How to Decompose Fractions into Unit Fractions?
Decomposing a fraction means breaking it into smaller parts. In this step-by-step guide, you will learn how to decompose fractions into unit fractions.
A Step-by-step guide to decomposing fractions into unit fractions
In order to decompose fractions, it merely involves taking them apart. The top fundamental method of decomposing fractions is to split them into unit fractions, which is whenever the numerator (the number on top) is \(1\).
Note: A fraction where the numerator is constantly \(1\) is known as a unit fraction.
Decomposing fractions into unit fractions involves breaking down a given fraction into a sum of smaller fractions with a numerator of 1. Here’s a step-by-step guide to decomposing fractions into unit fractions:
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Step 1: Choose the fraction to decompose
Select the fraction you want to decompose. For this example, let’s use 5/6.
Step 2: Simplify the fraction (if necessary)
If the fraction can be simplified, it’s helpful to do so before decomposing it. In our example, 5/6 is already in its simplest form, so we can move on to the next step.
Step 3: Identify the largest unit fraction
Find the largest unit fraction with the same denominator as the original fraction. In our example, the largest unit fraction is 1/6, as the denominator of the original fraction is 6.
Step 4: Subtract the largest unit fraction
Subtract the largest unit fraction from the original fraction to find the remaining fraction. In our example: 5/6 – 1/6 = 4/6
Step 5: Simplify the remaining fraction (if necessary)
Simplify the remaining fraction if possible. In our example, 4/6 can be simplified to 2/3.
Step 6: Repeat steps 3-5
Continue to subtract the largest unit fraction, simplify (if necessary), and repeat until the remaining fraction is a unit fraction. In our example: 2/3 – 1/3 = 1/3
Step 7: Write the decomposition
Combine the unit fractions found in the process to represent the original fraction. In our example, the decomposition of 5/6 is: 5/6 = 1/6 + 1/3 + 1/3
Step 8: Confirm the result
Verify that the sum of the unit fractions equals the original fraction. In our example, 1/6 + 1/3 + 1/3 = 5/6, confirming that our decomposition is correct.
Remember that there can be multiple valid decompositions for a given fraction. Practicing with different fractions and exploring various decompositions will help you build a solid understanding of the relationships between fractions and unit fractions.
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Decomposition of Fractions into Unit Fractions-Example 1:
Decompose fraction \(\frac{4}{9}\) into unit fractions.
Solution: To decompose \(\frac{4}{9}\) into unit fractions, we can split the numerator four and express the fraction as the sum of \(4\) one-ninths.
\(\frac{4}{9}=\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}\)
Exercises for Decomposition of Fractions into Unit Fractions
Decompose fractions using unit fractions.
- \(\color{blue}{\frac{7}{8}}\)
- \(\color{blue}{\frac{3}{4}}\)
- \(\color{blue}{\frac{5}{6}}\)
- \(\color{blue}{\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}}\)
- \(\color{blue}{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}\)
- \(\color{blue}{\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}}\)
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