Finding Sum of Fractions Using Decomposing Method? (Examples) – BYJUS

# Find the Sum of Fractions by Decomposing the Fractions

Fractions are obtained by dividing whole numbers into equal parts. Just like whole numbers, fractions can be further broken down or decomposed to get smaller fractions. Learn how to adopt this strategy to perform math operations like addition and subtraction easily....Read MoreRead Less

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## What is Meant by Decomposing a Fraction?

‘Splitting up’ or ‘dividing into smaller parts’ is what decompose means. To decompose a fraction, divide it into smaller fractions so that when all the smaller parts are added together, you get the original fraction.

For example, Let’s take apart a mixed fraction. A whole number and a proper fraction are represented together in a mixed fraction. It denotes a number that falls between two whole numbers.

Decomposing a mixed fraction $$3\frac{2}{4}$$ using models.

• We must first divide this fraction as $$3\frac{2}{4}=3+\frac{2}{4}$$.
• Splitting a mixed fraction generates a whole number and a proper fraction.
• As a result, the whole number 3 is represented by $$\frac{4}{4}$$, with all four blocks filled.
• Proper fraction $$\frac{2}{4}$$ is divided into two separate blocks, each represented by $$\frac{1}{4}+\frac{1}{4}$$, with one block filled out of four parts.

## How to Write a Fraction as a Sum of Unit Fractions?

A unit fraction is a fraction whose numerator is always 1.

Let’s understand with an example by writing $$\frac{5}{6}$$ as a sum of unit fractions.

We can use a tape diagram or a number line to understand this.

Tape Diagram: This model represents a whole. The whole is divided into 6 equal parts. Each part represents the unit fraction $$\frac{1}{6}$$, shade 5 parts to represent the fraction $$\frac{5}{6}$$.

So, $$\frac{5}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$$

Therefore, $$\frac{5}{6}$$ expressed as the sum of unit fractions.

Number line: We can also understand the same concept using a number line. First 0 and 1 are plotted and then 6 equal divisions are made such that each division represents, $$\frac{1}{6}$$. So, $$\frac{5}{6}$$ is marked and the difference between two consecutive divisions is $$\frac{1}{6}$$.

So that, $$\frac{5}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$$

Therefore, $$\frac{5}{6}$$ is expressed as the sum of unit fractions.

Fig: Using the number line.

## How to Write a Fraction as a Sum of Two Fractions?

Let’s learn to write a fraction as a sum of other fractions using an example. We can convert a fraction as a sum of fractions in two ways; by either writing the fraction as unit fractions or writing as two fractions.

For example, Writing $$\frac{7}{11}$$ as a sum of fractions.

First method: Writing the fraction $$\frac{7}{11}$$ as a sum of unit fractions.

The fraction $$\frac{7}{11}$$ represents the sum of 7 parts that are of each $$\frac{1}{11}$$ of the whole.

So, $$\frac{7}{11}=\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}$$

Second method: Writing the fraction $$\frac{7}{11}$$ as a sum of two fractions.

Break apart $$\frac{7}{11}$$ into 2 parts of $$\frac{1}{11}$$ and 5 parts of $$\frac{1}{11}$$.

So, $$\frac{7}{11}=\frac{2}{11}+\frac{5}{11}$$

## Solved Examples

Example 1: Write $$\frac{4}{5}$$ into the sum of unit fractions.

Solution: Writing the fraction $$\frac{4}{5}$$ as a sum of unit fractions.

The fraction $$\frac{4}{5}$$ represents 4 parts that are of each $$\frac{1}{5}$$ of the whole.

So, $$\frac{4}{5}=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}$$

Example 2: Write $$\frac{7}{9}$$ as a sum of unit fractions.

Solution: Writing the fraction $$\frac{7}{9}$$ as a sum of unit fractions.

The fraction $$\frac{7}{9}$$ represents 9 parts that are of each $$\frac{1}{9}$$ of the whole.

So, $$\frac{7}{9}=\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}$$

Example 3: Mary uses this equation to decompose the fraction $$\frac{3}{6}$$. $$\frac{3}{6}=\frac{1}{6}+\frac{?}{6}+\frac{1}{6}$$. What is the missing number in the equation?

Solution: Writing the fraction $$\frac{3}{6}$$ as a sum of unit fractions.

The fraction $$\frac{3}{6}$$ represents 6 parts that are of each $$\frac{1}{6}$$ of the whole.

So, $$\frac{3}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$$

Therefore, the number that belongs in the blank is 1.

Example 4: John, Joseph, and Steven went to a pizza shop and ate $$\frac{3}{5}$$ of a pizza together. Each had an equal amount of pizza. How much pizza did each person eat?

Solution:

First method: Writing the fraction $$\frac{3}{5}$$ as a sum of unit fractions.

The fraction $$\frac{3}{5}$$ represents 5 parts that are of each $$\frac{1}{5}$$ of the whole.

So, $$\frac{3}{5}=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}$$

Second method: Writing the fraction $$\frac{3}{5}$$ as a sum of two fractions.

Break apart $$\frac{3}{5}$$ into 2 parts of $$\frac{1}{5}$$ and 1 part of $$\frac{1}{5}$$.

So, $$\frac{3}{5}=\frac{2}{5}+\frac{1}{5}$$

Frequently Asked Questions

Let’s understand with the help of an example. We can see that $$\frac{3}{8}$$ is same as three times the unit fraction $$\frac{1}{8}$$. Hence we can decompose $$\frac{3}{8}$$ as $$\frac{1}{8}\times3=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}$$

A unit fraction is a fraction with a numerator of one. It represents 1 part out of all the equal parts.