Fractions are written to describe the part of the whole number. We can perform all arithmetic operations like addition, subtraction, multiplication, and division. On this page, we will discuss the fraction of a fraction which means multiplication of a fraction with another fraction.

Let us find the fraction of a fraction involving fractions like proper fraction, improper fraction, unit fraction, Equivalent fraction, Like and Unline fraction, and mixed fraction. Read the entire article to know how to solve a fraction of a fraction with example questions and answers provided.

**Do Refer:**

- Addition of Like Fractions
- Multiplication of Fractional Number by a Whole Number
- Subtraction of Fractions having the Same Denominator

### How to Solve a Fraction of a Fraction?

The fraction of a fraction is not like the addition and subtraction of the fractions where the denominator is the same. The method is different for the multiplication of fractions. Let us learn the steps to know how to find the fraction of a fraction.

**1. Multiplication of whole numbers and a fraction:** The whole number of a fraction is very easy. All you have to do is to multiply the whole number with the numerator of the fraction and then write the denominator as it is. Let us see one example for the multiplication of a whole number and a fraction.

Example: Multiply 3 and \(\frac{1}{5}\)

Solution:

3 × \(\frac{1}{5}\)

Multiply the whole number and numerator.

3 × 1 = 3

\(\frac{3}{5}\)

**2. Multiplication of proper fraction:** In this concept, you have to multiply the numerator with the numerator and the denominator with the denominator of the given fractions.

For example let us take \(\frac{2}{3}\) and \(\frac{4}{6}\)

Solution:

\(\frac{2}{3}\) × \(\frac{4}{6}\)

Multiply the numerators together: 2 × 4 = 8

Multiply the denominator together: 3 × 6 = 18

\(\frac{2}{3}\) × \(\frac{4}{6}\) = \(\frac{8}{18}\)

**3. Multiplication of Improper fraction:** The multiplication of improper fraction is similar to Multiplication of proper fraction. In this concept, we have to multiply the numerator with the numerator and the denominator with the denominator of the given fractions. And then convert the improper fraction to the mixed fraction.

**Example:** Multiply \(\frac{4}{3}\) and \(\frac{6}{5}\)

**Solution:**

\(\frac{4}{3}\) × \(\frac{6}{5}\)

Multiply the numerators together: 4 × 6 = 24

Multiply the denominator together: 3 × 5 = 15

\(\frac{4}{3}\) × \(\frac{6}{5}\) = \(\frac{24}{15}\)

Now convert this improper fraction to the mixed fraction.

\(\frac{24}{15}\) = 1 \(\frac{3}{5}\)

**4. Multiplication of Mixed Fraction:** The mixed fraction is a combination of a whole number and a fraction. In this concept, we can change the mixed fraction to the improper fraction and then find the multiplication of a fraction.

For example multiply 2 \(\frac{2}{3}\) and 3 \(\frac{1}{4}\).

**Solution: **

Change the given mixed fractions to the improper fraction

\(\frac{8}{3}\) × \(\frac{13}{4}\) = \(\frac{104}{12}\) = \(\frac{26}{3}\)

Now convert the improper fraction to the mixed fraction 8 \(\frac{2}{3}\)

Thus the product of 2 \(\frac{2}{3}\) and 3 \(\frac{1}{4}\) is 8 \(\frac{2}{3}\).

**Also Read Articles:**

- Worksheet on Multiplication on Fraction
- Multiplication of a Fraction by Fraction
- Word Problems on Multiplication of Mixed Fractions

### Fraction of a Fraction Solved Problems

**Example 1.**

Find the fraction of a fraction for \(\frac{1}{2}\) and \(\frac{1}{6}\).

**Solution:**

Given the fractions \(\frac{1}{2}\) and \(\frac{1}{6}\).

The two fractions are unit fractions.

Now find the product of two unit fractions.

Multiply the numerator and denominators of the unit fractions.

\(\frac{1}{2}\) × \(\frac{1}{6}\) = \(\frac{1×1}{2×6}\) = \(\frac{1}{12}\)

Thus the unit fraction of a fraction for \(\frac{1}{2}\) and \(\frac{1}{6}\) is \(\frac{1}{12}\).

**Example 2.**

Find the fraction of a fraction for \(\frac{4}{7}\) and \(\frac{2}{5}\).

**Solution:**

Given the fractions \(\frac{4}{7}\) and \(\frac{2}{5}\).

