22. Find the greatest common measure of 63 and 168, with whicn divide their sum, difference, and common multiple. A. 21. 23. Hence, if any number will measure each of two others, it will measure their sum, their difference, and their common multiple.

24. Since the common measure of two or more numbers becomes, when those numbers are multiplied together, or when each is multiplied into itself, a factor in every product, therefore

25. A common measure of two or more numbers will measure the squares, cubes, &c. of those numbers, also the continued product of those same numbers into each other.

19 49 151
T
12

26. What is the least common multiple of 2, 3, 4, 6, 7, 12, and 14? A. 84. 27. Reduce 25, 5, 127, and 54, to their equivalent improper fractions. A. ; ; ; 402. 28. Reduce to improper fractions 417, 123, 2751, and 374,53 A. 54; 2095; 4139; 38576. 29. Reduce to mixed numbers 13§ of 7}, 3 of 3 of 12, and 15,7 of 8 of 133. A. 991; 4; 1,8477.

17

103

A. 99 145.

31. Reduce to mixed numbers 3357; 18793; 3 of 4 of 63; of

8398

&

133

47

A. 30; 5275; 1; 278.

11050

109375

135795

32. Reduce to prime terms 29395; 35581; 10000000; 222210

61

840; 18

33. Find the least common denominator of 1, 3, 3, 4, and §.

34. Find the least common denominator of 24, 4 of 93, and

37 How many times is the sum of 5 and 3 greater than their difference?

39. Find the continued product of 361, 1238, 329, 324 1444, 441, and 211. A. 1.

40. Find the value of 3 of 4- of 34+ of 33. 2 5 41. Express 372 of a yard as the fraction of an inch, inch as that of a pole.

42. Required the sum and difference of of a pound, and 4 of a guinea, (21s.) A. £1. 2s. 8d.; 4s. 43. Reduce of a pound, of a guinea, and 4 of 3s. 91⁄2d. to fractions of the same denomination, and to the same denominator A. 7040 7056. 7007

1 21

22

1392 7302, 7392.

44. The aggregate of 3 of 3 of a sum of money is $133; what is that sum ? A. $332. 45. Find the fraction which, when multiplied by 3 of 4 of 31, gives a result equal to 7. A. 12. 46. A person having of a coal mine, sells of his share for $2,000; what is the whole mine worth? A. $6,666.663.

5

47. The third part of an army was killed, the fourth part taken prisoners, and 1,000 fled. How many were there in the army?— how many killed ?-how many taken prisoners?

A. 2,400; 800 killed; 600 prisoners.

2

48. A can do a job of work in 5 days, B in 6, and C in 7; how much can they jointly do in 2 days? A. 18 job. Tog 49. The owner of of a ship sold of of his share for $123;

4 17

21 41

what would of cost at the same rate?

A. $200.

50. If a cask be emptied by two taps in 4 and 6 hours respectively, in what time will it be emptied by both of them together, the rates of efflux remaining the same throughout? A. 2h. 24m.

51. A, B and C can perform a piece of work in 12 hours; also, A and B can do it in 16 hours, and A. and C in 18 hours; what part of the work can B and C do in 94 hours, and in what time would A do the whole? 4 A. : 28h. 48m. 52. What decimal fractions are equivalent to 25, 125, and 2š?0? A..16; .072; .006640625. A. .876,543,211.

4

53. From unity take .123456789.
54. Find the continued product of .275, 2.75, and 27.5..

17 2560

A. 20.796875.

55. Find the sum of the quotients of 1.68 by .024, of 971.7 by 123, and of 142.025 by .0437, and prove the results by vulgar frac tions. A. 3,327.9.

56. One man owns .6 of a bank, another, and Mr. Darby the rest; what is Darby's part, and what the value of each part, allowing the capital of the bank to be $100,000?

