4. In an army of 57068 men, 9503 are killed; 586 join the enemy; 4794 are prisoners; 1234 die of wounds; 850 are drowned how many return? Ans. 40101

5. On the first of the year a man is worth $123078; during the year he gains $8706; in January he spends $237, in February $301; in each of the remaining ten months, he spends as much as in the first two: how much had he at the end of the year? Ans. $125866.

6. In a building there are 72 rooms; in each room 4 windows, and in each window 24 lights: how many lights are there in the house?

Ans. 6912. 7. A merchant has 9 pieces of cloth, of 73 yards each: and 12 pieces, of 88 yards each: how many yards in all ? Ans. 1713 yards.

8. I spend 99 cents a day: how many cents will I spend in 40 years, of 365 days each? Ans. 1770615.

Ans.

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9. An Encyclopedia consists of 30 volumes; each volume has 774 pages of two columns cach; each column 67 lines; cach line 10 words; and every 10 words 47 letters: how many pages, lines, words, and letters, in the work? 30186 pages, 4044924 lines, 40449240 words, 190111428 letters. 10. The Bible has 31173 verses: in how many days can I read it, reading 86 verses a day? Ans. 3621 days. 11. I bought 28 horses for $1400; 3 died: for how much cach must I sell the rest, to incur no loss? Ans. $56. 12. How many times can I fill a 15 gallon cask, from 5 hogsheads of 63 gallons each? Ans. 21 times.

13. A certain dividend is 73900; the quotient 214; the remainder 70: what is the divisor?

Ans. 345.

14. Multiply the sum of 148 and 56 by their difference; divide the product by 23. Ans. 816.

15. How much cloth, at $6 a yard, will pay for 8 horses at $60 each, and 14 cows at $15 each? Ans. 115 yards.

16 A cistern of 360 gallons, has 2 pipes; one will fill it in 15 hours, and the other empty it in 20 hours. If both pipes are left open, how many hours will the cistern be in filling? Ans. 60 hours.

17. Two men paid $6000 for a farm; one man took 70 acres at $30 an acre, the other the remainder, at $25 an acre: how many acres in all? Ans. 226 acres.

SUGGESTION.-In the four following examples, obtain the required number by reversing the operations.

18. What is the number, from which, if 125 be subtracted, the remainder will be 222? Ans. 347. 19. What is the number, to which, if 135 be added, the sum will be 500?

20. Find a number, from which, if 65 and the remainder divided by 15, the be 45?

Ans. 365.. be subtracted, quotient will Ans. 740.

21. What is the number, to which if 15 be added, the sum multiplied by 9, and 11 taken from the product, the remainder will be 340?

Ans. 24.

22. If 98 be subtracted from the difference of two numbers, 27 will remain; 246 is the less number: what is the greater? Ans. 371

B

GENERAL PRINCIPLES O DIVISION.

ART. 57. The value of the quotient depends on the relative values of divisor and dividend. These may be changed by Multiplication and Division, thus:

1st. The Dividend may be multiplied, or the Divisor divided. 2d. The Dividend may be divided, or the Divisor multiplied. 3d. Both Dividend and Divisor may be multiplied, or both divided, at the same time.

ILLUSTRATIONS.-Let 24 be a dividend, and 6 the divisor; the quotient is 4: (246=4).

If the dividend (24), be multiplied by 2, the quotient will be multiplied by 2: for, 24X248; and 48÷68. which is the former quotient (4), multiplied by 2.

Now, if the divisor (6), be divided by 2, the quotient will we multiplied by 2; for, 6-2=3; and 2438, which is the former quotient (4), multiplied by 2. Hence,

PRINCIPLE 1.-If the dividend be multiplied, or the divisor be divided, the quotient will be multiplied.

ART. 58. Take the same Example, 24÷6=4.

If the dividend (24), be divided by 2, the quotient will be divided by 2: for, 242=12; and 12÷6=2, which is the former quotient (4), divided by 2.

And, if the divisor (6), be multiplied by 2, the quotient will be divided by 2; for, 6X2=12; and 24÷12=2, which is the former quotient (4), divided by 2. Hence,

PRINCIPLE II.—If the dividend be divided, or the divisor be multiplied, the quotient will be divided.

ART. 59. Take the same Example, 24÷6=4.

If the dividend (24), and divisor (6), be multiplied by 2, the quotient will not be changed; for, 24×2=48; and 6×2=12; 4812-4, the former quotient (4), unchanged.

