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24. A and B began to trade with equal sums of money. The first year A gained $40, and B lost $40; the second year A lost

of what he had at the end of the first, and B gained $40 less than twice what A lost; when it appeared that B had twice as much money as A. What sum did each begin with? Ans. $320.

25. What fraction is that, whose numerator being doubled, and denominator increased by 7, the value becomes ; but the denominator being doubled, and the numerator increased by 2, the value becomes? Ans..

26. A and B together can perform a piece of work in 8 days, A and C in 9 days, and B and C in 10 days. How many days would it take each person to perform the same work alone?

Let x, y, and z represent the number of days required for A, B, and C respectively;

1

Then is the part of the work that A could do in 1 day, &c.,

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and, by the conditions of the problem, the equations will be

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By subtracting the second equation from the first, we shall elimınate x, and then by adding the third equation we shall eliminate z Ans. A 1434 days, B 1723, C 2331.

27. From two places, which are 154 miles apart, two persons set out at the same time to meet each other, one traveling at the rate of 3 miles in 2 hours, and the other at the rate of 5 miles in 4 hours; in how many hours will they meet? Ans. 56 hours.

28. In a naval engagement, the number of ships captured was 7 more, and the number burned was 2 less, than the number sunk. Fif teen escaped, and the fleet consisted of 8 times the number sunk; of how many ships did the fleet consist? Ans. 32.

29. A and B together could have completed a piece of work in 15 days, but after laboring together 6 days, A was left to finish it alone, which he did in 30 days. In how many days could each have performed the work alone? Ans. 50, and 21 days. 30. On comparing two sums of money it is found, that of the first is $96 less than of the second, and that of the second is as of the first. What are the sums?

much as

Ans. $720, and $512. 31. A privateer, running at the rate of 10 miles an hour, discovers a ship 18 miles off, making way at the rate of 8 miles an hour. how many hours will the ship be overtaken?

Ans. 9 hours.

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32. In a composition of copper, tin, and lead, of the whole minus 16 pounds was copper, of the whole minus 12 pounds was tin, and of the whole plus 4 pounds was lead; what quantity of each was there in the composition? Ans. Copper 128, tin 84, lead 76 pounds.

33. The sum of $660 was raised for a certain purpose by four persons, the first giving as much as the second, the third as much as the first and second, and the fourth as much as the second and third. What were the several sums contributed?

Ans. $60, $120, $180, $300.

34. Two pedestrians start from the same point, and go in the same direction; the first steps twice as far as the second, but the second makes 3 steps while the first is making 2. How far has each one gone when the first is 300 feet in advance of the second?

Ans. 1200, and 900 feet.

35. A merchant has cloth at $3 a yard, and another kind at $5 a yard. How many yards of each kind must he sell, to make 100 yards which shall bring him $450? Ans. 25, and 75 yards.

36. In the composition of a quantity of gunpowder, the nitre was 10 pounds more than of the whole, the sulphur 4 pounds less than of the whole, and the charcoal 2 pounds less than of the nitre. What was the amount of gunpowder? Ans. 69 pounds.

37. Four places are situated in the order of the letters A, B, C, D. The distance from A to D is 34 miles; the distance from A to B is of the distance from C to D; and of the distance from A to B, plus of the distance from C to D, is 3 times the distance from B to C. What are the distances between A and B, B and C, C and D ? Ans. 12, 4, and 18 miles.

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38. A vintner sold at one time 20 dozen of port wine, and 30 of sherry, for $120; and at another time 30 dozen of port, and 25 of sherry, at the same prices as before, for $140. What was the price of a dozen of each sort of wine? Ans. $3, and $2.

39. A person pays, at one time, to two creditors, $53, giving to one of them of the sum due to him, and to the other $3 more than of his debt to him. At another time he pays them $42, giving to the first of what remains due to him, and to the other of what remains due to him. What were the debts? Ans. $121, and $36.

40. A farmer has 86 bushels of wheat at 4s. 6d. per bushel, with which he wishes to mix rye at 3s. 6d. per bushel, and barley at 3s. per bushel, so as to make 136 bushels, that shall be worth 4s. a bushel. What quantity of rye and of barley must he take?

Ans. 14, and 36 bushels.

