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22. Bought a pipe of wine for 84 dollars, and found it had leaked out 12 gallons; I sold the remainder at 12} cents a pint ; what did I gain or lose ?
Ans. I gained $30. 23. A gentleman bought 18 pipes of wine at 12s. 6d. (N. Jersey currency) per gallon; how many dollars will pay the purchase ?
Ans. $3780. 24. Bought a quantity of plate, weighing 15 lb. 11 oz. 13 pwt.
dollars will pay for it at the rate of 12s. 7d. (New York currency,) per ounce ?
Ans. $301,50 cts. 26 m. 25. A factor bought a certain quantity of broadcloth and drugget, which together cost £81 per yard, the quantity of broadcloth was 50 yards, at 18s. per yard, and for every 5 yards of broadcloth he had 9 yards of drugget; I demand how many yards of drugget he had, and what it cost him per yard? Ans. 90 yards, at 8s. per yard.
26. If I give 1 eagle, 2 dollars, 8 dimes, 2 cents and 5 mills, for 675 tops, how many tops will 19 mills buy?
Ans. 1 top:
27. If 100 dollars gain 6 dollars interest in a year, how much will 49 dollars gain in the same time ?
Ans. $2,94 cts. 28. If 60 gallons of water, in one hour, fall into a cistern containing 300 gallons, and by a pipe in the cistern, 35 gal. lons run out in an hour; in what time will it be filled ?
Ans. in 12 hours. 29. A and B depart from the same place and travel the same road; but A goes 5 days before B, at the rate of 15 miles a day; B follows at the rate of 20 miles a day; what distance must he travel to overtake A ?
Ans. 300 miles.
COMPOUND PROPORTION. Compound proportion, is a method of performing such operations in proportion, as require two or more stat. ings. It is sometimes called Double Rule of Three, be. cause its operations can be performed by two operations of the Rule of Three.
For example : If 56 lbs. of bread are sufficient for 7 men 14 days, how much bread will serve 21 men 3 days ?
Here the amount of bread consumed depends upon two circumstances, the number of days, and the number of men.
We will first consider the quantity of bread as depend. ing upon the number of men, supposing the number of days to be the same.
The proportion would then be this;
7 men : 21 men : : 56 lbs. to the number of lbs. re. quired.
Here we multiply the means together, and divide the answer by 7, and the answer is 168. That is, if the time was the same, viz. 14 days, the 21 men would eat 168 lbs. in that time.
We now make a second statement thus :
14 days : 3 days : : 168 lbs. : number of lbs. requir., ed.
The result of this statement is 36 lbs. which is the an.
In performing this operation, let the pupil notice that in the first statement, the 56 was multiplied by the 21 and the answer divided by 7. This gives the same answer as would be given, did we divide first, and then multiply.
That is, 56 multiplied by 21, and the product divided by 7, is the same as 56 divided by 7 and the quotient multi. plied by 21.
We divide by 7, to find how much one man would eat in the same time, or 14 days, and multiply by 21, to find how much 21 men would eat.
When we make the second statement, as we have found how much 21 men would eat in 14 days, we divided the quantity (168 lbs.) by 14, to find how much they would eat in one day, and then multiply by 3, to find how much they would eat in 3 days. But in this case also, the multiplication is done first.
Let the pupil also notice that the 56 lbs. was multiplied by 21 and divided by 7, and then that the answer to this (168 lbs.) was multiplied by 3 and divided by 14. Here 21 and 3 are used as multipliers, and 14 and 7 are used as divisors.
The answer will be the same (as may be found by trial)
if 56 is multiplied by the product of these multipliers, and the answer divided by the product of the divisors.
It is on this principle that the common rule in compound proportion is constructed, which is as follows.
RULE OF COMPOUND PROPORTION. Make the number which is of the same kind as the answer required, the THIRD term.
Take any two numbers of the same kind, and arrange them in regard to this third term, according the rule of proportion. Then take any other two numbers of the same kind, and
arrange them in like manner, and so on till all the numbers are used.
Then multiply the third term, by the product of the second terms, and divide the answer by the product of the first terms. The quotient is the answer.
EXAMPLES 1. If a man travel 273 miles in 13 days, travelling only 7 hours a day, how many miles will he travel in 12 days at the rate of. 10 hours a day?
Here the number, which is of the same kind as the answer required, is the 273 miles, and this is put as third term.
We now take two numbers of the same kind, viz. 13 days and 12 days, and placing them according to the rule of simple proportion, the question would stand thus.
