AQA GCSE Mathematics 8300: Fractions, decimals and percentages – Study Guide

This study guide teaches section 3.1.2, Fractions, decimals and percentages, of the AQA GCSE Mathematics (8300) specification, version 1.0, for teaching from September 2015 with exams from May/June 2017 onwards. It covers specification references N10, N11 and N12 in depth. Almost all of it is Foundation content, assessed at both tiers. One skill, changing recurring decimals into fractions and fractions into recurring decimals, sits in the specification’s “Higher content only” column and is labelled Higher tier only below.

Any of the three papers can test it. Paper 1 is non-calculator, so every conversion here must also be possible by hand.

Use this guide with the Fractions, decimals and percentages revision notes and the Fractions, decimals and percentages practice questions. For the whole of Topic 1, see the Number study guide, Number revision notes and Number practice questions. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free 10-minute diagnostics show where to start.

What this section covers

Spec	What you must be able to do	Tier
N10	Work interchangeably with terminating decimals and their corresponding fractions, including ordering	Both
N10	Change recurring decimals into their corresponding fractions and vice versa	Higher tier only
N11	Identify and work with fractions in ratio problems	Both
N12	Interpret fractions and percentages as operators, including percentage problems using a multiplier	Both

The specification links N11 to R8 (relating ratios to fractions) and N12 to R9 (percentages and percentage change). Percentage change, original-value problems and simple interest are taught under R9 in the ratio, proportion and rates of change revision notes, so this guide only uses them where they show an operator at work.

Terminating decimals and fractions (N10)

A terminating decimal stops after a finite number of digits, such as 0.4 or 2.875. Every terminating decimal can be written as a fraction with a denominator of 10, 100, 1000 and so on.

Decimal to fraction

Read the place value of the last digit, write that as the denominator, then simplify.

Worked example 1. Write 0.0875 as a fraction in its simplest form.

The last digit (5) is in the ten-thousandths column.
0.0875 = 875/10000
Divide top and bottom by 25:   35/400
Divide top and bottom by 5:    7/80

So 0.0875 = 7/80. Any common factor works, as long as you keep going until none is left.

A decimal greater than 1 works the same way: 2.45 = 245/100 = 49/20, or 2 9/20 as a mixed number.

Fraction to decimal

There are two methods. Use whichever is quicker for the numbers.

• Make the denominator a power of 10. 7/20 = 35/100 = 0.35, because 20 × 5 = 100.
• Divide the numerator by the denominator. Use short division, adding zeros after the decimal point.

Worked example 2. Without a calculator, write 13/40 as a decimal.

40 does not go neatly into 100, but it goes into 1000:  40 × 25 = 1000
13/40 = (13 × 25)/(40 × 25) = 325/1000 = 0.325

So 13/40 = 0.325.

A useful check: a fraction in its simplest form terminates only when its denominator’s prime factors are 2s and 5s (40 = 2³ × 5).

Percentages in the same family

A percentage is a number of parts per hundred, so it converts through the same routes. 17.5% = 17.5/100 = 0.175 = 175/1000 = 7/40.

Ordering (N10 notes: “including ordering”)

To order a mixed list, convert everything to decimals with the same number of decimal places, then compare.

Worked example 3. Write these in order of size, smallest first: 5/8, 0.61, 63%, 3/5, 0.627

5/8   = 0.625
0.61  = 0.610
63%   = 0.630
3/5   = 0.600
0.627 = 0.627

Smallest first: 0.600, 0.610, 0.625, 0.627, 0.630

Answer: 3/5, 0.61, 5/8, 0.627, 63%. Always give the answer in the original forms, not as your converted decimals.

Recurring decimals (N10, Higher tier only)

A recurring decimal has a block of digits that repeats forever. Dot notation puts a dot over the first and last digit of the block:

• 0.3̇ means 0.3333…
• 0.3̇6̇ means 0.363636…
• 0.83̇ means 0.83333… (only the 3 repeats)
• 0.2̇85714̇ means 0.285714285714…

Fraction to recurring decimal

Divide the numerator by the denominator and keep going until a remainder repeats. Once a remainder repeats, the digits repeat too.

4/11:  11 into 4.0000...  ->  0.3636...  =  0.3̇6̇
5/6:   6 into 5.0000...   ->  0.8333...  =  0.83̇
2/7:   7 into 2.000000... ->  0.285714285714...  =  0.2̇85714̇

Recurring decimal to fraction

The method removes the recurring part by subtraction.

1. Let x equal the decimal. Write out several repeats.
2. Multiply x by a power of 10 so that one full repeating block moves to the left of the decimal point.
3. If the repeat does not start straight after the point, also multiply x by a power of 10 that moves just the non-repeating digits across.
4. Subtract the two equations. The recurring tails cancel exactly.
5. Solve for x and simplify.

Worked example 4. Write 0.5̇1̇ as a fraction in its simplest form.

x    =  0.515151...
100x = 51.515151...       (two-digit block, so multiply by 100)
100x − x = 51.515151... − 0.515151...
99x  = 51
x    = 51/99 = 17/33

So 0.5̇1̇ = 17/33.

Worked example 5. Write 0.36̇ as a fraction in its simplest form.

x    =  0.36666...
10x  =  3.66666...        (moves the non-recurring 3 across)
100x = 36.66666...        (moves one repeat of the 6 across as well)
100x − 10x = 36.666... − 3.666...
90x  = 33
x    = 33/90 = 11/30

So 0.36̇ = 11/30. Lining up the tails as shown keeps both sides whole numbers.