41. A composition of copper and tin, containing 100 cubic inches, weighs 505 ounces. How many ounces of each metal does it contain, supposing the weight of a cubic inch of copper to be 51 ounces, and of a cubic inch of tin 4 ounces? Ans. 420, and 85 ounces.

42. A General having lost a battle, found that he had only onehalf of his army plus 3600 men left, fit for action; of his men plus 600 being wounded, and the rest, who were of the whole army, either slain, taken prisoners, or missing. Of how many men did his army consist? Ans. 24000.

43. Two pipes, one of them running 5 hours, and the other 4, filled a cistern containing 330 gallons; and the same two pipes, the first running 2 hours, and the second 3, filled another cistern containing 195 gallons. How many gallons did each pipe discharge per hour? Ans. 30 and 45 gallons.

44. After A and B had been employed on a piece of work for 14 days, they called in C, by whose aid it was completed in 28 days. Had C worked with them from the beginning, the work would have been accomplished in 21 days. In how many days would C alone have accomplished the work? Ans. 42 days.

45. Some smugglers discovered a cave which would exactly hold their cargo, viz., 13 bales of cotton and 33 casks of wine. A revenue cutter coming in sight while they were unloading, they sailed away with 9 casks and 5 bales, leaving the cave two-thirds full. How many bales or casks would it contain? Ans. 24 bales or 72 casks.

46. A gentleman left a sum of money to be divided among four servants, so that the share of the first was the sum of the shares of the other three; the share of the second of the sum of the other three; and the share of the third of the sum of the other three; and

it was also found that the share of $14. What was the whole sum?

the first exceeded that of the last by and the share of each?

Ans. Whole sum $120; shares $40; $30; $24; $26.

91

CHAPTER VI.

RATIO-PROPORTION-VARIATION.

RATIO.

(127.) The RATIO of one quantity called the antecedent to another of the same kind called the consequent, is the quotient of the former divided by the latter.

Thus the ratio of 12 to 4 is 3, since 12 is 3 times 4;

and the ratio of 5 to 13 is, since 5 is five thirteenths of 13. The antecedent and consequent together are called the terms of the ratio.

Sign of Ratio.

(128.) A colon (:) between two quantities denotes that the two quantities are taken as the antecedent and consequent of a ratio.

Thus 35, the ratio of 3 to 5; a:b, the ratio of a to b.

(129.) The value of a ratio may always be represented by making the antecedent the numerator, and the consequent the denominator of a Fraction.

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(130.) The direct ratio of the first of two quantities to the second, is the quotient of the first divided by the second; thus the direct ratio of 3 to 5 is .

The inverse ratio of the first quantity to the second, is the direct ratio of the second to the first;-in other words, it is the direct ratio of the reciprocals of the two quantities.

Thus the inverse ratio of 3 to 5 is §;

or it is the ratio of to, equal to

÷, equal to §, (127).

Hence inverse is often called reciprocal ratio. The term ratio used alone always means direct ratio.

Compound Ratio.

(131.) A compound ratio is the ratio of the product of two or more antecedents to the product of their consequents; and is equal to the product of all the simple ratios.

The compound ratio of a and b to x and y is

ab

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xy

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(132.) The ratio of the first to the last of any number of quantities, is equal to the product of the ratios of the first to the second, the second to the third, and so on to the last; that is, it is compounded of all the intervening ratios.

For example, take the quantities a, b, x, y. to the second, the second to the third, &c., are

The ratios of the first

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Duplicate and Triplicate Ratios.

(133.) The duplicate ratio of two quantities is the ratio of their squares, and the triplicate ratio is the ratio of their cubes.

Thus the duplicate ratio of a to b is the ratio of a2 to b2; and the triplicate ratio of a to b is the ratio of a3 to b3.

The subduplicate ratio of quantities is the ratio of their square roots, and the subtriplicate ratio is the ratio of their cube roots

Equimultiples and Equisubmultiples.

(134.) Equimultiples of two quantities are the products which arise from multiplying the quantities by the same integer, and equisubmultiples are the quotients which arise from dividing the quantities by the same integer.

Thus 3a and 36 are equimultiples of a and b, while, conversely, a and b are equisubmultiples of 3a and 36.

(135.) Equimultiples, or equisubmultiples, of two quantities have the same ratio as the quantities themselves, (81).

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