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5. The half-dollar of the United States coinage weighs 192 grains Troy, and consists of 9 parts pure silver and 1 part of copper. How much pure silver and how much copper in 20 half-dollars ? Ans. 7oz. 4pwt. silver ; 16pwt. copper.

6. Divide $ 600 between three men, so that the second man shall receive one third more than the first, and the third man shall receive two thirds more than the second.

7. A, B, and C freight a steamer; A puts on board 98 tons, B 86 tons, and C 64 tons. Owing to danger of being wrecked, there were thrown overboard while at sea 93 tons. What should be the number of tons lost by each?

Ans. A, 361 tons; B, 324 tons; and C, 24 tons. 8. A and B start together by railroad from Chicago for Galena; A travels by freight train at the rate of 15 miles per hour, and B by passenger train at the rate of 25 miles per hour. Cleaves Galena for Chicago at the same time by express train, whose velocity is at the rate of 32 miles

per

hour. Allowing the distance between the two places to be 160 miles, how far from Chicago will A and B each be, when C passes them ?

Ans. A, 51 miles; B 7019 miles.

COMPOUND PROPORTION.

5:75

340. A Compound Proportion is an expression of equality between a cor pound and a simple ratio. Thus,

4:3)

:: 60 : 63, is a compound proportion. Compound proportion is employed in the solution of such questions as would require two or more statements in Simple Proportion.

Ex. 1. If 8 men spend $ 32 in 13 weeks, what will 24 men spend in 52 weeks ?

Ans. $ 384.

In stating the

question, we make 8 men : 24 men

$ 32, which is of

: $ 32 : 13 weeks : 52 weeks

the same kind as

the required term, 3 4

the third term. 2 4 X 52 X 3 2

Then taking of the = 384. Ans. $ 384. 8 X 13

remainin? terms two of the same

Extreme.

OPERATION.
Mean.

Mean.

Extreme.

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kind, 8 men and 24 men, we inquire whether the answer depending on these alone must be greater or less than the third term; and since it must be greater, because 24 men will spend more than 8 men in the same time, we make 24 men the second term, and 8 men the first term. Again, we take the two remaining terms, and make 52 weeks the second term, and 13 weeks the first, since the same number of men would spend more in 52 weeks than in 13 weeks. We then find the continued product of the second and third terms, and divide by the product of the first terms.

BY ANALYSIS. - If 8 men spend $ 32 in 13 weeks, 1 man will spend $ 32 = $ 4 in 13 weeks, and 24 men will spend $ 4 X 24 $ 96 in 13 weeks. If 24 men spend $ 96 in 13 weeks, in 1 week they will spend $ is, and in 52 weeks, $ } $ x 52 $ 384, Ans.

BY RATIO. — The ratio of 8 : 24 = }; the ratio of 13 : 52 4; $32 = } x

$ 384, Ans. NOTE. — To have solved the question by simple proportion, two statements would have been required, which would have produced the following proportions : –

:: $ 32 : $ 96. 13 weeks : 52 weeks : : $96 : $384. RULE. — Make that number which is of the same kind as the answer required the third term of a proportion. Of the remaining numbers, take any two, that are of the same kind, and consider whether an answer depending upon these alone would be greater or less than the third term, and place them as directed in Simple Proportion.

Then take any other two, and consider whether an ansuer depending only upon them would be greater or less than the third term, and arrange them accordingly; and so on until all are used.

Multiply the product of the second terms by the third, and divide the result by the product of the first terms. The quotient will be the fourth term, or answer.

8 men

: 24 men

EXAMPLES

a day?

2. If a man can travel 117 miles in 30 days, travelling 9 hours a day, how far can he go in 20 days, travelling 12 hours

Ans. 104 miles. 3. If 6 men in 16 days of 9 hours each build a wall 20 feet long, 6 feet high, and 4 feet thick, in how many days of 8 hours each will 24 men build a wall 200 feet long, 8 feet high, and 6 feet thick ?

4. If $ 100 gain $ 6 in one year, how much would $ 500 gain in four months ?

Ans. $ 10. 5. If $ 100 gain $ 6 in one year, what must be the sum to gain $ 10 in 4 months ?

Ans. $ 500.

6. How long will it take $ 500 to gain $ 10, if $ 100 gain $ 6 in one year?

Ans. 4 months. 7. If $ 500 gain $ 10 in 4 months, what is the rate per cent ?

8. If 5 compositors in 16 days, 11 hours long, can compose 25 sheets of 24 pages in each sheet, 44 lines in a page, and 40 letters in a line, in how many days, 10 hours long, can 9 compositors compose a volume (to be printed on the same kind of type), consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters in a line ?

Ans. 12 days. 9. If 12 men can build a wall 30 feet long, 6 feet high, and 3 feet thick, in 15 days, when the days are 12 hours long, in what time will 60 men build a wall 300 feet long, 8 feet high, and 6 feet thick, when they work only 8 hours a day?

Ans. 120 days. 10. If 16 horses consume 84 bushels of grain in 24 days, how many bushels will suffice 32 horses 48 days ?

Ans. 336 bushels. 11. If the carriage of 5cwt. 3qr. 150 miles cost $ 24.58, what must be paid for the carriage of 7cwt. 3qr. 64 miles, at the same rate ?

