Educerie · SAT · Math
Problem-Solving and Data Analysis · PSDA.1 Ratios, rates, proportional relationships and units
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Solve a proportion | If 7 machines make 210 parts an hour, how many do 12 make? |
| Convert between units | Metres to kilometres, minutes to hours, lira per kg to lira per g |
| Work with a rate | Distance, speed and time; price per unit |
| Split a quantity in a ratio | …in the ratio 3 : 5 |
| Scale a recipe or a map | Similar shapes, scale drawings |
1The idea in one paragraph
A rate is a number with two units attached, and the whole skill is arranging the multiplication so that every unit you do not want cancels. Write the units into the working, not just the numbers. If the units come out right, the arithmetic is almost always right too, and if they come out wrong the answer is wrong however tidy the number looks.
Set up every proportion as two fractions with the same unit on top and the same unit underneath: parts/hour = parts/hour. Then cross-multiply.
2Proportions
Seven machines make 210 parts an hour. At the same rate, how many parts do twelve machines make in an hour?
The safeguard is consistency: machines on top both times, parts underneath both times. Mixing them is the commonest error and gives an answer that is plausibly sized, which is why it survives a glance.
3Unit conversion by cancelling
Chain the conversions for anything longer:
Write each conversion as a fraction equal to one, arranged so the unit you are removing sits opposite the one you are keeping. It is mechanical, and it does not care how many steps there are.
4Rates
Distance = speed × time, and the two rearrangements. Keep units consistent before you multiply: 45 minutes is 0.75 hours, not 45.
Average speed is total distance over total time, never the average of two speeds. Driving 60 km at 60 km/h and 60 km at 30 km/h takes 1 + 2 = 3 hours for 120 km, so the average is 40 km/h, not 45.
5Splitting in a ratio
Find the value of one part first. The wrong answers on these are the values you get by treating 3 : 5 as 3/5 of the total.
6Scale and similar figures
A scale of 1 : 250 means one unit on the drawing is 250 in life. Lengths scale by the factor, areas scale by its square, and volumes by its cube — a model at 1 : 10 has one hundredth of the surface area and one thousandth of the volume.
That squared-and-cubed rule appears in Geometry as well, and it is worth knowing in both places.
Where points are lost
- Inconsistent proportions — machines on top on one side, parts on the other.
- Not converting minutes to hours before using a rate per hour.
- Averaging two speeds instead of dividing total distance by total time.
- Taking a ratio part as a fraction of the whole: 3 : 5 is 3/8, not 3/5.
- Forgetting the square and the cube when scaling areas and volumes.
- Answering in the wrong unit — the question asked for grams and you gave kilograms.
Work it right
- Write down what unit the answer must be in.
- Write the rates as fractions with their units attached.
- Arrange so everything unwanted cancels.
- Multiply, then check the surviving unit is the one you wrote down first.
- For ratios, find one part before anything else.
Try it
Q1. A printer produces 84 pages in 3 minutes. At this rate, how many pages does it produce in 25 minutes?
A) 560 B) 630 C) 700 D) 2,100
Q2. A cyclist travels at 18 km/h. How many metres does she travel in 40 seconds? (Type your answer.)
Q3. A sum of 4,500 lira is divided between two funds in the ratio 4 : 5. How much does the larger fund receive?
A) 1,800 B) 2,000 C) 2,500 D) 3,600
Q4. A map has a scale of 1 : 40,000. Two towns are 8.5 cm apart on the map. What is the actual distance, in kilometres?
A) 0.34 B) 3.4 C) 34 D) 340
Q5. A model of a building is made at a scale of 1 : 20. The model's volume is 0.5 m³. What is the building's volume, in m³?
A) 10 B) 200 C) 2,000 D) 4,000
In one breath
Write the units into the working and arrange the multiplication so everything you do not want cancels — if the surviving unit is the one the question asked for, the arithmetic is almost certainly right. Keep proportions consistent, with the same quantity on top of both fractions, convert times before using a rate per hour, and find the value of one part before splitting anything in a ratio. Lengths scale by the factor, areas by its square, volumes by its cube.
Answers
Q1. C — 700. 28 pages a minute 84/3 = 28, and 28 × 25 = 700. A and B come from inconsistent proportions; D multiplies 84 by 25 and forgets the three minutes.
Q2. 200. convert to metres per second first
Answering 720 treats the 40 as minutes.
Q3. C — 2,500. nine parts, each 500 4,500 ÷ 9 = 500, so the larger fund gets 5 × 500 = 2,500. B is 4/9 of the total, the smaller share. D treats 4 : 5 as four fifths.
Q4. B — 3.4. cancel the units through
C stops at metres; A and D slip a factor of ten.
Q5. D — 4,000. volume scales by the cube
B scales by the square; A scales by the factor alone.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Problem-Solving and Data Analysis, skill/knowledge testing point "Ratios, rates, proportional relationships, and units"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
Check your understanding
The main ideas of this note. Tick each one you could do now, in an exam, without looking back up. Anything you cannot tick yet is the part to read again.