 11. For what sum must a note be drawn, payable in 15 days, so that when discounted at 8fo the proceeds may be $1257.25? 12. A speculator wishes to raise $5250 on an indorsed note, payable in 45 days, discounted at 7^. For what amount must he draw the note so that the proceeds may be exactly that sum? 13. For what amount must a note be made payable in 3 months, and discounted at l\% so that the proceeds may be $1875? 14. A merchant wishes to raise $500 at a bank by a note at 60 days. For what sum must the note be drawn that he may receive $500 in cash from the banker after paying the discount at 8% ? 15. For how large a sum must a note be drawn, payable in 90 days, that the net proceeds may be $15000 after de- ducting the bank discount at 8 fa ? PERCENTAGE. 245 REVIEW EXERCISES. 375. 1. What sum must be invested at 8 ft to yield an annual income of $1400? 2. A merchant bought a bill of goods amounting to $7825 on 90 days credit, but was offered a discount for cash of 4^. What was the difference in the offers, money being worth 9^? 3. A man bought a farm of 135 A. 25 sq. rd. for $62.50 per acre. He paid one-third of the purchase money in cash ; one-half the remainder in 6 mo., and the balance in 1 yr. 3 mo. Money being worth 6^, what would have been the cash price of the farm? 4. A will contained a bequest of $4500 to a charity hos- pital, to be paid in 1 yr. 3 mo. Money being worth 7%, what was the cash value of the bequest? 5. What is the amount of $3752 for 3 yr. 2 mo. 15 da., with compound interest at 7 % ? 6. $175. CINCINNATI, O., March 25, 1872. For value received, on demand I promise to pay J. H. Sheppard, or bearer, One Hundred Seventy-five Dollars, with interest at 8%. DAvn> R AsPELL . This note was paid April 15, 1876. How much was due upon it? 7. What is the difference between the true discount and the bank discount of $5728 for 90 days at 8% ? 8. On a note of $3729.75, dated Feb. 20, 1872, bearing 6^ interest, were the following indorsements; July 15, 1872, $525; Dec. 15, 1872, $478; Feb. 20, 1873, $25; May 17, 1873, $75; Sept. 28, 1873, $1000. What was due Jan. 15, 1874? 246 PERCENTAGE. 9. A coal dealer bought 8790 T. of coal at $3.75 per T. He sold 15 ft of it at a gain of 10^ on the cost, 40% of it at a gain of 5%, and the rest at a gain of 8 fa. He paid 3i^ of the cost for transportation. How much did he gain by the speculation ? 10. A grain speculator bought 10000 bushels of barley, at 85 cents per bushel cash. He sold it the same day at an advance of 4^ , receiving in payment a note at 30 days which he had discounted at a bank at 9^. What was his gain in cash? 11. A man bought a house, paying 62|^ of the price in cash, and the rest in notes to the amount of $3300. What was the cost of the house? 12. A merchant sold 15 ft of his stock of dry goods the first month, 10^ the second month and 25^ of the remainder the third month, when he took an inventory of his stock on hand, and found that he had remaining $5300 worth of goods. What was the original value of the stock ? 13. What number is that to which, if f of 25% of | of 480 be added, the sum will equal 25% of f of 50% of 324? 14. A speculator bought 1000 bbl. of flour at a given price per bbl. paying f of its value in cash and giving a bank- note at 60 days for the balance, which was discounted on the day it was given at 6% . The discount on the note was $31. 50. How much did he pay for the flour per bbl. ? 15. What is the difference between the simple and the com- pound interest of $4725.50 for 2 yr. 2 mo. 15 da. at 6%. 16. What is the difference between the amount of $3240 for 5 yr. 3 mo. 10 da. at 7% simple interest, and the amount of the same sum for the same time and rate, with interest payable annually? 17. A gentleman invested -|- of his annual income in mort- gages, paying 6^ annual interest. In 6 mo. 12 da. his in- terest from them was $640. What was his annual income? PROFIT AND LOSS. 247 PROFIT AND LOSS. 376. 1. When 25^ is gained what part is gained? 2. I sold a coat which cost me $12 for 25^ more than it cost. How much did I gain ? How much did I get for it? 3. Paid $15 for a ton of hay, and sold it at a loss of 20%. How much did I lose 2 How much did I get for the hay ? 4. If I sell land that cost me $50 an acre, at an advance of ^ , how much do I get per acre ? 5. If I paid $50 per acre for my land and sell it at $5 per acre less than I paid, what part of the cost do I lose ? What % ? 6. If I sell flour for $8 per bbl. that cost me $6, what part of the cost do I gain ? What per cent. ? 7. Sold boots at $5 a pair that cost $4, what part of the cost was gained ? What per cent. ? 8. By selling flour at a profit of $2 per bbl., 20^, or |, of the cost was gained. What was the cost ? 9. Sold wheat at a profit of $ .10 per bu. which was 5^ of its cost. What was its cost? 10. If I sell a cow that cost me $50, at an advance of 20^ on the cost, how much will my profit be ? How much do I get for her? 11. By selling flour at $10 per bbl., a profit of 25^ was made. What did it cost ? ANALYSIS. Since 25^ or \ of the cost was gained, the selling price must have been f of the cost; and since f of the cost was $10, J of the cost is \ of $10, which is $2 ; and since \ of the cost is $2, the cost is 4 times $2 or $8. 12. By selling dress goods at 66 cents per yd., a profit of was made. What was the cost? 13. By selling tea at $ .80 a pound, a loss of 20^ was in- curred. What was the cost ? 248 PERCENTAGE. 14. If I get $60 for a cow, and thereby gain 20^, or | of the cost, what did she cost me? 15. I bought a horse for $150 and sold him at an advance of 20^ . What did I get for him ? DEFINITIONS. 377. Profit and Loss are terms used to denote the gain or loss in business. 378. The processes in Profit and Loss involve the same elements as do the fundamental problems in Percentage. The corresponding terms are: 1. The Cost, or Sam Invested, is the Base. 2. The Rate Per Cent, of profit or loss is the Kate. 3. The Gain or Loss is the Percentage. 4. The Selling Price, when more than the cost, is the Amount. 5. The Selling Price, when less than the cost, is the Difference. PRINCIPLE. The gain or loss is reckoned at a certain rate per cent, on the cost or sum invested. WRITTEN EXERCISES. 379. 1. A paid $3500 for a house, and sold it at 10^ ad- vance. How much did he gain? How much did he get for it? PROCESS. ANALYSIS. Since the $3500 X .10 = $ 350, Gain. honse s sold at u an ad ' vance of 10 n tne cost > $3500 + $350 =f= $3850, Selling price. his gain was _i_o_ or A Cost X Rate = Gain. of $3500, which is $350; and the selling price is equal to the sum of the cost and gain, or $3850. PROFIT AND LOSS. 249 RULES. Since the same elements are involved as in thejun- damental problems in Percentage the rules are the same. FORMULAS. 1. Gain or loss Cost X Rate. 2. Rate = Gain or loss -+- Cost. 3. Cost Gain or loss -r- Rate. 4. Cost Selling price -f- ( 1 + Rate ). 5. Cost Selling price -f- ( 1 Rate ). 2. Mr. A. bought a piano for $275, and sold it at an ad- vance of 25^. How much did he receive for it? 3. A bookseller bought $3584 worth of books, and sold them at a gain of lO^J. How much did he gain? 4. A carriage maker sold a carriage at 25^ advance on the cost. It cost him $318.25. How much did he get for it? 5. A harness maker sold a set of double harness at a profit of 15^. It cost him $45. What did he get for it? 6. A manufacturer of tools sold 5 dozen axes at a profit of 12^. They cost him $9 per dozen. What was his profit? 7. A merchant sold a bill of goods at a profit of 15^. The goods cost him $84.25. What did he receive for them? 8. A speculator bought 50000 pounds of sole leather, which he sold at a profit of 8^. If it cost him $6000, what did he get for it? 9. A man sold his house at a profit of 15^. If he paid $3000 for it, how much did he get for it? 10. A drover sold a flock of sheep at a profit of 7^. If they cost him $1500, what did he get for them? 11. A poultry dealer bought a quantity of poultry, which he sold at a gain of 9^. He paid $250 for it. How much did he get for it? 12. Mr. A bought cloth at $2.15 per yard. At what price must he sell it to gain 250 PERCENTAGE. 13. A farm which cost $65 per acre was sold at a gain of For how much did it sell per acre? 14. A merchant bought 3950 yards of cotton at 9^- cents a yard. How much will he get for it if he sells it at a gain of 12^! 15. A merchant desires to mark goods that cost him $3.60 per yard so that he may gain 33^^. At what price must the goods be marked? 16. A bankrupt stock was sold at 35^ loss. What was the selling price of articles that cost 50c. ? $1? $1.50? $1.75? 17. What per cent, is lost by selling sugar at 10 cents per pound which cost 12 cents per pound? PROCESS. ANALYSIS. Since sugar that cost 12 < -^2 $ 10 = $ 02 cen ts was sold for 10 cents, there was a loss of 2 cents per pound. And since the * "^ ~^~ $ '1^ = 1* > 3/^ gain or loss is reckoned at a rate per cent. Loss -f- Cost = Rate. on the cost, we must find what per cent. 2 is of 12. 2 is J of 12; or, expressed as hundredths, is .16f of 12, or 16|y . 18. What per cent, is gained by selling tea at $1 which cost $.75? 19. What per cent, is lost by selling tea at $.75 which cost $1? 20. What per cent, is lost by selling cloth at $1.25 that cost $1.75? 21. Bought goods at 50 cents a yard and sold them at 60 cents a yard. What per cent, did I gain? 22. Goods that are selling at 12^- cents a yard cost 10 cents. What per cent, is gained by selling them at that rate? 23. A man bought a city lot for $4500 and sold it for $5000. What per cent, did he gain ? 24. Sold a quantity of potatoes for $850 which cost me $970. What per cent, did I lose? PROFIT AND LOSS. 251 25. Bought a quantity of crude petroleum at 5 cents per lion and sold it at 4-|- cents. What per cent, did I lose ? 26. A fruiterer bought 10 boxes oranges at $1.75 per box. Two of the boxes were worthless, but he sold the balance at nch price that he gained 5^ on the whole purchase. How luch did he sell them for per box? How much did he gain on the purchase? 27. I bought books at 10^ discount from the retail price, which was $1.50 per volume, and sold them at the retail price. What was my gain per cent. ? 28. An agent gets a discount of 40^ from the retail price of articles and sells them at the retail price. What is his gain per cent. ? 29. A merchant bought cloth at $3.25 per yard, and after keeping it 6 months sold it at $3.75 per yard. What was his gain per cent., reckoning 6^ per annum for the use of money ? 30. Which is more profitable, and how much per cent., to sell goods for cash, at once, at 25^ advance, or in 1 year at advance, money being worth 1% ? 31. Mr. A. gets a discount of 30^ from the retail or list price of goods. Mr. B. gets a discount of 30^ also, and 5^ off for cash. If both sell goods at the list price, what is each one's gain per cent. ? 32. A merchant bought tea at 20^ less than its market value, and received a discount of 5^ for cash. He sold it at an advance of 15^ above its market value. What was his gain per cent. ? 33. By selling cloth at a gain of 12 cents a yard, I real- ized a gain of Sf on the cost. What was the cost? PROCESS. $.12 -T- .08 = $1.50. Gain -f- Rate = Cost ANALYSIS. Since the gain, 12 cents, is g^ or y^ of the cost, J of 12 cents, or 1J cents, is T ^ of the cost, and the cost is 100 times 1 J cents, or $1.50. 252 PERCENTAGE. 34. I make 10^ by selling tea at a profit of 10 cents a pound. What did it cost me? What do I sell it for? 35. Flour was sold at a profit of $1.50 per bbl., which was 16|^ of the cost. What was the cost? 36. A merchant made 12^ by selling cloth at an advance of 12 cents a yard. What did it cost? 37. By selling butter at 8 cents a pound more than cost, a grocer made 20^. What did he pay for it? 38. A merchant sold cloth which was damaged by fire, at a sacrifice of 22 cents per yard, which was 40^ of the cost. What did the goods cost? 39. A farmer sold a yoke of cattle, to which he had fed $10 worth of grain, at an advance of $25, and still realized a profit of 15^. What did they cost? 40. A man sold a horse at an advance of $75, which was a gain of 25^. What was the cost of the horse? 41. If I sell a quantity of apples at an advance of 25 cents a barrel, and thereby realize 12^ profit, what was the cost? 42. By selling cloth at a gain of 23 cents per yard, I realize a profit of 20^. What did it cost? 43. A merchant asked 25% more for his goods than they cost him, but at last sold them at a reduction of 10^ from his asking price, thus realizing from the sale $4684 profit. What was the cost of the goods ? . 44. A gentleman sold a horse for $180 and gained 20 fo on him. What did the horse cost? PROCESS. ANALYSIS. Since 20^ of 100% +20% =120% ! he co ? t was gained > the sel1 - ing price must have been 20^, $180-5-1.20 = $150 m ore than the cost, or 120^ Selling Price -f- (1 +Rate) = Cost. of the cost - And > since 120 ^ of the cost is $180, lf c of the cost is T Jo of $180, or, $1.50, and the whole cost is 100 times $1.50, or $150. Therefore the horse cost the gentleman $150. PKOFIT AND LOSS. 253 45. A gentleman sold a carriage for $230, and thereby st 8% of the cost. What was the cost? PROCESS. ANALYSIS. Since 8^ of the lOQ^ 8^92^ St Was ^ St? ^ e se ^ n & P r ^ ce * must have been 8^ less than $230-r-.92 = $250 t ^ e costj or 92 ^ O f the cost. Selling Price -r-(l Rate) = Cost. And > since 92 /* of the cost was $230, lf c of the cost was jx of $230, or $2.50, and the whole cost was 100 times $2.50, or $250. 46. By selling apples at $.50 per bushel a grocer gained 25% on the cost. What was the cost? 47. A farm was sold for $38000, which was a loss of 5 ft of the cost. What was the cost? 48. A block of stores was sold for $185000, which was a gain of 15^. What did they cost? 49. A merchant lost 5 ft by selling calico at 9|- cents a yard. What did it cost? 50. A bankrupt stock was sold for $3582, which was a loss of 331^. What did it cost? 51. By selling molasses at 65 cents a gallon, a grocer gained 30^. What was its cost? 52. A man was compelled to sell his household furniture for $1250, which was a loss of 37^. What did it cost? 53. A boot and shoe dealer lost 9^ by selling boots at $3.75 a pair. What was the cost of the boots? 54. A stationer lost 26^ by selling paper at $2.22 a ream. What did he pay for it? t55. A druggist gained 125^ by selling alcohol for $3.50 r gallon. What did he pay for it? 56. Coal was sold at $4.56^- per ton, which was a loss of ^ of 1260 gal X 5 = 6 3 gal. leakage. the gross amou nt, 1260 gal. 6 3 gal. = 1 1 9 7 gal. net. or 63 gal., are de- ft .09X1197 $107.73 duty. ducted for leakage; and since the duty on 1 gal. is 9 cents, on 1197 gal. it is 1197 times 9 cents, or $107.73. 