Ratios, rates, proportions and units

Four ideas the exam treats as one skill: a ratio (a comparison of two quantities by division), a rate (a ratio of quantities measured in different units), a proportion (two ratios set equal to each other), and unit conversion (rewriting a quantity in different units without changing what it is). Items arrive as a short paragraph of context, sometimes attached to a table or a graph, and roughly a quarter of Math questions are grid-ins — so on a share of these there are no choices to check yourself against.

The arithmetic is deliberately easy. No item in this skill will ask you to do anything harder than multiply and divide, and the calculator is available on every question. The difficulty is manufactured almost entirely by units: a rate stated per 100 kilometers instead of per kilometer, a mass in grams and a price in kilograms, an area in square centimeters and a coverage figure in square meters. Take the units out of these questions and most of them become one line of arithmetic. That is the design, and it tells you where to spend your attention.

What is supplied and what is not. The on-screen reference sheet carries geometry formulas only — no conversion table of any kind. In released material an unusual conversion is stated inside the question (1 furlong = 220 yards), while everyday ones are assumed (60 seconds in a minute, 100 centimeters in a meter, 1,000 grams in a kilogram, 1,000 milliliters in a liter). That pattern is an observation about released items and the prep industry's reading of it, not a printed College Board guarantee — so treat the everyday metric and time conversions as things you already know cold, and do not count on a factor being handed to you.

If you can already turn any "per" statement into a fraction and set up a proportion without thinking about it, skip to the next block; nothing here will be new. If any of the four words in the title is vague, this is the highest-value block on the page, because these four ideas also carry the percentages, probability and data-analysis questions that come later.

A ratio compares two quantities by dividing them. "5 to 3", "5:3" and "5/3" all say the same thing: for every 5 units of the first quantity there are 3 of the second. Notice what a ratio does not tell you — the actual amounts. The ratio 5:3 fits 5 and 3, and 50 and 30, and 12.5 and 7.5 equally well.

The most useful move with any ratio is to count parts. A ratio of 5:3 describes 5 + 3 = 8 parts in total. If the total is 72 kilograms, one part is 72 ÷ 8 = 9 kilograms, so the two quantities are 45 and 27. Once you have the value of one part, every question about that mixture is a multiplication. And note the distinction the exam leans on hardest: 5/8 and 3/8 are the shares of the whole, while 5/3 is the comparison of the two parts. They are different numbers answering different questions.

A rate is a ratio whose two quantities are measured in different units — miles and hours, dollars and pounds, milligrams and square centimeters. The word "per" is a fraction bar and nothing else: "miles per hour" means miles divided by hours. A unit rate is that fraction rewritten with 1 in the denominator: 240 miles in 4 hours is 60 miles per 1 hour. Getting to the unit rate first makes almost every rate question shorter.

A proportion is a statement that two ratios are equal, and it is the workhorse for "if this much costs that much, what does this other amount cost." Three kilograms of a compound cost $7.50; what do 8 kilograms cost? Write 3 kg / $7.50 = 8 kg / $x, cross-multiply to get 3x = 60, so x = $20. Check it with the unit rate: $7.50 ÷ 3 = $2.50 per kilogram, and $2.50 × 8 = $20. ✓

The rule that actually matters in a proportion is not "cross-multiply" — it is that the same quantity must sit in the same position on both sides. Kilograms over dollars on the left means kilograms over dollars on the right. Writing 3/7.50 = x/8 instead is a different equation with a different, wrong answer, and it is the most common setup error there is. Percent is the same machinery with 100 fixed as the denominator, and it has its own lesson.

Treat a unit as a symbol that multiplies and divides exactly like a variable. Kilometers cancel kilometers the same way x cancels x. That single permission is the whole technique, and everything below is a consequence of it.

Every conversion statement hands you two fractions, and both of them equal 1. From "1 hour = 60 minutes" you get 60 min / 1 hr and you get 1 hr / 60 min. Neither is more correct than the other. You choose the one whose top or bottom cancels the unit you are trying to get rid of, and that is the only criterion.

The procedure, in full. Write the given quantity as a fraction with its units attached. Multiply by conversion fractions, each oriented so the unit you are leaving appears once above and once below. Cross out the cancelled units on the page. Multiply the numbers only at the very end, in one pass. Then read what units survived.

Worked in line: 90 kilometers per hour into meters per second. (90 km / 1 hr) × (1,000 m / 1 km) × (1 hr / 3,600 s). Kilometers cancel against kilometers; hours cancel against hours; meters and seconds are all that remain. Numbers last: 90 × 1,000 ÷ 3,600 = 25 meters per second. Sanity check in the other direction: 25 m/s × 3,600 s = 90,000 m = 90 km in an hour. ✓

Here is why the extra writing pays for itself. The surviving units are a verifiable prediction. If you asked for meters per second and the algebra leaves you holding kilometer-seconds per hour squared, you know you are wrong before doing a single multiplication, and the offending factor is the one whose unit failed to cancel — so you know not just that you are wrong but exactly where. Nothing else on the Math section tells you that for free.

