Mathematics: Algebra, Geometry, and Quantitative Reasoning
Core concepts
The mathematics section rewards clean fundamentals more than clever tricks. Master these clusters: arithmetic and number properties (divisibility, LCM/GCD, fractions, ratios, percentages, averages); algebra (linear equations and inequalities, systems of two equations, factoring, quadratic formula, exponent rules); geometry (angles on lines and in triangles, similar triangles, Pythagoras, areas and volumes, circle formulas); and data/word problems (rate–time–distance, work, mixtures, simple probability).
Traps to watch
- Percent change is not symmetric: a 20% rise then a 20% fall leaves 96%, not 100%. Always compute on the new base.
- Units: convert before calculating (minutes to hours, cm to m).
- Sign errors when moving terms across an inequality: multiplying or dividing by a negative flips the direction.
- "Average" vs "sum": average = sum ÷ count, so sum = average × count. Many questions hide the count.
- Geometry figures are illustrative, not to scale - never measure; use the stated values and theorems.
Rule of thumb: the substitution check
After solving any equation, plug your answer back into the original. For word problems, ask whether the number is physically sensible (a positive time, a plausible age).
Worked example
A shop raises a price of 400 by 25%, then discounts the new price by 20%. Final price?
- 25% of 400 = 100, so new price = 500.
- 20% of 500 = 100, so final = 400. The two changes cancel here - but only because the percentages were applied to different bases; do not generalize the cancellation.