Start with the order of operations
BODMAS gives the order: Brackets, Orders (powers and roots), Division and Multiplication (left to right), Addition and Subtraction (left to right). Most errors come from doing multiplication before division when division stands first, or from sign errors with brackets.
Worked examples
Example 1. 48 ÷ 6 × 2 + 5 = 8 × 2 + 5 = 21 (left to right, not 48 ÷ 12).
Example 2. 15 + 3 × (12 − 8)² ÷ 4 = 15 + 3 × 16 ÷ 4 = 15 + 12 = 27.
Example 3 (fractions). 3/4 + 5/6 − 1/3 = 9/12 + 10/12 − 4/12 = 15/12 = 5/4.
Example 4 (decimals). 0.6 × 0.5 + 1.2 ÷ 0.4 = 0.3 + 3 = 3.3.
Example 5 (surds). (√12 + √3)² = 12 + 3 + 2 × √36 = 15 + 12 = 27. Also √12 = 2√3, so (2√3 + √3)² = (3√3)² = 27 ✓.
Example 6 (approximation). 49.8 × 19.9 ÷ 9.95 ≈ 50 × 20 ÷ 10 = 100. Round sensibly and keep the order of operations.
Example 7 (identities). 997² = (1,000 − 3)² = 1,000,000 − 6,000 + 9 = 994,009.
Shortcuts worth learning
- (a + b)² and (a − b)² for squares near round numbers.
- a² − b² = (a + b)(a − b): 52² − 48² = 100 × 4 = 400.
- Multiplication by 5, 25, 125: × 5 = × 10 ÷ 2; × 25 = × 100 ÷ 4; × 125 = × 1,000 ÷ 8.
- Fractions of common percentages such as 12.5% = 1/8.
- Cancel before multiplying.
See mental maths techniques for more.
Common traps
- Ignoring left-to-right order for division and multiplication.
- Sign errors when removing brackets with a minus before them.
- Rounding too early in approximation, so that the error accumulates.
- Mishandling decimals: count decimal places carefully.
Check your answer before you mark it
- Estimate first. If the expression is about 48 × 19 ÷ 12, expect something near 76; an answer of 7.6 or 760 is a decimal slip.
- Check the last digit. The units digit of a product depends only on the units digits of the factors: 47 × 38 ends in 6 (7 × 8 = 56), so 1,786 is plausible and 1,788 is not.
- Use digit sums for a quick test of multiplication: the digit sum of the product equals the digit sum of the product of the digit sums, modulo 9. For 47 × 38 = 1,786: digit sums 11 and 11 give 2 × 2 = 4 modulo 9, and 1 + 7 + 8 + 6 = 22, which gives 4. ✓
These checks take under ten seconds and catch most slips in long expressions.
A practice routine
- Day 1: BODMAS with brackets and powers, 15 expressions.
- Day 2: fractions and decimals.
- Day 3: surds, identities and approximation.
- Day 4: a timed set of 15 at 40 seconds each.
A fully worked mixed expression
Evaluate: (36 ÷ 4 + 3²) × 2 − 5 × (14 − 6) ÷ 4. Brackets first: 36 ÷ 4 = 9; 3² = 9; so the first bracket is 18. The second bracket is 14 − 6 = 8. Now the expression is 18 × 2 − 5 × 8 ÷ 4. Multiplication and division from left to right: 18 × 2 = 36; 5 × 8 = 40; 40 ÷ 4 = 10. Then subtraction: 36 − 10 = 26. Writing each stage on its own line takes ten seconds and prevents the usual slips.
Where it appears
Simplification and approximation questions appear in banking quantitative aptitude (especially prelims), SSC CGL and railway mathematics. Memorise squares, cubes and roots to speed these up, and revise percentage.
Practice tests below are published by institutes and educators on TestMint.