Worked example 6. Write 0.12̇7̇ as a fraction in its simplest form.

x     =   0.1272727...
10x   =   1.272727...     (non-recurring 1 moved across)
1000x = 127.272727...     (one full block, 27, moved across as well)
1000x − 10x = 990x = 126
x = 126/990 = 7/55

So 0.12̇7̇ = 7/55.

There is a shortcut for pure recurring decimals: a one-digit block over 9, a two-digit block over 99, and so on (0.5̇1̇ = 51/99). Use it to check, but when a question says “show that” or “prove”, write out the algebra. A bare quoted result is not a method.

Fractions in ratio problems (N11)

A ratio compares parts with each other. A fraction compares a part with a whole. You need to move between the two in both directions.

• In the ratio a : b, the first quantity is a/(a + b) of the total and the second is b/(a + b) of the total.
• The first quantity is a/b of the second, and the second is b/a of the first.
• If A is p/q of B, then A : B = p : q.

For example, if juniors and seniors in a club are in the ratio 4 : 7, then juniors are 4/11 of the club and seniors are 7/11. The number of juniors is 4/7 of the number of seniors.

Worked example 7. Sana and Tom share some money. Sana gets 2/7 of it. Tom gets £45 more than Sana. How much money was shared?

Sana gets 2/7, so Tom gets 5/7.
Sana : Tom = 2 : 5        (7 parts in total)
Difference = 5 − 2 = 3 parts = £45
1 part = £45 ÷ 3 = £15
Total = 7 parts = 7 × £15 = £105

Answer: £105. Check: £75 − £30 = £45. The key step is turning “2/7 of the money” into 2 : 5, not 2 : 7.

Fractions and percentages as operators (N12)

An operator is something you do to a number. “3/8 of” means “multiply by 3/8”. “35% of” means “multiply by 35/100”, which is the same as multiplying by 0.35.

3/8 of 56  = 56 ÷ 8 × 3 = 7 × 3 = 21
35% of 240 = 10% is 24, so 30% is 72; 5% is 12;  72 + 12 = 84
           (or 0.35 × 240 = 84 with a calculator)

Multipliers

The specification notes say you should interpret percentage problems “using a multiplier”. The multiplier is the decimal you multiply by to get the answer in one step.

Instruction	Multiplier
Find 35% of	0.35
Increase by 35%	1 + 0.35 = 1.35
Decrease by 35%	1 − 0.35 = 0.65
Find 140% of	1.4
Increase by 2/5	1 + 2/5 = 7/5
Decrease by 1/4	1 − 1/4 = 3/4

A percentage greater than 100% gives a multiplier greater than 1, so the result is larger than the starting amount.

Worked example 8. (calculator) A monthly fee of £86.40 is increased by 12.5%. Find the new fee.

Multiplier = 1 + 0.125 = 1.125
86.40 × 1.125 = 97.2

New fee £97.20. Money answers need two decimal places, so write £97.20, not £97.2.

Combining operators

Two operators applied one after the other combine by multiplying the multipliers.

Increase by 1/4, then decrease by 1/5:   5/4 × 4/5 = 1     (back to the start)
Increase by 10%, then decrease by 10%:   1.1 × 0.9 = 0.99  (a 1% loss overall)

The 10% decrease acts on a larger amount, so it removes more than the increase added.

Fraction of a fraction

When a fraction acts on part of a group, multiply the fractions along the chain.

Worked example 9. In a choir, 3/5 of the members are girls and the rest are boys. 1/4 of the girls and 1/3 of the boys play an instrument. What fraction of the choir plays an instrument?

Boys = 1 − 3/5 = 2/5 of the choir
Girls who play = 1/4 × 3/5 = 3/20 of the choir
Boys who play  = 1/3 × 2/5 = 2/15 of the choir
Total = 3/20 + 2/15 = 9/60 + 8/60 = 17/60

Answer: 17/60. You cannot add 1/4 and 1/3 directly: they are fractions of different groups.

Undoing an operator

If you know the result after an operator, divide by the operator to get back. If 3/4 of a number is 45, the number is 45 ÷ 3/4 = 45 × 4/3 = 60. Original-value percentage problems use the same idea with a multiplier (R9).

Calculator and non-calculator skills

On Paper 1 you must convert, order and apply operators by hand: powers-of-10 conversions, percentages built from 10%, 5% and 1%, and (Higher tier only) short division for recurring decimals. On Papers 2 and 3, still write the multiplier or fraction you used, because method marks depend on it.

Common errors

• Writing 0.0875 as 875/1000 by miscounting decimal places. Count the digits after the point: four digits means ten-thousandths.
• Ordering by reading the digits as whole numbers, so 0.627 is put above 63% (0.63) because 627 is bigger than 63. Pad to the same number of decimal places first.
• (Higher tier only) Multiplying by 10 when the repeating block has two digits, so the tails do not cancel.
• (Higher tier only) In 0.36̇, treating both digits as recurring. The dot is only over the 6.
• Turning “Sana gets 2/7” into the ratio 2 : 7 instead of 2 : 5.
• Using 0.35 as the multiplier for a 35% increase. That finds 35% of the amount, not the new amount.
• Adding the percentage changes in a two-step problem instead of multiplying the multipliers.

Where to go next

Fix the facts in the revision notes, then try the practice questions.

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015, exams from May/June 2017, version 1.0, published by AQA – section 3.1.2 Fractions, decimals and percentages (N10, N11, N12).