1

A. $27,500; $60,000; $12,500. 57. In a certain school, .125 of the pupils study geography, .3 study grammar, arithmetic, and 18 learn to read. What is the number in each branch? A. Geography, 30; grammar, 72; arithmetic, 120; reading, 18. Whole number, 240.

2

58. A man whose annual income is $3,000, spends .12 of it; how many dollars will he have saved at the end of each year?

A. $2,640.

59. If a man in trading adds annually to his capital 20 per cent., how many years will be required for his capital to double.

A. 5 years. 60. What will be the value of 8cwt. 3qr. 14lb. of sugar at $5 per cwt.? A. $46.674. 61. What is the amount of $300 from Jan. 1st, 1830, to Nov. 19th, 1834? A. $387.90.

62. How much is the present worth of $1,000, discounted at bank, and payable in 4 months (which includes the 3 days grace) less than the present worth of the same sum for the same time, calculated as in LXXXIII.? A. 89c. 2m.

63. Suppose a minor, now 15 years old, is to receive a legacy, at 21, of $27,200; what is its present value? A. $20,000. 64. When a merchant wants a loan at bank of $3,958, for 60 days, what sum must be specified in the note to receive that sum?

A. $4,000.

65. What sum in cash, reckoning compound interest, is equivalent to $3,207.13%, due 20 years hence, without interest? A. $1,000. 66. A merchant bought cotton goods to the amount of $1,300 for cash; he kept them on hand 1 year and 8 months, then sold them for 15 per cent. advance on their cost, but on 4 months' credit; what was his profit ? A. $35.685+.

67. Suppose a bookseller purchases, at "trade sale," books to the amount of $500, the terms being indisputable paper for 6 months, or 4 per cent. discount for cash, and he chooses the latter, what does he make by advancing the cash? Discount made at the time of the purchase is usually computed like bank discount. A. $5.436.

68. A merchant buys on credit goods amounting to $2,060, but for cash down gets a deduction of 5 per cent. bank discount; keeps them on hand 60 days, then sells them on one year's credit, at 20 per cent. advance from the purchase price. What is the present worth of his clear profit at the time of sale? A. $355.50.

69. The bill for the union of the Canadas, in 1840, provided that the governor general should receive an annual salary of £7,000. How much does that exceed the salary of the president of the United States, which is $25,000? A. $6,111.1111/.

70. To find the equated time for the payment of bills due at different times.-Multiply each sum by the time it has to run, (computing from the date of the first,) and divide the sum of the products by the amount of the debt.

71. An agent sells goods for his employer, payable at the following dates $120 due 7th of January; $130 due 9th of February; $200 due 15th of March; $500 due 20th of May; $240 due 15th of August. Required the average time of payment. A. May 2d.

72. The prices of goods are often named in pounds, shillings, &c., while their amount is carried out in federal money, as follows :—

73. The notes referred to below are to be calculated according to the rules under which they are given as examples. In these operations, the pupil will derive much aid from the Table in Compound Interest.

FOR RULES SEE LXXXI.

74. What is the balance due at compound interest on Note 25? Results-357,304+; 115,730; 56,047; 12,240. A. $63.127+. 75. What is the balance due at compound interest on Note 26? Results-596,996+; 22,472; 43,778; 248,676. A. $79.822+. 76. What is the balance due at compound interest on Note 33? Results-2,472; 236,856; 210,476. A. $1,300.458, nearly. 77. What is the balance due at compound interest on Note 35? Results-64,236; 472,418; 433,641; 337,228. A. $197.057+. 78. What is the balance due at compound interest on Note 39? Results-81,265; 710,869; 662,824; 464,715. A. $479.423. 79. Find a 4th proportional to 35:::3; also to 125; .0145::35. A. 3 and .00406.

560

80. If two men, A and B, together, can finish a piece of work in 10 days, and A, by himself, in 18 days, what time will it take B to do the whole? A. 221 days.

81. Three agents, A, B, and C, can produce a given effect in 12 hours; also, A and B can produce it in 16 hours, and A and C in 18 hours; in what time can each of them produce it separately?

