# Multiplying Fractions - Multiplying Fraction by Whole Number and by its Reciprocal

Multiplying two fractions is very easy, and the rule is simple. Just to remind you, the denominator of a fraction is always non-zero integer.

### Note:

Let and be two fractions.

To multiply two fractions, multiply the numerators and then multiply the denominators.

Know that there are two other symbols of multiplication, using Dot or Parentheses.

### Note:

Let and be two fractions.

### Info!

• Note carefully that multiplying two fractions has nothing to do with whether the two fractions have the same or different denominators, as in the case of addition and subtraction.

• After multiplying two fractions, make sure your answer is reduced to lowest terms.

In some cases, the factor in the numerator of one of the two fractions is the same in the denominator of the other fraction. Here we resort to simplifying the operation before performing it.

Best Procedure:

• If possible, cancel the same factor from a numerator and a denominator.

• Multiply the remaining numerators and the remaining denominators.

These are some examples that show how to multiply two fractions, but before multiplying we always have to make sure that we cancel the common factor from the denominator and the numerator using reducing the factors rules.

#### Example1:

Multiply the following fractions:

1)

2)

3)

4)

5)

#### Solution:

1) The two fractions and are in the lowest term.

The result of the multiplication is already in its reduced form (the greatest common factor of 6 and 35 is 1).

2) The two fractions and are in the lowest term, but the gcf of 9 and 6 is 3.

The resulting fraction is in its reduced form since the greatest common factor of 14 and 15 is 1.

3) The fraction should be reduced to the lowest term:

Thus,

The resulting fraction is in its reduced form since the greatest common factor of 6 and 5 is 1.

4) The two fractions and are in the lowest term.

The result of the multiplication is already in its reduced form (the greatest common factor of 4 and 15 is 1).

5) In the same way, we multiple three fractions.

The three fractions , and are in the lowest term.

The resulting fraction is in its reduced form since the greatest common factor of 33 and 88 is 1.

## Multiplying Fraction by Whole Number

In the previous paragraph, we saw how to multiply two fractions, but how do we multiply two numbers, one of which is a fraction and the other is a whole number? This is what we will see in this paragraph.

Before we get to know the rule, let's first derive it from a simple example.

We all agree that means: goes 3 times, right?

We can express this mathematically as follows:

Now applying the rule for adding fractions with the same denominator, we get:

After the result obtained, can you derive the property of multiplying a fraction by a whole number without going through the addition step? There is no doubt that you can, it is very clear that we multiplied 3 by the numerator of the fraction, that is, we multiplied 3 by 2 to get 6 in the numerator of the resulting fraction.

### Note:

Let represent whole number and be fraction.

To multiply a fraction by the whole number, multiply the numerator by the whole number and keep the denominator.

#### Example2:

Multiply the following fractions:

1)

2)

3)

4)

#### Solution:

1) The fraction is in its reduced form.

The resulting fraction is in its reduced form since the greatest common factor of 12 and 5 is 1.

2) The fraction is in its reduced form. But, the greatest common factor of 12 and 4 is 4.

3) The fraction is in its reduced form.

The resulting fraction is in its reduced form since the greatest common factor of 21 and 2 is 1.

4) The fraction is in its reduced form. But, the greatest common factor of 15 and 25 is 5.

The resulting fraction is in its reduced form since the gcf of 6 and 5 is 1.

## Multiplying Fraction by Its Reciprocal

Perhaps we'll need to mention the reciprocal of the fraction in the next property.

### Note:

Let and represent two non-zero integers.

• The reciprocal of is

• The reciprocal of is

• The reciprocal of is

The reciprocal of a fraction is switcing the numerator and the denominator.

Multiplying a fraction by its reciprocal always gives us one as a result.

### Note:

Let and represent two non-zero integers.

Let's take some examples to further illustrate this important property that we will use often in dividing two fractions.

#### Example3:

Multiply the following number by its reciprocal.

#### Solution:

• The reciprocal of 3 is .
• The reciprocal of -5 is .
• The reciprocal of is 6.
• The reciprocal of is .
• The reciprocal of is .
• The reciprocal of 1 is itself.

• The number 0 has no reciprocal.