PSAT Data — rates, ratios and the units they hide in

More marks are lost in this domain to minutes against hours and centimeters against meters than to any idea in it. The arithmetic in these twelve questions is small enough to do in your head; the work is deciding what to convert, and whether the conversion multiplies or divides.

Three questions make the conversion the whole point. A machine making a hundred and twenty items every fifteen minutes, asked about three hours. A hiker at four miles an hour, walking for ninety minutes. A press using five hundred centimeters of roll for twenty posters, asked for an answer in meters. In each one, one of the wrong options is exactly what you get by leaving the unit alone, which is worth seeing written down.

Three are proportions, and one of them runs the other way: a large gear with forty teeth driving a small one with ten makes the small gear turn more often, not less. Proportions that behave in reverse catch people who have learned the setting-up as a ritual rather than as a statement about the situation, and the wrong options include the number you get by dividing where you should multiply.

Two divide a total in a stated ratio. Counting the parts first — two and seven make nine — and finding what one part is worth turns both into a single line of work. The wrong options here are all real quantities from the same problem: one part, one share, the total, which is why reading the question to its last word matters.

Two compare rates given in different units, and they cannot be answered at all until both are put into the same one: a press rated per eight minutes against one rated per hour, a tap giving liters per twenty seconds against one giving milliliters per second. Each offers an option that subtracts the bare numbers as they stand, which is the mistake the question exists to catch.

The last two are density and concentration: grams of salt per five hundred milliliters asked about two liters, and kilograms per liter asked for a volume, where the division runs the opposite way from the one people expect.

Every situation here was invented for the exercise, all values are exact, and nothing is reproduced from any published test.

• Convert the time or length inside a rate before doing anything else with it
• Decide whether a conversion multiplies or divides by naming the units on both sides
• Set up a proportion so that the units cancel, and check it against the situation rather than against a remembered pattern
• Handle a proportion that runs in reverse, where more of one thing means less of the other
• Divide a total in a stated ratio by counting the parts first and finding what one part is worth
• Compare two rates given in different units by putting both into a single unit
• Use density or concentration in either direction, from amount to volume and back

Every situation here was written for this exercise, and no two questions share one. Among them: a machine making a hundred and twenty items every fifteen minutes, asked about three hours; a hiker at four miles an hour for ninety minutes; five hundred centimeters of paper roll for twenty posters, answered in meters; three cups of orange juice to five of pineapple, scaled to twelve; a forty-tooth gear driving a ten-tooth one; eight hundred books split three to five; forty-five units of concrete mixed two parts to seven; a press doing two hundred forty pages in eight minutes against one rated at twenty-one hundred an hour; a tap giving three liters every twenty seconds against one giving five hundred milliliters a second; twenty grams of salt per five hundred milliliters, asked about two liters; and a density of one and a half kilograms per liter, asked for the volume holding twelve kilograms. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not an exam centre.