And if the dividend (24), and divisor (6), be divided by 2, the quotient will not be changed; for, 24÷2=12; and 6÷2=3; 12÷3=4; the former quotient (4), unchanged. Hence,

PRINCIPLE III.-If both dividend and divisor be multiplied or divided by the same number, the quotient will not be changed.

ART. 60. If a number be multiplied, and the product divided by the same number, the quotient will be the original number.

For, 24X2=48; and 48÷2=24, the original number, on the principle, that if the product be divided by the multiplier, the quotient will be the multiplicand. Also,

If a number be divided, and the quotient multiplied by the same number, the product will be the original number.

For, 24212; and 12X2=24, the original number, on the principle, that if the quotient be multiplied by the divisor, the product will be the dividend.

Hence, the operations of multiplication and division by the same number, destroy (cancel) each other.

REVIEW.-57. On what does the value of the quotient depend? How may the divisor and dividend be changed? If the dividend be multiplied, what effect on the quotient? If the divisor be divided?

58. If the dividend be divided, what effect on the quotient? If the divisor be multiplied?

59. If both divisor and dividend be multiplied by the same number, what effect on the quotient? If both be divided by the same number?

Most TEACHERS defer this subject until after Factoring.

1. I bought 3 oranges at 10 cents each, and paid for them with pears at 3 cts. each: how many pears did I give?

SOLUTION. Ten cents multiplied by 3, give 30 cents, the cost of the oranges.

Then it will take as many pears, as 3 cents are contained times in 30 cents; that is, 30 3 10, the number of pears.

OPERATION.

10

3

3)30

Ans. 10 pears.

Here, 10 is multiplied by 3 and the product divided by 3; but a number is not changed by multiplying it, and then dividing the product by the same number, (Art. 60); hence, multiplying by 3, and then dividing by 3, may be omitted, and 10 taken as the result; hence,

ART. 61. When a number is to be multiplied and then divided by the same number, both operations may be omitted, and a line drawn across the common multiplier and divisor, as in the margin.

OPERATION.

10 × 3

=10.

REM. In the above example, 10 and 3 form the dividend, and 3 the divisor. In arranging the numbers, place the dividend above a horizontal line, and the divisor below it.

2. How many barrels of molasses at $13 a barrel, will pay for 13 barrels of flour at $4 a barrel ?

Ans. 4. 3. If I buy 41 cows at $11 each, and pay in horses at $41 each, how many horses are required? Ans. 11. 4. If I buy 10 lemons at 3 cents each, and pay in oranges at 5 cts. each, how many oranges will I give?

SOLUTION.-Ten times 3 cents are 30 cents, the cost of the lemons: 30 cts. divided by 5 cents, equal 6, the number of oranges.

OPERATION.

3

10

5)30

Ans. 6.

But, as 10 is a composite number, whose factors are 5 and 2, (5×2=10), indicate the operation as in the margin on the left. 5X2X3 Since 5, 2, and 3 are to be 5 multiplied together, and their product divided by 5, omit 5 both as multiplier and divisor (Art. 60), draw a line across it, and only multiply 2 by 3.

6.

5. Multiply 17 by 18, and divide the product by 6.

(6X3=18), then cross the factor 6, which is common to both mul tiplier and divisor: after which, multiply 17 by 3.

6. In 15 times 8, how many times 4? 7. In 24 times 4, how many times 8? 8. In 37 times 15, how many times 5? 9. Multiply 36 by 40, and divide the multiplied by 8.

Dividend, 36×40

Divisor, 30 X 8

Ans. 30.

Ans. 12.

Ans. 111.

product by 30

SOLUTION.-Indicate the operation to be performed, then resolve (separate) the numbers into factors. Now the quotient will not be changed (Art. 59) by dividing both divisor and dividend by the same number, which is done by erasing the same factors in both.

mon factor 6, divide each by 6, and write the quotients 6 and 5 as in the operation: cancel 36 and 30. Next, canceling the common factor 5 in both dividend and divisor, the result is 6, as before.

10. In 36 times 5, how many times 15? 11. In 14 times 9, how many times 6?

Ans. 12.

Ans. 21.

ART. 616. The process of shortening the operations of arithmetic, by omitting equal factors from the dividend.

REVIEW.-60. If a number be multiplied, and the product divided by the same number, what will be the quotient? On what principle? If a number be divided and the quotient multiplied by the same number, what will the product be? On what principle?