2. What is the duty, at 5 cents a pound, on 3750 pounds of coffee, allowing 5^ for tare ? 3. What is the duty on 500 pounds of raisins, in boxes, valued at 10 cents a pound, allowing 15^ for tare, when the rate of duty is 6^ ad valorem ? 4. What will be the duty on $3000 worth of merchandise if the rate of duty is 15% ad valorem? 5. What is the duty, at 20^ ad valorem, on 7 tons of steel, of 2240 Ib. each, invoiced at 17 cents per Ib. ? 6. H. K. Thurber & Co., of New York, imported from Havana 75 hhd. molasses, 63 gal. each, valued at 35 cents per gal. ; 125 hhd. sugar, 500 Ibs. each, valued at 6 cents per Ib. ; 800 boxes cigars, valued at $8 per box. 7^ was allowed for leakage on the molasses; duty on same, 25^; tare on sugar, 45 Ibs. per hhd. ; duty on same, 30^ ; duty on cigars, 60^ . What was the amount of duties paid ? 7. A wine merchant imported 45 casks sherry wine, valued at $65 per cask; 56 casks Madeira wine, valued at $60 per cask; 38 casks German wines, valued at $37 per cask. If an allowance of 4^ be made for leakage, what will the duty be at 408. 1. Into how many shares can $100000 capital stock of a company be divided, if the shares are $100 each? Shares will be regarded as $100 each unless otherwise specified. 2. How much of the capital, or capital stock does a man own who has 30 shares? 25 shares? 3. How much stock is represented by a certificate entitling the holder to 40 shares ? 4. What is the selling price or market value of 10 shares of railroad stock, when stock is selling at its original value or at par f 5. What is the market value of 10 shares of stock, if it is sold at 105% of the original value or 5% above parf 6. What is the market value of 10 shares of stock, if it is sold at 95^ of the original value or 5% below parf 7. What will 10 shares of stock cost at 5% above par, if I pay a dealer in stocks or stock-broker \% of the par value of the stock for buying it? 8. What will 5 shares of stock cost at 5^ below par, if I pay a broker \% for buying or for brokerage? 9. What is the value of 15 shares of bank stock at of its par value? 10. What is the cost of 50 shares Chicago & Eock Island R. R. stock at 90% of its par value? 11. What is the cost of 100 shares Chicago & Alton R R. stock at 95% of its par value? (267) 268 PERCENTAGE. 12. What is the cost of 20 shares Pacific Mail stock at of its par value ? 13. A company whose capital stock was $50000, gained $5000 above expenses, which was divided among the stock- holders. What per cent, of the capital stock was the amount divided, or what was the dividend? 14. What amount will a man receive who owns 20 shares of stock, if a dividend of 5^ is- declared? 15. What amount will a man receive who has 30 shares of stock, if a 6^ dividend is declared ? 16. A company whose capital stock was $50000, lacked $5000 of meeting its obligations. What per cent, of the stock was the deficiency? 17. If the deficiency is made up by the stockholders, how much will be the assessment on a stockholder who owns 20 shares of the above stock? 18. How much will be the assessment on 15 shares of the above stock? 19. What will be the annual income on a written obliga- tion or bond for $5000 which yields 6^ interest annually? DEFINITIONS. 409. A Company is a number of persons associated together for the prosecution of some industrial pursuit. 410. A Corporate Company or Corporation is a company authorized by law to transact business as an individual. 411. A Charter is the legal document which defines the rights and obligations of a corporation. 412. Capital Stock is the property or money invested in the business of the company. STOCKS. 269 413. A. Share is one of the equal divisions of the capital stock of a company. The value of a share is different in different companies. Unless otherwise specified it will be regarded as $100. 414. A Certificate of Stock is a paper issued by a corporate company giving the number of shares to which the holder is entitled, and the original value of each. 415. Par Value is the value named in the certificate. When shares sell for the value named on their face the stock is said to be at par; when for more than their face, above par, or at a pre- mium; when for less than their face, below par, or at a discount. 416. The Market Value is the sum for which stocks sell. 417. An Installment is a portion of the capital stock paid by the stockholder. 418. A Dividend is a sum divided among the stock- holders as the profits of the business. 419. An Assessment is a sum which the stockhold- ers of a company are required to pay, to make up deficiencies or losses. The Government and Corporations frequently issue Bonds for the purpose of raising money. 420. A Bond is a written obligation under seal, securing the payment of a sum of money on or before a specified time. Interest is usually paid upon bonds at fixed dates, as an- nually or semi-annually. 421. Coupons are certificates of interest attached to bonds. They are cut off and presented for payment as often as the interest becomes due. 270 PERCENTAGE. 422. United States Government Securities are of two kinds, viz: bonds which are to be paid at a specified time, and bonds which are to be paid at a fixed date, or some earlier specified time at the option of the Government. Bonds are sometimes designated by combining the rate of interest and the time of payment. Thus, bonds that pay 6^? interest and are payable in 1881, are called 6's of '81. When the rate is uniform for a class of bonds it is omitted, and the time of redemption and payment only are given. Thus, bonds that may be redeemed in five years, and are payable in twenty years, are called 5-20's, those redeemable in ten years and payable in forty years are called 10-40's. The 4 -rr / f + Premium. 5. Market Value = Par Value < _. ( Discount. 2. Find the cost of 125 shares Union Pacific R. R. stock, at 68|, brokerage ^ ? 3. What will $8000 U. S. 5-20's, coupon bonds of '65, cost at 108J, brokerage \ income, the entire income from $4500 is 6f of $4500, which is $225. 15. How much income will I receive annually by investing $1299 in 6% stock purchased at 37%, allowing \% bro- kerage? 16. What will be the income from investing $4696.25 in Crawford Co. 6's at 45, brokerage \%. 17. Which is more profitable, and how much, to invest $5000 in 6^ stock purchased at 75%, or 5% stock purchased at 60%? 18. U. S. 5-20's pay 6% interest in gold. What will be my income in currency by investing $11212.50 at H2|-, when gold is quoted at 6|-% premium? 19. Which is more profitable, to buy 6% bonds, purchased at 90, interest payable in currency; or 5% bonds, purchased STOCKS. 275 at 95, interest payable in gold, when gold is quoted at premium? How much more profitable in currency is it on each $100 invested? 20. A. B. Howard sold a mill for $13850, which had been paying an annual profit of 5^ of that sum, and invested the proceeds in U. S. 10-40's at lllf, paying \ and ' the income from it is $ .06, the income is T f^, or 2 1 -, or 5^> of the amount of the investment. 276 PEECENTAGE. 27. What is the rate per cent, of income from bonds which pay 1% interest when they are bought at 105? 28. If stock which pays a semi-annual dividend of 5^% be bought at 10% premium, what rate per cent, of income does it pay? 29. Which affords the greater per cent, of income, bonds bought at 125 which pay 8%, or bonds which pay 6% bought at a discount of 10^ ? 30. Which is more profitable, and how much per cent., to buy New York 7's at 105^, or Louisiana 6's at 98^? 31. What per cent, of income does stock paying 9% divi- dends afford if it is bought at 112? 32. How much must I pay for New York 6's so that I may realize an income of 9^ on the investment? PROCESS. ANALYSIS. Since the income is 6^> $.06-=-$. 09 = . 66 2 f ever y dollar of the par value of the stock, if an income of 9^ on an invest- ment be desired, then 6^ of the par value of the stock must be 9^ of the sum paid for $1 of the stock, which is $ .66f , or, the stock must be bought for 66f every $1 of the draft $1.00^X5000 --^$5025 will cost $1.00}, and a draft for $5000 will therefore cost the purchaser 5000 times $1,00}, which is $5025. 286 PEKCENTAGE. 2. What will be the cost in New Orleans of a draft on New York, payable 60 days after sight, for $5000, exchange being at \\% premium ? ANALYsis.-Since the - PROCESS. exchange on New York there WIS 300 for 2 mo. = $1 for 600 mo. no term of credit for 400 for 4 mo.--=$l for 1600 mo. that sum . gince $1100 2200 mo. $300 was to be paid in 2 mo., the use of 2200 mo. -f-1 100 2 mo. Average term of credit, that sum for 2 mo. is equal to the use of $1 for 600 mo., and the use of $400 for 4 mo. is equal to the use of $1 for 1600 months. Hence, the credit of the whole debt, $1100, is equal to the credit of $1 for 2200 mo., or $1100 for TT Vo part of 2200 mo., which is 2 months, the average term of credit. RULE. Multiply each debt by its term of credit, and divide the sum of the products by the sum of the debts. The quotient will be the average term of credit. 2. H. B. Claflin & Co. sold a bill of goods amounting to $2300, on the following terms: $300 cash, $1200 due in 3 months, and the balance due in 4 months. What was the average term of credit. 3. Field, Leiter & Co. sold a bill of goods payable as fol- lows: $500 in 1 month, $500 in 2 months, and $800 in 4 months. What was the average term of credit? 4. Whitney & Co. sold a bill of lumber on the following terms: $1500 cash, $3000 payable in 30 days, and $2000 payable in 90 days. At what time will the debt be payable in one cash payment? 5. H. K. Thurber & Co. sold to F. N. Burt a bill of goods amounting to $2400, payable as follows : ^ in 30 days, \ the remainder in 60 days, and the balance in 4 months. What was the average term of credit? 6. Mr. Birge bought a bill of goods amounting to $3000, payable as follows: \ in 3 mo., \ in 2 mo., and the rest in 4 mo. What was the average term of credit ? AVERAGE OF PAYMENTS. 293 CASE II. 464. When the terms of credit begin at different dates. 1. Find the average time of payment of the following bills: Feb. 10, 1877, $400 due in 2 mo.; March 15, 1877, $350 due in 3 mo. ; and April 12, 1877, $300 due in 3 mo. PROCESS. ANALYSIS. - we o ~ tain the time 51000 -*- 1050 = 48^f days. when it is due, April 10 + 49 days = May 29, average term, and so we have $400 due April 10, $350 due June 15, $300 due July 12. The average time when the hills will he due will be either after the earliest date, or before the latest date, and so we may select either of these dates from which to compute the average time. Selecting the earliest date, we find that $350 was due 66 days after that time, and $300 was due 93 days thereafter. Averaging, as in Case I, we find the term of credit to be 48Jf, or a fraction more than 48 days, which must be 49 days. This, added to April 10, gives May 29, the average time of payment. RULE. Select the earliest date at which any debt becomes due for the standard date, and find how long after that date the other amounts become due. Find the average term of credit by multiplying each debt by the number of days from the standard date, and dividing the sum of the products by the sum of the debts. Add the average term of credit to the standard date, and the result will be the average term of payment. Instead of the earliest date, the first of the month may be used. 294 AVERAGE OF PAYMENTS. 2. What is the average time at which the following bills become due: Feb. 1, 1877, $200 on 1 mo. credit; March 10, 1877, $500 on 3 mo. credit; April 12, 1877, $275 on 2 mo. credit; and May 1, 1877, $400 on 4 mo. credit? 3. A merchant owes bills dated as follows: Jan. 1, 1877, $500 due in 2 mo.; Jan. 15, 1877, $850 due in 3 mo.; Feb. 20, 1877, $375 due in 3 mo.; and, Feb. 28, 1877, $650 due in 4 mo. What will be the average time of pay- ment? 4. A merchant purchased goods of Cragin Bros. & Co. as follows: Sept. 10, 1876, $300 on 4 mo. credit; Oct. 15, 1876, $400 on 6 mo. credit; Nov. 1, 1876, $750 on 2 mo. credit; and, Nov. 15, 1876, $300 on 1 mo. credit. What was the average time of payment? 5. Messrs. J. Rorbach & Son bought goods from George C. Buell & Co. as follows: Sept. 1, 1876, $600 on 3 mo. credit; Oct. 3, 1876, $400 on 4 mo. credit; Oct. 20, 1876, $250 on 2 mo. credit; and, Nov. 10, 1876, $375 on 1 mo. credit. What was the average time of payment? 6. Stevens & Shepard bought goods from the Russell Ir- win Manufacturing Co. as follows: Dec. 10, 1876, a bill of $460 on 4 mo. credit; Jan. 5, 1877,, a bill of $200 on 3 mo. credit; Jan. 30, 1877, a bill of $200 on 4 mo. credit; and, Feb. 25, a bill of $900 on 2 mo. credit. What was the average time of payment? 7. Bought goods of Carson, Pirie & Co. as follows: Jan. 25, 1877, $850 on 4 mo. credit; Feb. 15, 1877, $600 on 3 mo. credit; March 20, 1877, $500 on 4 mo. credit; and, April 10, 1877, $960 on 2 mo. credit. What was the aver- age time of payment? 8. May 1, 1877, Mr. S. purchased goods to the amount of $2400 on the following terms: \ payable in cash, \ payable in 2 months, and the balance in 6 months. When may the whole be equitably paid by one payment? AVERAGE OF ACCOUNTS. 295 AVERAGE OF ACCOUNTS. 465. 1. What should be the date of a note given to settle the following account? Dr. W. H. STEVENS. Or. 1877. 1877. May 5 To Mdse. 50 00 May 15 By Cash 25 00 June 7 " " 2 mo. 140 00 June 10 " Draft, 10 da. 100 00 June 20 " " 1 " 150 00 June 30 " " 100 00 PROCESS. (By Products.) Due. Amount. Days. Product. Paid. Amount. Days. Product. May Aug. July 5 7 20 $ 50 140 150 94 17 4700 2550 May June June 3150 15 23 30 . ^- $ 25 100 100 84 45 38 ft 2100 4500 3800 340 225 7250 225 10400 7250 115 L5 = 27 T 5 - 3150 Aug. 7 + 28 days = Sept. 4, the average time. ANALYSIS. From the dates at which the various amounts become due, we select the latest, which is Aug. 7, for the assumed time of set- tlement, and multiply each amount by the number of days intervening between that date and the time when each item of the account becomes due. The debit side of the account shows there is due $340 and the use of $1 for 7250 days, and the credit side shows that $225 has been paid, and that the debtor is entitled to the use of $1 for 10400 days, if the time of settlement is Aug. 7. Subtracting the amounts, there is shown to be $115 due, and the debtor is entitled to the use of $1 for 3150 days. Therefore, he should not be required to pay the account until the time when the use of $115 is equal to the use of $1 for 3150 days, which is 28 days. 28 days after Aug. 7 is Sept. 4. I 296 AVERAGE OF ACCOUNTS. RULE. Multiply each amount due by the number of days in- tervening between the time it becomes due and the latest date at which any sum on either side of the account becomes due. Divide the difference between the sum of the products of the debit and credit side of the account, by the balance due on the account. The quotient will be the average term of credit. 