Word Problems and Applied Quantitative Reasoning
What this domain tests
Applied quantitative reasoning asks you to convert a verbal scenario into an equation, rate, or proportion, then answer the exact quantity requested. Typical settings include work rates, mixtures, average speed, consecutive integers, percent applications, and unit conversions.
Core rules
- Define the variable in words before writing an equation.
- Rate problems multiply: distance = rate x time; work = rate x time. For combined work, add rates: 1/a + 1/b = 1/t.
- Average speed is total distance divided by total time, not the average of the speeds.
- Consecutive integers are n, n + 1, n + 2; consecutive even or odd integers differ by 2.
- Mixture problems track the amount of a substance: concentration x volume before mixing equals the total after mixing.
- Backsolving from the answer choices is efficient when the algebra is messy, but substitute into the original wording.
Worked example
One painter finishes a room in 6 hours; a second finishes it in 12 hours. Working together, their combined rate is 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 room per hour. The reciprocal gives 4 hours for one room. The answer is not the arithmetic mean of 6 and 12, which would be 9 hours.
Traps
Match units before computing: minutes with hours, centimeters with meters. A percent increase and a percent decrease at the same rate do not cancel. In work problems, add rates, not times. If a question asks how many items remain after a discount, do not stop at the discount amount. Re-read the final sentence to confirm whether it asks for a total, a difference, or a per-unit value.