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products obtained by multiplying together the extreme terms 4 and 3, and the mean terms 2 and 6.
These results prove, that, if four numbers be in proportion, the product of the two extreme terms is equal to the product of the two mean terms: a principle of great practical utility, and the foundation of the ancient Rule of THREE.
It follows from what has been said, that the order of the terms of a proportion may be changed, provided they be so placed, that the product of the extremes shall be equal to the product of the means; because, whenever the product of the extreme terms of four numbers is equal to the product of the mean terms, the numbers are proportionals.
Take, for example, the proportion 3:9=8: 24 3:9=8: 24, and observe the dif- 3 :8= 9: 24 ferent orders in which its terms may 24 :8=9: 3 be arranged.
24 :8=8: 3 That these changes do not disturb the proportion is evident; for the same numbers, which are the extreme terms in the first proportion, are the extreme terms in all the proportions; and the numbers, which are the mean terms in the first proportion, are the mean terms in all the proportions; therefore the products of the extremes and the products of the means must be the same in all the proportions.
Again, the order of the above pro 9: 3 = 24 :3 portionals may be so changed, that 8:3=24 : 9 the mean terms shall become the ex 9:24= 3:8 treme terms, and the extreme terms 8:24= 3:9 the mean terms. .
Since both the terms of a ratio may be multiplied or divided by the same number without altering the ratio, it follows, that all the terms of a proportion may be multiplied or divided by the same number without disturbing the proportion. Let us take, for example, the proportion 2:4=6:12, and multiply each of the terms by 2, and we shall have the proportion 4:8=12: 24. If, instead of multiplying, we divide the terms of the same proportion by 2, we shall have the proportion 1 : 2=3:6. Either the two antecedents or the two consequents, in two equal ratios, may be multiplied or divided by the same number without destroying the proportion; because the two ratios are increased or diminished alike, and therefore remain equal. Take the proportion 4 : 165 6:24, and multiply each of the antecedents by 2, and it will be 8:16=12 : 24; if, instead of the antecedents, we multiply the consequents by 2, we have the proportion 4 : 32=6:48; if, instead of multiplying, we divide each of the antecedents by 2, we have the proportion 2:16=3:24; if, instead of the antecedents, we divide the consequents, we have the proportion 4:8=6:12.
We may also multiply the antecedents and divide the conscquents at the same time, and vice versa, without destroying the proportion. If, for example, we take the proportion 3:=9:18, and multiply each of the antecedents by 3, and divide each of the consequents by the same number, we have the proportion 9:2=27 : 6; if we multiply the antecedents by 3, and divide the consequents by 2, we have the proportion 9:3=27 : 9; if we divide the antecedents by 3, and multiply the consequents by 2, we have 1:12=3 : 36.
Two or more proportions may be multiplied together, term by term, and the products will be proportionals; for it is the same as multiplying two equal fractions by two other equal fractions, the products of which will again be equal to each other. We give the following as an example.
3 : 4 = 6 : 8
6:12=48: 96 We may also divide one proportion by another, term by term, with equal correctness of conclusion; for this is only dividing two equal fractions by two other equal fractions, the quotients of which will again be equal. Take, for example, the proportion 24 : 32=27 : 36, and divide it by the proportion 6:2=9:3, term by term, and it gives 4:16 = 3:12.
A great variety of other changes may be made by differently multiplying, or dividing, or both; and such changes are frequently convenient in solving questions.
The magnitudes of proportionals are changed without destroying the proportionality, when either the antecedents, or consequents, or both, are respectively increased or diminished by quantities having the same ratio; or when the two terms of either or of both ratios are respectively increased or diminished by quantities in the same ratio with themselves. We will make a few such changes in the proportionals 8, 6, 20, and 15. 8:6
=20 : 15 8+6:6 =-20+15 :15, or 14:6=35 :15 8-6:6 =20—15 :15, or 2 :6= 5 : 15 8:8-6 =20 : 20-15, or 8 :2—20 : 5 8:8+6 =20 : 20+15, or 8:1420:35 8+6:3—6=20+15 : 20—15, or 14 : 2=35 : 5
Since the product of the extremes in every proportion is equal to the product of the means, one product may be taken for the other: now if we divide the product of the extremes by one extreme, the quotient is the other extreme; therefore, if we divide the product of the means by one extreme, the quotient is the other extreme: for the same reason if we divide the product of the extremes by one of the means, the quotient is the other mean; consequently, we can find any one term of a proportion, when we know the other three.