1. When the balances are both on one side of the account, the term of credit is to be counted backward from the date at which the first amount becomes due, but forward from that date if the balances are on opposite sides. 2. The average term of credit may also be found by reckoning interest upon each sum due for the number of days intervening be- tween the time it becomes due, and the earliest date at which any sum becomes due; then dividing the balance of the interest by the interest on the balance of the account for one day. This is called the Method by Interest. The result is the same whether the average term of credit is found by the method by products or by interest. 2. Find the average term of credit of the following account : Dr. OLMSTEAD & BISHOP. Or. 1877; 1877. Jan. 5 To Mdse., 2ino. $375 Jan. 30 By Cash $200 Feb. 15 u x u 200 Mar. 15 " 600 Feb. 25 4 " 800 Apr. 1 tt 200 Mar. 30 3 " 450 3. Find by both methods when the balance of the follow- ing account becomes due. Dr. HAMILTON & MATTHEWS. Or. 1876. 1877. Nov. 1 To Mdse., 4 mo. $1600 Jan. 15 By Accept' ce, 2 mo. $2000 Dec. 3 3 " 3800 Mar. 20 2 " 5000 1877. Jan. 15 4 " 5500 Mar. 1 3 ,* 1500 AVERAGE OF ACCOUNTS. 297 4. When should interest begin on the following account ? Dr. JAMES HOWARD, in ace't with HIRAM SIBLEY. Or. 1877. 1877. Apr. 10 To Mdse. $150 Apr. 12 By Cash $250 Apr. 30 400 May 1 " " 200 May 16 u 100 June 7 1C : 10. 6. 15 : 18 :: ( ) : 16. 14. 1 = f::(J : 15. 7. 17 : 19 :: 15 : () 15. 6 : | :: 5 : ( ) 8. 25 = ( ) : : 16 : 25. 16. 13 :7::( ):8. 17. 5 men : 7 men : : 8*: ( ) 18. $21.16 : $15.20 :: f : ( ). 19. 80.51 A. : 21.15 A. :: ( ) 20. 16 Ib. 3 oz. : 18 Ib. 2 oz. : 21. 5 gal. 3 qt. : ( ) : : 5 : 9. 22. 14 f : ( ) : : 6 : 15. : 2. () 7. SIMPLE PROPORTION. 490. A Simple Ratio is a ratio between any two numbers. Thus, 6: 8, $10: $8, 5 Ib. 6 oz.: 7 Ib. 3 oz., are simple ratios. 491. A /Sim%)le Proportion is an equality between two simple ratios. 492. A Direct Proportion is one in which each term increases or diminishes, as the one on which it depends increases or diminishes. Thus, proportions involving quantity and cost, men and work done, etc., are direct proportions, for as the quantity increases or diminishes, the cost increases or diminishes, and as the number of men increases or diminishes, the amount of work done will increase or diminish. PKOPORTION. 309 493. An Inverse Proportion is one in which each term increases as the term upon which it depends diminishes, or diminishes as it increases. Thus, in the problem, "If 6 men can mow a field of grass in 9 days, how long will it take 9 men to mow it," as the number of men increases, the number of days required to do the work decreases, and the proportion is an inverse proportion. WRITTEN EXERCISES. 494. 1. If 8 yd. of silk cost $24, what will 15 yd. cost? PROCESS. yd. yd. (1) 8 : 15 : yd. yd. (2) 15 : 8 : 24 : ( ) ( ) : 24 The term wanting (1) The term wanting (2) = $45 ANALYSIS. It is evi- dent that 8 yd. have the same relation to 15 yd. that the cost of 8 yd. has to the cost of 15yd. Hence we have the proportion, 8 yd. : 15 yd. :: $24, the cost of 8 yd. : the cost of 15yd., or 15 yd. : 8yd. :: the cost of 15 yd. : $24, the cost of 8 yd. To find the cost of 15 yards, the term wanting, we divide the product of the means by the extreme, as in (1) ; or the product of the extremes by the mean, as in (2). 2. If 5 men can cut a quantity of wood in 18 days, in how many days could 12 men do the same work? ANALYSIS. It is evident that exactly in proportion as the number of men is increased, the number of days required to do the work is diminished, and therefore 5 men : 12 men :: the PROCESS. men. men. days. days. (1) 5 : 12 :: ( ) : 18 men. men. days. days. (2) 12 : 5 :: 18 : ( ) Term wanting = -Lf4p. 7-J- da. days it will require 12 men to do the work : 18 the number of days required for 5 men to do the work. Or, 12 men : 5 men : : 18 days, the number of days it requires 5 men to do the work : the number of days 12 men require to do the work. 310 PKOPORTION. RULE. Express the ratio between the two numbers that are like numbers. Consider, from the conditions of the problem, whether the proportion is direct or indirect, and arrange the other number and the term wanted so that the two ratios will be equal. Divide the product of the extremes or means by the single ex- treme or mean. The result will be the term wanted. Problems in proportion are sometimes regarded as illustrations of cause and effect, in which two causes and their corresponding effects are compared, giving the following proportion: 1st cause : 2d cause : : 1st effect : 2d effect. 3. If 6 men earn $75 in one week, how much will 10 men earn in the same time? 4. If 16 yards of cloth cost $20, what will be the cost of 7 yards ? 5. A man can buy 45 sheep for $112.50. How much will 18 sheep cost at the same rate? 6. If 8 horses consume 15 tons of hay in 6 months, how much hay will 14 horses consume in the same time? 7. If 6 men can do a piece of work in 45 days, how many days will it take 11 men to do the same work? 8. If 10 men can do a piece of work in 6 days, in how many days can 13 men do the same work? 9. How many men will it require to build 60 rods of wall in the same time that 8 men can build 40 rods? 10. If 8 men can dig a ditch in 15 days, how many days will it take 13 men to dig it? 11. If 6 bushels of wheat can be bought for $7.32, how many bushels can be bought for $45 ? 12. How many barrels of apples can be bought for $2250, if 15 barrels cost $33.75? 13. If it requires 13 men to lay a certain number of bricks in 28 days, how many days will it take 9 men to lay the same number ? COMPOUND PROPORTION. 311 14. If 165 bushels of potatoes can be raised on 1^ acres of ground, how many bushels can be raised on 3^ acres ? 15. If it requires 1^ acres of ground to raise 405 bu. of carrots, how many acres will it require to raise 975 bu. ? 16. Five horses cost a man $626.25. What would be the cost of 13 horses at the same rate? 17. It required 26 men to build an embankment in 80 days. How long would it require 32 men to do the same work ? 18. It took 9 horses to move a stick of timber weighing 12590 pounds. How many pounds would a stick weigh which could be moved by 7 horses? 19. If an ocean steamer sails 1775 miles in 5 days, how many miles will she sail in 6^ days? 20. If a locomotive runs 96f miles in 3|- hours, how many miles will it run in 5 hours? 21. A dog is chasing a rabbit, which has 45 rods the start of the dog. The dog runs 19 rods while the rabbit runs 17. How far must the dog run before he catches the rabbit? 22. A cistern has 3 pipes. The first will fill it in 12 hours, the second in 16 hours, and the third in 18 hours. If all run together how long will it take them to fill it ? 23. If it requires 15 compositors 15 days to set up a book of 675 pages, how many days will they need to set up a book of 900 pages? COMPOUND PROPORTION. 495. A Compound Hatio is the product of two or more simple ratios. 496. A Compound Proportion is a proportion in which either ratio is compound. 497. PRINCIPLE. The product of two or more proportions .is a proportion. 312 PROPORTION. WRITTEN EXERCISES. 498. 1. If 6 men can mow 24 acres of grass in 2 days, by working 10 hours per day, how many days will it take 7 men to mow 56 acres, by working 12 hours per day?- PROCESS. ANALYSIS. A simple pro- (1) 7: 6:: 2 : (1-f- days.) portion is a proportion that (2) 24 : 56 : : 1* : (4 days.) has but one condition - A com ' )ov 10 -in A )oi ,1 pound proportion has more (3) 12 : 10 : : 4 : (3J days.) than one condition . The con . Q r ditions are introduced one at s n n ^ a time, therefore examples in compound proportion may be (Q) < Z4.0O > ..Z.Og- solved as several simple pro- I 19*10 I portions. The first condition Means 6 X 56 X 10 X 2 in this exam P le is : , If 6 men = 3^- can mow 24 acres of grass in Extremes 7 X 24 X 12 2 days, how long will it take 7 men to do the same work? This, solved by simple proportion, (1), gives If days. The second condition is: If the men can mow 24 acres of grass in If days, how long will it take them to mow 56 acres? This, solved by simple pro- portion, (2), gives 4 days. The third condition is: If the work can be done by the men in 4 days, by working 10 hours per day, how many days will it take to do the work if they work 12 hours per day? This, solved by simple proportion, (3), gives 3J days, the time it will take 7 men to mow 56 acres of grass, by working 12 hours per day, if six men can mow 24 acres in 2 days by wording 10 hours per day. Or, Since every simple proportion is an equality of ratios, the product of the three proportions, (1), (2), (3), 'will be an equality of ratios; and, since If and 4 appear in both antecedent and consequent, they may be omitted, and the simple proportions will assume the form of the compound proportion, (4). The problem may be stated, as in the second part of the process, by writing for the third term the term that is like the one sought, and by arranging the others in couplets, considering their relation to the ratio between the third term and the term sought. COMPOUND PROPORTION. 313 RULE. Solve by successive simple proportions, introducing the conditions one at a time. Or, Use for the third term the number which is of the same kind as the term required. Arrange the like numbers in couplets, as in simple proportion. The product of the means divided by the product of the ex- tremes will be the term required. Problems in compound proportion are readily solved by cause and effect. Example 1, stated by cause and effect, is as follows : Is* Cause. 2d Cause. 1st Effect. 2d Effect. 6 men ^ 7 men ^ C C 2 days > j ( ) days > : : < 24 acres : X 56 acres 10 hours) 12 hours } (_ (_ Means 6 X 2 X 10 X 56 _ Extremes 7 X 12 X 24 2. If 15 men can dig a ditch in 45 days by working 10 hours a day, how many days will it take 20 men to dig it, by working 12 hours a day? 3. If a block of granite 6 feet long, 3 feet wide, and 2 feet thick, weighs 5940 pounds, what will be the weight of a block of the same kind, which is 9 feet long, 4 feet wide and 3 feet thick? 4. If I place $1500 at interest for 18 months and receive $135 interest, what sum must I place at interest at the same rate, so that I may receive $275 interest in 8 months ? 5. If it cost $180 to support 5 grown persons and 3 children for 3 weeks, what will it cost to support 8 grown persons and 6 children for 7 weeks, allowing that it costs ^ as much to support a child as a grown person? 6. If 20 men working 8 hours a day, can dig a trench 65 feet long, 9 feet wide and 6 feet deep, in 25 days, how many days will it take 25 men, working 10 hours a day, to dig a trench 75 feet long, 8 feet wide, and 7 feet deep? 814 PBOPORTION. 7. If it costs $240 to board 16 persons 5 weeks, how much will it cost to board 9 persons 22 weeks? 8. If $800 placed at interest, amounts to $880 in 15 months, what sum must be placed at interest at the same rate, to amount to $975 in one year? 9. If it requires 275 yards of cloth f yd. wide to make 75 garments, how many yards of cloth 1J yd. wide, will it require to make 215 such garments? 10. If a bin which is 8 feet long, 6 feet wide and 8 feet deep, holds 309 bushels of wheat, how many bushels will a bin hold that is 14 feet long, 8 feet wide and 9 feet deep? 11. If 15 men, working 10 hours a day, can do a certain piece of work in 18 days, how many days will it require for 13 men to do the same work, by working 8 hours a day? 12. If 12 horses consume 40 bushels of oats in 8 days, how long will 140 bushels of oats last 16 horses? 13. If a regiment of 1025 soldiers consume 11500 pounds of bread in 15 days, how many pounds will 3 regiments of the same size, consume in 12 days ? 14. If the water that fills a vat, which is 8 feet long, 4 feet wide and 5 feet deep, weighs 10000 pounds, what will be the weight of the water required to fill a vat, which is 10 feet long, 5 feet wide and 6 feet deep ? 15. If 5 horses eat as much as 6 cattle, and 8 horses and 12 cattle eat 12 tons of hay in 40 days, how much hay will be needed to keep 7 horses and 15 cattle 65 days? 16. If 15 men working 6 hours a day, can dig a cellar 80 feet long, 60 feet wide and 10 feet deep in 25 days, how many days will it require 25 men working 8 hours a day, to dig a cellar 120 feet long, 70 feet wide and 8 feet deep? 17. If 52 men can dig a trench 355 feet long, 60 feet wide and 8 feet deep in 15 days, how long will a trench be, that is 45 feet wide and 10 feet deep, which 45 men can dig in 25 days? 499. 1. Of what number are 3 and 3 the factors? 4 and 4? 2. Of what number are 3 and 3 and 3 the factors? 4 and 4 and 4? 3. What is the product when 5 is used twice as a factor? 4. What is the product or power, when 6 is used twice as a factor? When 8 is used twice as a factor? 5. What is the product of f X f ? Of f X f? 6. What is the product when f is used twice as a factor ? When | is used three times as a factor? 7. What is the product of two 4's, or the second power of 4? What is the product of three 5's, or the third power of 5 ? What is the third power of 6 ? 8. What is the second power of f ? Of f ? Of f ? DEFINITIONS. 500. A Power of a number is the product arising from using the number a certain number of times as a factor, 501. The powers of a number are named from the number of times the number is used as a factor. Thus, when 2 is used as a factor twice, the product, 4, is called the second power of 2. 9 is the second power of 3. 27 is the third power of 3. The number itself is called the first power. (315) 316 INVOLUTION. 502. The number of times a number is used as a factor is indicated by a small figure called an Exponent, written a little above and at the right of the number. Thus, 3 2 means the second power of 3; 5 4 , the fourth power of 5, etc. Inasmuch as the area of a square is the product of two equal factors, and the volume of a cube is the product of three equal factors, the second power of a number is also called the square, and the third power the cube of the number. 