Multiplication of Fractions by Fractions.

It has been shown in a preceding section, that, to multiply a whole number by a fraction, we multiply by the numerator and divide by the denominator; and also, that a fraction is multiplied, either by multiplying its numerator, or dividing its denominator, and divided by multiplying its denominator, or dividing its numerator.

If a bushel of corn be worth of a dollar, how much is of a bushel worth?

Were the cost of 2 bushels required, we should multiply the price by the quantity; but as the quantity is less than 1 bushel, we multiply by the part taken-(see p. 95.) The question then is, how much is of ?

If the numerators be multiplied together, and also the denominators, we have the answer-Thus, X == or, by cancelling. Operation. 21

32 Ans.

In the first example, the price of a bushel is divided by 2, the denominator of the dividing fraction, and then multiplied by the numerator of the dividing fraction. In the second example, the process of multiplication and division is performed at once by cancelling equals on each side of the line. This process is, in fact, the same as multiplying a whole number by a fraction. To multiply a fraction by a fraction we have this Rule. Multiply the numerators together for a new numerator, and their denominators for a new denominator.

1. A man owning ; how much? Operation.

53

24

EXAMPLES.

of a ship, sold of his share of

As the numerators of the fractions are to be multiplied together for a new numerator, or dividend, they are placed on the right of the line, and the denominators, which are to be Ans. multiplied for a new denominator, The numor divisor, are placed on the left of the line. bers are cancelled and multiplied as in preceding examples. 2. What is the product of X; of 7X3?

103

Answer,, 36

QUESTIONS. 1. How do you multiply a fraction by a fraction? 2. Why, in example 1st, are the numerators of the fractions placed on the right of the line, and the denominators on the left?

3. Multiply by - by §. 4. A boy having of a dollar gave did the toys cost him?

of it for toys, what Ans., of a dollar. of a yard cost?

5. At of a dollar per yard, what will

Answer, & of a dollar.

6. At of a dollar per pound, what will

of tea cost?

7. At of a dollar a pound, what will coffee cost?

8. At 24 dollars per bushel, what wheat cost?

of a pound of a dollar. of a pound of of a dollar. bushels of

Answer,

Answer,
will 6

Answer, $133.

dollars, what is

9. If a house lot be worth 100 the lot worth?

10. If a flock of sheep be worth 75 of the flock worth?

Answer, $184.

Division of Fractions by Fractions.

of

1. If a bushel of corn cost of a dollar, how much may be bought for of a dollar?

If we divide of a dollar, the amount of money to be laid out, by of a dollar, the price of a bushel, we shall have the number of bushels which of a dollar will buy. By the Rule (see p. 96,) to divide a whole number by a fraction, we multiply the dividend, or whole number, by the denominator of the fraction, and divide by the numerator. The same rule is applicable in this case. Multiply the fraction to be divided by the denominator of the fraction by which you divide, and divide by the numerator, or invert the divisor, and proceed as in Multiplication. This process will be found the same as dividing a whole number by a fraction.

31

23

Operation.

According to the rule already given, the numerator of, the dividend, is placed on the right of the line, and the numerator of 3, the divisor, on the left-this is the same as inverting the divisor.

21 Ans.

2. If a bushel of potatoes cost of a dollar, bushels may be bought for of a dollar?

how many Ans., . .

3. If of a bushel of apples cost of a dollar, how much will 1 bushel cost?

Answer,

QUESTIONS. 1. How do you divide a fraction by a fraction?

4. If 9

16

of a yard of cloth cost of a dollar, how much is it per yard?

5. How many bushels of rye at § of a dollar may be bought for of a dollar?

6. If of a ton of hay cost cost per ton?

7. If 4 pounds of tea cost 34

Answer, 117 dolls. per bushel Ans., of a bu. of a dollar, what does it Answer, $9.

dollars, what is it per lb.? Answer, of a dollar.

8. If 13 of a dollar buy 1 pound of tea, how much will

31 dollars buy?

Answer, 4 pounds.

9. At 1 dollars per yard, how much carpeting can be bought for 15 dollars? Answer, 11 yards. 10. Divide 17 by 7-183 by . Ans., 2-561.

