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246. Compound Proportion is an expression of equality between a compound and a simple ratio.
It is employed in the solution of such questions as require two or more statements in Simple Proportion.
247. To state and solve questions in Compound Proportion.
$100 $600 12 mo.: 10 mo.
Ex. 1. If $100 will gain $8 in 12 months, what will $600 gain in 10 months?
600 X 10 X 8 100 X 12
as the required term, the third term. Then, taking of the remaining terms two of
the same kind, $100 and $ 600, we inquire if the answer, depending on these alone, must be greater or less than the third term; and since it must be greater, because $600 will gain more than $100 in the same time, we make $ 600 the second term, and $100 the first. Again, we take the two remaining terms, and make 10 mo. the second term, and 12 mo. the first, since the same sum would gain less in 10 mo. than in 12 mo. We then find the continued products of the second and third terms, and divide it by the product of the first terms, for the answer.
In stating this question, we make $8, the gain, which is of the same kind
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 answer, depending only upon them, would be greater or less than the third term, and arrange them accordingly; and so on until all are used.
246. What is compound proportion? For what is it employed? - 247. In stating the question, which of the numbers do you make the third term? Why? What do you do with the remaining terms? How do you know which of the two to take for the second term? Which for the first? After all the terms have been arranged, how do you find the answer? The rule for compound proportion?
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.
NOTE. Operations can often be much shortened by cancellation.
Ex. 2. If $100 will gain $6 in 12 months, what will $800 gain in 8 months?
12 × 10 Ø
BY ANALYSIS AND CANCELLATION. 4
6 × 8 × 800
If $6 is the gain of $100 in 12 mo., in 1 mo. the gain of $100 will be as much, or $2, and in 8 mo. 8 times as much, or $
6x 8 12 6 x 8 12
Again, if $100 gain $ in 8 mo., $1 will gain of it, or 6 X 8 and $800 will gain 800 times as much, or $6x8x800, and cancelling the common factors we obtain $ 32 for the answer.
We state the question according to the rule, and then write the second and third terms for a dividend and the first terms for a divisor, and cancel the common factors.
EXAMPLES FOR PRACTICE.
3. If $100 gain $ 6 in 12 months, in how many months will $800 gain $32? Ans. 8 months.
4. If $100 gain $6 in 12 months, how large a sum will it require to gain $32 in 8 months? Ans. $800.
5. If $800 gain $32 in 8 months, what is the per cent. ? per cent.
6. If 15 carpenters can build a bridge in 60 days when the days are 15 hours long, how long will it take 20 men to build the bridge when the days are 10 hours long?
Ans. 67 days.
247. How can operations often be shortened? How are questions stated for cancellation? Which terms are taken for the dividend? Which for the divisor? What are cancelled?
7. If a regiment of soldiers, consisting of 939 men, can eat 351 bushels of wheat in 3 weeks, how many soldiers will it require to eat 1404 bushels in 2 weeks? Ans. 5634 soldiers.
8. If 8 men spend $ 64 in 13 weeks, what will 12 men spend in 52 weeks? Ans. $384.
9. If 8 horses consume 42 bushels of grain in 24 days, how many bushels will suffice 32 horses 48 days?
Ans. 336 bushels.
10. 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 16 hours each will 24 men build a wall 200 feet long, 16 feet high, and 6 feet thick? Ans. 90 days.
11. If a man travel 117 miles in 15 days, employing only 9 hours a day, how far would he go in 20 days, travelling 12 hours a day? Ans. 208 miles.
12. If 12 men in 15 days can build a wall 30 feet long, 6 feet high, and 3 feet thick, when the days are 12 hours long, in what time will 30 men build a wall 300 feet long, 8 feet high, and 6 feet thick, when they work 8 hours a day? Ans. 240 days.
13. If the carriage of 5cwt. 3qr. 150 miles cost $ 24.58, what must be paid for the carriage of 7cwt. 2qr. 15lb. 32 miles, at the same rate? Ans. $6.97+.
14. A received of B $9 for the use of $600 for 6 months; now B wishes to hire of A $1800 until the interest shall amount to the same sum. For how long must he hire it?
Ans. 2 months.
15. If 15 oxen or 20 cows will eat 3 tons of hay in 8 weeks, how much hay will be sufficient for 15 oxen and 8 cows 12 weeks? Ans. 63 tons.
