Non-Calculator Techniques

Paper 1 of the AQA GCSE Mathematics exam is a non-calculator paper. This means you must be able to carry out all calculations by hand — including those involving fractions, percentages, surds, and trigonometric exact values. This lesson covers the mental and written methods you need to master for Paper 1, along with time management strategies.

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Why Non-Calculator Skills Matter

Paper 1 is worth 80 marks — one-third of your entire GCSE grade. Many students find Paper 1 the most challenging because they have become reliant on calculators. The key to success is practice: the more you work without a calculator, the faster and more accurate you become.

Even on Papers 2 and 3 (calculator papers), strong non-calculator skills help you:

• Check your calculator answers using estimation
• Spot errors when a calculator result looks wrong
• Save time on simple calculations instead of typing them into the calculator

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Mental Maths Strategies

Strong mental arithmetic is the foundation of Paper 1. Here are strategies to speed up your mental calculations.

Addition and subtraction

Strategy	Example	Method
Partitioning	67 + 28	67 + 20 = 87, then 87 + 8 = 95
Rounding and adjusting	347 + 199	347 + 200 = 547, then 547 - 1 = 546
Compensating	83 - 47	83 - 50 = 33, then 33 + 3 = 36
Counting on	1003 - 997	Count from 997 to 1003 = 6

Multiplication shortcuts

Strategy	Example	Method
Multiply by 5	48 x 5	48 x 10 = 480, then 480 / 2 = 240
Multiply by 25	36 x 25	36 x 100 = 3600, then 3600 / 4 = 900
Multiply by 9	17 x 9	17 x 10 = 170, then 170 - 17 = 153
Multiply by 11	34 x 11	34 x 10 = 340, then 340 + 34 = 374
Double and halve	14 x 16	7 x 32 = 224 (halve 14, double 16)

Division shortcuts

Strategy	Example	Method
Divide by 5	345 / 5	345 x 2 = 690, then 690 / 10 = 69
Divide by 4	148 / 4	148 / 2 = 74, then 74 / 2 = 37
Factor method	252 / 12	252 / 4 = 63, then 63 / 3 = 21

Exam Tip: Practise mental arithmetic for 10 minutes every day in the weeks before the exam. Speed and accuracy in basic calculations free up time for harder questions on Paper 1.

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Long Multiplication

When numbers are too large for mental maths, use the column method (long multiplication).

Method

1. Write the larger number on top.
2. Multiply by the units digit of the bottom number — write this row.
3. Multiply by the tens digit — write this row shifted one place left (or put a zero placeholder).
4. Add the rows together.

Worked Example 1

Calculate: 247 x 36

Step 1: 247 x 6

• 7 x 6 = 42 (write 2, carry 4)
• 4 x 6 = 24, plus 4 = 28 (write 8, carry 2)
• 2 x 6 = 12, plus 2 = 14

First row: 1482

Step 2: 247 x 30

• 7 x 3 = 21 (write 1, carry 2)
• 4 x 3 = 12, plus 2 = 14 (write 4, carry 1)
• 2 x 3 = 6, plus 1 = 7

Second row: 7410

Step 3: Add: 1482 + 7410 = 8892

Worked Example 2

Calculate: 508 x 73

508 x 3 = 1524

508 x 70 = 35560

1524 + 35560 = 37084

Exam Tip: Always estimate first to check your answer is reasonable. For 247 x 36, estimate 250 x 36 = 250 x 40 - 250 x 4 = 10000 - 1000 = 9000. Our answer of 8892 is close, so it is likely correct.

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Long Division

Long division is essential for Paper 1 when the numbers are too large for short division or mental methods.

Method

1. Divide, multiply, subtract, bring down — repeat.
2. Continue until you reach the required accuracy or the division is complete.

Worked Example 3

Calculate: 7896 / 24

• 24 into 78: 3 times (3 x 24 = 72). Remainder 6. Bring down 9 to make 69.
• 24 into 69: 2 times (2 x 24 = 48). Remainder 21. Bring down 6 to make 216.
• 24 into 216: 9 times (9 x 24 = 216). Remainder 0.

Answer: 329

Worked Example 4

Calculate: 5231 / 17

• 17 into 52: 3 times (3 x 17 = 51). Remainder 1. Bring down 3 to make 13.
• 17 into 13: 0 times. Bring down 1 to make 131.
• 17 into 131: 7 times (7 x 17 = 119). Remainder 12.

Answer: 307 remainder 12 or 307.7 (to 1 d.p.)

Exam Tip: If a question asks for an exact answer and long division gives a remainder, express your answer as a fraction or a decimal as appropriate. Check whether the question specifies the form of the answer.

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Fraction Arithmetic Without a Calculator

Fractions appear heavily on Paper 1. You need to be fluent in all four operations.

Addition and subtraction of fractions

1. Find a common denominator (the LCM of the denominators).
2. Convert each fraction to equivalent fractions with this denominator.
3. Add or subtract the numerators.
4. Simplify the result if possible.

