Converting Between Fractions, Decimals and Percentages

This lesson covers the methods for converting between fractions, decimals and percentages — a key skill for AQA GCSE Mathematics. You must be fluent in all six conversion directions and be able to recognise common equivalences. This topic also includes converting recurring decimals to fractions, which is a higher-tier skill.

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The Three Forms

Fractions, decimals and percentages are all different ways of representing the same value.

graph TD
    A[Fraction] -->|"Divide numerator by denominator"| B[Decimal]
    B -->|"Multiply by 100"| C[Percentage]
    C -->|"Divide by 100"| B
    B -->|"Write as a fraction and simplify"| A
    A -->|"Convert to decimal, then x100"| C
    C -->|"Write over 100 and simplify"| A

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Common Equivalences

You should memorise these common conversions:

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
2/5	0.4	40%
1/3	0.333...	33.3...%
2/3	0.666...	66.6...%
1/8	0.125	12.5%
3/8	0.375	37.5%
1/10	0.1	10%
1/100	0.01	1%

Exam Tip: Knowing these equivalences by heart saves valuable time in the exam. They also help you quickly estimate answers and check your work.

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Fraction to Decimal

Divide the numerator by the denominator.

Worked Example

Convert 7/8 to a decimal.

7 divided by 8 = 0.875

Worked Example with a Recurring Decimal

Convert 5/6 to a decimal.

5 divided by 6 = 0.8333... = 0.83 recurring (the 3 repeats)

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Decimal to Fraction

1. Write the decimal as a fraction over a power of 10.
2. Simplify.

Worked Example

Convert 0.35 to a fraction.

0.35 = 35/100

Simplify (divide by 5): 35/100 = 7/20

Worked Example

Convert 0.008 to a fraction.

0.008 = 8/1000

Simplify (divide by 8): 8/1000 = 1/125

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Fraction to Percentage

Method 1: Convert to a decimal first, then multiply by 100.

Method 2: Find an equivalent fraction with a denominator of 100.

Worked Example

Convert 7/20 to a percentage.

Method 1: 7 / 20 = 0.35, then 0.35 x 100 = 35%

Method 2: 7/20 = 35/100 = 35% (multiply numerator and denominator by 5)

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Percentage to Fraction

Write the percentage over 100 and simplify.

Worked Example

Convert 65% to a fraction.

65/100 = 13/20 (divide numerator and denominator by 5)

Worked Example

Convert 12.5% to a fraction.

12.5/100 = 125/1000 = 1/8

Exam Tip: For percentages that are not whole numbers (like 12.5% or 33.3%), multiply top and bottom by 10 (or an appropriate power of 10) to remove the decimal before simplifying.

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Decimal to Percentage

Multiply by 100 (or move the decimal point two places to the right).

Worked Examples

• 0.45 = 45%
• 0.7 = 70%
• 0.025 = 2.5%
• 1.3 = 130%

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Percentage to Decimal

Divide by 100 (or move the decimal point two places to the left).

Worked Examples

• 78% = 0.78
• 5% = 0.05
• 150% = 1.50
• 0.4% = 0.004

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Recurring Decimals to Fractions [H]

This is a higher-tier topic. A recurring decimal is one where one or more digits repeat infinitely.

Notation

• 0.333... is written as 0.3 with a dot above the 3
• 0.272727... is written as 0.27 with dots above both the 2 and the 7
• 0.1666... is written as 0.16 with a dot above the 6 only

Method: Using Algebra

Step 1: Let x = the recurring decimal. Step 2: Multiply x by a power of 10 to shift the recurring part. Step 3: Subtract to eliminate the recurring part. Step 4: Solve for x and simplify.

Worked Example 1: One Recurring Digit

Convert 0.777... to a fraction.

Let x = 0.777...

10x = 7.777...

Subtract: 10x - x = 7.777... - 0.777...

9x = 7

x = 7/9

Worked Example 2: Two Recurring Digits

Convert 0.363636... to a fraction.

Let x = 0.363636...

100x = 36.363636...

Subtract: 100x - x = 36.363636... - 0.363636...

99x = 36

x = 36/99 = 4/11 (simplify by dividing by 9)

Worked Example 3: Mixed Recurring Decimal

Convert 0.1666... to a fraction.

Let x = 0.1666...

10x = 1.666...

100x = 16.666...

Subtract: 100x - 10x = 16.666... - 1.666...

90x = 15

x = 15/90 = 1/6

Exam Tip: For recurring decimal questions, always show your algebra clearly. The method marks come from setting up the equations and subtracting. Identify how many digits recur to choose the correct power of 10.

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Comparing Fractions, Decimals and Percentages

When comparing values in different forms, convert them all to the same form — usually decimals.

Worked Example

Which is larger: 5/8 or 62%?

• 5/8 = 0.625 = 62.5%
• 62% = 0.62

Since 62.5% > 62%, 5/8 is larger.

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Summary

• Fraction to decimal: divide numerator by denominator.
• Decimal to fraction: write over a power of 10 and simplify.
• Fraction to percentage: convert to decimal then multiply by 100 (or find an equivalent fraction over 100).
• Percentage to fraction: write over 100 and simplify.
• Decimal to percentage: multiply by 100.
• Percentage to decimal: divide by 100.
• Recurring decimals to fractions [H]: use algebra — let x = the decimal, multiply by 10 or 100, subtract, and solve.
• Memorise the common equivalences table for speed.

Exam Tip: When a question asks you to "show that" a recurring decimal equals a specific fraction, you MUST use the algebraic method and show every line of working.

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Extended Worked Examples

Worked Example A: Fraction to Terminating Decimal

Convert $\tfrac{7}{40}$407​ to a decimal.