times the letter is to be taken, is called the coefficient of that letter. Thus, in the quantity 10x, 10, is the coefficient of x.

(13.) The solution of a problem, by Algebra, consists of two distinct parts:

1. To express the conditions of the problem algebraically; that is, to form the equation.

2. To find the value of the unknown quantity after the equation is formed; that is, to reduce the equation.

(14.) It is impossible to give a general rule which will enable us to translate every problem into algebraic language. This must be learned by practice But rules may be given for reducing the equation after it is formed.

After the preceding problems were reduced to equa. tions, the first step was to reduce all the terms containing the unknown quantity to a single term, which was done by adding the coefficients. The second step was to divide each member of the equation by the coefficient of the unknown quantity.

In a similar manner may the following equations be solved.

Prob. 10. A gentleman, having 36 shillings to divide between a man and a boy, wishes to give to the man twice as much as to the boy. How much must he give to each.

Ans. 12 shillings to the boy,

and 24 shillings to the man.

(15.) All examples of this kind admit of proof. The results are proved to be correct when they fulfill all

QUEST.--What is a coefficient? How is an algebraic problem solv. ed? How is an equation reduced?

the conditions of the problem. In the preceding prob lem there are two conditions; first, that the boy and man together receive 36 shillings; second, that the man receives twice as much as the boy. The numbers 12 and 24 fulfill both of these conditions. All the results in the following problems should be verified in a similar manner.

Prob. 11. A man having a horse and cow, was asked what was the value of each. He answered that the horse was worth three times as much as the cow, and together they were worth 72 dollars. What was the value of each? Ans. The cow was worth 18 dollars,

and the horse

54 dollars. Prob. 12. Said Thomas to Charles, my purse and money together are worth 15 dollars, but the money is worth four times as much as the purse. How much money was there in the purse, and what was the value of the purse?

Ans. The purse was worth 3 dollars,

and the money was worth 12 dollars. Prob. 13. Two men, A and B, trade in company; but B puts in five times as much money as A. They gain 900 dollars. What is each man's share of the gain?

Ans. A's share is 150 dollars, and B's share is 750 dollars.

Prob. 14. A gentleman, meeting two poor persons, divided 28 shillings between them, giving to the second six times as much as to the first. How much did he give to each?

Ans. He gave 4 shillings to the first,

and 24 shillings to the second. QUEST.-HOW may the answers obtained be verified?

Prob. 15. A gentleman bequeathed 200 dollars to two servants upon condition that one should receive seven times as much as the other. How much did each receive? Ans. One received 25 dollars,

and the other 175 dollars. Prob. 16. A draper bought two pieces of cloth, which together measured 144 yards. The second piece contained eight times as many yards as the first. What was the length of each?

Ans. The first piece contained 16 yards,

and the second

128 yards.

Prob. 17. A man being asked the price of his horse, auswered that his horse and saddle together were worth 120 dollars, but the horse was worth nine times as much as the saddle. What was each worth? Ans. The saddle was worth 12 dollars,

Prob. 18. A cask which held 132 gallons was filled with a mixture of brandy and water, and there was ten times as much brandy as water. How much was there of each?

and

Ans. There were 12 gallons of water, 120 gallons of brandy. The following problems are similar to the preceding, except that an additional term is introduced.

Prob. 19. A gentleman, meeting three poor persons, divided 72 cents among them; to the second he gave twice, and to the third three times as much as to the first. What did he give to each?

Let x= the sum given to the first.

Then 2x= the sum given to the second,

and 3x= the sum given to the third.

Then, by the conditions of the problem,

That is,

or,

x+2x+3x=72.

6x=72,

x=12, the sum given to the first. Therefore he gave 24 cents to the second, and 36 cents to the third.

The learner should verify this, and all the subsequent results.

Prob. 20. Three men, A, B, and C, found a purse of money containing 119 dollars, but not agreeing about the division of it, each took as much as he could get. A got a certain sum, B got twice as much as A, and C four times as much as A. How many dollars did each get?

Let x= the number of dollars A got.

Then 2x the number of dollars B got,

=

and 4x the number C got.

These, added together, must make 119 dollars, the whole sum to be divided.

Hence,

Prob. 21. Three men, A, B, and C, trade in company. A puts in a certain sum, B puts in three times as much as A, and C puts in five times as much as A. They gain 657 dollars. What is each man's share of the gain?

Let x A's share.

Prob. 22. A gentleman left 15,000 dollars to be divided between his widow, his son, and daughter. He directed that his son should receive three times as much as his daughter, and his widow six times as much as his daughter. Required the share of each. Let x represent the share of his daughter.

Then 3x will represent the share of his son, and 6x will represent the share of his widow.

Prob. 23. A farmer bought some oxen, some cows, and some sheep. The number of them all together was 48. There were three times as many cows as oxen, and four times as many sheep as oxen. How many were there of each sort?

Let x denote the number of oxen.

Then 3x will denote the number of cows,

and 4x will denote the number of sheep.