Simplification and approximation questions test fast, accurate arithmetic. They use the order of operations, fractions, decimals, powers, roots, and sensible rounding to reduce complex calculations.

Order of operations

Follow BODMAS or PEMDAS to evaluate an expression in the correct order. Complete brackets first, then orders such as powers and roots, followed by division and multiplication from left to right, and then addition and subtraction from left to right.

• B: brackets; O: orders; D: division; M: multiplication; A: addition; S: subtraction.
• Multiplication does not always come before division; solve whichever appears first from left to right.
• Likewise, solve addition and subtraction from left to right.

Fractions and decimals

• Convert mixed numbers to improper fractions before performing operations.
• Use cancellation before multiplying fractions to keep numbers small.
• Align decimal points for addition and subtraction.
• In decimal multiplication, count the total decimal places in the factors.
• In decimal division, make the divisor a whole number by shifting both decimal points equally.

Powers and roots

• a^m x a^n = a^(m + n) and a^m / a^n = a^(m - n).
• (a^m)^n = a^(mn) and (ab)^n = a^n b^n.
• A square root is a value which, when multiplied by itself, gives the original number.
• A cube root is a value which, when cubed, gives the original number.
• Memorise common squares up to at least 30 and cubes up to at least 10 for speed.

Useful identities

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• a^2 - b^2 = (a + b)(a - b)
• Use identities when numbers are near convenient bases such as 10, 50, or 100.

Approximation and rounding

Approximation gives a close value when an exact calculation is not required. Round numbers to nearby convenient values while preserving the scale of the calculation.

• For whole-number rounding, 5 or more rounds up; less than 5 rounds down.
• Round 49.8 to 50, 198 to 200, and 0.498 to 0.5 when estimation is appropriate.
• Estimate a quotient by rounding both dividend and divisor to compatible values.
• Do not round too early when several steps affect the final answer.

How to solve questions

1. Remove brackets and apply BODMAS carefully.
2. Convert fractions or decimals into the form that is easiest to calculate.
3. Use factor cancellation, identities, and common squares or cubes to reduce work.
4. For approximate values, round to compatible numbers and estimate the result.
5. Use the size and sign of the result to check whether the answer is reasonable.