Non-calculator synthesis

Objective. Use exact arithmetic and structure efficiently without a calculator.

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2. Worked example
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Learn: the key idea

Non-calculator synthesis combines exact fraction, surd and percentage arithmetic with structural shortcuts to solve problems quickly and accurately by hand. Look for ways to simplify before calculating, such as cancelling fractions early or recognising common factors, rather than working with large unsimplified numbers throughout. Keep answers in exact form (fractions, surds) unless a decimal is explicitly required, since rounding loses precision and may not match the mark scheme. For 3/4 × 8/9, cancel 4 into 8 first to get 3/1 × 2/9 = 6/9 = 2/3, avoiding large numerator and denominator multiplication. Efficient non-calculator work relies on recognising structure before brute-force computing.

Worked example

Without a calculator, work out 2/3 × 9/10, giving your answer as a fraction in simplest form.

1 $Cancel the 3 into the 9: 9 \div 3 = 3$
2 $Now compute 2/1 \times 3/10$
3 $This gives 6/10$
4 $Simplify to 3/5$

Answer: 3/5

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Common mistake. Students often multiply fractions fully before simplifying, creating large unwieldy numbers that are then hard to simplify correctly by hand, increasing the chance of arithmetic slips.

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