503. Involution is the process of finding the power of a number. WRITTEN EXERCISES. 504. 1. Find the third power of 15. PROCESS. ANALYSIS. To find the third icv/iK\/iK Q Q 7 K power of a number is to find the 1O/\1O/\1O O O i u . product, when the number is used 3 times as a factor. Therefore, the third power of 15 will be 15 X 15 X 15, which is equal to 3375. 2. Find the third power of 12. 23. 39. 24. 3. Find the second power of 47. 51. 29. 34. 4. What is the square of 15 ? 33 ? 24 ? 36 ? 25 ? 5. What is the cube of 28? 45? 18? 21? 41? 6. What is the third power of f ? A ns. -f- X -f X -f- = iff- 7. What is the cube of |? Off? f? T 6 T ? 8. What is the fourth power off? Cube of Find the value of the following : 9. 15*. 10. 25 3 . 11. 30 3 . 12. .05*. 13. .005 s . 14. 2.05 2 . 15. (if) 2 . 16. (M) 2 - 17. (4i) 8 . 18. (25|)V 19. (3.001) 2 . 20. (4.500}) 2 . 21. Eaise 10 to the fourth power; 8 to the third power; 3 to the 6th power. INVOLUTION. 317 505. To find the square of a number in terms of its parts. 1. Find the square of 35 in terms of its tens and units. PROCESS. ANALYSIS. If we square 35 or multiply 35 by itself and write 35 35 25 = w 2 15 15 9 =f every step in the process, we shall have for the first product 25, or the square of the units, for the next two products 15 tens, or two times the product of the tens and units, and for the third product 9 hundreds or the square of the tens. Hence, 506. PRINCIPLE. The square of any number consisting of tens and units, is equal to the tens 2 -f- 2 times the tens X the units -\- the units 2 . Thus, 25 = 20 + 5, and 25 2 = 20 2 + 2 (20 X 5) .+ 5 2 . The above principle is true into whatever two parts the number may be separated, and the principle stated in general terms would be, the square of any number consisting of two parts is equal to the first part 2 + 2 times the first part X the second + second part 2 . Thus, 14 = 8 + 6, and 14 2 = 8 2 + 2 (6X8) Express in terms of their tens and units the square of the following numbers: 2. 54. 5. 47. 8. 74. 11. 39. 3. 71. 6. 89. 9. 95. 12. 44. 4. 68. 7. 26. 10. 82. 13. 67, 14. Square 16 by squaring its parts 9 and 7. 15. Square 20 by squaring its parts 12 and 8. 16. Square 32 by squaring its parts 30 and 2. 17. Square 13 by squaring its parts 7 and 6. 18. Square 26 by squaring its parts 9 and 17. 19. Square 17 by squaring its parts 8 and 9. 318 INVOLUTION. 507. To find the cube of a number in terms of its parts. 1. Find the cube of 35 in terms of its parts. 25 *= s ANALYSIS. By multiplying the second power expressed as in Art. 5O5, by 35, and writing every step, we shall have the cube of the tens, plus the product of three times the square of the tens multi- plied by the units, plus the product of three times the tens multiplied by the square of the units, plus the cube of the units. Hence, 508. PRINCIPLE. The cube of any number consisting of tens and units is equal to the tens s + 3 times the tens 2 X the unite + 3 times the tens X the units 2 + the units*. Thus, 25 = 20 + 5, and 25 3 = 20 3 + 3 (20 2 X 5) + 3 (20 X 5 2 ) + 5 3 . The above principle may be stated in general terms thus: The cube of any number when separated into two parts is equal to the first part 3 + 3 times the first part 2 X second part + 3 times the first part multi- plied by. the second part of their like dimensions, the area of the 128 : x : : 256 : 144 first triangle (128 sq. rVl.) will be to the x __ 7 2 S q r( J area of the second triangle ( x ) as the square of the side of the first triangle (16 2 ) is to the square of the side of the second triangle (12 2 ). Solving the proportion, the area is 72 sq. rd. 2. If the area of a circle, whose diameter is 2 feet, is 6.2832 sq. ft. what will be the area of a circle whose diameter is 4 feet? 3. If the side of a rectangular field containing 25 acres is 40 rods, what is the side of a similar field containing 10 acres? ANALYSIS. Since the corresponding dimensions of similar surfaces are to each other as the square roots of their areas: !/25 : v/IO" :: 40 : x, or 5 : y'IO :: 40 : x. Extracting the square root of 10 and solving the proportion, x t or the corresponding side, is 25.296 rd. 4. If the side of a square field containing 40 acres is 80 rods, what will be the side of a similar field whose area is 25 acres? 5. If the area of a circle whose diameter is 20 feet is 314.16 square feet, what is the diameter of a circle whose area is 113. 0976 square feet? 6. A farmer has two rectangular fields similar in form: one, whose length is 120 rods and whose breadth is 12 rods, contains 9 acres, the other contains 6J acres. What are its length and breadth? 7. A horse tied to a stake by a rope 7.13 rods long can graze upon just 1 acre of ground. How long must the rope be that he may graze upon 5 acres? 330 EVOLUTION. 1ST PROCESS. 20 8 =. 3X20* = 1200 13-824(20+4=24 8000 3 X 4 X 20 = 4? = 240 16 1456 5824 5824 CUBE ROOT. 533. 1. What is the cube root of 13824, or what is the edge of a cube whose solid contents are 13824 units? ANALYSIS. Accord- ing to Pr. 2, Art. 51.9, the orders of units in the cube root of any number may be de- termined from the number of periods ob- tained by separating the number into pe- riods, containing three figures each, beginning at units. Separating the given number thus, there are two periods, or the root is composed of tens and units. The tens in the cube root of the number can not be greater than 2, for the cube of 3 tens is 27000. 2 tens, or 20 cubed, are 8000, which, subtracted from 13824, leave 5824; therefore the root, 20, must be increased by a number such that the additions will exhaust the re- mainder. The cube (A) already formed from the 13824 cubic units is one whose edge is 20 units. The additions to be made, keeping the figure formed a perfect cube, are 3 equal rect- angular solids, B, C and D; 3 other equal rectangular solids, E, F and G; and a small cube, H. Inasmuch as the solids, B, C and D, com- CUBE ROOT. 331 prise much the greatest part of the additions, their solid contents will be nearly 5824 cubic units, the contents of the addition. Since the cubical contents of these three equal solids are nearly equal to 5824 units, and the superficial contents of a side of each of these solids are 20 X 20, or 400 square units, if we divide 5824 by 3 times 400, or 1200, since there are 3 equal solids, we shall obtain the thickness of the addition, which is 4 units. Since all the additions have the same thickness, if their su- perficial contents, or area of each side, are multiplied by 4, the result will be the solid contents of these additions. Besides the larger additions there are three others, E, F, and G, that are each 20 units long and 4 units wide, or whose sides have an area of 80 units each, and the area of all 240 units, and a small cube whose sides have an area of 16 units. The sum of these areas, 1456, multiplied by 4, the thickness of the additions, gives the solid contents of the additions, which are 5824 units. Therefore the edge of the cube is 24 units in length, or the cube root of 13824 is 24. 2D PROCESS. 3J 2 = 20 2 X 3 = 1200 (20 X 4) X 3 = 240 ti a = 4x4= 16 13,824(24 8000 ANALYSIS. In the same manner as before, it may be shown that the root of the number contains only tens and units. The lens can not be great- er than 2, for 3 5824 5t 2 + Stu + n 2 = U5Q 5824 332 EVOLUTION. tens cubed would equal 27000. Cubing and subtracting, there is left 5824, which is composed of three times the tens 2 X the units + 3 times the tens X the units 2 + the units 3 , Art. 507. Since 3 times the tens 2 is much greater than 3 times the tens X the units 2 + the units 3 , 5824 is a little more than 3 times the tens 2 X the units. If, then, 5824 is divided by 3 times the tens 2 , or 1200, the trial divisor, the quotient 4, will be approximately the units of the root. Since 5824 is equal to the sum of 3 times the tens 2 multiplied by the units, 3 times the tens multiplied by the units 2 and the units 3 , the process may be shortened by adding together 3 times the tens 2 , 3 times the tens X the units and the units 2 , and multiplying this sum, 1456, by the units, 4. The product is 5824, which, subtracted from the num- ber, leaves no remainder. When the root consists of more than two orders of units the processes and analyses are similar to those already given. 2. What is the cube root of 48228544? 3X 3X300 300 s 300 2 X60 60 2 = 270000 = 54000 = 3600 48 27 228 000 544 000 300 60 4 21 , s 19 228 656 544 000 364 327600 3 X 360 2 = 388800 3x360x4= 4320 4 2 = 16 393136 1 572 544 1 572 544 RULE. Separate the number into periods of three figures each, beginning at units. Find the greatest cube in the left-hand period, and write its root for the first part of the required root. Cube this root, subtract the result from the left-hand period, and annex to the remainder the next period for a dividend. Take 3 times the square of the root already found for a trial CUBE ROOT. 333 divisor, and by it divide the dividend. The quotient or the quo- tient diminished will be the second part of the root. To this trial divisor add 3 times the product of the first part of the root by the second part, and also the square of the second part. Their sum will be the entire divisor. Multiply the entire divisor by the second part of the root and subtract the product from the dividend. Continue thus until all the figures of the root have been found. 1. When there is a remainder, after subtracting the last product annex decimal ciphers, and continue the process. The figures of the root obtained will be decimals. 2. Decimals are pointed off into periods of three figures each, by beginning at tenths and passing to the right. 3. The cube root of a common fraction is found by extracting the cube root of both numerator and denominator separately, or by reducing it to a decimal and then extracting its root. Extract the cube root of the following : 3. 74088. 4. 262144. 5. 166375. 6. 704969. 7. 185193. 8. 250047. 9. 5545233. 10. 2000376. 11. 153990656. 12. What is the cube root of 2 to 3 decimal places? 13. What is the cube root of 9 to 4 decimal places? 14. What is the cube root of .27 ? Of .64? 15. What is the cube root of f ? f ? APPLICATIONS OF CUBE ROOT. 534. 1. What is the length of the edge of a cubical box that contains 91125 cubic feet? 2. What are the dimensions of a cubical box that con- tains as much as a rectangular box, that is 2 feet 8 inches long, 2 feet 3 inches wide, and 1 foot 4 inches deep ? S34 EVOLUTION. 3. What is the depth of a cubical cistern whose contents are 2197 cubic feet? 4. What must be the depth of a cubical bin that will contain 1000 bushels? 5. What must be the depth of a cubical cistern that will hold 300 barrels of water? 6. A bin that contains 2000 bushels of grain, is just twice as long as it is wide or high. What is its length ? 7. What is the depth of a cubical box that will hold a bushel ? 8. What is the depth of a cubical box that will hold a barrel of water (31-J- gal.) ? 9. How much will it cost, at 30 cents per sq. yd., to plaster the bottom and sides of a cubical cistern that will hold 300 barrels ? 10. A miller wishes to make a wagon-box large enough to hold 100 bushels, having the length 3 times the width and height. What will be its dimensions? 11. Which has the greater surface, a cube whose solid contents are 13824 cubic feet, or a rectangular solid having the same solidity, whose height is half its length and whose width is three-fourths its length? How much? 12. If a cubic metre contains 61026.048 cubic inches, what is the length of a linear metre ? SIMILAR FIGURES. 535. The truth of the following principles can be shown by geometry: PRINCIPLES. 1. Similar solids are to each other as the cubes of their like dimensions. Hence, 2. The corresponding dimensions of similar solids are to each other as the cube roots of their volumes. CUBE HOOT. 335 1. If a globe 4 inches in diameter weighs 8 lb., what will be the diameter of a similar one that weighs 125 lb. ? PROCESS. ANALYSIS. Since the corre- A. a/S" 3/T9^ n\ spending dimensions of similar solids are proportional to the 4 : x : : 2 : 5 (2) cube roots of these volumes, we x or diameter is 10 in. have the . diam eter of the smaller globe 4 inches : the diameter of the larger globe x : : the cube root of the weight of the smaller globe ty 8 : the cube root of the weight of the other globe ^'125. (1). Ex- tracting the cube root of 8 and 125, and we have Prop. (2). Whence solving, the diameter is 10 inches. 2. If a ball 5 ft. in diameter weighs 800 lb., what will be the diameter of a similar ball which weighs 3 T. 4 cwt. ? 3. If a globe of gold 1 inch in diameter is worth $125, what will be the value of one 3 inches in diameter? 4. If a cubical bin 8 ft. long will hold 411.42 bu., what must be the dimensions of a similar bin, that will hold 1000 bushels ? 5. A ball 3 feet in diameter weighed 2000 lb. What will be the diameter of one that weighs 1000 lb. ? 6. The dimensions of a cubical bin were such that it would contain 1000 bushels of wheat. How would the dimensions of a similar bin that would hold 8000 bushels compare with the dimensions of such a bin? 7. The diameters of two spheres are respectively 4 and 12 inches. How many times the smaller sphere is the larger? 8. Three women own a ball of yarn 4 inches in diameter. How much of the diameter of the ball must each wind off. so that they may share equally? 9. A stack of hay in the form of a pyramid 12 ft. high, contained 8 tons. How high must a similar stack be, that it may contain 60 tons? 536. 1. How does each of the numbers 2, 4, 6, 8, 10, 12, compare with the number that follows it? 2. How may each of the numbers 4, 6, 8, etc., be obtained from the one that precedes it? 3. How does each of the numbers 2, 5, 8, 11, 14, 17, com- pare with the number that follows it? How with the one that precedes it? 4. Write in succession some numbers beginning with 3 having a common difference of 2. 5. Write a series of numbers beginning at 4, and having a common difference of 4. 6. Write a series of numbers beginning with 25, and de- creasing by the common difference 4. 7. How does each of the numbers 2, 4, 8, 16, 32, etc., com- pare with the one that follows it ? How may each be obtained from the one that precedes it? 8. Write a series of numbers beginning with 2 and increas- ing by a common multiplier 3. 