84. 1. How many feet are 12 feet less 6 feet, less 6 feet? 12 feet 4 feet 4 feet 4 feet? 12 feet

3 feet

3

2. How many times can 6 feet be taken from 12 feet? 4 feet from 12 feet? 3 feet from 12 feet?

3. How many 6's in 12?

4. How many times can 5

How many 4's? 3's? 2's ?

pounds of tea be taken from a

box containing 30 pounds? How many 5's in 30 ?

5. Into how many equal parts are 30 pounds separated? What is the size of each part? 5 pounds is contained in 30 pounds how many times?

6. Thirty is how many times 5? 6? 10? 3?

7. Distribute 30 pounds of tea equally among 6 families; how many pounds will each receive?

8. Can we say 6 families are contained in 30 pounds of tea 5 times? Why not?

9. Can we subtract 6 families from 30 pounds of tea 5 times? Why not?

10. How then shall we find one of 6 equal parts of 30 pounds of tea?

One of the 6 equal parts of 30 pounds of tea is 5 pounds, since six 5's or 6 times 5 make 30.

11. What is one of 6 equal parts of 30 pounds? Of 42 feet?

12. What is one of 3 equal parts of 30 pounds? Of 27 acres ?

13. When we say, 7 cents is contained in 63 cents, is the result denominate or abstract?

14. When we say, one of 7 equal parts of 63 cents, is the result denominate or abstract ?

85. The process of finding how many times the same number may be taken from a given number, or of finding one of the equal parts into which a number may be divided, is called Division, and the result obtained is called the Quotient.

Thus, if 24 marbles are divided equally among a number of boys, giving each boy 6 marbles, how many boys are there?

Here the whole number and one of the equal parts are given, to find the number of equal parts.

Again, if 24 marbles are divided equally among 4 boys, how many marbles will each boy receive?

Here the whole number and the number of equal parts are given, to find the size or value of one of the equal parts.

86. The number to be divided into equal parts of a known size, or which is to be separated into a given number of equal parts, is called the Dividend.

Thus, in the above examples, 24 marbles is the dividend.

87. One of the equal parts into which the dividend is to be divided, or the number of equal parts into which the dividend is to be divided, is called the Divisor.

Thus, in the first example, 6 marbles is the divisor; and in the second, 4 is the divisor.

88. Division may be regarded as the reverse of multiplication, since in multiplication both factors are given to find the product; while in division, one factor and the product (answering to the divisor and dividend) are given to find the other factor, called the Quotient.

Thus, 9 × 436, the factor 9 being taken 4 times gives the product 36; hence, there are four 9's in 36, or 9 is contained in 36, 4 times.

89. If anything remains after dividing the dividend, it is called the Remainder, and it must always be less than the divisor.

When there is no remainder, the division is said to be exact, and the dividend is the product of the divisor and quotient, and each expresses one of the equal parts into which the dividend may be separated.

Thus, 8 is contained in 32, 4 times; the divisor 8 expresses one of the four equal parts of 32, as also the quotient 4 expresses one of the eight equal parts of 32.

by.

90. The Sign of Division is. It is read, divided

When placed between two numbers it shows that the one on the left is to be divided by the one on the right.

Thus, 63÷7 = 9 is read, 63 divided by 7 equals 9, and signifies that there are nine 7's in 63, or 7 is contained in 63, 9 times, since 9 times 7 is 63.

91. Division is also indicated by writing the dividend above and the divisor below a short horizontal line.

92. 1. First, to find at sight how many times any number from 2 to 12 inclusive, used as a factor, is contained in the several products taken from the Multiplication Table, the result being the other factor.

Thus, begin with the products which have 2 as one factor, taken in any order, and find how many 2's in 6, 8, 12, 18, etc.

Written, 236 2)8 2) 12 2) 18 2) 20 2) 24, etc.

Write under the products the number of 2's each contains, which is the other factor, and say, 2's in 6, three; 2's in 8, four; 2's in 12, six, etc.

The number at the left of the curved line is the divisor, the one at the right the dividend, and the result obtained is the quotient.

Thus, 2)6 may be expressed, 6÷2.

2) 16 is the same as 16÷2 =8, read, 16 divided by 2 equals 8. 8

Erase and rewrite the quotients, until they can be promptly given from memory.

2. Practice in the same manner upon the products which have 3, 4, 5, 6, to 12 inclusive, as factors taken from the table.

3. Divide by 2 orally, from 2 in 2, to 2 in 24.

Thus, 2 in 2, once; 2 in 4, twice; 2 in 6, 3 times; 2 in 8, 4 times, etc.

Also the following:

4. 3 in 3, to 3 in 36. 5. 4 in 4, to 4 in 48. 6. 5 in 5, to 5 in 60. 7. 6 in 6, to 6 in 72. 8. 7 in 7, to 7 in 84.

9.

96.

9 in 108.

8 in 8, to 8 in 10. 9 in 9, to 11. 10 in 10, to 10 in 120. 12. 11 in 11, to 11 in 132. 13. 12 in 12, to 12 in 144.

14. Reverse the above; thus, 2 in 24, 12 times; 2 in 22, 11 times; 2 in 20, 10 times, etc.

15. Then combine; thus, 3 in 3, once; 3 in 6, twice; 2 in 6, 3 times; 3 in 12, 4 times; 4 in 12, 3 times, etc.

Express by the proper signs the following and their an

swers; thus, How many 3's in 12? 12 ÷ 3 = 4, or

12

3

=4

16. How many 6's in 24? In 48? In 54? In 60? In 36? 17. How many 8's in 40? In 64? In 72? In 96? In 80? 18. How many 9's in 36? In 45? In 90? In 63? In 81? 19. How many 10's in 40? In 70? In 50? In 100? 20. How many 12's in 48? 21. How many times 6 men are 48 men? 5 feet are 45

In 60? In 84?

feet? 7 days are 84 days? 8 cents are 72 cents?

In 96?

93. This last form of indicating division is often used to simplify two or more operations to be performed in the same example.

Thus,

Multiply 12 by 6 and divide the product by 8 may be ex

= 9, since 12 × 672, and 72÷÷8= 9.

To 10 times 9 add 6 and divide the sum by 12.

Express by signs each of the following:

1. Divide times 9 plus 3, by 11.

2. Divide the difference of 36 and 12, by 8.

3. Divide the product of 12 and 5, by 10.

4. Divide the sum of 4 times 8 and 5 times 8, by 6. 5. From the sum of 24 and 16 subtract 12, and divide the remainder by 7.

What is the value of the following expressions?

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

(11.)

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