Multiplication of Fractions
151. By the definition of multiplication, multiplying by a fraction is taking a part of the multiplicand, as many times as there are like parts of an unit in the multiplier. (Art. 88.) Now the denominator of a fraction shows into what parts the integral unit is supposed to be divided ; and the numerator showahow many of those parts belong to the given fraction. In multiplying by a fraction, therefore, the multiplicand is to be divided into such parts, as are denoted by the denominator ; and then one of these parts is to be repeated, as many times, as is required by the numerator.
Suppose a is to be multiplied by 3/4.
A fourth part of a is a/4.
This taken 3 times is a/4 + a/4 + a/4 = 3a/4.(Art. 145 )
Again, suppose a/b is to be multiplied by 3/4.
One fourth of a/b is a/4b. (Art. 135.)
This taken 3 times is a/4b + a/4b + a/4b = 3a/4b, the product required.
In a similar manner, any fractional multiplicand may be divided into parts, by multiplying the denominator; and one of the parts may be repeated, by multiplying the numerator. We have then the following rule:
152. TO MULTIPLY FRACTIONS, MULTIPLY THE NUMERATORS TOGETHER, FOR A NEW NUMERATOR, AND THE DENOMINATORS TOGETHER, FOR A NEW DENOMINATOR.
Ex. 1. Multiply 3b/c into; d/2m. Product 3bd/2cm.
2. Multiply (a + d)/y into 4h/(m - 2). Product (4ah + 4dh)/(my - 2y).
153. The method of multiplying is the same, when there are more than two fractions to be multiplied together.
Multiply together a/b, c/d, and m/y. Product acm/bdy.
For ix (a/b).(c/d) is, by the last article ac/bd, and this into m/y is acm/bdy.
2. Mult. (3 + b)/n, 1/h and d/(r + 2).
154. The multiplication may sometimes be shortened, by rejecting equal factors, from the numerators and denominators.
1. Multiply a/r into h/a and d/y. Product dh/ry.
Here a, being in one of the numerators, and in one of the denominators, may be omitted. If it be retained, the product will be adh/ary. But this reduced to lower terms, by Art. 142, will become dh/ry as before.
It is necessary that the factors rejected from the numerators be exactly equal to those which are rejected from the denominators. In the last example, a being in two of the numerators, and in only one of the denominators, must be retained in one of the numerators.
2. Multiply (a + d)/y into my/ah. Product (am + dm)/ah.
Here, though the same letter a is in one of the numerators, and in one of the denominators, yet as it is not in every term of the numerator, it must not be cancelled.
3. Multiply (am + d)/h into h/m and 3r/5a.
If any difficulty is found, in making these contractions, it mill be better to perform the multiplication, without omitting any of the factors; and to reduce the product to lower terms afterwards.
155. When a fraction and an integer are multiplied together, the numerator of the fraction is multiplied into the integer. The denominator is not altered; except in cases where division of the denominator is substituted for multiplication of the numerator, according to Art. 136.
Thus a.(m/y) = am/y. For a = a/1; and (a/1).(m/y) = am/y.
So r.(x/d(.(h + 1)/3 = (hrx + rx)/3d. And a.(1/b) = a/b. Hence,
156. A FRACTION IS MULTIPLIED INTO A QUANTITY EQUAL TO ITS DENOMINATOR, BT CANCELLING THE DENOMINATOR.
Thus (a/b).b = a. For (a/b).b = ab/b. But the letter b, being in both the numerator and denominator, may be set aside, (Art. 142.)
So 3m/(a - y).(a - y) = 3m.
On the same principle, a fraction is multiplied into any factor in its denominator, by cancelling that factor.
Thus (a/by).y = ay/by = a/b. And (h/24).6 = h/4.
157. From the definition of multiolication by a fraction, it follows that what is commonly called a compound fraction, is the product of two or more fractions. Thus 3/4 of a/b is (3/4).(a/b). For 3/4 of a/b is 1/4 of a/b taken three times, that is, a/4b + a/4b + a/4b. But this is the same as a/b multiplied by 3/4.(Art. 151.)
Hence, reducing a compound fraction into a simple one, is the same as multiplying fractions into each other.
Ex. 1. Reduce 2/7 of a/(b + 2). Ans. 2a/(7b + 14).
2. Reduce 2/3 of 4/5 of (b + h)/(2a - m). Ans. (8b + 8h)/(30a - 15m).
3. Reduce 1/7 of 1/3 of 1/(8 - d). Ans. 1/(168 - 21d).
158. The expression (2/3)a, (1/5)b, (4/7)y, &c. are equivalent to 2a/3, b/5, 4y/7. For (2/3)a is 2/3 of a, which is equal to (2/3).a = 2a/3.(Art. 155.)