Multiply the numerator with the numerator and denominator with the denominator of the given fractions.

\(\frac{4}{7}\) × \(\frac{2}{5}\)

Multiply the numerators together: 4 × 2 = 8

Multiply the denominator together: 7 × 5 = 35

\(\frac{4}{7}\) × \(\frac{2}{5}\) = \(\frac{8}{35}\)

Thus the fraction of a fraction for \(\frac{4}{7}\) and \(\frac{2}{5}\) is \(\frac{8}{35}\)

**Example 3.**

Find the fraction of a fraction for \(\frac{3}{8}\) and \(\frac{4}{9}\).

**Solution:**

Given the fractions \(\frac{3}{8}\) and \(\frac{4}{9}\).

Multiply the numerator with the numerator and denominator with the denominator of the given fractions.

\(\frac{3}{8}\) × \(\frac{4}{9}\)

Multiply the numerators together: 3 × 4 = 12

Multiply the denominator together: 8 × 9 = 72

\(\frac{3}{8}\) × \(\frac{4}{9}\) = \(\frac{12}{72}\) = \(\frac{1}{6}\)

Thus the fraction of a fraction for \(\frac{3}{8}\) and \(\frac{4}{9}\) is \(\frac{12}{72}\) or \(\frac{1}{6}\).

**Example 4.**

Find the fraction of a fraction for \(\frac{7}{2}\) and \(\frac{9}{5}\).

**Solution:**

Given the fractions \(\frac{7}{2}\) and \(\frac{9}{5}\).

Multiply the numerator with the numerator and denominator with the denominator of the given fractions.

We have to multiply the numerator with the numerator and the denominator with the denominator of the given fractions.

And then convert the improper fraction to the mixed fraction.

\(\frac{7}{2}\) × \(\frac{9}{5}\)

Multiply the numerators together: 7 × 9 = 63

Multiply the denominator together: 2 × 5 = 10

\(\frac{7}{2}\) × \(\frac{9}{5}\) = \(\frac{63}{10}\)

Now convert this improper fraction to the mixed fraction.

\(\frac{63}{10}\) = 6 \(\frac{3}{10}\)

Therefore the fraction of a fraction for \(\frac{7}{2}\) and \(\frac{9}{5}\) is \(\frac{63}{10}\) or 6 \(\frac{3}{10}\)

**Example 5.**

Find the fraction of a fraction for 1 \(\frac{1}{4}\) and 2 \(\frac{1}{5}\).

**Solution:**

Given the fractions 1 \(\frac{1}{4}\) and 2 \(\frac{1}{5}\).

Change the given mixed fractions to the improper fraction.

We have to multiply the numerator with the numerator and the denominator with the denominator of the given fractions.

And then convert the improper fraction to the mixed fraction.

1 \(\frac{1}{4}\) = 4 × 1 + 1 = 5 = \(\frac{5}{4}\)

2 \(\frac{1}{5}\) = 5 × 2 + 1 = 11 = \(\frac{11}{5}\)

Now multiply the fractions.

\(\frac{5}{4}\) × \(\frac{11}{5}\)

Multiply the numerators together: 5 × 11 = 55

Multiply the denominator together: 4 × 5 = 20

\(\frac{5}{4}\) × \(\frac{11}{5}\) = \(\frac{55}{20}\)

Now convert this improper fraction to the mixed fraction.

\(\frac{55}{20}\) = 2 \(\frac{3}{4}\)

Therefore the fraction of a fraction for 1 \(\frac{1}{4}\) and 2 \(\frac{1}{5}\) is \(\frac{55}{20}\) or 2 \(\frac{3}{4}\)

### FAQs on Fraction of a Fraction

**1. How do I find a fraction of a fraction?**

I can find the fraction of a fraction by multiplying the top numbers of the two fractions and multiply the bottom numbers of the fractions and then write the result.

**2. Write an example of the multiplication of a unit fraction with a unit fraction.**

Suppose let us consider two unit fractions \(\frac{1}{3}\) and \(\frac{1}{10}\).

**Solution:**

Now find the product of two unit fractions.

Multiply the numerator and denominators of the fractions.

\(\frac{1}{3}\) × \(\frac{1}{10}\) = \(\frac{1×1}{3×10}\) = \(\frac{1×1}{30}\)

**3. What are six kinds of fractions?**

There are 6 types of fractions. They are proper fraction, improper fraction, mixed fraction, like fraction, unlike fraction and equivalent fraction.