To apply these principles to practice, let it be asked If 64 yards of cloth cost 304 dollars, what will 36 yards cost? In the first place, the ratio of the two pieces of cloth is 64 : 36; and secondly, the prices are in the same ratio; that is, 304 dollars must have the same ratio to the price of 36 yards, that 64 yards have to 36 yards. Now, if we put A. instead of the answer, we shall have the following proportion, 64 : 36=304 : A, in which the product of the means is 10944, which, being divided by 64, one of the extremes, gives the quotient 171, the other extreme, which was the term sought; therefore, 171 dollars is the price of 36 yards.
Of the four numbers, which constitute a proportion, two are of one kind, and two of another. In the preceding example, two of the terms are yards, and two are dollars.
If there are different denominations in the two first terms, they must both be reduced to the lowest denomination in either of them; and the third term must be reduced to the lowest denomination mentioned in it. Thus, if 4 yards cost 18 shillings and 6 pence, what will 3 yards 1 quarter 2 nails cost? Nails being the lowest denomination in the two first terms, they must both be reduced to nails; pence being the lowest denomination in the third term, this term must be reduced to pence; and when thus reduced, the terms will make the following proportion; 64 : 54=222 : A. The answer, when obtained, being in pence, must be reduced to shillings and pounds. In this question the answer is 15 s. 7 d.
From the principles of ratio and proportion, which have been explained, we deduce the following rule for solving questions.
RULE. Make the number, which is of the same kind with the answer, the third term; and, if from the nature of the question, the third term must be greater than the fourth term or answer, make the greater of the two remaining terms the first term, and the smaller the second; but, if the third term must be less than the fourth, make the less of the two remaining terms the first term, and the greater the second: then multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term, or answer.
1. If I buy 871 yards of cotton cloth for 78 dollars 39 cents, what is the price of 29 yards of the same? 871 : 29=78.39 : A. The statements of this 29
question may be read thus 70551
—The ratio of 871 to 29 15678
is equal to the ratio of
78.39 to the answer. Or 871)2273.31(2.61 Ans.
thus—As 871 yd. is to 29 1742
yd., so is $78.39 to the 5313
answer. The operation 5226
amounts to nothing more 871
than the multiplication of 12* 871
78.39 by '11
2. If 14 yard of cotton cloth cost 42 cents, what will 871 yards cost ?
3. If I can buy 11 yard of cotton cloth for 61 pence, how many yards can I buy for £10 6 s. 8 d.?
4. If I buy 54 barrels of four for 297 dollars, what must I give for 73 barrels, at the same rate ?
5. If 7 workmen can do a piece of work in 12 days, how many can do the same work in 3 days?
6. If 20 horses eat 70 bushels of oats in 3 weeks, how many bushels will 6 horses eat in the same time?
7. If a piece of cloth containing 76 yards cost 136 dollars 80 cents, what is that per ell English ?
8. If a staff 4 feet long cast a shadow 7 feet in length, on level ground, what is the height of a steeple, whose shadow at the same time measures 198 feet ?
9. How many, yards of paper 21 feet wide, will hang a room, that is 20 yards in circuit, and 9 feet high?
10. A certain work having been accomplished in 12 days by working 4 hours a day, in what time might it have been done by working 6 hours a day?
11. If 12 gallons of wine are worth 30 dollars, what is the value of a cask of the same kind of wine, containing 311 gallons ?
12. If 83 yards of cloth cost 4 dollars 20 cents, what will 13 yards cost, at the same rate ?
13. How many yards of cloth ) yard wide, are equal to 30 yards 14 yard wide ?
14. If 7 pounds of sugar cost 75 cents, how many pounds can I buy for 6 dollars ?
15. If 2 pounds of sugar cost 25 cents, and 8 pounds of sugar are worth 5 pounds of coffee, what will 100 pounds of coffee cost?? : 16. A merchant owning of a vessel, sold of his share for 957 dollars. What was the vessel worth at that rate ?
17. A merchant failing in trade, owes 62936 dollars · 39 cents; but his property amounts to only 38793 dollars 96 cents, which his creditors agreed to accept, and discharge him. How much does the creditor receive, to whom he owes 2778 dollars 63 cents ?.