Multiplication of Fractions

We have discussed the concept of fractions and studied their types. We have studied some operations using fractions as well. In this discussion, we study the multiplication of fractions in detail with solved examples to clear the concept.

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Fractions

Fractions are represented in the “p/q” format and this form is most preferred for carrying out multiplication of 2 or more fractions.

(Source: Quora)

Multiplication of Fractions

Multiplication of fractions is a simple operation. Multiply the numerator of all the fractions separately and multiply the denominators of all the fractions separately to obtain the final numerator and denominator. For example, consider 3/5 × 4/9

= (3 × 4)/(5 × 9) = 12/45

Now, in the above question, the numerator and the denominator have a common factor i.e. 3. Reduce the numerator and denominator to their lowest forms to obtain the final result. So,

12/45 = (3 × 4)/(3 × 3 × 5) = 4/15

A shortcut is to cancel the common terms while multiplying initially rather than dividing the answer. But that process requires some practice and can be developed over a period of time. Now, consider

2/6 × 3/4 × 5/7

Now here, we see that the first fraction is reducible to a smaller form, so we do it beforehand

2/6 = 1/3

so the question changes to

1/3 × 3/4 × 5/7 = (1 × 3×  5)/(3 × 4× 7)

again, we see that the numerator and denominator both have a common factor i.e.3, which we cancel from both the numerator or denominator. We basically divide 3 by 3. So we get,

( 1× 5)/(4 × 7)

= 5/28

Note that for multiplication it is not necessary to have the same denominator in both the fractions, a rule mandatory in addition and subtraction of fractions.

Need for multiplication

Multiplication of fractions is particularly important to obtain ratios. For example, if we need 1/3 of  9/5 we multiply 1/3 to 9/5 to obtain the answer. So 1/3 rd of 9/5 is

1/3 × 9/5 = 1/3 × (3 × 3)/5

= 3/5

Multiplication of a Fraction with a Whole number

Suppose we have to obtain 3/4th  of 7 to obtain this we consider the whole number as a fraction itself.

“Any whole number “x” can be represented as x/1″

It basically means that we divide the number by 1. any number divided by 1 is the number itself. Now the multiplication becomes

3/4 × 7/1

Using the rules of multiplication we get,

(3 × 7)/(4 × 1)= 21/4

Multiplication of Mixed Fractions

A mixed fraction is one which has a whole number and a fraction together. For example, 2³/₅  or 7¹/ 9. To multiply them we convert each mixed fraction into a simple fraction and then multiply these fractions.

A mixed fraction of form x^y/_z is converted to normal by multiplying the denominator by the whole number and adding the numerator to obtain the numerator. For example, in the above form, it is (z.x+ y)/z as a normal fraction.

Consider the multiplication, 2 ³/₅  ×   7 ¹/₉

we first convert them to normal fractions,
= (5 × 2 + 3)/5  and (9× 7 +1)/9
= 13/5 and 64/9

now following rules of multiplication of fractions,
13/5 ×  64/9
= (13 × 64)/(5 × 9)
= 832/ 45

Solved Question for You

Q1. Multiply 2¹/₅ and 15/4

Solution: Converting the mixed fraction to normal we get 11/5.
Now,  11/5 × 15/4
= (11 × 15)/(5 × 4)
Canceling common terms,
=( 11×  5 × 3)/ (5× 4)
= 33/4

Q2. Of the land owned by a farmer, 90 percent was cleared for planting. Of the cleared land, 40 percent was planted with soybeans and 50 percent of the cleared land was planted with wheat. If the remaining 720 acres of cleared land was planted with corn, how many acres did the farmer own?

1. 5832
2. 6420
3. 7200
4. 8000

Solution: D. Land used for planting wheat and soybeans = 90 percent
Remaining land = 10 percent
So the total area of cleared land must be 7200 acres.
So 7200 is 92 percent of the total land.
Therefore the total land area must be 8000 acres.