LESSON II.

1. JAMES and John together have 18 cents, and John has twice as many as James; how many cents has each?

If x represents the number of cents James has, what will represent the number John has? What will represent the number they both have? If 3x is equal to 18, what is x equal to? Why?

NOTE. If the pupil does not readily perceive how to solve a question, let the instructor ask questions similar to the preceding.

2. A travels a certain distance one day, and twice as far the next, in the two days he travels 36 miles; how far does he travel each day?

3. The sum of the ages of Sarah and Jane is 15 years, and the age of Jane is twice that of Sarah; what is the age of each?

4. The sum of two numbers is 16, and the larger is 3 times the smaller; what are the numbers?

5. What number added to 3 times itself will make 20?

6. James bought a lemon and an orange for 10 cents, the orange cost four times as much as the lemon; what was the price of each? 7. In a store-room containing 20 casks, the number of those that are full is four times the number of those that are empty; how many are there of each?

8. In a flock containing 28 sheep, there is one black sheep for each six white sheep; how many are there of each kind?

9. Two pieces of iron together weigh 28 pounds, and the heavier piece weighs three times as much as the lighter; what is the weight of each?

10. William and Thomas bought a foot-ball for 30 cents, and Thomas paid twice as much as William; what did each pay? 11. Divide 35 into two parts, such that one shall be four times the other.

12. The sum of the ages of a father and son is equal to 35 years, and the age of the father is six times that of his son; what is the age of each?

13. There are two numbers, the larger of which is equal to nine times the smaller, and their sum is 40; what are the numbers?

14. The sum of two numbers is 56, and the larger is equal to seven times the smaller; what are the numbers?

15. What is x+2x equal to?

16. What is x+3x equal to?

17. What is x+4x equal to ?

LESSON III.

1. THREE boys are to share 24 apples between them; the second is to have twice as many as the first, and the third three times as many as the first. If x represents the share of the first, what will represent the share of the second?. What will represent the share of the third? What is the sum of x+2x+3x? If 6x is equal to 24, what is the value of x? What is the share of the second? Of the third?

VERIFICATION.-The first received 4, the second twice as many, which is 8, and the third three times the first, or 12; and 4 added to 8 and 12, make 24, the whole number to be divided.

2. There are three numbers whose sum is 30, the second is equal to twice the first, and the third is equal to three times the first; what are the numbers?

3. There are three numbers whose sum is 21, the second is equal to twice the first, and the third is equal to twice the second. If x represents the first, what will represent the second? If 2x represents the second, what will represent the third? What is the sum of x+2x+4x? What are the numbers?

4. A man travels 63 miles in 3 days; he travels twice as far the second day as the first, and twice as far the third day as the second; how many miles does he travel each day?

5. John had 40 chestnuts, of which he gave to his brother a certain number, and to his sister twice as many as to his brother; after this he had as many left as he had given to his brother; how many chestnuts did he give to each?

6. A farmer bought a sheep, a cow, and a horse, for 60 dollars; the cow cost three times as much as the sheep, and the horse twice as much as the cow; what was the cost of each?

7. James had 30 cents; he lost a certain number; after this he gave away as many as he had lost, and then found that he had three times as many remaining as he had given away; how many did he lose?

8. The sum of three numbers is 36; the second is equal to twice the first, and the third is equal to three times the second; what are the numbers?

9. John, James, and William together have 50 cents; John has twice as many as James, and James has three times as many as William ; how many cents has each?

10. What is the sum of x, 2x, and three times 2x?

11. What is the sum of twice 2x, and three times 3x?

LESSON IV.

1. If 1 lemon costs x cents, what will represent the cost of 2 lemons? Of 3? Of 4? Of 5? Of 6? Of 7?

2. If 1 lemon costs 2x cents, what will represent the cost of 2 lemons? Of 3? Of 4? Of 5? Of 6?

3. James bought a certain number of lemons at 2 cents a piece, and as many more at 3 cents a piece, all for 25 cents; if x represents the number of lemons at 2 cents, what will represent their cost? What will represent the cost of the lemons at 3 cents a piece? How many lemons at each price did he buy?

4. Mary bought lemons and oranges, of each an equal number; the lemons cost 2, and the oranges 3 cents a piece; the cost of the whole was 30 cents; how many were there of each?

5. Daniel bought an equal number of apples, lemons, and oranges for 42 cents; each apple cost 1 cent, each lemon 2 cents, and each orange 3 cents; how many of each did he buy?

6. Thomas bought a number of oranges for 30 cents, one-half of them at 2, and the other half at 3 cents each: how many oranges did he buy? Let x= one-half the number.

