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15. How many pounds of meat, at 12 cents a pound, will cost as much as 9 pounds of cheese at 8 cents a pound?

16. If a stage run 45 miles in 5 hours, how far will it run in 7 hours? In 9 hours? In 12 hours?

17. What cost 9 quarts of milk, if 4 quarts cost 24 cents? 18. How many bags will be required to hold 108 bushels of wheat, if 4 bags hold 9 bushels?

19. If 5 barrels of flour are worth $60, how many cords of wood at $4 a cord will pay for 3 barrels?

20. If 12 yards of cloth cost $40, for how much must it be sold a yard to gain $20?

21. 1f 8 yards of cloth cost $35, for how much a yard must it be sold to gain $13?

22. A man received $50 for 5 barrels of pears, and paid all but $14 for 4 chairs. What did each chair cost?

23. What is the value of 5 tons of hay, at the rate of $48 for 4 tons?

24. If 6 bushels of wheat are worth $12, how many bushels of wheat must be given for 9 tons of hay, worth $10 a ton? 25. If 15 days' work will pay for 10 cords of wood, at $3 a cord, what is the price of 1 day's labor?

26. If a man receive 16 pounds of sugar in exchange for 20 pounds of cheese at 8 cents a pound, what is the price of the sugar per pound?

27. A newsboy sold 24 papers at 4 cents each, and thereby gained 48 cents. At what rate did he buy the papers?

28. If a woman pay 60 cents for some lemons, at the rate of 10 cents for 6, and sell them at the rate of 9 for 20 cents, how many cents will she gain?

29. If a man earn $5 while a boy earns $2, how many dollars will the man earn while the boy is earning $18 ?

30. A tailor bought 11 yards of one kind of cloth for $55, and 9 yards of another kind for $36. What was the difference

in the price per yard?

68. Two or more operations, to be performed in the same example, are often indicated by the use of signs. Thus,

1. To the product of 6 and 10 add 4, and divide the sum by 8.

Written,

10 × 6+4

8

= 8, for 10×6+4 = 64, and 64÷8 = 8.

Copy, read, and give the value of the following expressions:

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The operations of multiplication and division, indicated by signs, must be performed before those of addition and subtraction, unless otherwise indicated.

The distinction between terms and factors must always be observed. Terms may be added or subtracted, but factors never. To illustrate, tak · examples 2 and 3. Each expression is composed of two terms, one a single number, the other the product of two numbers. In Ex. 2, the term 8 is not to be subtracted from the factor 9, but from the product of 8 and 9, or 72, which is the other term. In Ex. 3, 6 is not to be added to 12, but to the product of 12 and 7, or 84.

If the same are written thus, 8×9-8, and (6+12) × 7, the addition and subtraction must be performed before multiplication, and the expressions will contain but one term each, composed of two factors, and equivalent to 8 × 1 or 8, and 18 × 7 or 126.

X

This law or principle becomes more obvious if we substitute the letters a, b, and c for the figures, and express algebraically thus: a ×b-c, and a + bxc, or ab-c, and a + bc. We can no more subtract 8 from 9, or add 6 to 12 in the above examples, than we can subtract c from b, or add a to b in the algebraic expressions.

The law is the same when the sign of division is used.

In Ex. 8, the dividend is composed of two terms, 8 × 12 or 96, and 27+3 or 9, and is equivalent to 96-9. If expressed thus, (8 × 12-27)÷3, then 27 would be subtracted from the product of 8 and 12, and the remainder divided by 3.

By observing these few hints, and also, that all the numbers included by a parenthesis or a vinculum must be resolved into one number before they can be used with any other, much of the difficulty experienced by the pupil in the use of signs will be obviated.

Complete the following equations:

9. 8x0+6x4÷8 = ? 10. 10 × 12-5x6÷6? 11. 9 x 11-54÷6+20 = ? 12. 10+4 7+20= ? 13. (8x9-12)÷ 5 = ? 14. 40-8÷4×7 = ? 15. (25+10)+7+9 = ?

16. 72÷12+50—16 = ?

17. 48+12-10÷? = 5. 18. 1089-4x? = 72. 19. (14+7÷3) × ? = 84. 20. (14412-3) x 11 = ? 21. 24+20-480÷÷÷? 22. 567x8 = 70-?

