MULTIPLICATION OF VULGAR FRACTIONS.

TO MULTIPLY A FRACTION BY A WHOLE NUMBER.

Art. 95. To multiply a fraction by a whole number is to repeat the size or value of the fraction as many times as there are units in the multiplier or whole number.

The size or value of every fraction depends upon the proportion which its numerator bears to its denominator. The denominator of a fraction not only expresses the number of parts into which a unit or integer is supposed to be divided, but it also indicates the size of those parts. As the numerator indicates the number of parts expressed by the fraction, if the numerator be made any number of times greater, while the denominator remains the same, the fraction is made the same number of times greater.

If the denominator be made any number of times smaller, while the numerator remains the same, the fraction is made the same number of times greater.

Let it be required to multiply by 3. Multiplying the numerator, 4 fifteenths, by 3, the product is 12 fifteenths, which we write over the denominator, thus: X3 = 13 = $⋅ Dividing the denominator, 15 fifteenths, by 3, the quotient is 5 fifths, which we write under the numerator, thus: ✩÷3 , as above.

From the above remarks and illustration we deduce the following rules for multiplying a fraction by a whole number. RULE I. Multiply the numerator by the whole number, and write the product over the denominator.

RULE II. Divide the denominator by the whole number, and write the quotient under the numerator.

If the given fraction be compound or complex, reduce it to a simple fraction. Mixed numbers may be reduced to improper fractions, and then multiplied, or the whole number and fraction may be multiplied separately.

1. If of a yard of cloth will make 1 vest, how many yards will make 4 vests?

2. If 1 pound of tea be worth of a dollar, how many dollars are 6 pounds worth?

3. If 1 yard of silk be worth 4. Suppose of a yard of of a dollar, what are 8 yards silk will make a bonnet, how worth? much will make 9 bonnets ? 5. What will be the product of of multiplied by 12?

Ans. 7.

6. What will be the product of 87 multiplied by 32?

Ans. 18+.

7. What will be the product of 12517 multiplied by 25?

Ans. 3142.

TO MULTIPLY A WHOLE NUMBER BY A FRACTION.

Art. 96. To multiply a whole number by a fraction is to find such a part or parts of the whole number, for a product, as the fraction indicates.

To multiply any whole number by, we find one half of the number for the product; to multiply any whole number by, we find three fourths of the number for the product.

Suppose we wish to multiply 24 by §. If we multiply 24 by the numerator, 5, the product is 120; if we divide 120 by the denominator, 8, the quotient is 15, which is 5 eighths of 24. Again, if we divide 24 by the denominator, 8, the quotient is 3; if we multiply 3 by the numerator, 5, the product is 15, which is 5 eighths of 24.

The reason why these two operations produce the same result is, that 1 eighth of 5 times any given number is equal to 5 times 1 eighth of the same number. Hence we derive the following rules for multiplying a whole number by a fraction.

RULE I. Multiply the whole number by the numerator of the fraction, and divide the product by the denominator.

RULE II. Divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator.

1. If a quire of paper is worth 24 cents, what is of a quire worth?

2. If linen is worth 40 cents a yard, what is § of a yard worth?

4. If 60 be multiplied by what will be the product? 6. If an acre of land is worth 30 dollars, what is the value of of an acre?

3. If you multiply 50 by , what will be the product?, 5. If a yard of silk is worth 56 cents, what will be the value of of a yard?

7. If 63 be multiplied by , what will be the product?

8. What will be the product of 144 multiplied by?

9. What will be the product if 1728 be multiplied by +? Ans. 612.

10. If a ton of hay is worth 16 dollars, what is of of a ton worth? Ans. 9 dollars.

of

11. What will be the product if 3456 be multiplied by

47

Ans. 2527.

TO MULTIPLY A FRACTION BY A FRACTION.

Art. 97. To multiply a fraction by a fraction is to find such a part or parts of any given fraction as is indicated by the fractional multiplier.

Let it be required to multiply by 2.

