SAT Math — ratios, rates, units and scale

This part of the test rewards a habit more than a technique: keeping track of what the units are and what they must become. A learner who writes the units down beside the numbers rarely sets a proportion upside down, rarely multiplies by a conversion factor that should have divided, and never reports a rate where a total was wanted.

Three questions set up a proportion from a situation — a recipe scaled up, a distance read off a map, a machine working at a steady rate. The explanations show two ways through each: finding the amount for one unit first, which is usually quicker, or writing both ratios the same way round. The wrong options come from scaling by the wrong factor and from reporting the number of cookies as a quantity of flour, which the units alone rule out on sight.

Three more are conversions, and they get harder in a deliberate order. One converts a quantity between two units of weight, where the useful check is whether the answer should be a bigger or smaller number than you started with — a smaller unit always gives a larger count, and an answer that violates that is wrong before any arithmetic is examined. One converts a time so that a rate per minute can be applied to a span given in hours. And one converts a RATE, which means converting both of its units in opposite directions at once: multiplying by the seconds in an hour and dividing by the feet in a mile. That the units cancel to miles per hour is the proof that it was set up correctly, and a wrong option is built from doing both factors the other way round.

Two questions concern scale, one about a model and the real object it stands for, one about a drawing that must keep the shape of the park it represents. Both have a plausibility check worth performing: a model car twelve feet long in reality is believable, one measuring a quarter of an inch is not.

The last two are the pair this set exists for. They both divide fifty by twelve and reach the same number, and their answers differ, because one asks how many crates can be filled COMPLETELY and the other how many buses are NEEDED so that nobody is left behind. The arithmetic is identical; the situation decides the direction of the rounding, and each explanation points at the other to make that plain.

Every firm, farmer and town is invented, sums of money are written in words, and nothing is drawn from any official publication.

• Set up a proportion with both ratios the same way round, or find the value for one unit first
• Decide whether a conversion multiplies or divides by asking whether the answer should be a larger or a smaller number
• Convert a rate by converting both its units in opposite directions, and check that they cancel
• Compare two package sizes by reducing each to a price per unit, even when the larger total is the better value
• Apply a scale factor in the right direction and sanity-check the result against the real world
• Round a whole-number answer in the direction the situation requires rather than the direction arithmetic suggests

Written for this catalogue, with no source document. The content is the ratios and rates material named in the College Board's own public description of the Problem Solving and Data Analysis domain — ratios, rates, proportional relationships and units; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test also asks some questions in a student-produced response format, which a multiple-choice bank cannot reproduce. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.