the third term; and, because it will require more men to do a piece of work in 4 days than in 8 days, the least of the remaining terms is made the first term and the other the second term.

In the second example, days are required; therefore, 40 days are made the third term; and, because 10 men will require longer time to do the work than 16, the least of the remaining terms is made the first term and the other the second term.

The 1st of the above examples is direct, and the 2d inverse.

After a question is stated, the work may often be abridged by a person acquainted with the nature of proportion. To assist the student in accomplishing this important object, the following observations will very much contribute:

CONTRACTIONS.

1. When the second or third term can be divided by the first, or if any number occur to the memory, which will divide the first and second, or first and third term, without a remainder, it will shorten the work to divide these terms by that number, and use the quotients thence arising, instead of the original

terms.

2. When the first term can be divided by the second, divide the third term by the quotient and the last quotient will be the

answer.

3. When the first term can be divided by the third, divide the second by the quotient, and the last quotient will be the an

swer.

The three following examples will illustrate these different modes of contraction.

EXAMPLE I.

If 27 yards of cloth cost £54, what will 81 yards cost?

Yds.
27

: 81 :: 54

1st Contraction or 5427 = 2 × 81 = £162 81 273 x 54 = £1622 Ans.

EXAMPLE II.

If 81 yards of cloth cost £162, what will 27 yards cost?

If £162 be paid for 81 yards of cloth, how many yards of the same may be bought for £54?

1. If 16 men do a piece of work in 12 days, in how many days will 6 men do it?

2. If 8 men do a piece of work in 18 days, how many men will do it in 12 days?

3. If 12 men consume a certain quantity of provisions in 15 days, how long will the same quantity serve 20 men at that rate?

4. How many barrels, of 32 gallons each, will hold as much as 48 barrels of 42 gallons each?

5. A farmer had his crop cut down last year by 15 men in 24 days; how many men must he engage this year to cut it down in 18 days?

6. There are two numbers to one another, as 8 is to 11; the greater is 44, what is the less?

7. If 8cwt. be carried 51 miles for a certain sum, how far ought 34 cwt. to be carried for the same money?

8. If the interest of £100 for a year be £4, what sum will produce £500 of interest in the same time?

9. If 6 horses eat 21 bushels of oats in a week, how many bushels will serve 30 horses the same time?

10. How many hours will a person require to count £245000, supposing him to count £250 in 3 minutes?

11. How many crowns are equal in value to 100 half-guineas?

1800

12. If there be 18+ children born in a town, and 12 girls born for 13 boys, how many boys and how many girls are born?

13. A steeple projected a shadow to the distance of 35 yards, when a 4-foot staff, perpendicularly erected, cast a shadow 5 feet, required the height of the steeple?

14. How many yards of carpet (yard wide,) will cover a room, 25 feet long and 18 feet wide?

15. It is between 4 and 5 o'clock, and the hour and minute hands are exactly together; what is the precise time?

16. Suppose the arms of a deceitful balance to be to each other as 12 to 114, what weight will be required at the end of the shorter arm to counterbalance a weight of 461b. suspended from the end of the longer arm?

17. A traveller walks 24 miles a day, and after he had advanced 42 miles, another follows him, who walks 32 miles a day; in what time will he overtake him?

18. Janet can spin a certain quantity of yarn in 12 days, Margaret an equal quantity in 16 days; in what time will it be finished, if both work together?

19. A cistern, which would be filled in 3 hours, by 2 pipes running into it, would be filled in 12 hours by one of the pipes alone; in what time would it be filled by the other pipe alone?

20. If 21 oxen eat up 8 acres of grass in six weeks, and 18 oxen eat up the same in 9 weeks; how many oxen will it maintain for 18 weeks, if the grass grow uniformly during that time?

COMPOUND PROPORTION.

WHEN a proportion depends on several circumstances, connected with each other, it may be divided into as many simple proportions as these circumstances require, or it may be performed by one operation.

Thus, it may be asked, if 12 men consume 32 lbs. of bread in 18 days, how much will 24 men consume in 36 days?

Here the quantity of bread required depends, partly on the number of men and partly on the time; and, therefore, two distinct operations would be necessary to obtain the answer by simple proportion.

This, and all similar questions, forms what is called a Compound Proportion, which may be solved, by resolving it into as many simple proportions as there are particulars required in the question; or, what is much better, by the following rule:

RULE.

Place that number, which is of the same kind with the one required, as the third term; then state the several numbers, on which the question depends, as so many simple proportions, directly under each other, attending to the nature of the question, to discover whether the answer ought to be greater or less than the third term; then multiply all the numbers in the first row together, for a divisor, and all those in the second row together, and then proceed in every respect as directed in the simple rule of proportion.

1. After stating the proportion, if the same number occur in both rows, it may be struck out of both; and if any number present itself to the mind, that will divide any number, in both rows, without a remainder, strike out the original numbers, and use the quotients in their stead.

2. Though it is in general better to multiply the second and third terms together, before dividing by the first; yet, if any of these terms can be divided by the first, without a remainder, this should be done before multiplying them together.

EXAMPLE.

If 10 horses consume £30 value of oats, in 8 weeks, when the oats cost 48s. per quarter, how many horses will consume £54 value, in 14 weeks, when the price is 27s. per quarter?