Do not decide multiply-or-divide by feel. "Bigger unit to smaller unit means multiply" survives one-step conversions and collapses on the two-step and mixed-direction chains this skill actually asks for. Orientation-by-cancellation never has that failure mode, and it costs about eight seconds of writing.

Two quantities are proportional when y = kx for some fixed k: the ratio y/x is the same at every point, and the graph passes through the origin. That, and only that, is the license to say "double the input, double the output" or to scale from a single data point.

A fixed component destroys proportionality. With cost = 4 + 2d, the ratio cost/distance is $2.80 per mile at 5 miles and $2.40 per mile at 10 — it changes at every distance, so there is no single per-mile number to scale with. Whenever a scenario contains a flat fee, a deposit, a base charge, a setup cost, or a starting amount, scaling from one data point is wrong. This is the same distinction as slope versus intercept, wearing different clothes.

Inverse proportion is the other family: xy = k, so one quantity rises exactly as fast as the other falls and their product stays fixed. Five machines take 12 hours, so the job is 60 machine-hours; eight machines take 60 ÷ 8 = 7.5 hours. The trap is handling it as a direct proportion and getting 12 × 8/5 = 19.2 — more machines producing more hours, which the situation itself rules out. Read the direction off the story before setting up any proportion.

Scale factors and dimensions. If lengths are scaled by a factor k, areas scale by k squared and volumes by k cubed. A 1:20 model has 1/400 of the surface area and 1/8,000 of the volume of the real thing. This is the same fact as the squared-conversion rule, and it appears as map scale, model scale, and "if the radius doubles, what happens to the area" — one idea, three costumes.

At 1500 the method is not what is costing points. Six things are: rates whose denominator is not one, squared units hidden inside a rate, chains that convert in two directions at once, expression-selection items with no numbers to sanity-check against, the average-of-rates family, and a short list of execution errors that survive knowing all of the above.

Rates whose denominator is not one. "1.8 liters per 100 kilometers." "$54 for every 4 posters." "3 grams per 6 meters of filament." These are already complete rates; the error is reading the second number as a conversion factor to be applied somewhere else. Two clean ways through: reduce to a unit rate before anything else (0.018 L/km), or carry the fraction whole and let cancellation handle it (1.8 L / 100 km). Both work. Starting one way and finishing the other is where the factor of 100 goes missing.

Squared and cubed units hidden inside a rate. Milligrams per square centimeter, dollars per square foot, grams per cubic centimeter, people per square mile. Converting one of these means two different powers in the same chain — the numerator's factor to the first power, the denominator's to the second or third — and the item is built so that applying one power to both produces a listed answer. When a rate has an area or a volume in its denominator, write the denominator's conversion out as a product of identical factors rather than reaching for a remembered number.

Chains that convert in two directions at once. Grams going up to milligrams while square meters come down to square centimeters; dollars per gallon into cents per liter. Every extra factor is another opportunity to be right about orientation in one place and wrong in another, and the two errors can partially hide each other in the final magnitude. Writing the units and striking them out is not a beginner's crutch on these items; it is the only check that exists.

Expression-selection items — "which expression gives the number of days" — hand you letters instead of numbers, which removes the rough estimate you would normally use to sniff-test an answer. The substitute is to pick your own numbers: assign each letter a small value you can compute with in your head, work the answer out arithmetically, then evaluate all four expressions at those same values and keep the one that matches. Choose values that cannot coincide — avoid 1, which hides every multiplication, and avoid 2, where doubling and squaring give the same result.

Averages of rates. Average speed for a journey is total distance divided by total time, full stop. Drive 60 miles out at 30 mph and the same 60 miles back at 60 mph: 2 hours out, 1 hour back, 120 miles in 3 hours, so 40 mph — not the 45 you get by averaging the two speeds. The mean of the rates is correct only when the two legs take the same amount of TIME, which is almost never what an item describes, and 45 will be sitting there as a choice.

Execution errors, which is where the remaining points at this level actually live. First and largest: answering the right question about the wrong quantity — reporting the new total when the question asked how much was added, the part when it asked for the whole, the per-unit figure when it asked for the total. It is beaten by rereading the last eight words of the question after computing, not before. Second: dropping a factor of 1,000 between grams and kilograms, milliliters and liters, meters and kilometers. This is the single most common slip in the domain, and it is also the most visible — an answer that is 1,000 or 1,000,000 away from your estimate is a unit error, never an arithmetic one, and the credited answer is usually the choice sitting exactly that many orders of magnitude away from the one you computed.

Three more that are pure careless error rather than reasoning. Reading "45 minutes" as an hour, or "during the first three years" as "in the third year." Rounding mid-chain: two thirds carried through three more multiplications as 0.67 will miss a grid-in band that 2/3 would have hit, so round once, at the end, or not at all. And the final unit itself — if the question asks for dollars per hour and the unit still standing at the end of your work is dollars per minute, everything else being right does not help. Write the units you are being asked for at the top of the scratch area before you start, and compare against them last.

One note on tooling. Desmos will do the arithmetic of an ugly proportion faster than you can cross-multiply — put each side on its own line and read the intersection. What it cannot do is notice that you set the proportion up with kilograms opposite pounds. Conversion is the one part of this domain that no calculator checks for you.