ALGEBRA.

=

SECTION I.

Questions in Simple Equations.

THIS sign is called plus, or more, and signifies, that the number or quantity before which it stands is to be added. This is called the sign of equality; and signifies, that the numbers or quantities between which it stands are equal. If we wished to express by signs that 2 and 3 are equal to 5, we write it 2 + 3 5, and read it 2 plus 3 are equal to 5. When a number or quantity has no sign, or the sign + before it, it is called a positive number or quantity.

1. A boy bought an apple and an orange for 21 cents; but the orange came to twice as much as the apple. How much did each come to ?

Suppose the apple cost x cents; then the orange must cost 2x cents. The 2 is called the coefficient of x.

2. A man bought a cow and a sheep for $15; but the cow came to twice as much as the sheep. What did each cost? x= what the sheep cost.

Let

Then 2x what the cow cost.

And x+2x= 15

15 proof.

3. A man gave his two sons 48 cents; but to one he gave three times as much as to the other. How much did he give to each?

4. Two numbers make 25; one is four times the other. What are the numbers?

5. Two numbers are 18; but one is five times the other. What are the numbers?

6. A man bought a horse, saddle and bridle for $82. The horse cost nine times as much as the saddle, and the saddle cost four times as much as the bridle. What did each cost? Let x the number of dollars the bridle cost. Then 4x the number of dollars the saddle cost. And 36 x = the number of dollars the horse cost. Hence x+4x+36x=82

proof:

All the parts must be equal to the whole.

As the answers are not given to the questions, it would be well for the learner to prove them.

7. There are three numbers, the sum of which is 42; the second is twice, and the third three times as large as the first. What are the numbers?

8. A man bought four articles for $64; for the second he gave three times, for the third five times, and for the fourth seven times, as much as for the first. What did he give for each?

9. Three men, A, B, and C, commenced trade. A put in a certain sum, B put in twice as much as A, and C put in as much as A and B both. The whole stock was $36. What did each put in?

10. A man bought five acres of land for $64; for the first and second he gave the same sum, and for each succeeding acre as much as for all the preceding. What did he give for each acre?

11. A man traveled 93 miles in five days, traveling a certain distance the first day, and each succeeding day twice as far as the day before. How many miles did he travel each day?

12. Four boys had 144 marbles; the second had twice as many as the first, the third as many as the first and second, the fourth as many as all the others. How many had each?

13. Six men had $72; the second had three times as many as the first, and the difference between what the first and second had, is the same as the difference between what the second and third had, and also the same as the difference between the third and fourth, the fourth and fifth, and fifth and sixth had. many had each?

How

SECTION II.

Simple Equations.

1. A man bought an equal number of apples and oranges for 50 cents; the apples at 2 cents a-piece, and the oranges at 3 cents a-piece. How many of each kind were there?

Let x = the number of each; then if the apples had been 1 cent a-piece, they would have come to x cents; but, as they were 2 cents each, they must come to twice as many cents, or 2 x cents. And, as the oranges were 3 cents each, they would come to three times as many cents as there were oranges; but, there were x oranges; therefore the oranges would cost 3 x cents. Hence 2x + 3 x = 50

2 X 10 = 20 = cost of the apples.
3 X 10: =30= cost of the oranges.
50 Proof.

2. A man bought some wheat and flour for $56. For the wheat he gave $1 per bushel, and for the flour $ 6 per barrel. There were as many bushels of wheat as there were barrels of flour. How many of each did he buy?

3. Two men built 45 rods of fence; the first built 5 rods per day, and the second 4. How many days did they work?

4. A man bought flour for $ 33. For one half he gave $5 per barrel, and for the other $6 per barrel. How many barrels did he buy?

If x = one half; then another x would be equal to the other

* This is the sign of multiplication.

half; 5x price of one half, and 6x= price of the other half, &c.

5. Two men at different times paid 99 dollars; at each payment one gave $4 and the other 5. How many payments did they make?

6. Three men at different times paid 75 dollars. The first paid 4, the second 5, and the third 6 at each payment. How many payments did they make?

7. From two towns 50 miles apart, two men set out with a design to meet. One traveled 4 miles per hour and the other 6. In how many hours did they meet?

8. There are two numbers, one of which is twice the other; and three times the smaller added to the larger will make 35. What are the numbers?

9. There are three numbers; the second is twice, and the third three times as large as the first: and their sum is 102. What are the numbers?

10. A man bought some flour, sugar, and coffee for 144 cents; the flour at 3 cents, the sugar at 6, and the coffee at 12 cents per pound. There was twice as much sugar as coffee; and twice as much flour as sugar. How many pounds were there of each?

11. A boy bought two dozen of apples and three dozen of eggs for 84 cents; the eggs came to four times as much per dozen as the apples, How much did each cost per dozen?

12. Three boys, A, B and C, talking of their money,-A says to B, I have three times as many cents as you; says C to A and B, I have as many as the difference between yours. Now A's money added to twice B's and twice C's, make 63 cents. How many cents had each?

13. The sum of two numbers added to their difference makes 36. One number is six times as much as the other. What are the numbers?

SECTION III.

Simple Equations.

1. Two boys had 20 apples; but one had 4 more than the other. How many had each?

If one had x, then the other must have x +4, and 2x+4

20. Now it appears that 2 x is not so much as 20 by 4; because 4 are added to 2 to make them equal to 20; hence 2x=16.

2. Two numbers make 25, and their difference is 5. What are the numbers?

For

Any expression containing the sign of equality is called an equation. All on the left of the sign = is called the first member. All on the right is called the second member. example 2x+4=20, is called an equation. 2x+4 is called the first member of the equation; and 20 is called the second member. The quantities 2x, 4, and 20, are called terms. 2x is called the first term of the first member, and 4 is the second term of the first member of the equation. 20 is a term of the second member of the equation.

3. A and B paid 100 dollars; but A paid 10 more than B. What did each pay?

4. John and James had 60 marbles; but John had 20 more than James. How many had each?

5. Two men, A and B, make a joint contribution which amounts to 36 dollars; but A's contribution was twice as much as B's and 3 dollars more. What did each contribute?

6. Two towns are unequally distant from Louisville, and on the same road. The distance between the two towns is 20 miles, and the sum of their distance from Louisville is 120 miles. What are their distances?

7. Four men paid 120 dollars; the second paid 2 more than the first; the third, 4 more than the second; and the fourth 6 more than the third. How many did each pay?

8. Two boys divide 20 apples between them; the second received twice as many as the first and 8 more. How many did each receive?