Worked Example 5

Calculate: 2/3 + 3/4

LCM of 3 and 4 = 12

2/3 = 8/12

3/4 = 9/12

8/12 + 9/12 = 17/12 = 1 and 5/12

Worked Example 6

Calculate: 5/6 - 1/4

LCM of 6 and 4 = 12

5/6 = 10/12

1/4 = 3/12

10/12 - 3/12 = 7/12

Answer: 7/12

Multiplication of fractions

Multiply numerators together and denominators together. Simplify.

Worked Example 7

Calculate: 3/5 x 4/7

(3 x 4) / (5 x 7) = 12/35

Answer: 12/35

Division of fractions

Flip the second fraction (take the reciprocal) and multiply.

Worked Example 8

Calculate: 2/3 divided by 5/6

2/3 x 6/5 = 12/15 = 4/5

Answer: 4/5

Mixed numbers

Convert mixed numbers to improper fractions before multiplying or dividing.

Worked Example 9

Calculate: 1 and 2/3 x 2 and 1/4

Convert: 5/3 x 9/4

Multiply: 45/12 = 15/4 = 3 and 3/4

Exam Tip: Always simplify your final answer. An unsimplified fraction such as 12/15 will often lose the final accuracy mark. Cross-cancel before multiplying if you can — it makes the arithmetic simpler.

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Percentage Calculations Without a Calculator

Percentages are tested on Paper 1 every year. Master these methods for finding percentages without a calculator.

The 10% method (build-up method)

This method works by finding 10% first, then building up to the required percentage.

Percentage	How to Find It
10%	Divide by 10
5%	Half of 10%
1%	Divide by 100
20%	Double 10%
25%	Divide by 4 (or half of 50%)
50%	Divide by 2
15%	10% + 5%
17.5%	10% + 5% + 2.5%

Worked Example 10

Find 35% of £460

10% of £460 = £46

30% = £46 x 3 = £138

5% = £46 / 2 = £23

35% = £138 + £23 = £161

Worked Example 11

Find 12.5% of 240

10% of 240 = 24

2.5% = 24 / 4 = 6

12.5% = 24 + 6 = 30

Alternatively: 12.5% = 1/8, so 240 / 8 = 30.

Percentage increase and decrease

To increase by a percentage: find the percentage amount and add it.

To decrease by a percentage: find the percentage amount and subtract it.

Worked Example 12

Increase £350 by 15%

10% of £350 = £35

5% of £350 = £17.50

15% = £35 + £17.50 = £52.50

New amount = £350 + £52.50 = £402.50

Fraction-percentage equivalents you must know

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%
1/10	0.1	10%
1/100	0.01	1%

Exam Tip: AQA often asks "non-calculator" percentage questions that look like they need a calculator but don't. Recognising that 12.5% = 1/8 or that 37.5% = 3/8 can turn a multi-step build-up into a single division.

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Converting Between Fractions, Decimals, and Percentages

Being able to move fluently between these representations is essential on Paper 1.

Fraction to decimal

Divide the numerator by the denominator.

3/8 = 3 divided by 8 = 0.375

Decimal to percentage

Multiply by 100.

0.375 x 100 = 37.5%

Percentage to fraction

Write over 100 and simplify.

37.5% = 37.5/100 = 375/1000 = 3/8

Fraction to percentage

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

Method 2: Find an equivalent fraction with denominator 100.

7/20 = 35/100 = 35%

Recurring decimals to fractions [H]

For Higher tier students, you need to convert recurring decimals to fractions algebraically.

Worked Example 13

Convert 0.272727... to a fraction.

Let x = 0.272727...

100x = 27.272727...

100x - x = 27.272727... - 0.272727...

99x = 27

x = 27/99 = 3/11

Answer: 3/11

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Surd Manipulation [H]

Surds are square roots that cannot be simplified to a whole number. On Paper 1, you may need to simplify, add, subtract, multiply, or rationalise surds without a calculator.

Simplifying surds

Look for the largest perfect square factor.

Worked Example 14

Simplify: sqrt(72)

72 = 36 x 2

sqrt(72) = sqrt(36) x sqrt(2) = 6 sqrt(2)

Answer: 6 sqrt(2)

Adding and subtracting surds

You can only add or subtract surds with the same radicand (the number under the root).

Worked Example 15

Simplify: 3 sqrt(5) + 2 sqrt(5)

= (3 + 2) sqrt(5) = 5 sqrt(5)

Worked Example 16

Simplify: sqrt(50) - sqrt(18)

sqrt(50) = sqrt(25 x 2) = 5 sqrt(2)

sqrt(18) = sqrt(9 x 2) = 3 sqrt(2)

5 sqrt(2) - 3 sqrt(2) = 2 sqrt(2)

Answer: 2 sqrt(2)

Multiplying surds

sqrt(a) x sqrt(b) = sqrt(ab)

Worked Example 17

Simplify: sqrt(3) x sqrt(12)