9. Write a series of numbers beginning with 5, and increas- ing by a common multiplier 5. DEFINITIONS. 537. A Series of numbers is numbers in succession, each derived from the preceding according to some fixed laws. (336) ARITHMETICAL PROGRESSION. 337 538. The first and last terms of a series are called the extremes, the intervening terms the means. Thus, in the series 2, 4, 6, 8, 10, the numbers 2 and 10 are the ex- tremes and the others are the means. 539. An Ascending Series is one in which the num- bers increase regularly from the first term. Thus, 2, 5, 8, 11, 14, 17, 20, etc., is an ascending series. 540. A Descending Series is one in which the num- bers decrease regularly from the first term. Thus, 48, 24, 12, 6, 3, is a descending series. ARITHMETICAL PROGRESSION. 541. An Arithmetical Progression is a series of numbers which increase or decrease by a constant common difference. Thus, 5, 9, 13, 17, 21, etc., is an arithmetical progression of which the common difference is 4. 542. 1. The first term of an arithmetical series is 3 and the common difference is 2. What is the 7th term? PROCESS. ANALYSIS. S i n c e the common difference Com. diff. , 2 X 6 : is 2> the second term is First term, 3 -j- 12 = 15, the 7th term. equal to the first plus once the common differ- ence, the third term is equal to the first plus twice the common differ- ence, the fourth term is equal to the first term plus three times the com- mon difference. Hence, the seventh term will be equal to the first term plus six times the common difference, which is 15. .J.ra/ term of an arithmetical progression is equal to the first term, increased by the common difference multiplied by a number one less than the number of terms. 22 338 PROGKESSIONS. 2. The first term is 10 and the common difference 5. What is the 10th term? Prove it. 3. The first term is 6 and the common difference is 8. What is the 25th term ? 4. A boy agreed to work for 50 days at 25 cents the first day, and an increase of 3 cents per day. What were his wages the last day ? 5. A body falls ISy 1 ^ feet the first second, 3 times as far the second second, 5 times as far the third second. How far will it fall the seventh second ? 6. An arithmetical series has 1000 terms, the first term of which is 75 and the common difference 5. What is the last term? 7. Find the sum of an arithmetical series of which the first term is 2, the common difference 3, and the number of terms 7. PROCESS ANALYSIS. By examining the series 2, 2 4. ( 6 y 3 ^ = 2 5 ' 8> n ' 14 ' 17 ' 2 ' h is evident that the average term is 11, for if half the sum 2 -\- 20 2 2 o f an y t wo terms equidistant from the 2 2 -r- 2 = 11 extremes be found it will be 11, and in 1 1 X 7=77 general in any arithmetical progression the average term is equal half the sum of the extremes or any two terms equidistant from the extremes. Since the first term is 2 and the common difference 3, the last term is found by the previous rule to be 20. The sum of the extremes is therefore 22, which, divided by 2, gives the average term. And since there are 7 terms, the sum will be 7 times the average term, or 77. RULE. To find the sum of an arithmetical series : Multiply half the sum of the extremes by the number of terms. 8. What is the sum of an arithmetical series composed of 50 terms, of which the first term is 2 and the common differ- ence 3 ? 9. What is the sum of a series in which the first term is Y 1 ^ the common difference, y 1 ^-, and the number of terms 100? GEOMETKICAL PROGRESSION. 339 10. A man walked 15 miles the first day, and increased his rate 3 miles per day for 10 days. How far did he walk in the eleven days? 11. How many strokes does a clock strike in 12 hours? 12. A person had a gift of $100 per year from his birth until he became 21 years old. These sums were deposited in a bank and drew simple interest at 6^. How much was due him when he became of age? GEOMETRICAL PROGRESSION. 543. A Geometrical Progression is a series of numbers which increase or decrease by a constant multiplier or ratio. Thus, 5, 10, 20, 40, 80; etc., is a geometrical progression, of which the multiplier or ratio is 2. WRITTEN EXERCISES. 544. 1. The first term of a geometrical series is 3 and the multiplier or ratio is 2. What is the 5th term? PROCESS. ANALYSIS. Since the multiplier is 2, the 2*= 1 6 second term will be 3 X .2, the third 3X2X2 o v i * _ 4 o or 3 X 2 L> , the fourth 3 X 2 2 X 2 or 3 X 2 3 and the fifth 3 X 2 3 X 2 or 3 X 2 4 , that is, the fifth term is equal to the first term multiplied by the ratio raised to the fourth power. RULE. Any term of a geometrical progression is equal to the first term, multiplied by the ratio raised to a power one less than the number of the term. 2. The first term of a geometrical progression is 10, the ratio 3. What is the 6th term? 340 PROGRESSIONS. 3. The first term of a geometrical progression is 10, the ratio 4, and the number of terms 6. What is the 6th term? 4. If a farmer should hire a man for 10 days, giving him 5 cents for the first day, 3 times that sum for the second day, and so on, what would be his wages for the last day? 5. If the first term is $100 and the ratio 1.06, what is the 6th term ? Or, what is the amount of $100 at compound interest for 5 years at 6% ? 6. What is the amount of $520 for 6 years, at 5^ com- pound interest? 7. What is the sum of a geometrical series, of which the first term is 5, the ratio 3, and the number of terms 5 ? PROCESS. ANALYSIS. Since in this series the first term is 5 X 8 1 = 4 5, the 5th term. ^ ^ ^ s> and the 3X405 _ 5 number of terms 5, their 3 _ 1 =605, the sum. Bum may be obtained by the following process, which illustrates the formation of the rule : Series 5 + 15 + 45 + 135 + 405 3 times Series 15 + 45 + 135 + 405 + 1215 2 times Series = 1215 5 Series = RULE. The sum of a geometrical series is equal to the differ- ence between the first term 9 and the product of the last term by the ratio, divided by the difference between the ratio and 1. 8. The extremes of a geometrical progression are 4 and 1024, and the ratio 4. What is the sum of the series? 9. The extremes are ^ and fff and the ratio 2-|-. What is the sum of the series? 10. What is the sum of the series in which the first term is 2, the last term 0, and the ratio -^; or what is the sum of the infinite series 2, 1, ^, ^, , ^ -^-, etc? l7.laKEiU;ftliPKi DEFINITIONS. 545. Mensuration treats of the measurement of lines, surfaces, and solids. 546. A Line is that which has length only. 547. A Straight Line is a line that does not change its direction. 548. A Curved Line is a line that changes its direction at every point. 549. Parallel Lines are such as are* equidistant throughout their whole extent. Straight Line. Curved Lines. Parallel Lines. 550. A Plane Surface is a surface such that a straight line joining any two points of it is wholly in the surface. 551. A Curved Surface is a surface such that no part of it is a plane surface. 552. An Angle is the divergence of two lines that meet. 553. A Right Angle is the angle formed when one straight line meets another making the adjacent angles equal. The lines are perpendicular to each other when a right angle is formed. Angle. Two Right Angles. (341) 342 MENSURATION. Acute Angle. Obtuse Anglo. Triangle. Kec tangle. Circle. 554. An Acute Angle is an angle which is less than a right angle. 555. An Obtuse Angle is an angle which is greater than a right angle. 556. The Vertex of an angle is the point where the sides meet. 557. A Triangle is a figure which has three sides and three angles. 558. A Quadrilateral is a figure bounded by four sides. 559. A Parallelogram, is a quadri- lateral whose opposite sides are parallel. 560. A Rectangle is a parallelogram whose angles are right angles. 561. A Polygon is a plane figure bounded by straight lines. 562. A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center. 563. The Circumference is the line which bounds the circle. 564. A Hadius of a circle is a straight line drawn from the center to the circum- ference. 565. A Diameter of a circle is a straight line drawn through the center and terminating at both ends in the circumfer- ence. MENSURATION. 343 Base. 566. The Hase of a figure is the side on which it is assumed to stand. 567. The Altitude of a figure is the perpendicular distance between the base and the highest point opposite it. 56S. A Diagonal of a figure is a straight line joining the vertices of two angles not adjacent. 569. The Perimeter of a figure is the length of the lines that bound it. 570. The Area of a surface is the definite amount of surface it contains. MEASUREMENT OF LINES. 571. It can be shown by geometry that the circumference of a circle is 3.1416 -f- times its diameter. For ordinary measurements it is sufficiently accurate to consider the circumference 3f times the diameter. RULE. 1. The circumference is equal to the diameter multi- plied by 3.1416. 2. The circumference divided by 3. 1416 is equal to the diameter. WRITTEN EXERCISES. 572. 1. What is the circumference of a circle 10 feet in diameter? 2. What is the circumference of a circle 45 feet in diameter ? 3. How far is it around a circular lake that is 300 rods in diameter ? 4. What is the circumference of a circle whose radius is 20 rods? 344 MENSURATION. 5. What is the circumference of a circle whose radius is 5 feet 6 inches ? 6. What is the diameter of a circle whose circumference is 318.5 rods? 7. What is the radius of a circle whose circumference ia 1284 rods? MEASUREMENT OF SURFACES. . 573. To compute the area of a parallelogram. The truth of the following principle has been shown already : PRINCIPLE. The area of any rectangular figure is equal to the product of its length by its breadth or altitude. By examining the figure A, B, C, D, it will be seen that it is equal to E, F, D, C, and that any oblique parallelogram is equal to a rectangu- lar parallelogram of the same base and altitude. Therefore, RULE. The area of any parallelogram is equal to the prod- uct of the base multiplied by the altitude. WRITTEN EXERCISES. 574. 1. How many square feet are there in a parallelo- gram, whose length is 40 feet and altitude 13 feet? 2. What is the area of a parallelogram whose base meas- ures 7 feet and whose altitude is 3 feet 8 inches ? 3. What is the area of a field in the form of a parallelo- gram, whose length is 30 rods and the perpendicular distance between the sides is 24 rods? 4. What is the area of a parallelogram whose length is 35 feet and whose altitude is 15 feet? MENSURATION. 345 575. To compute the area or a triangle. If C E be drawn parallel to the base of the triangle, and B E be drawn parallel to A C, the parallelogram A B E C will be formed, of which the original triangle is one-half. In the same manner it can be shown that every triangle is one-half of a parallelogram of the same base and altitude. Therefore, RULE. The area of a triangle is equal to one-half the prod- uct of the base by the altitude. When the three sides are given, the following is the rule: RULE. From half the sum of the three sides subtract each side separately. Multiply together the half sum and the three remain- ders, and extract the square root of the product. The result will be the area of the triangle. WRITTEN EXERCISES. 576. 1. What is the area of a triangle whose base is 24 feet, and whose altitude is 18 feet? 2. What is the area of a triangle whose base is 21 feet and whose altitude is 12 feet? 3. What is the cost, at $850 per acre, of a triangular piece of ground, the three sides of which are in a ratio of 5, 6 and 8, and whose shortest side is 120 feet? 4. What is the area of a triangle, the three sides of which are respectively 180 feet, 150 feet, and 80 feet? 5. A house is 32 feet wide, and the rafters are 20 feet long on each side, exclusive of any projections. What will the lumber cost at $22.50 per M, which will inclose both gable ends of the house? 6. What is the area of a triangle whose base is 300 feet, and whose altitude is 100 feet? 346 MENSURATION. 577. To compute the area of a polygon. Since any figure may be divided into tri- angles, its area will be the area of the triangles which compose it. Therefore, RULES. I. The area of a trapezium is equal to the diagonal, multiplied by half the sum of the perpendiculars drawn from the vertices of the opposite angles to the diagonal. II. Tlie area of a trapezoid is equal to the sum of the parallel sides multiplied by half the altitude. III. The area of a regular polygon is equal to the perimeter of the polygon mul- tiplied by one -half the perpendicular dis- tance from the center to one of the sides of ^ ie polygon. WRITTEN EXERCISES. 578. 1. What is the area of a trapezium, the diagonal of which is 110 feet, and the perpendiculars to the diagonal are 40 feet and 60 feet respectively? 2. The parallel sides of a trapezoid are respectively, 10 rods and 8 rods, and the altitude 6 rods. What is its area ? 3. A regular octagon has a perimeter of 96 ft. ; the perpen- dicular from the center to one side is 12 ft. What is its area ? 4. What is the cost, at $125 per acre, of a piece of ground in the form of a trapezoid, whose parallel sides are respect- ively, 40 rods and 30 rods, and whose altitude is 20 rods? 5. I paid $110 per acre for a piece of ground in the form of a -trapezium. A diagonal line crossing it was 120 rods long, and the perpendiculars drawn to the diagonal were, respectively, 30 rods and 20 rods. What did it cost me ? MENSURATION. 347 579. To compute the area of a circle. From the accompanying figure it is evident that a circle may be regarded as composed of a large number of tri- angles, the sum of whose bases forms the circumference of the circle, and whose altitude is the radius of the circle. Hence, RULE. 1. The area of a circle is equal to the circumference, multiplied by half the radius; or, 2. Multiply the square of the diameter by ,7854, WRITTEN EXERCISES. 580, 1. What is the area of a circle whose diameter is 5 feet? 2. What is the area of a circle whose diameter is 8 feet ? 3. What is the area of a circle whose circumference is 120 rods? 4. What is the area of a circle whose circumference is 100 feet? 5. A gentleman discovered that the distance around a cir- cular pond was 320 rods. What was its area ? 6. If a horse is tethered to a stake by a rope 15 rods long, over how much surface can he graze? 7. How long must a rope be that a horse can graze on just an acre? 8. The area of a circle is 113.0976 square rods. What is its diameter? 9. The round-house of the P. and S. Railroad is 350 feet in diameter. How much land does it cover? 