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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 compositoi-s 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 i Ans. $ 14.135-f-.

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 Is. 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, 337^ feet long, 5f 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,

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.

antecedents. Then, to find the terra 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 o f 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 term 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.

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. 9t,r25 acres.

3. If 101b. of cheese are equal in value to 71b. of butter, and 111b. 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

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

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EXAMPLES.

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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 measure?

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 Davenport? 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. 83^ 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 2001b. 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, 1470001b.; spoiled, 210001b.

5. Two thousand soldiers, having lost 105 barrels of bread, weighing 2001b. each, were obliged to subsist on 12 ounces a day for 12 weeks; but had none been lost, they might have had 14 ounces a day for the same time. What was the whole weight, including what was lost, and how much had they left to subsist on?

Ans. The whole weight, 1470001b.; left to subsist on, 1260001b.

6. Bought threescore pieces of Hollands for three times a^ many dollars, and sold them again for four times as many dollars; but if they had cost me as much as I sold them for, for what should I have sold them to gain at the same rate?

Ans. $ 320.

7. Bought 20 pounds of tea at the rate of 1J pounds for a dollar, and 62 pounds more at the rate of 15 pounds for $ 12; and sold the whole at the rate of 3 pounds for $ 4. How much did I gain, or lose?

8. Each of the four sides of a certain field appeared to be 2 furlongs 30 rods and 3 yards in length when measured by a line supposed to be 4 rods long; but the line was found to have been only 64 feet in length. Required the true distance round the field. Ans. 7074j$ feet.

9. If 5 oxen or 7 cows eat 3^*^- tons of hay in 87 days, in what time will 2 oxen and 3 cows eat the same quantity of hay? Ans. 105 days.

10. If 360 men be placed in a garrison, and have provisions for 6 months, how many men must be sent away at the end of 4 months that the remaining provisions may last the garrison 8 months longer? Ans. 270 men.

11. My tailor informs me it will take 10£ square yards of cloth to make me a full suit of clothes. The cloth I am about to purchase is 1| yards'wide, and on sponging it will shrink ^ in width and length. How many yards of the above cloth must I purchase for my " new suit"?

Ans. 6Tg§3yd.

12. If the price of a farm of 130A. 2R. 20p. be $6537.50, what will be the price of another, containing 100A. Oil. 30p., if 4 acres of the latter be worth 5 of the former?

Ans. $ 6267.71+.

13. How much each of copper, tin, zinc, and lead will be required to make a bell weighing 192 tons 17cwt. 161b., the weight of that at Moscow, bell-metal being composed of 800 parts copper, 101 tin, 56 zinc, and 43 lead?

Ans. Copper, 308572f 1b.; tin, 38957/^1b.; zinc, 21600#j1b.; lead, 16585 J1b.

14. The relative heating power per cord of white-oak to white-pine being as 81 to 42, and that of white-pine to pitchpine as 7 to 8; how many cords composed of white-pine and pitch-pine in equal parts will be required to produce as much heat as 20 cords of white-oak? Ans. 36 cords or 18 of each.

15. If a cask of wine, containing 85 gallons, cost $116.95, what would be the value of a hogshead containing 63 gallons, and composed of four parts of the same wine and one part of water? Ans. $ 69.34ff.

16. If 8 horses in 30 days consume 3£ tons of hay, how long will 4fv tons last 10 horses, 15 cows, and 7 sheep, each cow consuming £• as much as a horse, and each sheep £ as much as a cow?

17. The relative nutritious matter per pound of potatoes to beets being as 25 to 14, that of beets to carrots as 7 to 5, and that of carrots to turnips as 5 to 2; how many pounds of turnips are equal in nutritious matter to 2000 pounds of potatoes? and allowing 60 pounds to a bushel, how much cheaper or dearer relatively per bushel are potatoes at 80 cents, and turnips at 20 cents, than either beets at 35 cents or carrots at 25 cents.

Ans. 12500 pounds; potatoes 17£cts. dearer; turnips lOcts. dearer.

18. A and B set out from the same place, and in the same direction. A travels uniformly 18 miles per day, and after 9 days turns and goes back as far as B has travelled during those 9 days; he then turns again, and, pursuing his journey, overtakes B 22£ days after the time they first set out. It is required to find the rate at which B uniformly travelled.

Ans. 10 miles per day.

19. Two men in Boston hire a carriage for $ 25, to go to Concord, N. H., and back, the distance being 72 miles, with the privilege of taking in three more persons. Having gone 20 miles, they take in A; at Concord, they take in B; and when within 30 miles of Boston, they take in C. How much shall each man pay?

Ans. First man, $7.609f#|; second man, $7.609igf; A $5.873XV„; B, $ 2.864-^; C, $ 1.041TV

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