**In this page, we will learn:**

- How to multiply fractions which are both proper and improper fractions
- How to multiply fractions and a whole number
- How to multiply mixed numbers

In each of the above three types of how to multiply fractions, apply the following rule:

## Type 1: How to multiply fractions which are proper fractions.

Proper fractions are those in which numerators are less than denominators.

*So, (3/5) × (4/7) = (3 × 4)/ (5 × 7) = 12/35*

**Note:**

Divide both numerator and denominator by the highest common factor (hcf) that they may have.

**For example:**

In *(3/5) × (10/8)* which is equal to 30/40, in the final product 30/40, the numerator 30 and the denominator 40 have common factors. Divide each by their hcf to reduce the final product to the simplest terms. Now *30 = 3 × 10, and 40 = 4 × 10. *Therefore, divide them both by this hcf, 10 to reduce the fraction into the simplest form.

## Type 2: How to multiply fractions which are improper fractions

Fractions in which numerators are bigger than denominators are called improper fractions. Apply the same rule.

**Example:**

*(6/5) × (9/8) = (6 × 9)/ (5 × 8) = 54/40*

**Now, what is the HCF of 54 and 40?**

*54 = 2 × 27 = 2 × 3 × 3 × 3, and 40 = 2 × 20 = 2 × 2 × 10 = 2 × 2 × 2 × 5 =*

Now, the hcf of 54 and 40 is 2

*So, 54/40 = (2 × 27)/ (2 × 20) = 27/20*

**Note:**

In writing 54/40 as 27/20, one of the two below is done:

Divide both 54 and 40 by their hcf, 2 as:

*(54 ÷ 2/40 ÷ 2) = 27/20, or in (2 × 27)/ (2 × 20),* we say “the common factor 2 is cancelled”, which step is shown as: *(2 × 27)/ (2 × 20) = 27/20*

## Type 3: How to multiply fractions and a whole number

**Example:**

*3 × (4/15) =*

We can rewrite the whole number 3 as an improper fraction 3/1, but this is not always necessary.

*(3/1) × (4/15) = (3 × 4)/ (1× 15) = 12/15*

**We could also straight away write:**

*3 × (4/15) = (3 × 4)/15 = 12/15*

**Once again, what is the hcf of 12 and 15?**

*12 = 3 × 4 and 15 = 3 × 5, s*o *12/15 = (3 × 4)/(3 × 5) = 4/5*

## Type 4: How to multiply fractions obtained from mixed numbers.

**Example:**