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, \(\frac{3}{5}=\frac{2}{5}+\frac{1}{5}\)
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.
