If, then, one of the extremes be wanting, divide the product of the means by the extreme given; or, if one of the means be wanting, divide the product of the extremes by the means given, and the result will be the term sought.

To apply this, we will take the following question. If 5 yards of cloth cost $15, what will 7 yards cost? It is evident that twice the quantity of cloth would cost twice the sum, and that three times the quantity, three times the sum, &c.; that is, the price will be in proportion to the quantity purchased. We then have three terms of a proportion given, one of the extremes and the two means, to find the other extreme.

Thus, 57:: 15. Therefore, to find the other extreme by the rule above stated, we multiply the two means, 7 and 15, and divide their product by the extreme given, and the quotient is the extreme required. 7 × 15105. 105 ÷ 5 = 21 dollars, the answer required.

To perform this question by analysis, we reason thus. If 5 yards cost 15 dollars, 1 yard will cost one fifth as much, which is 3 dollars; and if 1 yard cost 3 dollars, 7 yards will cost 7 times as much, which is 21 dollars.

RULE.*. State the question by making that number which is of the same name or quality of the answer required the third term; then, if the answer required is to be greater than the third term, make the second term greater than the first; but if the answer is to be less than the third term, make the second less than the first.

Reduce the first and second terms to the lowest denomination mentioned in either, and the third term to the lowest denomination mentioned in it.

Multiply the second and third terms together, and divide their product

* This rule was formerly divided into the Rule of Three Direct, and the Rule of Three Inverse. The Rule of Three Direct included those questions where more required more and less required less; thus, If 5lb. of coffee cost 60 cents, what would be the value of 10lb.? would be a question in the Rule of Three Direct, because the more coffee there was the more money it would take to purchase it. But if the question were thus: :- If 4 men can mow a certain field in 12 days, how long would it take 8 men?- it would be in Inverse, because the more men the less would be the time to perform the labor, that is, more would require less.

The method for stating questions was this: :- To make that number which is the demand of the question the third term, that which is of the same name the first, and that which is of the same name as the answer required, the second term.

If the question was direct, the second and third terms must be multiplied together, and their product divided by the first; but if it was inverse, the first and second terms must be multiplied together, and their product divided by the third.

by the first, and the quotient is the answer, in the same denomination to which the third is reduced.

If any thing remains after division, reduce it to the next lower denomination, and divide as before.

If either of the terms consists of fractions, state the question as in whole numbers, and reduce the mixed numbers to improper fractions, compound fractions to simple ones, and invert the first term, and then multiply the three terms continually together, and the product is the answer to the question. Or the fractions may be reduced to a common denominator; and their numerators may be used as whole numbers. For when fractions are reduced to a common denominator, their relative value is as their numerators.

NOTE 1. In the Rule of Three, the second term is the quantity whose price is wanted; the third term is the value of the first term; when, therefore, the second term is multiplied by the third, the answer is as much more than it should be, as the first term is greater than unity; therefore, by dividing by the first term, we have the value of the quantity required. Or, multiplying the third by the number of times which the second contains the first will produce the answer.

NOTE 2.-The pupil should perform every question by analysis, pre. vious to his performing it by Proportion.

EXAMPLES.

1. If a man travel 243 miles in 9 days, how far will he travel in 24 days?

Ans. 648 miles.

As the answer to the question must be in miles, we make the third term miles (243); and from the nature of the question we know that he will travel farther in 24 days than in 9 days; we therefore place 24, as the larger of the two remaining terms, in the second place, and the remaining number, 9 days, in the first place.

CANCELLING.

27

To perform this question by analysis, we proceed thus: If he travel 243 miles in 9 days, he will in one day travel of 243 miles, which is 27 miles; then if he travel 27 miles in

one day, in 24 days he will travel 24 times as far, which is 648 miles, the answer, as before.

2. If 17 yards of broadcloth cost $102, what will 7 yards cost?

yd. yd. $
17:7:: 102

7
17)714($ 42
68

34

34

Ans. $42.

As the answer is to be in dollars, we make the third term dollars (102); and as 7 yards will not cost so much as 17 yards, we make the second term (7) less than the first (17).

CANCELLING.

6

To perform this question by analysis, we reason thus. If 17 yards cost $102, one yard will cost of $102, which is $6; and if one yard cost $ 6, 7 yards will cost 7 times as much, which is $42, the answer, as before.

3. If 3 men drink a barrel of beer in 24 days, how long would it last 9 men? Ans. 8 days.

Men. m. da. 9:3:24

3

9)72

8 days.