Ans. $ 14.135+: 12. If 7 (oz. of bread be bought for 4 d. when corn is 4s. 2d. per bushel, what weight of it may be bought for ls. 2d. when the price per bushel is 5s. 6d. ?

13. If 496 men, in 5 days of 11 hours each, dig a trench of 7 degrees of hardness, 465 feet long, 3 wide, 2} deep, in how many days of 9 hours long will 24 men dig a trench of 4 degrees of hardness, 3371 feet long, 53 wide, and 3] deep?

Ans. 132 days.

CONJOINED PROPORTION.

341. A Conjoined Proportion is a proportion which has each antecedent of a compound ratio equal in value to its consequent.

The first term of each pair of equivalent terms is an antecedent, and the term following a consequent. NOTE. — This kind of proportion is sometimes called the Chain Rule. Ex. 1. If 10 barrels of apples will pay for 5 cords of wood,

OPERATION.

Antecedents.

and 20 cords of wood for 4 tons of hay, how many barrels of apples will it take to purchase 50 tons of hay ? Ans. 500.

We arrange the an

tecedents at the left Consequents. 10 barrels - 5 cords.

of the sign of equal

ity, and the conse20 cords 4 tons.

quents at the right of 50 tons barrels.

the same; and, as the 10 x 20 x 50

answer is to be of the = 500 barrels, Ans. same kind as the first 5 X 4

term,we place the odd

term in the column of antecedents. Then, to find the term corresponding in value with the odd term, we divide the continued product of the 10, 20, and 50 of the column of antecedents, by the product of the 5 and 4 of the column of consequents; and obtain 500 barrels, Ans.

BY ANALYSIS. — If 4 tons of hay equal in value 20 cords of wood, 1 ton equals in value 5 cords of wood. If 5 cords of wood equal 10 barrels of apples, then 1 ton of hay equals 10 barrels of apples, and 50 tons equal 50 times the value of 10 barrels of apples, or 500 barrels, answer as before.

RULE. Place the antecedents in one column and the consequents in another, on the right, with the sign of equality between them. Divide the continued product of the terms in the column containing the odd term by the continued product of the other column, and the quotient will be the answer.

NOTE 1. The odd term will always belong to that column which does not contain the other terin of the same kind.

Note 2. — Conjoined proportion is nothing else than simple or compound proportion, exhibited in a different, and in many cases a more convenient form; and, to show the correctness of the rule, the following examples may also be performed by either simple or compound proportion.

EXAMPLES.

2. If 100 acres in Bradford be worth 120 in Haverhill, and 50 in Haverhill be worth 65 in Methuen, how many acres in Bradford are equal to 150 in Methuen ? Ans. 96 *s acres.

3. If 10lb. of cheese are equal in value to 7lb. of butter, and 11lb. of butter to 2 bushels of corn, and 11 bushels of corn to 8 bushels of rye, and 4 bushels of rye to one cord of wood, how many pounds of cheese are equal in value to 10 cords of wood ?

Ans. 432 lb. 4. If 12 men can do as much work as 25 women, and 5

women do as much as 6 boys, how many men would it take to do the work of 75 boys ?

Ans. 30 men. 5. If 6 gallons liquid measure equal 5 English imperial gallons, and if 10 English imperial gallons equal 6 velts of France, and if 26 velts of France equal 16 Russian vedros of wine, how many vedros of wine equal 63 gallons liquid meas

ure ?

port?

6. If 7 bushels of corn in Boston be worth 8 bushels in Buffalo, and 10 bushels in Buffalo be worth 14 bushels in Chicago, and 21 bushels in Chicago be worth 25 bushels at Davenport, how many bushels in Boston are worth 1200 bushels at Daven

Ans. 630 bushels. 7. If 24 shillings in Massachusetts are equal to 32 shillings in New York, and if 48 shillings in New York are equal to 45 shillings in Pennsylvania, and if 15 shillings in Pennsylvania are equal to 10 shillings in Canada, how many shillings in Cannada are equal to 100 shillings in Massachusetts ?

Ans. 831 shillings.

MISCELLANEOUS EXAMPLES IN PROPORTION.

1. A sets out on a journey, and travels 27 miles a day ; 7 days after, B sets out and travels the same road 36 miles a day: in how many days will B overtake A ?

Ans. 21 days. 2. If I sell coffee at 2s. 3d. per pound and gain 35 per cent, what did I give per pound?

3. A detachment of 2000 soldiers were supplied with bread sufficient to last them 12 weeks, allowing each man 14 ounces a day; but on examination they find 105 barrels, containing 200lb. each, wholly spoiled; how much a day may each man eat, that the remainder may supply them 12 weeks?

Ans. 12oz. 4. In consequence of having a seventh part of their bread spoiled, 2000 soldiers were put on an allowance of 12 ounces of bread per day for 12 weeks; what was the whole weight of their bread (good and bad), and how much was spoiled ?

Ans. The whole weight, 147000lb.; spoiled, 21000lb. 5. Two thousand soldiers, having lost 105 barrels of bread, weighing 200lb. each, were obliged to subsist on 12 ounces a

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