X4=

One fourth of is, which is obtained by multiplying the denominator of the multiplicand by the denominator of the multiplier; thus, x4-2. Three fourths of is 3 times, or 18, which is obtained by multiplying the numerator of the multiplicand by the numerator of the multiplier; thus, 3-8=3.

From the preceding illustration we derive the following rule for multiplying a fraction by a fraction.

RULE. Reduce compound and complex fractions to simple fractions, and mixed numbers to improper fractions.

Multiply the numerator of the multiplicand by the numerator of the multiplier for the numerator of the product, and the denominator of the multiplicand by the denominator of the multiplier for the denominator of the product.

DIVISION OF VULGAR FRACTIONS.

TO DIVIDE A FRACTION BY A WHOLE NUMBER.

Art. 98. To divide a fraction by a whole number is to find such a part of the fraction as is indicated by the whole number, or it is to find what part of one time the fraction will contain the whole number.

The operations in dividing a fraction by a whole number must be the reverse of those in multiplying a fraction by a whole number; hence it follows that as many times as the numerator is made smaller, while the denominator remains the same, so many times is the fraction made smaller; also, that as many times as the denominator is made greater, while the numerator remains the same, so many times is the fraction made smaller, because the number of parts in the denominator having been made any number of times greater, each of the parts is the same number of times smaller.

Let it be required to divide the fraction by the whole number 3; that is, to find one third of g, or to find what part of 1 time the fraction will contain the whole number 3.

One third of is, which is obtained by dividing the numerator of the fraction by the whole number 3; thus, 3. The same result may be obtained by multiplying the denominator of the fraction by the whole number 3; thus, x3.

X3 =

From the above illustration we derive the following rules for dividing a fraction by a whole number.

RULE I. Divide the numerator by the whole number, and write the quotient over the denominator.

RULE II. Multiply the denominator by the whole number, and write the product under the numerator.

divided among 4 girls, what part of a dollar will each of them receive?

1. If of a bushel of 2. If a dollar be equally apples be equally divided among 3 boys, what part of a bushel will each boy receive? 3. If be divided by 5, what will be the quotient?

5. If of a yard of silk be cut into 7 pieces of equal length, what will be the length of each piece?

7. What part of 1 time is 9 contained in ?

4. If be divided by 6, what will be the quotient?

6. If of a dollar be equally divided among 8 boys, what part of a dollar will each of them receive?

8. What part of 1 time is 11 contained in

9. If you have of a ton of hay, and distribute it equally among 75 horses, what part of a ton will each horse receive? Ans. 1 of a ton. 10. If be divided by 25, what will be the quotient?

Ans. 375. 11. If of a cwt. of sugar be equally divided among 64 1 soldiers, what part of a cwt. will each receive?

Ans. of a cwt. 12. What will be the quotient if the fraction by 19?

13. What part of 1 time will the fraction whole number 144?

13 be divided Ans.

contain the Ans. 1728

14. If of be divided by 12, what will be the quotient?

15. If 5 yards of cloth be worth is the value of 1 yard?

16. If 15 yards of silk be worth value of 1 yard?

Ans.

of of a dollar, what Ans. of a dollar. 127 dollars, what is the Ans. 198 of a dollar.

TO DIVIDE A WHOLE NUMBER BY A FRACTION.

Art. 99. To divide a whole number by a fraction is to find the number of times, or part of time, a fraction is contained in a whole number.

It is obvious that the operations in dividing a whole number by a fraction must be the reverse of those in multiplying a whole number by a fraction.

Let it be required to divide the whole number 15 by the fraction §.

The fraction being only 5 eighths of a unit, it is plain that § is contained as many times in any whole number as there are units in that number. If we multiply the whole number 15 by the denominator 8, the product will be eighths; if we divide this product by the numerator, 5 eighths, the quotient will be the number of times that is contained in 15; thus, 15 × 8—120, and 120÷5—24.

Again, if we divide 15 by the numerator 5, and then multiply the quotient by the denominator 8, the product will be the number of times that is contained in 15; thus, 15÷5 3, and 3 × 8=24.

=

From these illustrations we deduce the following rules for dividing a whole number by a fraction.