69. When the divisor is not greater than 12. 1. Divide 952 by 7.

EXPLANATION.-Write the divisor at the left of the dividend, with a line between them.

Beginning with the left-hand or highest order of the dividend, proceed thus:

7 is contained in 9 hundreds, 1 hundred times, with a remainder. Write the 1 hundred at the right of the dividend for the first figure of the quotient.

Multiply the divisor 7 by the 1 hundred of the quotient, and write the product 7 hundreds under the hundreds of the dividend; subtract, and to the remainder, 2 hundred, annex the 5 tens of the dividend, making 25 tens.

Divisor. Dividend. Quotient. 7) 952 (136

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25

21

42

42

PROOF.

136

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952 (Prin. III).

7 is contained in 25 tens, 3 tens times, with a remainder. Writing the 3 tens in the quotient, multiply and subtract as before, and to the remainder annex the 2 units of the dividend, making 42 units.

7 is contained in 42 units, 6 times, which write in the quotient. Multiplying and subtracting, as before, nothing remains.

The solution of the preceding example may be abbreviated, as follows:

7) 952

136 Quotient.

EXPLANATION.-7 is contained in 9, once, and 2 remainder. Writing 1, the quotient figure, under the part of the dividend used, prefix the remainder 2 to the next lower order of the dividend, 5, making 25. 7 is contained in 25, 3 times and 4 remainder; prefix the remainder 4 to the next lower order 2, making 42. 7 is contained in 42, 6 times, and no remainder.

Observe, that the quotient figure is always of the same order as the lowest order of the dividend used.

The principles in the two preceding operations are the same, the difference being, that in the first all the work is written ; while in the second only the quotient is written, the other steps being performed mentally. The first is usually termed long division, the second, short division. The pupil should be taught always to use the latter, when the divisor does not exceed 12.

In like manner, by short division find the quotient, and prove the following:

(2.)

(3.)

6) 34806 tons.

8) 206752 days.

(4.) 7) 44058 rods.

5. Divide 83762 by 7; 79884 by 6; 3263 by 8.

EXPLANATION.-Since 8 is not contained

8) 3263

4073 Quotient.

in 3 thousands, unite the 3 thousands and 2 hundreds, making 32 hundreds. 8 is contained in 32 hundreds, 4 hundreds times, which write in the hundreds' place in the quotient.

Next, 8 is not contained in 6 tens, so write a cipher in tens' place in the quotient, and unite the 6 tens and 3 units. 8 is contained in 63 units, 7 times and 7 units remainder, which write over the divisor and annex as a part of the quotient. Hence the quotient is 4077.

PROOF.-407 × 8+7 = 3263 (PRIN. HI).

6. Divide 403076 by 8; 120050 by 7; by 9.
7. Divide 3761201 by 9; 897063 by 6; by 7.
8. Divide 72947 by 7; by 8; by 5; by 6; by 3.

9. Divide 213064 by 4; by 9; by 8; by 5; by 6.

10. How many tons of coal, at $7 a ton, can be bought for $87605 ?

11. If 36314 bushels of grain be put into 6 bins of equal size, how many bushels must each bin contain?

12. A gentleman divided his estate, worth $42641, equally among his wife and 5 children. How much did each receive? 13. If 75000 bushels of grain are put into 8 bins of equal size, how many bushels does each bin contain?

14. How many times are 8 cents contained in $48.56?

8 cts.) 4856 cts.

607 times.

Or,

$.08 ) $48.56

607 times.

EXPLANATION.-1. Eight cents may be written $.08 (35).

2. When the divisor and dividend are like numbers, the quotient is an abstract number (PRIN. I). Hence, 8 cents are contained in $48.56, 607 times.

3. When the divisor and dividend are both concrete numbers, they must be of the same name

Hence, if one is dollars and the other cents, or dollars and cents, before dividing change, so that both may be cents.

15. Find one of 8 equal parts of $48.56.

EXPLANATION.-When the divisor is an abstract number, the dividend and quotient are like numbers (PRIN. II).

Solve and prove,

8) $48.56 $6.07

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