As the answer is to be in days, we make 24 days the third term, and because 9 men will drink the beer in less time than 3 men, the second term will be less than the first.

To perform this question analytically, we would say, that if 3 men drink a barrel of beer in 24 days, it would take one man 3 times 24 days, which is 72 days; and if one man drink a barrel of beer in 72 days, 9 men would drink it in ¦ of 72 days, which is 8 days, as before.

4. If 12 yards of cloth cost $48, what will 15 yards Ans. $60.

cost?

5. If 17 pounds of sugar cost $1.19, what will 365 pounds cost? Ans. $25.55. 6. If 16 acres of land cost $720, what will 197 acres cost? Ans. $8865. 7. If $8865 buy 197 acres, how many acres may be bought for $720? Ans. 16. 8. What will 84hhd. of molasses cost, if 15hhd. can be purchased for $175.95 ? Ans. $985.32.

9. If $100 gain $6 in 12 months, how much would it gain in 40 months? Ans. $20. 10. If a certain vessel has provisions sufficient to last a crew of 10 men 45 days, how long would the provisions last if the vessel were to ship 5 new hands? Ans. 30 days. 11. If 7 and 9 were 12, what, on the same supposition, I would 8 and 4 be?

Ans. 9.

12. If 9 men can perform a certain piece of labor in 17 days, how long would it take 3 men to do it? Ans. 51 days. 13. If 3 men can perform a piece of labor in 51 days, how many must be added to the number to perform the labor in 17 days? Ans. 6. 14. How much in length, that is 5 rods in breadth, is sufficient for an acre? Ans. 29 rods. 15. If 2 barrels of flour cost $12, what will 24 barrels cost? Ans. $144. 16. If 5 quintals of fish cost $16.25, what is the value of 75 quintals? Ans. $243.75. 17. If 2 cords of wood cost $11.50, what will 17 cords cost? Ans. $97.75. 18. If 7cwt. of iron cost $56.85, what will 49cwt. cost?

Ans. $397.95.

19. If 5 acres of land be valued at $375.75, what would be the value of 35 acres? Ans. $2630.25. 20. If 7 pairs of shoes cost $10.50, how many pairs will $ 52.50 buy? 21. If $4.75 be paid for 191b. of salmon, how many pounds will $25.50 buy?

Ans. 35.

Ans. 102lb. how far will he Ans. 192 miles.

22. If a man travels 48 miles in 6 hours, travel in 24 hours?

23. If 8 men eat a barrel of flour in 24 days, how long would it last 3 men?

Ans. 64 days. 24. If 7 ounces of silver are sufficient to make 17 teaspoons, how many may be made from 42 ounces?

Ans. 102.

25. If $100 gain $6 in a year, how much will $850 gain? Ans. $51. 26. If $100 gain $ 6 in a year, how much would be sufficient to gain $32 in a year? Ans. $533.334. 27. If 20 gallons of water weigh 1671b., what will 180 gallons weigh? Ans. 1503lb. 28. If a staff 3 feet long cast a shadow 2 feet, how high is that steeple whose shadow is 75 feet? Ans. 112 feet. 29. If 5 cwt. be carried 36 miles for $4.75, how far might it be carried for $160? Ans. 12121 miles. 30. If 100 workmen can finish a piece of work in 12 days, how many men are sufficient to finish the work in 8 days?

Ans. 150.

31. If of a yard cost of a dollar, what will of a yard cost?

Ans. $0.48.

To perform this question by analysis, we would proceed thus: :- -If of a yard cost of $, which is ; and if 2, which is of a dollar. dollar, of a yard will cost

=

of a dollar, would cost cost, will cost 12 times And if one yard cost of a of a dollar, and will cost 4

times as much, that is, it will cost 4 times, which is 12= $0.48, as before.

32. If of a yard cost of a £., what will of a yard Ans. 1£. 1s. Od.

cost?
33. If 4 yards cost $ 9.75, what will 13 yards cost?

34. How much in length, that is 2 a square foot ? 35. If

Ans. $29.25. inches wide, will make Ans. 573 inches. of her worth? Ans. 10£. 18s. 6 d.

of a ship cost 51£., what are

36. A merchant bought a number of bales of velvet, each containing 12917 yards, at the rate of $7 for 5 yards, and sold them out again at the rate of $11 for 7 yards, and gained $200 by the bargain; how many bales were there? Ans. 9 bales.