Quantitative Reasoning: Arithmetic, Algebra, Geometry and Data Interpretation
The Quantitative Reasoning domain tests numerical fluency rather than advanced mathematics: percentages, ratio and proportion, averages and rates, linear and quadratic algebra, basic geometry, and the interpretation of tables and graphs with elementary statistics. Most items are solvable in under a minute once the correct relationship is written down, so the real skill is translating the wording into an equation before calculating.
Percentages, ratio and rates
- Percentage change = (new − old) ÷ old × 100. Successive changes multiply: +20% then −20% gives 0.8 × 1.2 = 0.96, a net fall of 4%, not zero.
- A ratio a : b = 3 : 5 means the quantities are 3k and 5k; fix the total to find k.
- Work rates add: if A finishes in 6 hours and B in 3, together they do 1/6 + 1/3 = 1/2 of the job per hour.
- Average speed = total distance ÷ total time, never the mean of two speeds.
Algebra and geometry
Transpose linear equations by doing the same operation to both sides; an inequality flips when you multiply or divide by a negative. Geometry high-yields: angle sums in triangles and straight lines, exterior angles, Pythagorean triples (3-4-5, 5-12-13), the 30-60-90 ratio x : x√3 : 2x, the 45-45-90 ratio x : x : x√2, triangle area ½ × base × height, circle area πr², circumference 2πr, and sector area as a fraction of the circle.
Data interpretation and statistics
Mean = total ÷ count, so total = mean × count; a combined mean is weighted by group size. The median requires sorting first, and with an even count it is the average of the two middle values. Outliers shift the mean but barely move the median; the range ignores everything between the extremes.
Worked example
A shirt is marked 25% off, and a further 20% is taken off the reduced price. What single discount is equivalent? Paying 75% then 80% means 0.75 × 0.80 = 0.60 of the original price, so the equivalent single discount is 40%, not 45%.
Traps that cost marks
- Adding percentages instead of multiplying them.
- Confusing percentage points with percent change.
- Treating a geometry diagram as drawn to scale.
- Mixing units, such as minutes and hours, in rate problems.