Table of Contents

Last modified on July 18th, 2024

To multiply any type of fraction, we follow the following three steps:

- Multiplying the given numerators
- Multiplying the given denominators (we often perform step 1 and step 2 together)
- Simplifying or reducing the product to its lowest terms if needed.

Let us multiply ${\dfrac{2}{5}}$ and ${\dfrac{3}{8}}$

${\dfrac{2}{5}\times \dfrac{3}{8}}$

**Multiplying the Numerators**

2 × 3 = 6

**Multiplying the Denominators**

5 × 8 = 40

Now, the product is ${\dfrac{2\times 3}{5\times 8}}$

= ${\dfrac{6}{40}}$

**Simplifying**

The GCF of 6 and 40 is 2

Now, by dividing the numerator and the denominator by 2, we get

${\dfrac{6\div 2}{40\div 2}}$

= ${\dfrac{3}{20}}$

Here, the fractions with different denominators are multiplied. However, we can also multiply fractions with the same denominators.

**Problem: **Multiplying fractions with the** SAME DENOMINATOR**

**E.g.1. **

**Multiply: ${\dfrac{2}{5}\times \dfrac{3}{5}}$**

Solution:

Given, ${\dfrac{2}{5}\times \dfrac{3}{5}}$

On multiplying the numerators and the denominators,

= ${\dfrac{2\times 3}{5\times 5}}$

= ${\dfrac{6}{25}}$, which can not be simplified or reduced further.

**Problem: **Multiplying **THREE FRACTIONS **

**Multiply: ${\dfrac{3}{5}\times \dfrac{15}{7}\times \dfrac{14}{25}}$**

Solution:

Given, ${\dfrac{3}{5}\times \dfrac{15}{7}\times \dfrac{14}{25}}$

On multiplying the numerators and the denominators,

= ${\dfrac{3\times 15\times 14}{5\times 7\times 25}}$

= ${\dfrac{630}{875}}$

Simplifying the product, we get

= ${\dfrac{18}{25}}$

Thus, the product is ${\dfrac{6}{25}}$

Sometimes, we simplify the fractions first and then follow the same multiplication rules to obtain the final product more easily.

If ${\dfrac{12}{15}\times \dfrac{3}{5}}$

Simplifying the fractions, we get

${\dfrac{4}{5}\times \dfrac{3}{5}}$

On multiplying the numerators and the denominators, we get

${\dfrac{4\times 3}{5\times 5}}$

= ${\dfrac{12}{25}}$, which can not be simplified or reduced further.

Thus, the product is ${\dfrac{12}{25}}$

Now, when multiplying ${\dfrac{2}{5}}$ by an improper fraction ${\dfrac{12}{4}}$, we follow the same steps as discussed above.

**Multiply the given fractions: ${\dfrac{2}{5}}$ and ${\dfrac{12}{4}}$**

Solution:

Given, ${\dfrac{2}{5}\times \dfrac{12}{4}}$

By multiplying the numerators and the denominators, we get

${\dfrac{2\times 12}{5\times 4}}$

= ${\dfrac{24}{20}}$

Simplifying the product, we get

= ${\dfrac{6}{5}}$

**Find the product: ${\dfrac{20}{7}}$ and ${\dfrac{4}{5}}$**

Solution:

Given, the two fractions are ${\dfrac{20}{7}\times \dfrac{4}{5}}$

Multiplying the Numerators and the denominators, we get

= ${\dfrac{20\times 4}{7\times 5}}$

= ${\dfrac{80}{35}}$

Simplifying the product, we get

= ${\dfrac{16}{7}}$

The result is that of a simplified fraction and thus is the product.

To multiply a fraction by a whole number, we multiply the numerator with the given whole number, keeping the denominator the same.

Let us multiply the fraction ${\dfrac{2}{3}}$ by a whole number ${5}$

By multiplying the numerator and the whole number,

${\dfrac{2\times 5}{3}}$

= ${\dfrac{10}{3}}$, which is in the simplest form.

Thus, the product is ${\dfrac{10}{3}}$

Now, representing this example with the visual method,

**Multiply ${\dfrac{2}{21}}$ by the whole number ${7}$**

Solution:

Given, ${\dfrac{2}{21}\times 7}$

By multiplying the numerators and the denominators, we get

= ${\dfrac{2\times 7}{21}}$

= ${\dfrac{14}{21}}$

Simplifying the product, we get

= ${\dfrac{2}{3}}$

Thus, the product is ${\dfrac{2}{3}}$

To multiply a fraction with a mixed number, we first convert the mixed number into an improper fraction and then follow the same steps.

If ${\dfrac{12}{7}\times 1\dfrac{2}{3}}$

Converting ${1\dfrac{2}{3}}$ into an improper fraction,

${1\dfrac{2}{3}}$ = ${\dfrac{\left( 1\times 3\right) +2}{3}}$ = ${\dfrac{5}{3}}$

Now, we have ${\dfrac{12}{7}\times \dfrac{5}{3}}$

On multiplying the numerators and the denominators,

${\dfrac{12\times 5}{7\times 3}}$

= ${\dfrac{60}{21}}$

Simplifying,

${\dfrac{20}{7}}$

Thus, the product is ${\dfrac{20}{7}}$

**Multiply: ${\dfrac{7}{8}\times 1\dfrac{1}{2}\times 2\dfrac{3}{4}}$**

Solution:

Given, ${\dfrac{7}{8}\times 1\dfrac{1}{2}\times 2\dfrac{3}{4}}$

Converting the mixed numbers into improper fractions, we get

= ${\dfrac{7}{8}\times \dfrac{3}{2}\times \dfrac{11}{4}}$

By multiplying the numerators and the denominators, we get

= ${\dfrac{7\times 3\times 11}{8\times 2\times 4}}$

= ${\dfrac{231}{64}}$, which is in the simplest form.

Thus, the product is ${\dfrac{231}{64}}$

Here is a summary of what we learned.

Last modified on July 18th, 2024