10. What is the area of a railroad turn-table 35 feet in diameter? 348 MENSURATION. MEASUREMENT OF SOLIDS. DEFINITIONS. 581. A Solid or Body is that which has length, breadth and thickness. The planes which bound a solid are called its /aces, and their inter- sections its edges. Triangular Quadrangular Parallelopipcdon. Cylinder. Prism. Prism. 582. A Prism is a solid, having its two ends equal poly- gons, parallel to each other, and its sides parallelograms. Prisms are named from the form of their bases triangular, quadrangular, pentagonal, etc. 583. A Parallelopipedon is a solid whose opposite faces are equal and parallel parallelograms. 584. A Cylinder is a regular solid bounded by a uniformly curved surface, and having for its ends two equal circles, parallel to each other. The face of any section of a cylinder parallel to the base is a circle equal to the base. 585. A Pyramid is a solid whose base is a polygon and whose faces are triangles, meeting at a point called the vertex of the pyramid. MEASUREMENT OF SOLIDS. 349 586. A Cone is a solid, whose base is a circle and whose surface tapers uniformly to a point called the vertex. Pyramid. Frustum of Cone. Frustum of Pyramid. 587. A Frustum of a pyramid or cone is the portion remaining, after the top has been cut off by a plane parallel to the base. 588. A Sphere is a solid, every point of whose surface is equally distant from a point within, called the center. 589. The Diameter of a sphere is a straight line passing through the center, and terminating in the surface at both ends. Sphere. 590. The Radius of a sphere is one-half the diameter, or the distance from. the center to the surface. 591. The Circumference of a sphere is the greatest distance around the sphere. 592. The Altitude of a solid is the perpendicular dis- tance from its highest point to the plane of the base. CONVEX SURFACE OF SOLIDS. 593. The Convex Surface of a solid, is all its surface except its base or bases. The entire convex surface includes the area of the bases also. 350 MENSURATION. 594. To find the convex surface of a prism or cylinder. It is evident that if a prism or cylinder were 1 inch high, its convex surface would contain as many square units of surface as there were units in the perimeter of the base; and if it were 2 inches, 3 inches, or 4 inches high, the convex surface would contain 2, 3, or 4 times the number of units in the perimeter of the base. Hence the following RULE. Multiply the perimeter of the base by the altitude. WRITTEN EXERCISES. 595. 1. What is the convex surface of a cylinder whose diameter is 2 feet and length 5 feet? 2. What is the convex surface of a quadrangular prism whose sides are each 2^ feet and whose height is 4 feet ? 3. What is the convex surface of a triangular prism whose sides are each 6 feet and whose altitude is 8 feet? 4. What is the entire surface of a cylinder which is 5 feet in length, and whose base is 2 feet in diameter? 5. What is the convex surface of a piece of timber in the form of a triangular prism, which is 18 feet long and the sides of whose base are 10 inches, 14 inches, and 18 inches? 596. To lifiicl the convex surface of a pyramid or cone. It is evident that the convex surface of any pyra- mid is composed of triangles, and the convex surface of a cone may also be assumed to be made up of an infinite number of triangles. The bases of these triangles form the perimeter of the solid, and their height is the slant height of the solid. Therefore the following is the rule: RULE. Multiply the perimeter of the base by one-half the slant height. MEASUREMENT OF SOLIDS. 351 WRITTEN EXERCISES. 597. 1. What is the convex surface of a quadrangular pyramid, whose base is 15 feet square and the slant height 18 feet? 2. What is the convex surface of a cone whose diameter at the base is 12 feet and whose slant height is 20 feet? 3. What is the convex surface of a cone whose base is 20 feet in diameter and whose slant height is 20 feet? 4. What is the cost of painting a church steeple, the base of which is an octagon 6 feet on each side, and whose slant height is 80 feet, at $.30 per square yard? 5. How many feet of convex surface are there on a cone, the base diameter of which is 6 feet and whose slant height is 9-J- feet? 6. How many feet of convex surface are there on a pyra- mid whose base is 10 feet square and whose slant height is 20 feet? 7. How many feet of convex surface are there on a cone whose base is 8 feet in diameter and whose slant height is 6 feet? 8. What is the convex surface of a cone whose base is 10 feet in diameter and whose slant height is 10 feet? 598. To find the convex surface of a frustum of a pyramid or cone. It is evident that the convex surface of a frustum of a pyramid is composed of trapezoids, the sum of whose parallel sides forms the perimeter of the bases, and whose altitude is the slant height of the frustum; and the convex surface of a cone may be assumed to be made of an infinite number of trapezoids. Hence, RULE. Multiply half the sum of the perimeter of the two bases by the slant height. 352 MENSUEATION. WRITTEN EXERCISES. 599. 1. How many feet of convex surface are there in the frustum of a cone whose slant height is 8 feet, the diameter of whose lower base is 12 feet and upper base 8 feet? 2. What is the convex surface of the frustum of a pyramid the slant height of which is 25 feet, whose lower base is 40 feet square, and whose upper base is 20 feet square ? 3. What did it cost, at $ .15 per sq. yd., to paint the con- vex surface of a vat which was 10 feet in diameter at the bottom and 8 feet at the top, the slant height of which was 12 feet? 4. What is the convex surface of a vat, the base of which is 9 feet square whose top is 8 feet square and whose slant height is 10 feet? 5. What would the lumber cost at $40 per M, to build such a vat if the sides were of 1|- inch plank, and the bottom was 2 inch plank ? 600. To find the convex surface of a sphere. The convex surface of a sphere is computed, according to geomet- rical principles, by the following rule: EULE. 1. Multiply the diameter by the circumference. 2. Multiply the square of the diameter by 3.1416. EXER CISES. 601. 1. What is the convex surface of a sphere whose diameter is 15 inches? 2. What is the convex surface of a spherical cannon-ball 8 inches in diameter? 3. What is the convex surface of a base-ball whose circum- ference is 9|- inches? MEASUREMENT OF SOLIDS. 353 4. What is the convex surface of a sphere whose circum- ference is 12 feet? VOLUME OF SOLIDS. 602. The Volume of any body is the number of solid units it contains. ' 603. To find the volume of a prism or cylinder. It is evident that if a prism or cylinder were 1 inch high, it would contain as many cubic inches as there were square inches in the area of the base; and if it were 2 inches, 3 inches, or 4 inches high, the volume would be 2 or 3 or 4 times as much. Hence the fol- lowing is the rule: RULE. Multiply the area of the base by the altitude. WRITTEN EXERCISES. 604. 1. What are the solid contents of a prism whose base is 12 inches square and whose height is 2 feet ? 2. What is the volume of a cylinder whose diameter is 1|- feet and whose length is 4 feet? 3. What would be the cost of a piece of timber 20 feet long, 18 inches wide and 12 inches thick at $.30 per cubic foot? 4. What will be the capacity in bushels of a square bin the base of which was 8 feet square, and the height of which was 9 feet on the inside ? 5. How many gallons of water will a vat in the form of a cylinder hold, whose inside dimensions are base 8 feet in diameter, height 7 feet? 6. How much would the wheat be worth at $1.85 per bushel, which would just fill a bin the base of which is 15 feet square, and the height of which is 12 feet? 23 354 MENSURATION. 605. To find the volume of a pyramid or cone. It can be shown by geometry that a pyramid or cone is one-third of a prism or cylinder of the same base and altitude. Hence the fol- lowing is the RULE. Multiply the area of the base by one-third of the altitude. WRITTEN EXERCISES. 606. 1. What are the solid contents of a cone, the diam- eter of whose base is 6 feet and whose altitude is 9 feet ? 2. What are the solid contents of a pyramid whose base is 30 feet square and whose altitude is 60 feet? 3. If a cubic foot of granite weighs 165 lb., what is the weight of a granite cone the diameter of whose base is 6 feet and whose altitude is 8 feet ? 4. What is the weight of a marble pyramid whose base is 4 feet square and whose altitude is 8 feet, if a cubic foot of marble weighs 171 pounds ? 607. To find the volume of a frustum of a pyramid or cone. It can be shown by geometry that the frustum of a pyramid or cone is equal to three pyramids or cones, having for their bases, re- spectively, the upper base of the frustum, its lower base, and a mean proportional between the two bases. Hence the following is the RULE. To the sum of the areas of the two ends add the square root of the product of these areas, and multiply the result by one- third of the altitude. EXER CISES. 608. 1. What is the volume of a frustum of a pyramid the lower base of which is 20 feet square, the upper base 10 feet square and the altitude 20 feet? MEASUREMENT OF SOLIDS. 355 2. What are the solid contents of the frustum of a cone whose upper base is 5 feet in diameter, whose lower base is 8 feet in diameter, and whose altitude is 7 feet? 3. A tree was 3 feet in diameter at the butt and its diam- eter at a height of 40 feet was 1 foot. What were the cubical contents of that portion of the tree? 4. A vat whose inside measurements were as follows diameter of the bottom 12 feet, diameter of the top 10 feet, height 9 feet was filled with water. How many gallons did it contain? 609. To find the volume or contents of a sphere. A sphere may be regarded as composed of pyra- mids whose bases form the surface of the sphere, and whose altitude is the radius of the sphere. Hence the following is the KULE. 1. Multiply the convex surface by one -third of the radius; or, 2. Multiply the cube of the diameter by .5236. EXERCISES. 610. 1. The diameter of a sphere is 5 feet. How many cubic feet does it contain? 2. Find the contents of a sphere whose diameter is 8 feet. 3. The circumference of a sphere is 9.4248. What are its cubical contents? 4. A cubic foot of cast-iron weighs about 450 pounds. What is the weight of a cannon-ball whose diameter is 18 inches? 5. What are the cubical contents of a spherical vessel the diameter of which is 2|- feet? 6. How many cubic feet are there in a spherical body whose diameter is 25 feet ? 356 MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 611. 1. If 5 men can do a piece of work in 12 days, how long will it take 6 men to do the same work? 2. If 5 barrels of apples cost $7.50, what will 8 barrels cost at the same rate? 3. It required 20 men to load a vessel in 6 days, how many men would it require to load it in 1^- days ? 4. A steamboat sailed 42|- miles in 2|- hours. How far did she sail in 20 minutes? 5. If 6 men can dig 28 rods of ditch in 1 day, how many men will it require to dig 56 rods in f of a day? 6. If | of a yard of broadcloth cost $3f , what will of a yard cost? 7. If it costs $50 to support a family of 8 persons for 2^- weeks, what will it cost to support 10 persons 3 weeks? 8. If 3 pounds of tea are worth 14 pounds of coffee, and 5 pounds of coffee are worth 18 pounds of sugar, and 21 pounds of sugar are worth 60 pounds of flour, how many pounds of flour are equal in value to 7 pounds of tea ? 9. A farmer sold 12 firkins of butter, each containing 56 pounds, for 23 cents a pound, and received in payment 5 pounds tea at 85 cents per pound, 60 pounds sugar at 13 cents per pound, 15 yards cloth at $1.121 per yard, and the balance in money. How much money did he receive? 10. A regiment of soldiers consisting of 1100 men, was furnished with bread sufficient to last it 8 weeks, allowing each man 15 oz. per day. If ^ of it was found to be unfit for use, how many ounces per day shall each man receive so that the balance may last 8 weeks? 11. A man being asked how many sheep he had, replied, " If I had 3 times as many as I have and 5 sheep, I would have 185." How many had he? MISCELLANEOUS EXAMPLES. 357 12. A man paid \ of his money on a debt, \ of the remain- der for a suit of clothes, \ of the remainder for provisions, and lost \ of the remainder, when he had $5 left. How much had he at first? 13. Three men engage to reap a field of wheat. A can do it in 15 days, B in 18 days and C in 20 days. In what time can they do it together ? 14. A farmer was offered $1.45 per bu. for his wheat, but determined to have it ground and sell the flour. It cost to take it to the mill 2^ cents per bu. ; the miller took \ for grinding; it took 4| bu. to make a barrel of flour; he paid 45 cents apiece for barrels, and it cost 25 cents per barrel commission to sell it. 75 bbl. sold for $550 and 25 bbl. for $165. If the refuse was sold for $100, did he make or lose, and how much per hundred barrels? 15. A farmer being asked how many apple-trees he had, replied, " If I had 3 times as many and 5 trees more, I would have 1358." How many had he? 16. \ of A's money is equal to of B's, and the difference is $8. How much has each ? 17. A, B and C hire a pasture for $170. A puts in 70 sheep for 6^ months, B 24 cattle for 4^- months, C 10 cattle and 35 sheep for 5|- months. If 2 cattle eat as much as 7 sheep, how much should each pay? 18. If a pole 10 feet long, casts a shadow 13 feet long, what is the length of a pole which will cast a shadow 62^- feet long at the same time? 19. A's weight is f that of B, and C's weight is as much as A's and B's together. The sum of their weights is 490 pounds. What is the weight of each ? 20. f of A's money is equal to f of B's, and the difference is $5. How much money has each ? 21. The ages of A, B and C, are to each other as 3, 4 and 5, and their sum is 136 years. What is the age of each? 358 MISCELLANEOUS EXAMPLES. 22. A boy bought a certain number of apples at the rate of 4 for 5 cents, and sold them at the rate of 3 for 4 cents. He gained 60 cents. How many did he buy ? 23. A, B and C agree to build a house. A and B can do the work in 32 days, B and C in 28 days, and A and C in 26 days. How long will it take them to do it by working together? How long would it take each to do it alone? 