Number and algebra
Number and algebra
Number questions reward fast, correct arithmetic and proportional reasoning more than formal algebra. You need: the order of operations (brackets, indices, then × and ÷ left to right, then + and −); confident work with directed numbers; conversion between fractions, decimals and percentages; ratio and rate problems (speed, unit price, dosage-style rates); direct proportion; substitution into simple formulas; and one- and two-step linear equations.
Know the difference between finding a percentage of an amount and expressing a change as a percentage (change ÷ original × 100). Learn reverse percentage early: if a price after a 15% discount is $68, the original was 68 ÷ 0.85 = $80, not 68 × 1.15.
Common traps: adding then subtracting the same percentage does not restore the original value; dividing by a fraction smaller than 1 makes the result larger; (−3)² = 9 but −3² = −9; rates need matching units, so convert minutes to hours before dividing; and rounding mid-calculation distorts the final answer, so round only at the end.
Worked example. A jacket is discounted by 20%, then a further 10% is taken at the register. The final price is $108. What was the original price?
Multipliers multiply: 0.8 × 0.9 = 0.72, so $108 is 72% of the original. Original = 108 ÷ 0.72 = $150. Note the total discount is 28%, not 30%, because the second 10% applies to the already-reduced price.