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2x + 2 is equal to 10, 2x will be equal to 10 – 2, or 8; or x will be equal to 4. Hence we write, Let
x = the number of John's marbles. Then
x + 1 = the number after 1 is added, and
2(x + 1) = 2 x + 2 = 10. Hence
2x = 10 – 2,
VERIFICATION. 2(4 + 1) = 2 x 5 = 10. 4. Write x + 2 multiplied by 3.
Ans. 3(x + 2). What is the product ?
Ans. 3x + 6. 5. Write x + 4 multiplied by 5.
Ans. 5(x + 4). What is the product ?
Ans. 5x + 20. 6. Write x + 3 multiplied by 4. Ans. 4(x + 3). What is the product ?
Ans. 4x + 12. 7. Lucy has a certain number of books. Her father gives her two more, when twice her number is equal to 14. How many has she ?
8. Jane has a certain number of roses in blossom. Two more bloom, and then three times the number is equal to 15. How many were in blossom at first?
9. Jane has a certain number of handkerchiefs, and buys four more, when five times her number is equal to 45. How many had she at first?
10. John has one apple more than Charles, and three times John's added to what Charles has make 15. How many has each?
Let x denote Charles's apples. Then x + 1 will denote John's; and x + 1 multiplied by 3, added to x, or 3x + 3 + x, will be equal to 15, what they both had : hence 4x + 3 equals 15; and 4x equals 15 – 3, or 12; and x = 4. Write and verify.
11. James has two marbles more than William, and twice his marbles plus twice William's are equal to 16. How many has each?
12. Divide 20 into two such parts that one part shall exceed the other by 4.
13. A fruit-basket contains apples, pears, and peaches. There are 2 more pears than apples, and twice as many peaches as pears. There are 22 in all. How many of each sort?
14. What is the sum of x+3x+ 2(x+1)?
17. The sum of two numbers is 11, and the second is equal to twice the first plus 2. What are the numbers ?
18. John bought 3 apples, 3 lemons, and 3 oranges, for which he paid 27 cents. He paid one cent more for a lemon than for an apple, and 1 cent more for an orange than for a lemon. What did he pay for each?
19. Lucy, Mary, and Ann have 15 cents. Mary has 1 more than Lucy, and Ann twice as many as Mary. How many has each?
1. If x denote any number, and 1 be subtracted from it, what will denote the difference?
Ans. x — 1. If 2 be subtracted, what will denote the difference? If 3 be subtracted ? 4 ? etc.
2. John has a certain number of marbles. If 1 be taken away, twice the remainder will be equal to 12. How many has he?
Let x denote the number. Then x - 1 will denote the number after 1 is taken away; and twice this number, or 2(x - 1) = 2 x — 2, will be equal to 12. If 2x diminished by 2 is equal to 12, 2 x is equal to 12+ 2, or 14: therefore x equals 14 divided by 2, or 7. Hence we write, — Let
x = the number. Then X – 1 = the number which remained, and 2(x - 1) = 2 x – 2 = 12. Hence 2x = 12 + 2, or 14;
x = 14 = 7. VERIFICATION. 217 – 1) = 14 – 2= 12;
also 2(7 – 1) = 2x6 = 12. 3. Write 3 times x – 1.
Ans. 3 (2 — 1). What is the product equal to ?
Ans. 3x — 3. 4. Write 4 times x — 2.
Ans. 4(x - 2). What is the product equal to ?
Ans. 4x – 8. 5. Write 5 times x -- 5.
Ans. 5(x - 5). What is the product equal to ?
Ans. 5x – 25. 6. If x denotes a certain number, will x – 1 denote a greater, or less, number? How much less ? 7. If x – 1 is equal to 4, what will x be equal to ?
Ans. 4+1, or 5. 8. If x — 2 is equal to 6, what is x equal to ?
9. James and John together havė 20 oranges. John has 6 less than James. How many has each ?
10. A grocer sold 12 pounds of tea and coffee. If the tea be diminished by 3 pounds, and the remainder multiplied by 2, the product is the number of pounds of coffee. How many pounds of each?
11. Ann has a certain number of oranges. Jane has 1 less, and twice her number added to Ann's make 13. How many has each?
VERIFICATION. 5–4=1; and 2 x 4 +5 = 13.
12. Charles and John have 20 cents, and John has 6 less than Charles. How many has each ?
13. James has twice as many oranges as lemons in his basket, and if 5 be taken from the whole number, 19 will remain. How many had he of each ?
14. A basket contains apples, peaches, and pears, 29 in all. If 1 be taken from the number of apples, the remainder will denote the number of peaches, and twice that remainder will denote the number of pears. How many are there of each sort ?
15. If 2x – 5 equals 15, what is the value of x?
18. The sum of two numbers is 32, and the greater exceeds the less by 8. What are the numbers?
19. The sum of two numbers is 9. If the greater number be diminished by 5, and the remainder multiplied by 3, the product will be the less number. What are the numbers ?
20. There are three numbers such that 1 taken from the first will give the second, the second multiplied by 3 will give the third, and their sum is equal to 26. What are the numbers ?
21. John and Charles together have just 31 oranges. If 1 be taken from John's, and the remainder be multiplied by 5, the product will be equal to Charles's number. How many has each?
22. A basket is filled with apples, lemons, and oranges; in all, 26. The number of lemons exceed the number of apples by 2, and the number of oranges is double that of the lemons. How many are there of each ?
LESSON XI. 1. John has a certain number of apples, the half of which is equal to 10. How many has he ?
Let x denote the number of apples. Then x divided by 2 is equal to 10. If one half of x is equal to 10, twice one half of x, or x, is equal to twice 10, which is 20: hence x is equal to 20.
NOTE. — A similar analysis is applicable to any one of the fractional units. Let each question be solved according to the analysis.
2. John has a certain number of oranges, and one third of his number is 15. How many has he?
3. If one fifth of a number is 6, what is the number?
5. What number added to one half of itself will give a sum equal to 12?
Denote the number by x. Then æ plus one half of x equals 12. But x plus one half of x equals three halves of x: hence three halves of x equal 12. If three halves of x equal 12, one half of x equals one third of 12, or 4. If one half of x equals 4, x equals twice 4, or 8: therefore x equals 8. Hence we write,