24. A can build a wall in 10 days, by working 12 hours a day, B can build it in 9 days, by working 10 hours a day. In how many days can both build it, by working 8 hours a .day? 25. A pair of horses is sold for $390. One of them is worth |- as much as the other. What is the value of each? 26. A hind wheel of a carriage 4 feet 6 inches high, re- volved 720 times in going a certain journey. How many revolutions did the fore wheel make, which was 4 feet high ? 27. The shadow of a pole 6 feet long is 9 inches, and the shadow of a steeple at the same time is 9 feet long. What is the height of the steeple ? 28. What is the bank discount on a note for $245.30, due in 90 days, if discounted at 6^ ? 29. If a man takes 2 steps of 30 inches each in 3 seconds, how long will it take him to walk 10 miles? 30. It cost $150 to support 4 grown persons and 3 children 8 weeks. What will it cost to support 3 grown persons and 8 children for the same time, if 3 children cost as much as 2 grown persons? 31. A man bought 20 bushels of wheat and 15 bushels of corn for $36, and 15 bushels of wheat and 25 bushels of corn for $32.50? What did he pay per bu. for each? 32. A fox has 120 rods the start of a hound. If the hound runs 30 rods while the fox runs 26, how far will the hound run before he overtakes the fox? MISCELLANEOUS EXAMPLES. 359 33. A starts on a journey at the rate of 3 miles an hour. 6 hours afterward, B starts after him at the rate of 4 miles an hour. How far will B travel before he overtakes A? 34. One-fourth of a certain number is 10 more than -J- of it. What is the number? 35. If to a certain number you add ^ of itself and -^ of itself, the sum will be 105. What is the number? 36. If to a certain number you add 15 more than | of itself, the sum will be 40. What is the number? 37. How many days will it take 30 men to do a piece of work, which 20 men can do in 45 days? 38. If a man can earn -| of a dollar in f of a day, how much can he earn in f of a day? 39. How many yards of silk f yard wide, will it take to line 4^ yards of broadcloth If yards wide? 40. If 14 ounces of wool make 2J yards of cloth 1 yard wide, how much will it take to make 6|- yards 1^ yards wide? 41. How many tiles 14 inches long, will it take to make a drain which is -|- of a mile long ? 42. If $300 placed at interest yields an income of $18 in 9 months, how much must be placed at interest at the same rate to yield an income of $115 in 6 months? 43. If to a certain number you add \ of itself, the result will be 20 less than double the number. What is the number ? 44. At what time between 5 and 6 o'clock will the hour and minute hands of a clock be exactly together ? 45. Two soldiers start together for a certain fort. One, who travels 12 miles per day, after traveling 9 days, turns back as far as the other had traveled during those 9 days. He then turns and pursues his way toward the fort, where both arrive together 18 days from the time they set out. At what rate did the other travel? 46. A man agreed to execute a piece of work in 60 days, 360 MISCELLANEOUS EXAMPLES. and employed 30 men to perform the labor. At the end of 40 days it was only half finished. How many additional laborers was he obliged to employ to perform the work within the time agreed upon? 47. A person, being asked the time of day, replied that it was past noon, and that f of the time past noon was equal to f of the time to midnight. What was the time ? 48. A gentleman wishes his son to have $3000 when he is 21 years of age. What sum must be deposited at the son's birth, in a savings bank, which pays compound interest at the annual rate of 6^, so that the deposit shall amount to that sum when the boy becomes of age? 49. A note for $100 was due on Sept. 1st, but on Aug. llth the maker proposed to pay as much in advance as will allow him (?0 days after Sept. 1st to pay the balance. How much must be paid Aug. llth, money being worth 6%? 50. What sum must a person save annually, commencing at 21 years of age, so that he may be worth $25000 when he is 40 years old, if he gets 6^ compound interest on his money? 51. If a merchant sells f of an article for what -J of it cost, what is his gain per cent. ? 52. If goods are sold so that -f of the cost is received for half the quantity of goods, what is the gain per cent. ? 53. A man sold a horse and carriage for $597, gaining by the sale 25^ on the cost of the horse and 10^ on the cost of the carriage. If f of the cost of the horse equaled -| of the cost of the carriage, what was the cost of each ? 54. If 300 cats can kill 300 rats in 300 minutes, how many cats can kill 100 rats in 100 minutes? 55. A party of 8 hired a coach. If there had been 4 more the expense would have been reduced $1 for each person. How much was paid for the coach? 56. I sold goods at a gain of 20%. If they had cost me MISCELLANEOUS EXAMPLES. 361 $250 more than they did, I would have lost 20% by the sale. How much did the goods cost me? 57. A laborer agreed to work for $1.25 per day and his board, paying $ .50 per day for his board when he was idle. At the end of 25 days he received $19. How many days was he idle? 58. A is 20 years of age; B's age is equal to A's and half of C's; and C's is equal to A's and B's together. What is the age of each? 59. A and B were partners in a profitable enterprise. A put in $4500 capital and received -f of the profits. What was B's capital? 60. A man spent $4 more than half his money traveling, one-half what he had left and $2 more for a coat, $6 more than half the remainder for other clothing, and had $2 left. How much money had he at first? 61. A boy bought at one time 5 apples and 6 pears for 28 cents, and at another time 6 apples and 3 pears for 21 cents. What was the cost of each kind of fruit ? 62. A and B can do a piece of work in 20 days. If A does f as much as B, in how many days can each do it ? 63. A man bought a farm for $5000, agreeing to pay prin- cipal and interest in 5 equal annual installments. What will be the annual payment, including interest at 6%? 64. A carriage maker sold two carriages for $300 each. On one he gained 25^ ; on the other he lost 25^. Did he gain or lose by the sale? How much, and how much per cent. ? 65. If a ladder placed 8 feet from the base of a building 40 feet high, just reached the top, how far must it be placed from the base of the building that it may reach a point 10 feet from the top? 66. Mr. A. is 35 years of age and his son is 10. How soon will the son be one-half the age of the father? - t 362 MISCELLANEOUS EXAMPLES. 67. A person in purchasing sugar found that if he bought sugar at 11 cents he would lack 30 cents of having money enough to pay for it, so he bought sugar at 10^- cents and had 15 cents left. How many pounds did he buy? 68. A farmer had his sheep in three fields, f of the num- ber in the first field was equal to f of the number in the second field, and -f of the number in the second field was f of the number in the third field. If the entire number was 434, how many were there in each field? 69. A and B can do a piece of work in 10 days, B and C can do it in 12 days, and A and C in 15 days. How long will it take each to do it? 70. A, B and C pasture an equal number of cattle upon a field of which A and B are the owners A of 9 acres and B of 15 acres. If C pays $24 for his pasturage, how much should A and B each receive? 71. How many acres are there in a square tract of land containing as many acres as there are boards in the fence inclosing it, if the boards are 11 feet long and the fence is 4 boards high? 72. What is the greatest number which will divide 27, 48, 90, and 174, and leave the same remainder in each case? 73. A and B invested equal sums in business. A gained a sum equal to 25^ of his stock, and B lost $225. A's money at this time was double that of B's. What amount did each invest? 74. A man at his marriage agreed that if at his death he should leave only a daughter, his wife should have f of his estate; and if he should leave only a son, she should have \. He left a son and a daughter. What fractional part of the estate should each receive, and how much was each one's portion, if the estate was worth $6591 ? TEST QUESTIONS. 363 TEST QUESTIONS. 612. Define a unit; a number. Explain the necessity for a uni- form system of grouping objects,, In how many ways may numbers be represented? Name them. Define numeration; notation; Arabic notation; Roman notation. Give the first principle of Arabic nota- tion. Illustrate it. What is meant by "units of first order," etc.? Give the general principles of Arabic notation. What is meant by a period of figures? Give the names of the first seven periods. Give the rule for notation ; for numeration. State how cents and mills are written in notation of U. S. money. What characters are em- ployed in Roman notation? Give the principles of Roman notation. Define addition; sum, or amount; equation; like numbers. De- scribe the sign of addition; the sign of equality. How many cases are there in addition? Show the truth of the principles of addition. Repeat the rule for addition. Why do we begin at the right to add? Why are the numbers of the same order written in the same column? Define subtraction; minuend; subtrahend; remainder; difference. What is the sign of subtraction? W T hat is it called? State the prin- ciples of subtraction. Show that they are true. Explain what is to be done when some figure of the subtrahend expresses more than the corresponding figure of the minuend. Define multiplication; multiplicand; multiplier; product; factors of the multiplier ; abstract number. Describe the sign of multipli- cation. Give the principles of multiplication. Show that they are true. Show that multiplication is a special case of addition. Repeat the rule for multiplication. What steps in the process are for con- venience? How may you multiply when there are ciphers on the right of either or both factors? Define division; dividend; divisor; quotient; remainder. What is the sign of division? In how many ways is division indicated? State the principles of division. Show that they are true. Show that division is a special case of subtraction. In how many ways may the remainder be expressed? Illustrate each way by an example. W T hat is a fraction? What is meant by long division? What is meant by short division? Which should precede the other? Why? What steps in the process of division are for convenience? What are necessary? How may you proceed when there are ciphers on the 364 TEST QUESTIONS. right of either divisor or dividend? State the principles governing the relation of dividend, divisor, and quotient. Illustrate each by an example. Define analysis. Illustrate the process. Describe the parenthesis and vinculum, and show their uses. Define and illustrate what is meant by an integer; exact divisor; factor; a prime number; a composite number; an even number; an odd number. Give eleven facts relating to exact divisibility of num- bers. Illustrate each statement by an appropriate example. What is meant by factoring? Prime factors? What is an exponent? State the principles relating to the prime factors of numbers. Illustrate the truth of these principles by appropriate examples. Give the rule for finding the prime factors of a number. Explain the process of multi- plying by factors. Show the use of this process. Show how to divide by factors. Explain how to find the true remainder in division by factors. Give the rule for dividing by the factors of a number. What is meant by cancellation? Upon what principle is the pro- cess based? Illustrate the process. Define what is meant by a common divisor; the greatest common divisor; numbers that are prime to each other. What is the princi- ple underlying the greatest common divisor? Give the ordinary method of finding the greatest common divisor when the numbers are small. Solve an example, and give the analysis when the num- bers can not be readily factored. What is a multiple? Define what is meant by a common multi- ple; the least common multiple. State the principle upon which the processes in least common multiple are based. Solve an example showing the truth of the principle. Define and illustrate what is meant by the terms fraction ; unit of a fraction; fractional unit ; the denominator; the numerator; the terms of a fraction; a proper fraction ; an improper fraction; a mixed number; a common fraction; a decimal fraction. How are fractional expres- sions read? Interpret the expression f-. What is meant by reduction of fractions? What is Case I? When is a fraction reduced to larger or higher terms? Upon what princi- ple does the process in Case I depend? What is Case II? What is meant by reducing a fraction to smaller or lower terms? To smallest or lowest terms? Upon what principle is the process in Case II based? What is Case III in reduction? Solve an example illustrat- ing the process. What is Case IV? Solve an example illustrating TEST QUESTIONS. 365 the process. What is meant by similar fractions? Dissimilar frac- tions? When have similar fractions their least common denominator? Give the principles relating to the common and least common denom- inator of fractions. What is the rule for finding the least common denominator of several fractions? What kind of fractions only can be added? Why? What must be done with dissimilar fractions before they can be added? How should mixed numbers be added? What kind of fractions only can be subtracted? What must be done to dissimilar fractions before they can be subtracted? How could mixed numbers be subtracted? What is Case I in multiplication of fractions? What principle un- derlies the process? Demonstrate the truth of the principle. What is Case II? What is the principle? What is Case III? What is the general rule for multiplication of fractions? Solve and explain the following: Multiply f by f. What is Case I in division of fractions? What principle underlies the process? Show by an example that the principle is true. What is Case II? Give the rule for dividing an integer by a fraction. What is Case III? Solve the following: What is the value of | -4-f ? Give an analysis and explanation of the process. Give the general rule for division of fractions. Describe what are included among fractional forms, How are they simplified? What is Case I in frac- tional relation of numbers? What is the principle upon which rela- tion of numbers is based? What is Case II? Illustrate each case by an example. What is a decimal fraction? From what is the word decimal derived? How are decimal fractions expressed? How are decimals distinguished from integers? State the principles of decimal frac- tions. Show each to be true. What is the decimal point? What other name has it? What is a pure decimal? What is a mixed deci- mal? What is a complex decimal? Name the orders of decimals as far as ten-millionths. How does the place occupied by any order of decimals compare with that occupied by integers of the correspond- ing name? How are decimals reduced to a common denominator? Explain the process. How are common fractions reduced to decimals? Analyze the process. If a common fraction can not be exactly re- duced to a decimal, what is done? How do addition and subtraction of decimals compare with the same processes in integers? 