The first part of this process consists in clearing the numerators and denominators of fractions. This is done by multiplying all the terms of the mixed fraction by the denominators. It is plain that the value of the expression is not changed in doing this, because all the terms are multiplied by the same number.

7. Reduce of of to a simple fraction.

51 81

8. Reduce of of to a simple fraction.

V

Ans..

Ans. Toz

TO MULTIPLY A WHOLE NUMBER BY A FRACTION.

Multiply the whole number by the numerator, and divide by the denominator; or divide by the denominator, and multiply by the numerator.

The last process may be analyzed thus: one fourth of 87 is 21; and of 87 are 3 times 21, or 651.

The first process amounts to this: one fourth of 261 is equal to of 87. As of 1 are equal to 1×3÷÷4=; or 14X3; so 87X3÷4=651; or 87÷4X3-651.

2. Multiply 8756 by f

3. Multiply 45 by

4. Multiply 75 by .

5. Multiply 84 by g

Ans. 7164.

Ans. 31.

Ans. 324.

Ans. 58

VI.

TO DIVIDE A WHOLE NUMBER BY A FRACTION.

Multiply the whole number by the denominator of the frac tion, and divide the product by the numerator.

In example first, I wish to ascertain how many times is contained in 6. It is evident that it is contained in 6 as many times as there are halves in 6. Therefore, 6X2=12, Ans.

In the second example, the divisor is . Had the divisor been, then 8×4, or 32, would have been the quotient. But as the divisor is, 32 are three times as large as the true quotient. Therefore, divide 32 by 3, and the quotient, 10%, is the answer.

Again. is contained in 8 as many times as there are fourths in 8. And are contained in 8 as many times as there are in 8.

In 8 there are 32 fourths; as are three times as much as, therefore, there will be as many three fourths as there are one fourths in 8.

Therefore, 8X4÷3, show how many times are contained in 8.

TO MULTIPLY A FRACTION BY A WHOLE NUMBER.

Multiply the numerator by the whole number, or divide the denominator by the whole number, when this can be done without a remainder.

1. Multiply by 12.

144÷÷12=12, denominator.

Or thus:

1=1, Ans.

13×12=156, numerator.
1=1, Ans.

In the first example, it is plain that are multiplied by dividing the denominator by 12, because the number of parts into which the unit is divided, is diminished; and therefore their magnitude is increased. Or if the denominator be considered a divisor, then, if 144 be divided by 12, the value of the fraction is increased in the same proportion.

Again; 12 times are 1; therefore, multiplying the numerator by 12, increases the value of the fraction twelve times.

2. Multiply by 7. 3. Multiply

by 7.

Ans. 56-63.
Ans. 333.

4. If one ton of hay cost $10, what will 13 tons cost?

5. If a man pays $2 for one week's board, how much must he pay for 11 weeks' board?

Ans. 271.

6. If a man builds 3 rods of wall in one day, how many rods of wall can he build in 7 weeks, allowing that he does not work on the Sabbath? Ans. 147.

7. If a man pays of a mill for the use of one dollar one day, how much must he pay for the use of $847 one day? Ans.,1414.

VIII.

TO DIVIDE A FRACTION BY A WHOLE NUMBER.

Multiply the denominator by the whole number, or divide the numerator by the whole number, when it can be done without a remainder.

1. Divide by 8.

2. Divide by 7.

Ans.. Ans. T

In each of these examples, the division is performed by dividing the numerator by the whole number. That is 8 times less than 8, is plain; because the denomination of parts in the fraction remains unchanged, while the number of parts is diminished 8 times.

3. Divide by 5.

Ans. +5.

In this example, the division is performed by multiplying the denominator by the whole number. That this process divides the fraction is evident, because the parts of the fraction, are 5 times less than the parts of the fraction }, while the number of parts is the same in both.

4. Divide by 6.

Ans.

5. If 13 tons of hay cost $144, what is the price of

one ton?

Ans. $11.

12. Divide 889900112233445577667788885 by 1243576. Ans. 71559768943228 85246105608119

124215431904528

13. Divide 875625375887 by 125375.

Ans. 698444861874.

14. Divide 6251878 by 375875.

Ans. if

4675633

15. Divide 1875 by 892756.

16. Divide 178488235825838 by 1155.

IX.

TO MULTIPLY ONE FRACTION BY ANOTHER.

Multiply the numerators together for a new numerator, and the denominators for a new denominator.

1. Multiply by 2.

OPERATION.

X, Ans.

This process may be explained by referring to the definition of multiplication. Multiplying by a whole number consists in taking the multiplicand as many times as there are units in the multiplier. Multiplying by a fraction is taking a certain part of the multiplicand as many times as there are like parts of a unit in the multiplier. Multiplying by, is taking of the multiplicand once. One half of is found by multiplying the denominator by 2. are

of

are taken once.

Nineteen thirtieths of are taken once, in example 2; or one thirtieth of nineteen times. One thirtieth of & is found by multiplying the denominator 9 by 30, and writing the product under 8, thus, . Nineteen thirtieths are 19 times as much as one thirtieth; therefore by multiplying by 19, we have the true answer, 19=78.