AQA GCSE Mathematics 8300: Fractions, decimals and percentages – Revision Notes

These notes condense section 3.1.2, Fractions, decimals and percentages (N10, N11 and N12), of the AQA GCSE Mathematics (8300) specification, version 1.0, for teaching from September 2015 with exams from May/June 2017 onwards. Everything here is Foundation content, assessed at both tiers, except converting between recurring decimals and fractions, which is Higher tier only. Any of the three papers can test this section, and Paper 1 is non-calculator.

For full explanations and worked examples, use the Fractions, decimals and percentages study guide. Then test yourself with the practice questions. The topic-level Number revision notes cover the rest of Topic 1, and percentage change and original-value problems (R9) are in the ratio, proportion and rates of change revision notes. Course hub: AQA GCSE Mathematics. Every statement is on the printable checklist. Not sure where you stand? Try the free 10-minute diagnostics.

Key words

• Terminating decimal: stops after a finite number of digits (0.125).
• Recurring decimal: a digit or block repeats forever (0.1666…). (Higher tier only for conversions)
• Dot notation: a dot over the first and last digit of the repeating block. 0.1̇6̇ = 0.161616…, but 0.16̇ = 0.1666…
• Operator: something you do to a number. “2/3 of” means “× 2/3”.
• Multiplier: the single number you multiply by to apply a percentage or fraction change in one step.

Equivalents to know by heart (N10)

You save time on Paper 1 if these are instant.

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/8	0.125	12.5%
3/8	0.375	37.5%
1/10	0.1	10%
1/20	0.05	5%
1/25	0.04	4%
1/40	0.025	2.5%
1/3	0.3̇	33 1/3%
2/3	0.6̇	66 2/3%
1/6	0.16̇	16 2/3%
1/9	0.1̇	11 1/9%
1/11	0.0̇9̇	9 1/11%

Build others from these. 7/8 = 7 × 0.125 = 0.875. 3/40 = 3 × 0.025 = 0.075.

Converting: method in steps

Decimal → fraction (terminating)

1. Count the decimal places: 1 place → /10, 2 places → /100, 3 places → /1000.
2. Write the digits over that power of 10.
3. Cancel by common factors until none is left.

Fraction → decimal

1. If the denominator divides into 10, 100, 1000 or 10000, scale up to it.
2. Otherwise divide the numerator by the denominator (short division, add zeros after the point).
3. If a remainder repeats, the digits will repeat: you have a recurring decimal.

Check: in simplest form, a fraction terminates only when its denominator has no prime factors other than 2 and 5.

Percentage ↔ decimal ↔ fraction

• Percentage → decimal: divide by 100 (move the point two places left). 4.5% = 0.045.
• Decimal → percentage: multiply by 100. 1.3 = 130%.
• Percentage → fraction: put over 100 and simplify. 85% = 85/100 = 17/20.

Ordering (N10)

1. Convert every value to a decimal.
2. Pad with zeros so all have the same number of decimal places.
3. Compare column by column from the left.
4. Write the answer using the original values.

Negative values reverse the order: −0.45 is smaller than −0.4.

Recurring decimals to fractions (Higher tier only)

Method in steps

1. Let x = the decimal and write out at least two repeats.
2. Multiply by 10ⁿ, where n = the number of digits in the repeating block, so that one block crosses the point. If there are non-repeating digits after the point, multiply x by another power of 10 that moves only those across.
3. Line up the two equations so the recurring tails match. Subtract.
4. Solve for x and simplify.

Decimal	Multiply to get	Subtract	Fraction
0.4̇	10x = 4.444…	9x = 4	4/9
0.7̇2̇	100x = 72.7272…	99x = 72	8/11
0.23̇	100x and 10x	90x = 21	7/30

Small worked reminder. 0.23̇ = 0.2333…

10x  =  2.333...
100x = 23.333...
90x  = 21     ->   x = 21/90 = 7/30

Fraction → recurring decimal: divide until a remainder repeats. 5/9 = 0.5̇. 7/30 = 0.2333… = 0.23̇.

For a “show that” question, the full algebra must be written. Quoting “two digits recurring goes over 99” with no working does not show anything.

Fractions in ratio (N11)

For a ratio a : b:

You want	Fraction
First quantity as a fraction of the total	a/(a + b)
Second quantity as a fraction of the total	b/(a + b)
First as a fraction of the second	a/b
Second as a fraction of the first	b/a

Going the other way: if X is 4/5 of Y, then X : Y = 4 : 5. If X is 4/9 of a total, then X : rest = 4 : 5.