A. A, 28 h.; B, 36h.; C, 48h.

82. Distribute $200 among A, B, C, and D, so that B may receive as much as A; C as much as A and B together, and D as much as A, B, and C, together. A. A's, $25; B's, $25; C's, $50; D's, $100.

83. At what time between 2 and 3 o'clock are the hour and minute hands of a clock together? At two o'clock the hour hand is two of the portions called hours of one hand, and five minutes of the other, in advance of the minute hand; and their rates being as 1 : 12, the minute hand gains 55 in 60, or 11 in 12, upon the hour hand; hence, 11: 2 :: 12: 2h. 101 m. A. 101 m. past 2 o'clock.

84. A person, on looking at his watch, was asked the "time of day." He replied, that it was between 4 and 5, and the minute and hour hand were together; what was the exact time?

A. 21m. 49 sec. past 4.

85. Two clocks point out 12 at the same instant; one of them gains 7sec. and the other loses 8sec. in 12 hours; after what interval will one have gained half an hour of the other, and what o'clock will each then show?

A. 60 days; 14m. past 12; 16m. of 12.

86. A father divided his estate among his three sons, giving to A $10 as often as to B $6, and to C but $3 as often as to B $7, and yet C's dividend was $4,800. What did the whole estate amount to? A. $34,6663.

87. If 1 dollar or 6 shillings in New England be equal in value o 8 shillings in New York; £1. 8s. in New York to £1. 6s. 3d. r New Jersey; £1. 7s. 6d. in New Jersey to £1. 9s. 4d. in North Carolina; £1. 8s. in North Carolina to 17s. 6d. in Canada; and £2. 10s. in Canada to £2. 5s. sterling; how many dollars in New England are equal to £450 sterling? A. $2,000. 88. A person, by disposing of goods for $182, loses at the rate of 9 per cent.; what ought they to have been sold for to realize a profit of 7 per cent.? A. $214.

89. A stationer sold quills at 11s. a thousand, by which he cleared of the money, and he afterwards raised them to 13s. 6d. a thousand; what did he clear per cent. by the latter price?

A. £96. 7s. 3d. 1qr. 90. At what price must a commodity, purchased at the rate of £14. 5s. per cwt., be sold to gain 21 per cent., and how much must be sold to clear £100?

A. £17. 4s. 10 d. per cwt.; 33cwt. 1qr. 16lb. 11,0z. 91. Divide $64 among A, B, and C, so that A may have three times as much as B, and C may have one third of what A and B together have? A. A's $36; B's $12; C's $16.

92. A person paid a tax of 10 per cent. on his income; what must his income have been, when, after he had paid the tax, there was $1,250 remaining? A. $1,388.8888.

93. A hare starts 40 yards before a greyhound, and is not perceived by him till she has been up 40 seconds; she scuds away at the rate of 10 miles an hour, and the dog pursues her at the rate of 18 miles an hour; how long will the course last, and what distance will the hare have run? A. 60sec.; 490yd.

94. If 5 men or 7 women can perform a piece of work in 35 days, in what time can 7 men and 5 women do the same? A. 1641 days. 95. If 15 men, 12 women, and 9 boys, can complete a piece of work in 50 days, what time would 9 men, 15 women, and 18 boys, take to do twice as much, the parts done by each in the same time being as the numbers 3, 2, and 1? A. 104 days.

96. If A by himself can do a piece of work in 5 days, B twice as much in 7 days, and C four times as much in 11 days, in what time can A, B. and C, together, do 3 times the said work?

26 10

A. 3 days 12h. 46,36 m. 97. If A can do a piece of work by himself in 1 hour, B in 3 hours, C in 5 hours, and D in 7 hours, in what time can they do three times as much, all working together? A. 1h. 47m. 23sec. 98. If 27 men can do a piece of work in 14 days, working 10 hours in a day, how many hours a day must 24 boys work, in order to com