Fractions, Decimals and Percentages

This lesson covers converting between fractions, decimals and percentages, ordering them, and the Higher-tier skill of converting recurring decimals to fractions. These topics appear throughout the Edexcel GCSE Mathematics (1MA1) specification and are fundamental to the Number strand.

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Key Vocabulary

Term	Meaning
Fraction	A part of a whole, written as numerator/denominator
Decimal	A number written using the decimal point and place value
Percentage	A fraction out of 100, shown with the % symbol
Equivalent	Having the same value in a different form
Recurring decimal	A decimal where one or more digits repeat forever
Terminating decimal	A decimal that ends (has a finite number of digits)

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Converting Fractions to Decimals

Divide the numerator by the denominator.

Worked Example 1

Convert 3/8 to a decimal.

$3 \div 8 = 0.375$3÷8=0.375

Answer: 0.375

Worked Example 2

Convert 5/11 to a decimal.

$5 \div 11 = 0.454545$5÷11=0.454545... = 0.4̇5̇ (recurring)

Notation: Edexcel uses dots above the first and last repeating digits. A single repeating digit gets one dot: 0.3̇ means 0.333... Two repeating digits: 0.4̇5̇ means 0.4545...

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Converting Decimals to Fractions

Terminating Decimals

Write the decimal as a fraction with a power-of-10 denominator, then simplify.

Worked Example 3

Convert 0.375 to a fraction.

0.375 = 375/1000

Simplify by dividing numerator and denominator by 125:

375/1000 = 3/8

Answer: 3/8

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Converting Fractions to Percentages

Multiply the fraction by 100.

Worked Example 4

Convert 7/20 to a percentage.

$7/20 \times 100 = 700/20 = 35$7/20×100=700/20=35

Answer: 35%

Alternatively, convert to a decimal first: $7 \div 20 = 0.35$7÷20=0.35, then $0.35 \times 100 = 35\%$0.35×100=35%.

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Converting Percentages to Fractions

Write the percentage over 100 and simplify.

Worked Example 5

Convert 65% to a fraction in its simplest form.

65/100 = 13/20

Answer: 13/20

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Converting Decimals to Percentages (and vice versa)

• Decimal $\to$→ Percentage: Multiply by 100 (move decimal point 2 places right).
• Percentage $\to$→ Decimal: Divide by 100 (move decimal point 2 places left).

Worked Example 6

• 0.045 as a percentage: $0.045 \times 100 =$0.045×100= 4.5%
• 12.5% as a decimal: $12.5 \div 100 =$12.5÷100= 0.125

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Common Equivalents to Memorise

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	0.3̇	33.3̇%
2/3	0.6̇	66.6̇%
1/8	0.125	12.5%
3/8	0.375	37.5%
1/10	0.1	10%
1/100	0.01	1%

Edexcel Exam Tip: Learning these common equivalents saves valuable time on Paper 1 (non-calculator). Examiners often set questions requiring you to compare fractions, decimals and percentages — having these memorised lets you order them quickly.

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Ordering Fractions, Decimals and Percentages

To compare numbers given in mixed forms, convert them all to the same form (usually decimals).

Worked Example 7

Put these in order from smallest to largest: 3/8, 0.4, 35%, 2/5

Convert everything to decimals:

• 3/8 = 0.375
• 0.4 = 0.4
• 35% = 0.35
• 2/5 = 0.4

Ordering: 0.35, 0.375, 0.4, 0.4

Answer: 35%, 3/8, 0.4 = 2/5

Note: 0.4 and 2/5 are equal, so either order is acceptable for those two.

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Recurring Decimals to Fractions [H]

This is one of the most commonly examined Higher-tier topics on Edexcel papers. The algebraic method is always required.

Method

1. Let x = the recurring decimal.
2. Multiply both sides by 10, 100 or 1000 (so the repeating block lines up).
3. Subtract to eliminate the repeating part.
4. Solve for x and simplify.

Worked Example 8 [H]

Convert 0.7̇ to a fraction.

Let x = 0.777...

10x = 7.777...

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

9x = 7

x = 7/9

Answer: 7/9

Worked Example 9 [H]

Convert 0.1̇8̇ to a fraction.

Let x = 0.181818...

100x = 18.181818...

Subtract: 100x - x = 18.181818... - 0.181818...

99x = 18

x = 18/99 = 2/11

Answer: 2/11

Worked Example 10 [H]

Convert 0.23̇6̇ to a fraction.

Let x = 0.2363636...

Here the non-recurring part is "2" (1 digit) and the recurring block is "36" (2 digits).

10x = 2.363636...

1000x = 236.363636...

Subtract: 1000x - 10x = 236.363636... - 2.363636...

990x = 234

x = 234/990 = 13/55

Answer: 13/55

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Terminating vs Recurring Decimals

A fraction in its simplest form produces a terminating decimal if and only if the denominator has no prime factors other than 2 and 5.

Denominator	Prime factors	Terminates?
8	$2 \times 2 \times 2$2×2×2	Yes
20	$2 \times 2 \times 5$2×2×5	Yes
6	$2 \times 3$2×3	No — recurring
11	11	No — recurring
25	$5 \times 5$5×5	Yes

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Common Mistakes and Misconceptions

Mistake	Correction
Writing 1/3 = 0.3	1/3 = 0.3̇ (recurring). 0.3 = 3/10
Forgetting to simplify fractions	Always check if the fraction can be simplified
Multiplying by the wrong power of 10 for recurring decimals	Match the power of 10 to the length of the repeating block
Thinking 0.9̇ $\neq 1$=1	0.9̇ = 1 exactly (let x = 0.999..., 10x = 9.999..., 9x = 9, x = 1)
Confusing percentages and decimals	5% = 0.05, NOT 0.5

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Practice Problems