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 a 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 a 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 is used, it is to be read minus. 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 a represents the number in the class in Grammar, what will represent the number in the class in Geography? In the class in Arithmetic? What What is it equal to? is 6x-x represented by? 5. The age of Sarah is three times the age of Jane, and the difference of their ages is 12 years; what is the age of each? 6. The difference of two numbers is 28, and the greater is equal to eight times the less; what are the numbers? 7. Daniel has four times as many cents as William, and Joseph has twice as many as both of them; but if twice the number of Daniel's cents be taken from Joseph's, the remainder is only 16; how many cents has each? 8. Susan bought a lemon, an orange, and a pine-apple; the orange cost twice as much as the lemon, and the pine-apple three times as much as both the lemon and the orange; the pine-apple cost 14 cents more than the orange; what was the cost of each? 9. James bought 1 lemon and 2 oranges; an orange cost twice as much as a lemon, and the difference between the cost of the oranges and the lemon was 6 cents; what was the cost of each? 10. Charles bought 2 lemons and 3 oranges; an orange cost twice as much as a lemon, and the difference between the cost of the lemons and the oranges was 8 cents; what was the cost of each? 11. A man bought a cow, a calf, and a horse; the cow cost twice as much as the calf, and the horse twice as much as the cow; the difference between the price of the horse and that of the calf was 30 dollars; what was the cost of each? 12. There are three numbers, of which the second is three times the first, and the third is twice as much as both the first and second, while the difference between the second and third is 10; what are the numbers? LESSON VII. 1. James-and John together have 11 cents, and John has 3 more than James; how many has each? If James has x cents, then John has x+3, and they both have x+x+3, or 2x+3 cents; hence, 2x+3 are equal to 11; hence, if 2x and 3 are equal to 11, 2x must be equal to 11 less 3, which is equal to 8; then, if 2x is equal to 8, one x, or x, must be equal to 4. 2. William and Daniel together have 9 apples, and Daniel has one more than William; how many has each? If x represents the apples William has, what will represent the apples Daniel has? What will represent the number they both have? 3. In a class containing 13 pupils, there are three more boys than girls; how many are there of each? 4. In a store-room containing 40 barrels, the number of those that are empty exceeds the number filled by 10; how many are there of each? 5. In a flock of fifty sheep, the number of those that are white exceeds the number that are black, by 30; how many are there of each kind? 6. Two men together can earn 60 dollars in a month, but one of them can earn 10 dollars more than the other; how many dollars can each earn? 7. The sum of two numbers is 25, and the larger exceeds the smaller by 15; what are the numbers? 8. Sarah and Jane bought a toy for 25 cents, of which Jane paid 5 cents more than Sarah; how much did each pay? 9. The difference between two numbers is 4, and their sum is 16; what are the numbers? If a represents the smaller number, what will represent the larger? 10. The difference between two numbers is 5, and their sum is 35; what are the numbers? LESSON VIII. 1. James and John together have 15 cents, and John has twice as many as James, and 3 more; how many has each? If x represents the number James has, then 2x+3 will represent the number John has, and x+2x+3, or 3x+3, what they both have. If 3x+3 is equal to 15, then 3x must be equal to 15 less 3, or 12; hence x is equal to 4, the number James has; then John has 11. 2. William bought a lemon and an orange for 7 cents; the orange cost twice as much as the lemon and 1 cent more; what was the cost of each? 3. There are two numbers whose sum is 35; the second is twice the first and 5 more; what are the numbers? 4. In an orchard containing apple-trees and cherry-trees, the number of apple-trees is three times that of the cherry-trees, and 7 more; the whole number of trees in the orchard is 51; how many are there of each kind? |