# Multiplying Fractions

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

1. Multiplying the given numerators
2. Multiplying the given denominators (we often perform step 1 and step 2 together)
3. 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

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 ${\dfrac{12}{25}}$

## With Improper Fractions

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.

## With Whole Numbers

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

## With Mixed Numbers

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.