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10. If a staff 4 feet long cast a shade on level ground 7 feet long, what is the height of a steeple whose shade at the same time is 198 feet? Ans. 1134.

11. An express who rides from Washington city had been dispatched at the rate of 60 miles per day for five days, when a second was sent to overtake him, in order to do which, he must travel 75 miles a day, in what time will he overtake the former? Ans. in 20 days.

12. A prize of $2,329 was divided between two persons, A. and B. whose shares therein were in proportion as 5 to 12, what was the share of each?

Ans. A. $685, and B. $1644. 13. When a man's yearly income is $949, how much is it per day? Ans. $2.60. 14. Bought a stove weighing 4 cwt. 3 qrs. 24 lbs. at $2.10 per cwt. and 27 lbs. of pipe at 183 cts. per lb. with 2 elbows at 50 cts. each, what is the price of the stove pipe and elbows? Ans. $16.48.

15. Bought 4 pieces of linen, viz: No. 1 and 2, each contained 27 yards, No. 3 and 4, contained each 25 yards at 62 cts. per yard. what was the cost?

Ans. $66.561. 16. If a person's salary be $1,333 per annum, and his daily expences $2.14, how much will he save? Ans. $551.90. 17. What will 4 pieces of sattinet, containing 23, 24, 25, and 27 yards come to at 72 cents

per yard?

Ans. $71.28.

18. A farmer upon measuring his corn produced by a certain field, found he had but 48 bushels. It appeared that it yielded one third more than was sown, how much was that? Ans. 36.

19. A bookseller sold 10 books at a certain price, and afterwards 15 more at the same rate; now at the latter sale he received $2.25 more than at the former, what did he receive for each book?

Ans. 45c.

CONTRACTED OPERATIONS.

CASE I.

If the first term be a multiple or part of the second, the third will be a multiple or part of the fourth.

Example 1.-If 6 yards cost $18, how much will 24 yards cost? Ans. $72.

CASE II.

If a part of the first be added to or subtracted from the first so as to be equal to the second, a like multiple must be added to or subtracted from the third.

Example 1.-If 12 yards of cloth cost $15, what will 20 yards cost?

Ans. $25.

2. If 6 yards cost $9, what will 24 yards cost?

Ans. $36.

SOLUTION.-Agreeably to Case II. the difference between the first and second terms is 3, and as 3, is the half of 6, of the third term must be added to it. Thus, 24 +24 $36 as above. 3. If 8 yards cost $12, what will 48 yards cost?

Ans. $72.

Ans. $110.

4. If 45 yards cost $30, what will 165 yards cost?

5. If 3 yards cost $2, what will 27 yards cost?

Ans. $18.

6. If 9 yards cost $6, what will 30 yards come to?

Ans. $20.

7. If 3 yds cost $6, what will 10 yds. cost?" $20. 8. Bought 6 yards for $8, how much did 30 yards cost? Ans. $40.

INVERSE PROPORTION.

It is generally admitted by writers on Arithmetic that more requires less, or less requires more; more requires less, when the third term is greater than the first, and the fourth less than the second. 2d. Less requires more when the third term is less than the first, and the fourth

term greater than the second. These definitions may be illustrated by the following questions, viz:

1. If in 10 days 8 men can perform a piece of work, in what time could 40 men do the same.

Illustration-8 men at work 10 days perform as much as 80 men in 1 day. 80 men perform the same in 1 day, 40 2 days.

uniformly would

Consequently, if men will do it in

Q. E. D.

2. If 8 men can perform a piece of work in 10 days, how many men would be required to perform the same in 2 days? Ans. 40.

2 : 40.

days. men. days. Solution-As 10 8 Notwithstanding, these statements are made to correspond with the above definitions, and bring correct answers. Yet, it is manifest, they are wrong!!! if the doctrine of proportion be recognized, for agreeably, to Prop. B. book 5 Euclid, in Inverted Proportion, the third is to the second, as the first is to the fourth. In solution 1st, we find that more requires less for 10 days require 2 days, and in solution 2d, less requires more-for 8 men require 40 men.

Now the

proper method of stating is thus: as 40 men, : 10 days: 8 men = 2 days.

Solution 2d-2 days: 8 men: 10 days =40 men. Hence, 40 x 2 = 10 × 8,

And 2 × 40 = 8 × 10-for by Prop. 16, 6 Euclid, the rectangle of the extremes will be equal to that of the means, which is not the case with the first and second statements; therefore, it follows, that although custom and the opinion of most writers, sanction the truth of the 'first and second' statements they are evidently mistaken.

3. Suppose 450 soldiers are in a garrison, and their provisions are calculated to last them but 5 months, how many must leave the garrison, that the same allowance may be sufficient for those who remain 9 months.

Ans. 200.

4. If a man perform a piece of work in 15 days of 12 hours long, in how many days of 10 hours long can he perform the same? Ans. 18 days.

5. There is a cistern having a pipe

that will empty it in 6 hours, how many pipes of the same capacity will empty it in 20 minutes? Ans. 18 pipes.

COMPOUND PROPORTION.

Is that, in which 5 terms are given to find the sixth. Three of those terms are a supposition and two a demand. RULE. Arrange the terms so that two terms, one side of the statement shall be of the same name and denomination with two terms on the other side. the demand will be in the middle term.

Consequently,

Example. If 10 men in 18 days earn $56, how many dollars can 20 men earn in 36 days?