Number
Number
Number assesses exact and approximate calculation: fractions, decimals, percentages, ratio and proportion, standard form, indices, surds, bounds and rounding. Questions often combine two ideas, so choose a method before calculating.
Core rules
- Convert between fractions, decimals and percentages using equivalent fractions; a percentage change multiplies by a scale factor, such as +15% = ×1.15 and −10% = ×0.90.
- For repeated proportional change, multiply scale factors; do not add percentages.
- Standard form is a × 10^n with 1 ≤ a < 10. Adjust the index when multiplying or dividing powers of 10.
- For a rounded value to a given unit, the upper bound is halfway up and the lower bound is halfway down; use bounds for maximum/minimum calculations.
- Index laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn).
Worked example
A coat costs $240 before a 15% increase and a later 10% reduction. The multiplier is 1.15 × 0.90 = 1.035, so the final price is 240 × 1.035 = $248.40. The net change is +3.5%, not +5%, because the second percentage acts on the increased price.
Traps
- Adding percentage changes instead of multiplying scale factors.
- Rounding too early in multi-step work; keep full accuracy until the final answer.
- Using the wrong base for reverse percentages: if $84 is after a 20% increase, solve 1.20x = 84, so x = $70.
- Mixing up lower and upper bounds when subtracting.