81. A problem is a question proposed, requiring a solution.

82. The solution of a problem is the operation of finding a quantity, or quantities, that will satisfy the given conditions.

The solution of a problem consists of two parts: the statement, and the solution of the equation, or equations of the problem.

The statement consists in translating the conditions of the problem into algebraic language, the resulting equations being called the equations of the problem.

The solution of the equations is made by the general rules for solving equations.

The statement is made by representing the unknown quantities of the problem by some of the final letters of the alphabet, and then operating upon these so as to comply with the conditions of the problem. The method of stating a problem will be best learned from practical examples.

1. What number is that to which if its fifth part be added, the sum will be equal to 24?

Xx

Let a denote the number; then will denote its fifth part.

From the conditions of the problem,

5

2. The sum of two numbers is 30, and their difference 6. What are the numbers ?

3. Two couriers start from points distant 200 miles,

and travel towards each other.

per hour, and the second 8

The first travels 9 miles

miles per hour. How

long before they will meet, and how far will each have traveled ?

Let x denote the number of hours; then will 9x denote the distance the first travels, and 8x denote the distance the second travels. From the conditions of the problem, we have,

Solving,

9x+8x = 200.

..

x = 11, the number of hours required.

9x=

81x =

102, the number of miles the first travels; and 974, the number of miles the second travels.

4. A hare starts 50 leaps before a dog, and makes 4 leaps to the dog's 3; but 2 of the dog's leaps are equal to 3 of the hare's. How many leaps must the dog make to overtake the hare?

Let x denote the number of leaps that the dog makes: then 4.x denote the number that the hare makes in the same time;

will

3

and if we take the length of the hare's leap as the unit of distance, the whole distance to be passed over by the dog, will be denoted by

4x

+ 50: but the dog passes over a distance of units at each leap; 3 hence, in a leaps, he will pass over a distance denoted by 3. From the conditions of the problem, these two quantities are equal; hence, we have,

5. A. can do a piece of work in 9 days, and B. can do the same work in 10 days. In how many days can they both do it together?

Let x denote the number of days required. If we denote the 1 of it in 1 day, and in x days he can do

work by 1, A. can do 10

x

9

1

of it. B. can do of it in 1 day, and in x days he can do 10

x

10

of it. But, from the conditions, the sum of these two will be equal to the entire work; that is,

6. If to a certain number its half and its eighth part be added, the sum will be equal to 78. What is the number? Ans. 48.

7. A man bought a horse, harness, and wagon, for 250 dollars: he gave for the harness one fourth as much as for the horse, and for the wagon as much as for the horse and harness together. What did he give for each?

Ans. $100 for the horse, $25 for the harness, and $125 for the wagon.

8. A drover sold from a flock of sheep one half, and two more; he then sold half that remained, and