(10.) When a problem is proposed for solution, it is generally required to find one or more quantities which are unknown. It is convenient to have signs to represent these unknown quantities, so that all the operations which are required to be performed may be presented at once in a single view.

The signs generally employed to represent these unknown quantities are some of the last letters of the alphabet; as, x, y, z, etc.

These principles will be best understood after attending to a few practical examples.

Problem 1. A boy bought an apple and an orange for 6 cents; for the orange he gave twice as much as for the apple. How much did he give for each ?

Let x represent the number of cents he gave for the apple, then 2x will represent the number of cents he gave for the orange. Now these, added together, must make the sum given for both, which was 6 cents; that is, x+2x=6.

But twice x, added to once x, makes three times x; that is,

3x=6;

and if three times x is equal to 6, once x must be equal to 2; that is,

Therefore the apple cost 2 cents, and the orange 4 cents, the sum of which is 6 cents, according to the conditions of the problem.

Prob. 2. 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

QUEST.-How are unknown quantities represented?

together they were worth 60 dollars. What was the value of each?

Let x represent the number of dollars equal to the value of the cow, then 3x will represent the value of the horse. These, added together, must make 60, according to the conditions of the problem; that is,

x+3x=60.

But three times x, added to once x, makes four times x; that is,

4x=60;

and if four times x is equal to 60, once x must be equal to 15; that is,

60
x=- =15.
4

Therefore the cow was worth 15 dollars, and the horse, being worth three times as much as the cow, amounted to 45 dollars. The sum of 15 and 45 is 60, according to the conditions of the problem.

Prob. 3. Said Charles to Thomas, my purse and money together are worth 10 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?

Let x represent the value of the purse.

Then 4x will represent the value of the money it contained. Then, by the problem, we must have x+4x=10, 5 =10

or,

Therefore the purse was worth 2 dollars, and the money 8 dollars, the sum of which is 10 dollars.

Prob. 4. Two men, A and B, trade in company.

B

puts in five times as much money as A. They gain 660 dollars. What is each man's share of the gain? Let x represent A's share.

Then 5x will represent B's share.

Therefore A's share is 110 dollars, and B's share is 550 dollars, the sum of which is 660 dollars.

Prob. 5. A gentleman, meeting two poor persons, divided 21 shillings between them, giving to the second six times as much as to the first. How much did he give to each?

or,

Let x= the shillings he gave to the first.

Then 6x= the shillings he gave to the second.
Therefore,

Hence,

x+6x=21,

7x=21,

Therefore he gave 3 shillings to the first and 18 shillings to the second, the sum of which is 21 shillings.

Prob. 6. A gentleman bequeathed 144 pounds to two servants upon condition that one should receive seven times as much as the other. How much did each re. ceive?

or,

Let x the smallest share.

Then 7x the share of the other.

Therefore,

x+7x=144,

8x=144.

Therefore one received 18 pounds, and the other 126, the sum of which is 144 pounds.

Prob. 7. A draper bought two pieces of cloth, which together measured 171 yards. The second piece contained eight times as many yards as the first. What was the length of each?

or,

Let x= the number of yards in the first piece. Then 8x= the number of yards in the second piece. Therefore, x+8x=171,

Hence,

Therefore the length of the first piece was 19 yards, and that of the second was 152 yards, the sum of which is 171 yards.

Prob. 8. A man being asked the price of his horse, answered that his horse and saddle together were worth 90 dollars, but the horse was worth nine times as much as the saddle. What was each worth?

Therefore the saddle was worth 9 dollars, and the horse was worth 81 dollars, the sum of which is 90 dollars

Prob. 9. A cask which held 143 gallons was filled with a mixture of brandy and water, and there was

How much was

ten times as much brandy as water.
there of each?

Let x= the gallons of water.
Then 10x= the gallons of brandy.
Therefore, x+10x=143,

Therefore there were 13 gallons of water and 130 gallons of brandy, the sum of which is 143 gallons.

(11.) The pupil will observe that when a problem is proposed for solution, the first thing to be done is to find an expression which shall contain the unknown quantity, and which shall be equal to a given quantity. Then, from this expression, by arithmetical operations, we deduce the value of the unknown quantity.

This expression of equality between two quantities is called an equation. Thus, x+10x=143; is an equation.

The quantity or quantities on the left side of the sign of equality are called the first member of the equation; those on the right, the second member of the equation.

Thus, x+10x is the first member of the above equation, and 143 is the second member.

(12.) Quantities connected by the signs + and — are called terms. Thus, x and 10x are terms in the above equation.

A number written before a letter, showing how many

QUEST.--What is the course pursued in solving a problem? What is an equation? What are the members of an equation? What are terms?