Multiplication and Division of Fractions.

MULTIPLICATION.

DIVISION.

1. A man owned of a 2 A man sold 2 of a house, sold of his share, house, which was

what part of the house he sell?

3. If a bushel of salt 19 of a dollar, what will

a bushel cost?

5. If a peck of coal 6 of a dollar, what will a peck cost?

of his did share. What part of the house did he own?

7. If I cord of wood cost 8. If of a cord of wood 22 of a dollar, how much will cost 319 dollars, how much of a cord cost?

is it per cord?

9. If 1 foot of hammered 10. If of a foot of hamstone cost of a dollar, what mered stone cost of a dolwill of a foot cost? lar, what will one foot cost?

The simple rule may now be repeated for solving any question which may arise in Multiplication and Division of Fractions by Whole numbers-Multiplication and Division of Whole Numbers by Fractions-Multiplication and Division of Fractions by Fractions.

RULE.

Place all those numbers which are to be multiplied together for a numerator, or dividend, on the right of the perpendicular line, and those numbers which are to be multi

plied together for a denominator, or divisor, on the left of the line, and proceed to cancel as before directed.

PROMISCUOUS EXAMPLES.

1. A man owning of 3 of 3 of 4 of 7 of a ship, sold of his share. What part of his share did he

of

sell?

of

Thus-21

32

54

87

73

38

32

152=Ans.

Answer,

Fractions connected by the word of are called compound fractions. They are reduced to simple fractions, by multiplying all the numerators together for a new numerator, and all the denominators for a new denominator. By cancelling, the process of multiplying and reducing the fraction, is performed at once.

2. Reduce of 2 of § of 3 of 1⁄2 of to a simple frac

tion.

3. A man owning of

of of of his share.

sell?

Answer, 2.
of of of a factory, sold
What part of the factory did he
Answer,

4. What simple fraction is equivalent to 3

of of of of 9, of of 18, of of 2?
용
18

multiply by of 3 of 3 of 2, divide by

of 6.

12. A man owns of a farm, sells his half, what part of the farm does he sell?

13. Multiply 12 by of 3, divide by 4 of 1, multiply by of 6, divide by 4 of 14, multiply by

of 27.

Addition of Fractions.

1. What is the amount of +3. If we add the nume

QUESTIONS. 1. What is the rule for the addition of fractions? 2. What is the first method given for reducing fractions of different denominators to frac tions having a common denominator?

rators, and under their sum write the denominator, we have , the answer. We have, then, this RULE for adding fractions. Add together their numerators, and, under the sum, write the denominator.

2. What is the amount of

3. A man gave away at one

+0+1+10+10?
Answer, 18-1 8

time

of a bushel of corn,

at another, at another §, at another, how many bushels did he give away in all? Answer, 2021.

4. What is the amount of and ? We have before seen, that, to multiply or divide the terms of a fraction by any one figure, does not alter its value; therefore, if we multiply both the numerator and denominator of by 2, we have, which may be added to 1+1=§.

1

5. What is the amount of +2+24? If we multiply the terms of the fraction have, and if we multiply the terms of

by 2, we

by 4, we

have We can now add the fractions, 24+2 4+24 =24, Answer.

6. What is the amount of, 12, 24 4 8 ? If we divide the terms of by 2, we have , and the terms of by 4, we have, which when added, +&+3=§ Answer.

24

When fractions can be reduced to a common denominator by multiplying or dividing the terms of one or more of the fractions, the preceding mode is often convenient, but, the following is the more general RULE.

Multiply all the denominators together for a new denominator, and each numerator by every denominator, except its own, for a new numerator.

1. Reduce, and to fractions of equal value, having a common denominator.

Operation.

denominators-3×4×5=60 com. denominator.
1st numerator-2×4×5-40 new numerator.

The new fractions are 48, 15, 24. In this example, we first multiply all the denominators together, and obtain 60 for a new denominator. It is now necessary that each numerator should be multiplied into the same numbers by which its denominator has been multiplied, that the value of the fraction may be retained. Taking the first fraction ; the denominator 3 has been multiplied into 4 and 5, the QUESTIONS. 3. What is the general rule for finding a common denominator?