7. Two men are 40 miles apart; if they travel toward each other at the rate of 4 miles an hour each, in how many hours will they meet?

8. Two men are 28 miles asunder; if they travel toward each other, the first at the rate of 3, and the second at the rate of 4 miles an hour, in how many hours will they meet?

9. Two men travel toward each other, at the same rate per hour, from two places whose distance apart is 48 miles, and they meet in six hours; how many miles per hour does each travel?

10. Two men travel toward each other, the first going twice as fast as the second, and they meet in 2 hours; the places are 18 miles apart; how many miles per hour does each travel?

11. James bought a certain number of lemons, and twice as many oranges, for 40 cents; the lemons cost 2, and the oranges 3 cents a piece; how many were there of each?

12. Two men travel in opposite directions; the first travels three times as many miles per hour as the second; at the end of 3 hours they are 36 miles apart; how many miles per hour does each travel?

13. A cistern, containing 100 gallons of water, has 2 pipes to empty it; the larger discharges four times as many gallons per

hour as the smaller, and they both empty it in 2 hours; how many gallons per hour does each discharge?

14. A grocer sold 1 pound of coffee and 2 pounds of tea for 108 cents, and the price of a pound of tea was four times that of a pound of coffee: what was the price of each?

If x represents the price of a pound of coffee, what will represent the price of a pound of tea? What will represent the cost of both the tea and coffee?

15. A grocer sold 1 pound of tea, 2 pounds of coffee, and 3 pounds of sugar, for 65 cents; the price of a pound of coffee was twice that of a pound of sugar, and the price of a pound of tea was three times that of a pound of coffee. Required the cost of each of the articles.

If x represents the price of a pound of sugar, what will represent the price of a pound of coffee? Of a pound of tea? What will represent the cost of the whole?

LESSON V.

1. James bought 2 apples and 3 peaches, for 16 cents; the price of a peach was twice that of an apple; what was the cost of each? If x represents the cost of an apple, what will represent the cost of a peach? What will represent the cost of 2 apples? Of 3 peaches? Of both apples and peaches?

2. There are two numbers, the larger of which is equal to twice the smaller, and the sum of the larger and twice the smaller is equal to 28; what are the numbers?

3. Thomas bought 5 apples and 3 peaches for 22 cents; each peach cost twice as much as an apple; what was the cost of each?

4. William bought 2 oranges and 5 lemons for 27 cents; each orange cost twice as much as a lemon; what was the cost of each?

5. James bought an equal number of apples and peaches for 21 cents; the apples cost 1 cent, and the peaches 2 cents each; how many of each did he buy?

6. Thomas bought an equal number of peaches, lemons, and oranges, for 45 cents; the peaches cost 2, the lemons 3, and the oranges 4 cents a piece; how many of each did he buy?

7. Daniel bought twice as many apples as peaches for 24 cents; each apple cost 2 cents, and each peach 4 cents; how many of each did he buy?

8. A farmer bought a horse, a cow, and a calf, for 70 dollars; the cow cost three times as much as the calf, and the horse twice as much as the cow; what was the cost of each?

9. Susan bought an apple, a lemon, and an orange, for 16 cents; the lemon cost three times as much as the apple, and the orange as much as both the apple and the lemon; what was the cost of each?

10. Fanny bought an apple, a peach, and an orange, for 18 cents; the peach cost twice as much as the apple, and the orange twice as much as both the apple and the peach; what was the cost of each?

LESSON VI.

1. James bought a lemon and an orange; the orange cost twice as much as the lemon, and the difference of their prices was 2 cents; what was the cost of each?

If x represent the cost of the lemon, what will represent the cost of the orange? What is 2x less x represented by?

2. What is 3x less x represented by? What is 3x less 2x represented by?

What is 4x less x represented by? What is 5x less 2x represented by?

for

The word minus, is used instead of less; and the sign the sake of brevity, is used to avoid writing the word minus. Thus, if we wish to take the difference between 3x and x, we may say,

3x less x,

or 3x minus x; which may be written 3x-x.

When the sign

3. Thomas bought a lemon and an orange; the orange cost three times as much as the lemon, and the difference of their prices was 4 cents; what was the price of each? If x represents the cost of the lemon, what will represent the cost of the orange? What is 3x-x represented by?

4. In a school containing classes in Grammar, Geography, and Arithmetic, there are three times as many studying Geography as Grammar, and twice as many studying Arithmetic as Geography; there are 10 more in the class in Arithmetic than in that in Grammar; how many more are there in each class? If x represents the number in the class in Grammar, what will represent the number