16. If 5 men, by laboring 10 hours a day, can mow a field of 30 acres in 10 days, how long will it require 8 men and 7 boys, provided each boy can do as much as a man, to mow a field containing 54 acres? Ans. 73 days.
17. If 2 men can build 12 rods of wall in 6 days, how long will it take 18 men to build 247 rods? Ans. 14 days.
18. If 248 men, in 5 days of 11 hours each, dig a trench of 7 degrees of hardness, and 232 feet long, 3 feet wide, and 24 feet deep, in how many days of 9 hours each will 24 men dig a trench of 4 degrees of hardness, and 3374 feet long, 5 feet wide, and 34 feet deep? Ans. 132 days.
PROFIT AND LOSS.
248. Profit and Loss is the process by which merchants and other traders estimate their gain or loss in buying and selling goods.
Gains and losses are usually reckoned on the prime or first cost of articles.
249. To find the RATE PER CENT. of profit or loss when the cost and selling price are given.
Ex. 1. If I buy flour at $4 per barrel, and sell it at $5 per barrel, what is the gain per cent.? Ans. 25 per cent.
$5-$4-$1; } 1.004
.25, or 25 per cent.
By subtracting the cost from the selling price, we find the gain per barrel to be $ 1. Now, if the gain is $1 on $4, it is and .25, or 25 per cent.
of the cost,
OPERATION BY PROPORTION.
$5-$4 $1; $4: $1:: 1.00: .25, that is, 25 2. If I buy flour at $5 per barrel, and sell it at $4 per barrel, what is the loss per cent.? Ans. 20 per cent.
By subtracting the selling price from the cost, we find the loss per barrel to be $ 1. Now, if the loss is $ 1 on $ 5, it is of the cost, and
.20, or 20 per cent. From this analysis, and that of the preceding example, it is seen that the operation is equivalent to making the gain or loss the numerator of a fraction, and the cost the denominator, and then reducing this fraction to a decimal; or, in short, to simply dividing the gain or loss by the cost.
RULE 1. - Divide the gain or loss by the cost, and the quotient will be the gain or loss per cent. Or,
OPERATION BY PROPORTION.
$1; $5: $1 :: 100 per cent.: 20 per cent.
RULE 2.. As the cost is to the sum gained or lost, so is 100 per cent. to the per cent. gained or lost.
248. What is profit and loss? or loss in buying or selling goods?
What is the first rule for finding the profit
EXAMPLES FOR PRACTICE.
3. Bought 40 yards of broadcloth at $5.40 per yard, and I sell of it at $6 per yard, and the remainder at $7 per yard; what do I gain per cent.? Ans. 15 per cent.
4. A merchant purchased for cash 50 barrels of flour, at $5 per barrel, and immediately sold the same on 8 months' credit, at $5.98 per barrel; what does he gain per cent?
Ans. 15 per cent.
5. A grocer bought a hogshead of molasses, containing 100 gallons, at 30 cents per gallon; but 30 gallons having leaked out, he disposed of the remainder at 40 cents per gallon. Did he gain or lose, and how much per cent.?
Ans. Lost 63 per cent.
6. A gentleman in Rochester, N. Y., purchased 3000 bushels of wheat, at $1.12 per bushel. He paid 5 cents per bushel for its transportation to New York city, and then sold it at $1.37 per bushel; what did he gain per cent.?
Ans. 17 per cent.
7. J. Morse bought, in Lawrence, a lot of land 7 rods square, for $5 per square rod. He sold the land at 5 cents per square foot; what did he gain per cent.?
Ans. 172 per cent.
250. To find the SELLING PRICE, when the cost and the gain or loss per cent. are given.
Ex. 1. If I buy flour at $4 per barrel, for how much must I sell it per barrel to gain 25 per cent.? Ans. $5.
$4.25 $1.00; then $4 +$1 = $5, Ans.
If I sell the flour for 25 per cent. gain, I sell it for .25 more than it cost. Therefore, if I add to the cost .25 of the cost, the sum will be the price per barrel for which the flour must be sold.
OPERATION BY PROPORTION.
1.00.25 1.25; 1.00 1.25 $4 $5, Ans.
2. If I buy flour at $5 per barrel, for what must I sell it per barrel to lose 20 per cent.?
250. Explain how you find the selling price when the cost and the gain of loss per cent. are given.