Small worked reminder. Flour and sugar are mixed in the ratio 7 : 3. Flour is 7/10 of the mix. Sugar is 3/7 of the amount of flour.

Method in steps for “share” problems that give a fraction

1. Turn the fraction of the total into a ratio (2/9 of the total → 2 : 7).
2. Find what one part is worth from the information given (a difference, a total or one share).
3. Scale up to the quantity asked for.

Fractions and percentages as operators (N12)

“Of” means multiply.

2/9 of 63  = 63 ÷ 9 × 2 = 14
160% of 35 = 1.6 × 35   = 56

Multipliers

Change	Multiplier
Increase by r%	1 + r/100
Decrease by r%	1 − r/100
Increase by a fraction p/q	1 + p/q
Decrease by a fraction p/q	1 − p/q

Examples: decrease by 4.5% → × 0.955. Increase by 2/3 → × 5/3. Decrease by 3/5 → × 2/5.

Successive changes: multiply the multipliers, never add the percentages.

Increase by 2/3, then decrease by 2/5:   5/3 × 3/5 = 1   (no overall change)

Reading a multiplier back: a multiplier of 1.07 means a 7% increase. A multiplier of 0.92 means an 8% decrease. A multiplier of 2.5 means 250% of the original, which is a 150% increase.

Undoing an operator: divide by it. If 2/5 of a number is 18, the number is 18 ÷ 2/5 = 45.

Fraction of a fraction: when a fraction acts on part of a group, multiply along the chain, then add the separate branches with a common denominator.

Must-know distinctions

• “Percentage of” vs “increase by a percentage”. 30% of 50 is 15 (× 0.3). Increase 50 by 30% is 65 (× 1.3).
• Fraction of the total vs fraction of the other part. In 3 : 5, the first is 3/8 of the total but 3/5 of the second.
• 0.1̇6̇ vs 0.16̇. Dots on both digits means both repeat. One dot means only that digit repeats.
• Terminating vs recurring. 3/16 = 0.1875 terminates. 3/14 recurs, because 14 has a factor of 7.
• Rounded vs exact. 2/3 written as 0.67 is a rounded value. On a “give your answer as a fraction” question, a rounded decimal loses the accuracy mark.

Quick self-test

1. Write 0.85 as a fraction in its simplest form.
2. Write 3/16 as a decimal.
3. Write these in order, smallest first: 0.3, 7/25, 29%, 0.295
4. (Higher) Write 5/9 as a recurring decimal.
5. (Higher) Write 0.7̇2̇ as a fraction in its simplest form.
6. (Higher) Write 0.23̇ as a fraction in its simplest form.
7. Boys and girls in a club are in the ratio 7 : 3. What fraction of the club are boys? The girls are what fraction of the boys?
8. P is 4/5 of Q. Write P : Q.
9. Work out 2/9 of 63.
10. Write down the multiplier for a decrease of 4.5%.
11. Work out 160% of 35.
12. A price is increased by 2/3 and then the new price is decreased by 2/5. What is the overall effect?

Answers

1. 85/100 = 17/20
2. 3/16 = 1875/10000 = 0.1875
3. 7/25 = 0.280, 29% = 0.290, 0.295, 0.300 → 7/25, 29%, 0.295, 0.3
4. 0.5̇ (0.5555…)
5. 100x − x = 72, so x = 72/99 = 8/11
6. 100x − 10x = 23.333… − 2.333… = 21, so x = 21/90 = 7/30
7. Boys 7/10; girls are 3/7 of the boys
8. 4 : 5
9. 63 ÷ 9 × 2 = 14
10. 1 − 0.045 = 0.955
11. 1.6 × 35 = 56
12. 5/3 × 3/5 = 1, so no overall change

Where marks are usually lost

• Miscounting decimal places, so 0.045 becomes 45/100 instead of 45/1000.
• Not cancelling fully when the question says “simplest form”.
• Ordering 0.3 below 0.295 because 295 is bigger than 3 as a whole number.
• Leaving the ordered answer as converted decimals rather than the values given.
• (Higher) Multiplying by 10 for a two-digit repeating block, so the tails do not cancel.
• (Higher) Misreading the dot notation and making every digit recur.
• (Higher) Quoting a shortcut on a “show that” question without the algebra.
• Turning “A gets 3/8” into the ratio 3 : 8 instead of 3 : 5.
• Using 0.3 as the multiplier for a 30% increase.
• Adding percentage changes in a two-step problem instead of multiplying the multipliers.

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).