13 : 12 : : 273 We next take two other numbers of the same kind, viz. 10 hours, and 7 hours, and arrange them under the former proportion according to the same rule, thus :
13 : 12
:: 273 We now multiply the 273 by the product of 12 and 10, and divide by the product of 13 and 7, and the quotient is the answer.
We can explain this process analytically, thus.
We divide by 13, to find how much the man would tra. vel in one day, at the rate of 7 hours per day.
What is compound proportion? What is the rale ?
We multiply by the 12, to find how much he would travel in 12 days, at the same rate.
We divide by 7 to find how much he would travel in one hour, and multiply by 10 to find how much he would travel in 10 hours.
Let the pupils explain the following in the same man.
2. If £100 in one year gain £5 interest, what will be the interest of £750 for 7 years ? Ans. £262 10s.
3. What principal will gain £262 10s. in 7 years at 5 per cent. per annum ?
Ans. £750. 4. If a footman travel 130 miles in 3 days, when the days are 12 hours long; in how many days, of 10 hours each, may he travel 360 miles ?
Ans. 983 days. 5. If 120 bushels of corn can serve 14 horses 56 days, how many days will 94 bushels serve 6 horses?
Ans. 1021 days. 6. If 7 oz. 5 pwts. of bread be bought at 44d. when corn is at 4s. 2d. per bushel, what weight of it may be bought for ls. 2d. when the price of the bushel is 5s. 6d. ?
Ans. 1 lb. 4 oz. 3439 pwts. 7. If the carriage of 13 cwt. I qr. for 72 miles be £2. 10s. 6d., what will be the carriage of 7 cwt. 3 qrs. for 112 miles ?
Ans. £2 5s. 11d. 115. q. 8. A wall, to be built to the height of 27 feet, was raised to the height of 9 ft. by 12 men in 6 days; how many men must be employed to finish the wall in 4 days at the same rate of working ?
Ans. 36 men. 9. If a regiment of soldiers, consisting of 939 men, can eat 351 quarters of wheat in 7 months ; how many soldiers will eat 1464 quarters in 5 months, at that rate ?
Ans. 5483-23 10. If 248 men, in 5 days of 11 hours each, dig a trench 230 yards long, 3 wide and 2 deep ; in how many days of 9 hours each, will 24 men dig a trench of 420 yards long, 5 wide and 3 deep?
Ans. 2887627 11. If 6 men build a wall 20 ft. long, 6 ft. high, and 4 ft. thick, in 16 days, in what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft. thick ? Ans. 80 days.
12. If the freight of 9 hhds. of sugar, each weighing 12 cwt., 20 leagues, cost £16, what must be paid for the freight of 50 tierces, each weighing 21 cwt. 100 leagues ?
Ans. £92 11s. 102d. 13. If 4 reapers receive $11.04 for 3 days' work, how many men may be hired 16 days for $103.04 ?
Ans. 7 men. 14. If 7 oz. 5 pwt. of bread be bought for 44d. when corn is 4s. 2d. per bushel, what weight of it may be bought for 1s. 2d. when the price per bushel is 5s. 6d. ?
Ans. 1 lb. 4 oz. 3474 pwts. 15. If $100 gain $6 in 1 year, what will 400 gain in 9 months ?
16. If $100 gain $6 in 1 year, in what time will $400 gain $18?
17. If $400 gain $18 in 9 months, what is the rate per cent. per annum?
18. What principal, at 6 per cent. per ann., will gain $18 in 9 months ?
19. A usurer put out $75 at interest, and, at the end of 8 months, received, for principal and interest, $79; I de. mand at what rate per cent. he received interest.
Ans. 8 per ct. 20. If 3 men receive £886 for 194 days' work, how much must 20 men receive for 1004 days'?
Ans. £305 Os. 8d. 21. If 40 men in 10 days, can reap 200 acres of grain, how many acres can 14 men reap in 24 days?
Ans. 168 acres. 22. If 14 men in 24 days, can reap 168 acres of grain ; how many acres can 40 men reap in 10 days ?
Ans. 200 acres. 23. If 16 men in 32 days, can mow 256 acres of grass ; in how many days will 8 men mow 96 acres ?
Ans. 24 days. 24. If 4 men mow 96 acres in 12 days; how many acres can 8 men mow in 16 days?
Ans. 256. 25. If a family of 16 persons spend $320 in 8 months ; how much would 8 of the same fanıily spend in 24 months ?
Ans. $480. 26. If a family of 8 persons in 24 months spend $480 ;