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can be seen that the answer is obtained by multiplying the denominator of by the divisor 4. This is a method which can always be pursued in dividing any fraction by a whole number, viz: "multiply the denominator by the di visor."

But there is another method which is sometimes more convenient.

Let be divided by 4.

Now the quotient of 8 units divided by 4, is 2 units. Of course the quotient of 8 sixteenths divided by 4, is 2 sixteenths. In this case we have divided the numerator by the divisor 4. This can be done in all cases where the numerator can be divided without remainder.

But when a remainder would be left, it is best to divide, by multiplying the denominator. The answer is of the same value either way, though the name is different.

For example; in dividing by 2, we are to find how many times 2 is contained in . Divide by multiplying the denominator by 2, and we find that it is contained not once, but of once. By dividing the numerator by 2, we find also that it is contained not once, but & of once. Now and is the same value, by a different name. For if a thing is divided into eighteen parts, and we take four of them, we have the same value as if it were divided into nine parts and we took two of them.

Divide the following by both methods, and explain them as above.

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RULE FOR DIVISION WHERE THE DIVIDEND IS A
FRACTION.

Divide the numerator of the Fraction by the Divisor, or,

What other method is there?

(if this would leave a remainder,) multiply the denominator by the Divisor.

EXAMPLES FOR THE SLATE.
In the following examples, divide the numerator by the
divisor.
Divide

14 by
4 Divide

by 11
5

16 8

7 10

75 12

66

132 520 4.68 366 56 369 150 .359

15 25 80 32 64 50 96 144 250

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In the following examples multiply the denominator by the divisor.

66

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10

66

91

80

Divide by 4 | Divide ef by 5
6

7
8

70 9 12

i

12 24

61 In the following examples divide the numerator by the divisor. Divide a by 3 | Divide them by 6

8 8

9 66 7

6 4

9

36 460

6 12

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42 32 127 6_4 912 4.9 320 28 54

92 42

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66

EXAMPLES FOR MENTAL EXERCISES. 1. If you have of an orange, and wish to divide it equally between 2 children, what part do you give each? 2. If you have of a load of hay, and divide it equally

а among 6 horses, how much do you give each?

3. If you have ii of a yard of muslin, and divide it into 3 equal parts, what part of a yard is each part ?

4. If you have li of an ounce of musk, and divide it into 12 equal portions, what part of an ounce is each por. tion ?

What is the rule for Division when the dividend only is a fraction ?

5. If you divide of a dollar into 4 equal parts, what part of a dollar will each part be?

6. If a man owns 1 of a cargo, and divides it equally among 4 sons, how much does he give each ?

DIVISION OF ONE FRACTION BY ANOTHER.

When one fraction is to be divided by another, the same principle is employed, as when whole numbers are divided by a fraction.

For example, if the whole number 12 is to be divided by, we first multiply by the denominator 4, to find how often one fourth is contained in 12, and then divide by 3, to find how often three fourths are contained in it.

In like manner, if we wish to find how many.times, or parts of a time, is contained in, we first find how often one fourth is contained in it, by reasoning thus:

One unit would be contained in, two twelfths of one time.

One fourth would be contained four times as often, or of one time.

We thus find how often one fourth is contained in 2, by multiplying it by 4, thus:

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But three fourths would be contained only one third as often, and we find a third of by multiplying its denomi nator by 3. For when we wish to divide a fraction by 3, we multiply its denominator, and thus make the parts rep. resented by the denominator, three times smaller, thus:

÷ 3 = .

Here the twelfths are changed to thirty-sixths; and a - thirty-sixth is a third of one twelfth.

It will be found by examining the foregoing process, that in dividing one fraction by another, the fraction which is the dividend has its numerator multiplied by the denom

Is any different principle employed in dividing a fraction by a whole number? Explain the process. In the above example why was the numerator of the dividend multiplied by the denominator of the divisor?

inator of the divisor, and its denominator multiplied by the numerator of the divisor.

Let another example be taken and observe thus.
Let be divided by 4.

if divided by one unit would contain it not once but ៖ But if divided by one sixth it would contain it 6 times as often or 6 times, which is 12.

of once.

Here the numerator of the dividend (3) has been multiplied by the denominator of the divisor (#), and we have thus found how often one sixth is contained.

Four sixths would be contained only one fourth as often, and we therefore divide 12 by 4 by multiplying its denomi nator and the answer is 1, and here the denominator of the dividend (4) has been multiplied by the numerator of the divisor (4).

We therefore multiplied the numerator of the dividend by the denominator of the divisor to find how often one sixth was contained, and multiplied the denominator of the dividend by the numerator of the divisor to find how often four sixths were contained.

Let the following be performed and explained as above. Divide by | Divide

by

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13 66 6

20 1/3

This process corresponds with that used in dividing a whole number by a fraction.

For if we divide 12 by we first multiply it by 4 to find how many one fourths there are in 12, and then divide the answer by 3 to find how many three fourths there are. So in dividing by we first multiply it by 4 to find how many times one fourth is contained thus (8), and then divide it by 3 to find how many times three fourths are contained thus, (,).

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Why was the denominator of the dividend multiplied by the numerator of the divisor? Explain how this process corresponds with that used in dividing whole numbers.

We invert a fraction when we exchange the places of the numerator and the denominator.

Thus inverted is †, and 3 inverted is and 3 in. verted is , &c.

Now it appears, as above, that if we wish to divide by we are to multiply its numerator (3) by the denominator (6) and its denominator (4) by the numerator (2). This is more easily done, if we invert the divisor, thus §.

When the divisor is thus inverted we can multiply the numerators together for a new numerator and the denomi nators for a new denominator and the process is the same. Thus let us divide & by 3.

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Inverting the divisor the two fractions would stand together thus. We now multiply the numerators and denominators together and the answer is 13, and it is the same process, as if we had not inverted the divisor, but multiplied the numerator of the dividend by the denomi. nator of the divisor and its denominator by the numerator of the divisor.

This method therefore is given as the easiest rule, but it must be remembered that in this process we always mul. tiply the dividend by the denominator of the divisor and di vide it by the numerator, as we do in case of whole numbers.

COMMON RULE FOR DIVIDING ONE FRACTION BY ANOTHER.

Invert the Divisor, and then multiply the numerators and denominators together.

EXAMPLES FOR THE SLATE.

Divide by 2.

Invert the divisor and the fractions stand thus 24.
Multiply them together, and the answer is 1.

How is a fraction inverted? What is the common rule for dividing one fraction by another?

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