366 TEST QUESTIONS. What is the principle upon which multiplication of decimals is based? Show that it is true. How may a decimal be multiplied by 1 with any number of ciphers annexed? What is the principle upon which the process of division of decimals is based? How may a deci- mal be divided by 1 with any number of ciphers annexed? How may we multiply by a number that is a little less than a unit of the next higher order? How may we multiply when one part of the multiplier is a factor of another part? How may we multiply by a number that is a part of some higher unit? What is an aliquot part of a number? What are the common aliquot parts of 10? What of 100? How is the cost found when the quantity and price per 100 or 1000 are given? What is a debt? Define what is meant by a credit; a debtor; a creditor; an account; the balance of an account; a bill; the footing of a bill. State some of the more common abbreviations used in business correspondence. Tell what a concrete number is; an abstract number; a denominate number; a simple denominate number; a compound denominate num- ber; a standard unit; a scale. Illustrate each of the preceding by an appropriate example. How many kinds of numerical scales are there? What is money? Of how many kinds is it? What is coin, or specie? What is paper money? Give the table and denominations of the currency of the United States. What are the ordinary coins? What are the denominations and coins of Canada? Give the table of English money and the coins in common use. What are the cur- rency and coins of France? What is meant by reduction of denominate numbers? What is reduction descending? Give the rule. What is reduction ascending? Give the rule. Define and illustrate what is meant by space, a line, a surface, a solid. For what are linear measures used? Repeat the table of Linear Measure, and of Surveyor's Linear Measure. What is an angle? A square? A square inch? A rectangle? What is the area of a surf ace ? How is the area of a rectangular surface computed? Repeat the table of Square Measure, and of Surveyors' Square Meas- ure. What is a solid? A cube? A cubic inch? A cubic foot? The volume or solid contents? How is the volume of a rectangular solid computed? Repeat the tables of Cubic Measure, and Wood and Stone Measure. TEST QUESTIONS. 367 What are the measures of capacity? Kecite the table of Liquid Measure. In estimating the capacity of cisterns, etc., how many gal- lons are considered a barrel? How many a hogshead? How many cubic inches are there in a gallon? Repeat the table of Apothecaries' Fluid Measure. For what is Dry Measure used? Repeat the table. How many cubic inches are there in a bushel? What is weight? For what is Avoirdupois Weight used? Repeat the table. How many pounds are there in the long ton? How many grains are there in an avoirdupois pound? For what is Troy Weight used? Repeat the table. How many grains are there in a Troy pound? For what is Apothecaries' Weight used? Repeat the table. How many grains are there in a pound Apothecaries' Weight? Repeat the table of Measures of Time. Explain how often leap year occurs. What is a circle? What is the circumference of a circle? An arc of a circle? A degree of the circumference? What is the measure of an angle? Repeat the table of Circular Measure. What a quadrant? A sextant? Give the Stationers' Table and the table of Counting. Give the cases in Reduction of Denominate Fractions. Solve an example illustrative of each case and explain the process. How do the fundamental processes in Compound Denominate Num- bers compare with the same processes in Simple Numbers? How does the number of degrees apparently passed over by the sun compare with the number of hours occupied in passing that distance? The number of minutes of space with the number of minutes of time? The seconds of space with the seconds of time? Repeat the table showing the relation between longitude and time. What is a merid- ian? What is longitude? Give the rule for finding the difference in time when the difference in longitude of two places is given. Give the rule for finding the difference in longitude of two places when their difference in time is given. What is the unit of length in the Metric System of measures? To what is it nearly equal? What is the metric unit of area? What the unit of solidity? What the unit of capacity? What the unit of weight? Define per cent. What is the commercial sign of per cent.? Of what does Percentage treat? How may per cent, be expressed? What elements are involved in problems in Percentage? What is meant by the base? The rate? The percentage? The amount? The difference? What are the five fundamental problems or cases in 368 TEST QUESTIONS. Percentage? Solve an example illustrating each, and give a rule for each case. What is interest? Define the terms principal; amount; rate of interest ; legal interest ; usury ; a note or promissory note. Give three methods for computing interest. What is compound interest? Give the rule for computing compound interest. How is the compound interest table formed? What is meant by annual interest? Give the rule for computing annual interest. In what respect does compound interest differ from annual interest? What are partial payments? What is an indorsement? What is the Mercantile Kule for computing the amount due when partial pay- ments have been made? When is the Mercantile Kule used? What is the principle upon which the United States Rule is based? Give the United States Kule. When the principal, rate, and interest are given, how is the time found? When the principal, time, and interest are given, how is the rate found? When the rate, time, and interest are given, how is the principal found? What is a promissory note? Who is the maker or drawer? Who is the payee? Who is the holder? Who is the indorser? In how many ways may he indorse? What is the face of a note? When is a note negotiable? When is a note not negotiable? What are days of grace? Write a negotiable note and transfer it by indorsement. What is discount? What is commercial discount? What is net price? What is the cash value of a bill? In cases where there is a discount of some per cent., as 20 ^ off and 5^, off for cash, upon what sum is the 5^ computed? What is true discount? Define present worth. Give the rule for solving problems in true discount. What is a bank? A check? Bank discount? The proceeds or avails of a note? The maturity of a note? The term of discount? How is the bank discount computed? Is it right or wrong in principle? How can we find how large to make a note that we may have a certain sum left after paying the discount at a bank? What elements in Profit ans; T 8 7 %. 6. $1423.30 loss. be assessed; 31. 8Jg%. 7. $465 loss. $1168406+, value 33. 214f%. of property. 34. 25%. Page 286. 35. 70%. 3. $1005. Page 266. 36. 85f%. 4. $3037.50. 2. $178.125. 37. 225%. 5. $4978.75. 3. $2.55. 38. $9941.20. 6. $1471. 4. $450. 39. $7812.03. 7. $4987.50. 5. $533.12. 8. $3003.75. 6. $5248.246. Page 277. 9. $4928.75. 7. $3322.512. 40. $4694.84. 41. $6888.36. Page 287. Page 273. 42. 61%. 10. $1486.25. 2. $8593.75. 43. 111%. 11. $4952.50. 3. $8670. 44. Gained $36.73. 14. $5710.72. 4. $1595. 45. Loss $26.19. 15. $1506.40. 388 ANSWERS. 16. $1213.04. Page 300. 3. A, $640; Page 288. 2. A, $2100 ; B, 2100; B, $840; C, $840. 17. $10012.51. C, $1800. 4. B, $705; 18. $3514.93. 19. $1747.81. 3. A, $600 ; B, $1200 ; C, $800. C, $740.25 ; D, $1057.50. 5. A, $426.505 ; Page 289. 4. A, $2000 ; B, $621.987; 2. 571 4s. 6Jd. 3. 732 6s. 2fd. 4. 1149 6s. 7d. 5. 1061 7s. 5|d. 6. $668.87. 7. $1832.25. ' B,'$1600; C, $2400. 5. A, $200; B, $160; C, $280. 6. A, $600; B $750- C, $426.505. 6. A, $2550 ; B, $3400; C, $2550. 7. A, $1080; B, $1600; C, $1820. Page 29O. 8. 7621.67 fr. 9. $1169.77. 10. $4650. 11. $3063.14. 12. $4484.11. C,$675; D, $975. 7. D, $1600; G, $2000; L, $1800. 8. E, $862.50 ; . F, $575; 8. G, $.561. 702; L, $702.127; F, $936.170. Page 31O. 3. $125. 4. $8.75. Page 292. G, $862.50. 5. $45. 9. A, $2744.78 ; 6. 26 T. 2. 2 mo. 29 da. B, $2299.33 ; 7. 24 T 6 T . 3. 2 mo. 18 da. C, $1446.63. 8. 4 T V 4. 1 mo. 12 da. 10. A, $74.86; 9. 12 men. 5. 2 mo. 10 da. B, $86.84; 10. 9 T \ da. 6. 3J mo. C, $104.81 ; 11. 36ff bu. Page 294. D, $203.63. 12. 1000 bbl. 13. 40 J da- 2. June 20, 1877. Page 301. 3. May 2, 1877. 4. Jan. 24, 1877. 5. Dec. 22, 1876. 6. April 23, 1877. 7. June 7, 1877. 8. Aug. 16, 1877. 11. A, $1200 gain; B, $1600 gain; C, $7000 stock. 12. A, $335.365; B, $402.439 ; C, $536.585 ; Page 311. 14. 520 bu. 15. 2J A. 16. $1628.25. 17. 65 da. 18. 9792f Ib. Page 296. D, $670.731 ; E, $804.878. 19. 2307.V mi. 20. 162 T V. T mi. 2. Aug. 22, 1877. 13. A, $750; 21. 427|rd. 3. Mar. 21, 1877. B, $1000; 22. 4fJM C, $1250. 23. 20 da. Page 297. 4. June 19, 1877. Page 3O2. Page 313. 5. June 15, 1877. 2. A, $2880 ; 2. 28J da. 6. July 5, 1877. B, $3600 ; 3. 17820 Ib. 7. $1198.60. C, $2880. 4. $6875. ANSWERS. 389 5. $710.76 + . 9. 821. Page 333. 6. 19 T y T da. 10. 886. 3. 42. 11. 969. 4. 64. Page 314. 12. 2424. 5! 55*. 7. $594. 13. 3546. 6. 89. 8. $902.77. 14. 5555. 7. 57. 9. 473 vd. 15. 472; 3375. 8. 63. 10. 8111 bu. 16. .874; .5555. 9. 177. 11. 25|f. 10. 126. 12. 21 da. Page 325. 11. 536. 13. 27600 Ib. 17. .306. 12. 1.259+. 14. 18750 Ib. 18. .315. 13. 2.0800+. 15. 21J-T. 19. 23^ 14. .6463+; .8617+. 16. 15f da. 20. fff. 15. .8739+; .849+; 17. 546i 2 3 ft. O1 942 AV Page 316. 22*. 707*10 + . 23. .86602 + . 1. 45 ft. 2. 1728; 12167; 24. .79056 + . 2. 24 in. 59319; 13824. 3. 2209 ; 2601 ; 25. .9486 + . Page 334. 841; 1156. 1. 25 ft. 3. 13 ft. 4. 225; 1089; 576; 2. 45 rd. 4. 10.75 ft. 1296; 625. 3. 52 ft. wide ; 5. 10.82 ft. 5. 21952; 91125; 104 ft. long. 6. 21.50 ft. 5832; 9261; 68921. 4. 480 rd. 7. 12.9 + in. ' 12Y ) T2~l> ) 7~2 i) J 5. 240 rd. 8. 19.37 in. 6. $44. 9. $19.51. 8. i^Vr*' A 2 A- 10. 3.46+ ft, width; 9! 50625. Page 327. 10.38+ ft, leng h 10. 15625. 2. 25 ft. 11. Rectangle ; 11. 27000. 3. 113.137 ft. 143.55 + sq. ft. 12. .00000625. 4. 40 ft. 12. 39.37 in. 13. .000000125. 5. 122.474 ft. 14. 4.2025. 6. 140.584 ini. Page 335. 15. J44. 2. 10ft. 2 8 9* Page 328. 3. $3375. 17! 9lf. 7. 75 rd. 4. 10.75 ft. 18. 641J. 8. 119.482ft. 5. 2.38. 19. 9.0200V 9. 205.704 ft. 6. Twice as great. 20. 20.251285||. 10. 172.046 ft. 7. 27 times. 21. 10000; 512; 729. 11. 386.003 ft. 8. 1st, .506; 2d, .721; Page 324. Page 329. 3d. 2.773. 3. 53. 2. 25.1328. 9. 23.48. 4. 63. 4. 63.245. 5. 66. 5. 12 ft. Page 338. 6. 96. 6. 100 rd. length; 2. 55. 7. 266. 10 rd. breadth. 3. 198. 8. 344. 7. 15.94268 rd. 4. $1.72. 390 ANSWERS. 5. 209 T V ft. Page 347. 6. $4013.837. 6. 5070. 1. 19.635 sq. ft. 8. 3775. 2. 50.265 sq. ft. Page 354. 9. 505. 3. 1145.9 sq. rd. 1. 84.8232 cu. ft. 10. 330 mi. 4. 795.77 sq. ft. 2. 18000 cu. ft. 11. 78. 5. 50.929 A. 3. 12440.736 Ib. 12. $3360. 6. 706.86 sq. rd. 4. 7296 Ib. 2. 2430. 7. 7.136 rd. 8. 12 rd. 1. 4666f cu. ft. Page 34O. 9. 2 acres 33.3939 Page 355. 3. 10240. 4. $984.15. 5. $133.82+. 6. $696.849. 8. 1364. sq. rd. 10. 962.115 sq. ft, Page 350. 1. 31.416 sq. ft. 2. 236.405+ en. ft 3. 136.136+ cu. ft. 4. 6415.28 gal. 1. 65.45 cu. ft. 9. 4JfJ. 10. 4. Page 343. 2. 40 sq. ft. 3. 144 sq. ft. 4. 37.6992 sq. ft. 5. 63 sq. ft. 2. 268.0832 cu. ft. 3. 14.1372 cu. ft. 4. 795.217+ Ib. 5. 8.181 cu. ft. 1. 31.416 ft. 2. 141.372 ft. Page 351. 6. 8181.25 cu. ft. 3. 2 mi. 302.48 rd. 1. 540 sq. ft. Page 356. 4. 125.664 rd. 2. 376.992 sq. ft. 1. 10 da. 3. 628.32 sq. ft. 2. $12. Page 344. 4. $64. 3. 80 men. 5. 34.557ft. 6. 101.38 rd. 7. 204.354 rd. 5. 89.5356 sq. ft. 6. 400 sq. ft. 7. 75.3984 sq. ft. 8. 157.08 sq. ft. 4. 5| mi. 5. 16 men. 6. $5k 7. $75. 1. 520 sq. ft. 2. 25 1 sq. ft. Page 352. 8. 336 Ib. 9. $125.635. 3. 720 sq. rd. 1. 251.328 sq. ft. 10. 13Joz. 4. 525 sq. ft. 2. 3000 sq. ft. 3. $5.654. 11. 60 sheep. Page 345. 4. 340 sq. ft. Page 357. 1. 216 sq. ft. 5. $26.88. 12. $80. 2. 126 sq. ft. 3. $168.376. 4. 5935.85 sq. ft. 5. $8.64. 6. 15000 sq. ft. 1. 4.908 sq. ft. 2. 1.396 sq. ft. 3. 26.50+ sq. in. Page 353, 13. 5J}. 14. $40.33 loss. 15. 451 trees. 16. $48, A's money; $40, B's money. Page 346. 4. 45.836 sq. ft. 17. $65, A's; $50, B's ; 1. 5500 sq. ft. 1. 2 on. ft. $55, C's. 2. 54 sq. rd. 2. 7.0686 cu. ft. 18. 48^ ft. 3. 576 sq. ft. 3. $9. 19. A's, 105 Ib.; 4. $546.875. 4. 462.857+ bu. B's, 1401b.; 5. $2062.50. 5. 2632.089+ gal. C's, 245 Ib. ANSWERS. 391 20. A's, $32; Page 35>. 58. B's age, 60 yr.; B's, $27. 33. 72 mi. C's age, 80 yr. 21. 34, A's age ; 34. 200. 59. $3000. 45^, B's age ; 35. 57 T 3 T . 60. $80. 56f , C's age. 36. 15. 61. 2 cts., apples; 37. 30 da. 3 cts., pears. Page 358. 38. $2. 82. A, 46| da.; 22. 720 apples. 23. 18f f da., time in 39. 15 T V yd. 40. 50| oz. B, 35 da. 63. $1186.98. which all can do 41. 566 tiles. 64. $40 loss; it; 42. $2875. 6J%. 58|f da., time in 43. 26}. 65. 27.64 ft. which A can do 44. 27 T 3 y min. past 5. 66. 15 yr. it; 70Jf da., time in 45. 6 mi. per day. 46. 30 men. Page 362. which B can do 67. 90 Ib. it; Page 36O. 68. 162 in No.l; 46 T 7 2 4 5 da., time in 47. 20 min. past 5. 144 in No. 2; which C can do 48. $882.46. 128 in No. 3. it. 49. $74.07. 69. A, in 24 da.; 24. 6f da. 50. $740.52+. B, in 17f da.; 25. $240, one; 51. 16f % gain. C, in 40 da. $150, other. 52. 42f%. 70. A's share, $3; 26. 810 revolutions. 53. $270, carriage ; B's share, $21. 27. 72 ft. $240, horse. 71. 92160 A. 28. $3.80. 54. 300 cats. 72. 21. 29. 8 hr. 48 min. 55. $24. 73. $600. 30. $208.33i. 56. $500, cost. 74. $4563, son's; 31. $1.50, wheat; $1521, widow's; $ .40, corn. Page 361, $507, daughter's. 32. 900 rd. 57. 7 da. UNIVEESITY OF CALIFORNIA LIBRARY, BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to a fine of 50c per volume after the third day overdue, increasing to $1.00 per volume after the sixth day. Books not in demand may be renewed if application is made before expiration of loan period. 50m-8,'28 I D I 438005 Ms-4 UNIVERSITY OF CALIFORNIA LIBRARY lTHMETIC ^