To verify this result, double 6, which makes 12, and diminish it by the half of 6 or 3; the result is 9, according to the conditions of the problem.

Prob. 3. The sum of two numbers is 14, and their difference is 4. What are those numbers?

Let x the least number.

Then x+4 will be the greater number.

The sum of these is 2x+4, which is required to equal 14. Hence we have

2x+4=14.

x=5, the least number.

By transposition,

2x=14-4=10,

and

Also,

Verification.

x+4=9, the greater number. 9+5=14) according to the condi9-5= 4

tions.

The following is a generalization of the preceding problem :

Prob. 4. The sum of two numbers is a, and their difference b. What are those numbers?

Let x represent the least number..

Then x+b will represent the greater number. The sum of these is 2x+b, which is required to equal a. Hence we have

F

(103.) As these results are independent of any particular value attributed to the letters a and b, it follows that

Half the difference of two quantities, added to half their sum, is equal to the greater; and

Half the difference subtracted from half the sum is equal to the less.

b

The expressions+and

a b
2 2

are called formulas,

Thus, let

because they may be regarded as comprehending the solution of all questions of the same kind; that is, of all problems in which we have given the sum and difference of two quantities.

a=14

b= 4)

as in the preceding problem.

a+b 14+4

Then

2

=9, the greater number,

2

QUEST.-How may two quantities be determined from their sum and difference? What is a formula?

Prob. 5. From two towns which are 63 miles distant, two travelers set out at the same time with an intention of meeting. One of them goes 4 miles, and the other 5 miles per hour. hour. In how In how many hours will they meet?

Let x represent the required number of hours.

Then 4x wil represent the number of miles one traveled,

and 5x the number the other traveled;

and since they meet, they must together have traveled the whole distance.

Proof. In 7 hours, at 4 miles an hour, one would travel 28 miles; the other, at 5 miles an hour, would travel 35 miles. The sum of 28 and 35 is 63 miles, which is the whole distance.

The following is a generalization of the preceding problem.

Prob. 6. From two points which are a miles apart, two bodies move toward each other, the one at the rate of m miles per hour, the other at the rate of ʼn miles per hour. In how many hours will they meet?

Let x represent the required number of hours.

Then mx will represent the number of miles one

body moves,

and nx the miles the other body moves,

and we shall obviously have

This is a general formula, comprehending the solution of all problems of this kind. Thus:

We see that an infinite number of problems may be proposed, all similar to Prob. 5, but they are all solved by the formula of Prob. 6.

Prob. 7. A bookseller sold 10 books at a certain price, and afterward 15 more at the same rate. Now at the last sale he received 25 dollars more than at the first. What did he receive for each book?

the

Ans. Five dollars. The following is a generalization of the preceding problem.

Prob. 8. Find a number such that when multiplied successively by m and by n, the difference of the prod ucts shall be a.

Let x represent the required number.
Then, by the conditions of the problem,

This formula comprehends the solution of an infinite number of problems all similar to Prob. 7. Thus:

Prob. 9. A gentleman dying bequeathed $900 to three servants. A was to have twice as much as B, and B three times as much as C. What were their respective shares?

Ans. A received $540, B $270, and C $90. The following is a generalization of the preceding problem.

Prob. 10. Divide the number a into three such parts that the second may be m times as great as the first, and the third n times as great as the second.

Ans.

1+m+mn' 1+m+mn' 1+m+mn' Prob. 11. Four merchants entered into a specula tion, for which they subscribed $9510, of which P paid three times as much as A, C paid as much as A and B, and D paid as much as C and B. What did each pay?

Let x the number of dollars A paid.
Then we shall have

x+3x+4x+7x=9510;

whence

x=634 dollars A paid;