Quantitative Reasoning: Ratios, Rates, Percents, and Number Sense
Quantitative Reasoning questions ask you to compare quantities, scale them up or down, and judge whether an answer is reasonable. The core tools are ratios and proportions, unit rates, percent relationships, fraction–decimal–percent equivalence, order of operations, absolute value, and estimation.
Set up proportions by matching labels: if 3 pounds cost $7.50, write dollars over pounds on both sides - $7.50/3 = x/5 - and cross-multiply. Unit rates are just proportions with a denominator of 1, which is why "per hour" or "per ounce" problems are best solved by dividing before you multiply.
Percent problems come in three flavors: finding a part (0.35 × 80), finding the whole (part ÷ rate), and finding the rate (part ÷ whole). Percent change always compares the change to the original amount: change ÷ original × 100.
Traps to watch. (1) Chained percents do not add - a 20% discount followed by 10% off is not 30% off. (2) "Percent more than" vs. "percent of" use different bases. (3) Averaging two speeds is not the same as averaging the times. (4) Estimating is often faster than exact computation; answer choices are usually far enough apart that rounding exposes the right one.
Worked example. A jacket is listed at $80, marked down 25%, then taxed at 8%. Sequential multipliers make this clean: 80 × 0.75 = $60, then 60 × 1.08 = $64.80. Note that 80 × 0.83 = $66.40 is wrong - the tax applies to the reduced price, not the original.
Rule of thumb: multiply by (1 ± the rate) for each successive change, and keep every step labeled with its units.