Fractions and Recurring Decimals (Advanced)

This lesson extends the work on fractions and recurring decimals to cover complex fraction operations and the algebraic proof that recurring decimals are rational. These are Higher-tier topics in the Edexcel GCSE Mathematics (1MA1) specification and are commonly tested on all three papers.

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Key Vocabulary

Term	Meaning
Proper fraction	A fraction where the numerator is less than the denominator, e.g. 3/5
Improper fraction	A fraction where the numerator is greater than or equal to the denominator, e.g. 7/4
Mixed number	A whole number and a fraction combined, e.g. 1 3/4
Reciprocal	The reciprocal of a/b is b/a; the reciprocal of n is 1/n
Rational number	Any number that can be expressed as p/q where p, q are integers and $q \neq 0$q=0

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Adding and Subtracting Fractions

To add or subtract fractions, they must have a common denominator.

Worked Example 1

Calculate 3/4 + 2/5

LCD of 4 and 5 is 20.

3/4 = 15/20

2/5 = 8/20

15/20 + 8/20 = 23/20 = 1 3/20

Answer: 23/20 or 1 3/20

Worked Example 2

Calculate 5/6 - 3/8

LCD of 6 and 8 is 24.

5/6 = 20/24

3/8 = 9/24

20/24 - 9/24 = 11/24

Answer: 11/24

Worked Example 3 (Mixed Numbers)

Calculate 3 2/5 - 1 3/4

Convert to improper fractions:

• 3 2/5 = 17/5
• 1 3/4 = 7/4

LCD of 5 and 4 is 20:

• 17/5 = 68/20
• 7/4 = 35/20

68/20 - 35/20 = 33/20 = 1 13/20

Answer: 1 13/20

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Multiplying Fractions

Multiply numerators together and denominators together. Simplify where possible.

Worked Example 4

Calculate $3/8 \times 4/9$3/8×4/9

$= (3 \times 4)/(8 \times 9) = 12/72 = 1/6$=(3×4)/(8×9)=12/72=1/6

Tip: Cross-cancel before multiplying to keep numbers smaller. 3 and 9 share factor 3; 4 and 8 share factor 4: $(1 \times 1)/(2 \times 3) = 1/6$(1×1)/(2×3)=1/6

Worked Example 5

Calculate $2 1/3 \times 1 4/5$21/3×14/5

Convert: $7/3 \times 9/5 = 63/15 = 21/5 = 4 1/5$7/3×9/5=63/15=21/5=41/5

Answer: 4 1/5

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Dividing Fractions

To divide by a fraction, multiply by its reciprocal (flip the second fraction).

Worked Example 6

Calculate $5/6 \div 2/3$5/6÷2/3

$= 5/6 \times 3/2 = 15/12 = 5/4 = 1 1/4$=5/6×3/2=15/12=5/4=11/4

Answer: 1 1/4

Worked Example 7

Calculate $3 1/2 \div 1 1/4$31/2÷11/4

Convert: $7/2 \div 5/4 = 7/2 \times 4/5 = 28/10 = 14/5 = 2 4/5$7/2÷5/4=7/2×4/5=28/10=14/5=24/5

Answer: 2 4/5

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Fraction of a Quantity

To find a fraction of a quantity, divide by the denominator then multiply by the numerator.

Worked Example 8

Find 3/7 of 84 kg.

$84 \div 7 = 12$84÷7=12

$12 \times 3 = 36$12×3=36

Answer: 36 kg

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Complex Fraction Problems

Worked Example 9

Calculate $(2/3 + 1/4) \div (5/6 - 1/3)$(2/3+1/4)÷(5/6−1/3)

Step 1: Numerator: 2/3 + 1/4 = 8/12 + 3/12 = 11/12

Step 2: Denominator: 5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2

Step 3: $(11/12) \div (1/2) = 11/12 \times 2/1 = 22/12 = 11/6 = 1 5/6$(11/12)÷(1/2)=11/12×2/1=22/12=11/6=15/6

Answer: 1 5/6

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Proving Recurring Decimals Are Rational [H]

This is one of the most commonly examined Higher topics on Edexcel papers. You must use algebra to prove that a recurring decimal can be written as a fraction.

Method

1. Let x = the recurring decimal.
2. Multiply both sides by the appropriate power of 10.
3. Subtract to eliminate the recurring part.
4. Solve for x and simplify.

Worked Example 10 [H]

Prove that 0.4̇5̇ = 5/11

Let x = 0.454545...

100x = 45.454545...

Subtract: 100x - x = 45.454545... - 0.454545...

99x = 45

x = 45/99 = 5/11

Therefore 0.4̇5̇ = 5/11 ✓

Worked Example 11 [H]

Prove that 0.2̇7̇ = 3/11

Let x = 0.272727...

100x = 27.272727...

99x = 27

x = 27/99 = 3/11

Therefore 0.2̇7̇ = 3/11 ✓

Worked Example 12 [H]

Convert 0.16̇ to a fraction.

Let x = 0.1666...

10x = 1.666...

100x = 16.666...

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

90x = 15

x = 15/90 = 1/6

Answer: 1/6

Worked Example 13 [H]

Prove that 0.41̇8̇ = 23/55

Let x = 0.4181818...

The non-repeating part ("4") has 1 digit; the repeating block ("18") has 2 digits.

10x = 4.181818...

1000x = 418.181818...

1000x - 10x = 418.181818... - 4.181818...

990x = 414

x = 414/990

Simplify: 414/990. Divide both by 2: 207/495. Divide both by 3: 69/165. Divide both by 3: 23/55.

x = 23/55

Check: 23/55 = 0.41818... = 0.41̇8̇ ✓

Answer: 23/55

Edexcel Exam Tip: The command word "prove" means you must show full algebraic working. Simply writing the answer as a fraction earns no marks. Always start with "Let x = ...", show the multiplication, the subtraction, and the simplification. This is typically worth 3-4 marks on the exam.

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Recurring Decimal Proofs — Special Cases [H]

Proof that 0.9̇ = 1 [H]

Let x = 0.999...

10x = 9.999...

10x - x = 9.999... - 0.999...

9x = 9

x = 1

Therefore 0.9̇ = 1 exactly. This is not an approximation — they are the same number.

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The Reciprocal

The reciprocal of a number n is 1/n. Equivalently, for a fraction a/b, its reciprocal is b/a.

Number	Reciprocal
5	1/5 = 0.2
3/4	4/3
0.25 = 1/4	4
-2	-1/2

Note: The reciprocal of a number multiplied by the number itself always equals 1: $n \times (1/n) = 1$n×(1/n)=1. Zero has no reciprocal.

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Common Mistakes and Misconceptions

Mistake	Correction
Adding fractions by adding numerators and denominators	You must find a common denominator first: $1/3 + 1/4 \neq 2/7$1/3+1/4=2/7
Forgetting to convert mixed numbers to improper fractions	Always convert before multiplying or dividing
Not simplifying the final fraction	Always reduce to simplest form
In recurring decimal proofs, subtracting the wrong equations	The subtraction must align the repeating blocks
Thinking 0.9̇ < 1	0.9̇ = 1 exactly

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Practice Problems