Ratios, rates, and proportions form the backbone of the Problem Solving and Data Analysis portion of the PSAT/NMSQT, and they show up more often than almost any other topic on the test. At their core, all three ideas describe the same relationship — how one quantity compares to another — but the PSAT tests that relationship from many different angles: unit rates (miles per hour, bushels per acre, dollars per ounce), part-to-part ratios split across three or more categories, scale drawings and maps, and proportional reasoning where you set two fractions equal to each other and cross-multiply to solve.
The biggest source of errors on these questions isn't the arithmetic — it's setting up the relationship correctly. A ratio like "quarters to dimes to nickels is 2:4:7" needs to be translated into actual quantities (2k, 4k, 7k) before you can use it, and a rate like "20 machines produce 1,240 printers a day" needs to be reduced to a per-unit rate (printers per machine) before you can scale it up or down. Many PSAT ratio problems also ask a follow-up question that requires an extra step beyond finding the basic ratio or rate — such as finding how many more units are needed, or working backward from a total to find one part of a three-way split.
Below are 20 PSAT/NMSQT-style practice problems covering unit rates, part-to-part and part-to-whole ratios, proportions, scale drawings, and rate-based real-world scenarios like gas mileage and unit pricing. Work through each problem, select your answer, and check whether you got it right — every problem includes a complete, step-by-step explanation. Once you've worked through these, keep building your skills with thousands more official-style questions in our free PSAT/NMSQT Math QBank.
If 20 machines produce 1,240 printers in a day, how many more machines are needed to produce 1,984 printers in a day?
Find the unit rate: printers produced per machine per day. 1,240 ÷ 20 = 62 printers per machine.
Find the total number of machines needed to reach 1,984 printers: 1,984 ÷ 62 = 32 machines.
The question asks how many more machines are needed, not the total: 32 − 20 = 12.
The correct answer is A.
If 3/4 quart of lemonade concentrate is mixed with 6 2/3 quarts of water to make lemonade for 40 people, how many quarts of lemonade concentrate are needed to make the lemonade for 24 people?
The amount of water isn't needed here — everything in the recipe (including the concentrate) scales proportionally with the number of people served. Find the scale factor from 40 people down to 24 people:
24/40 = 3/5
Multiply the original concentrate amount by this scale factor: (3/4) × (3/5) = 9/20.
The correct answer is D.
A machine produced 735 tapes in 5 1/4 hours. What fraction of the 735 tapes was produced in one hour?
Convert the mixed number to an improper fraction: 5(1/4) = 21/4 hours.
Find the tapes-per-hour rate: 735 ÷ (21/4) = 735 × (4/21) = 140 tapes per hour.
The question asks for the fraction of the total 735 tapes made in one hour: 140/735. Both numbers are divisible by 35: 140/735 = 4/21.
The correct answer is B.
A 32-acre field yields 768 bushels of corn each year. How many more acres are needed to yield 960 bushels of corn each year?
Find the yield rate per acre: 768 ÷ 32 = 24 bushels per acre.
Find the total acres needed for 960 bushels: 960 ÷ 24 = 40 acres.
The question asks how many more acres are needed: 40 − 32 = 8.
The correct answer is B.
The length of a rectangle is 8 inches longer than the width. If the ratio of the length to perimeter is 5:16, what is the area of the rectangle?
Let w be the width, so the length is L = w + 8. The perimeter is 2(L + w) = 2(2w + 8) = 4w + 16.
The ratio of length to perimeter is 5:16, so L / (4w+16) = 5/16. Cross-multiply:
16L = 5(4w + 16) → 16(w+8) = 20w + 80 → 16w + 128 = 20w + 80
128 − 80 = 20w − 16w → 48 = 4w → w = 12. Then L = 12 + 8 = 20.
Area: L × w = 20 × 12 = 240 in². The correct answer is C.
If 12 grams of coffee costs x dollars and each gram makes y cups of coffee, what is the cost of one cup of coffee in terms of x and y?
First find the cost of a single gram of coffee: 12 grams costs x dollars, so one gram costs x/12 dollars.
Each gram makes y cups, so the cost per cup is the cost per gram divided by the number of cups per gram: (x/12) ÷ y = x/(12y).
The correct answer is D.
Keep the momentum going.
Explore 2,500+ Free PSAT/NMSQT Math ProblemsOn a map, 1 inch represents 5 miles. If a certain state is represented on a map by a rectangle 10 inches by 7.2 inches, what is the area of the state in square miles?
Convert each map dimension to real-world miles using the scale (1 inch = 5 miles) before multiplying, not after:
10 in × 5 mi/in = 50 mi and 7.2 in × 5 mi/in = 36 mi
Area = 50 × 36 = 1,800 mi². A common mistake is finding the map's area first (72 in²) and multiplying by 5, which incorrectly scales area by a linear factor instead of squaring it.
The correct answer is D.
Together there are 754 students and teachers in the meeting. If the ratio of students to teachers is 27:2, how many teachers are there?
A ratio of 27:2 means the total is divided into 27 + 2 = 29 equal parts.
Each part represents 754 ÷ 29 = 26 people.
Teachers make up 2 of those parts: 26 × 2 = 52.
The correct answer is B.
Concrete is made by mixing cement, sand, and gravel in the ratio 5:9:13. How much cement is needed to make 324 ft³ of concrete?
The ratio 5:9:13 divides the total into 5 + 9 + 13 = 27 equal parts.
Each part represents 324 ÷ 27 = 12 ft³.
Cement makes up 5 of those parts: 12 × 5 = 60 ft³.
The correct answer is B.
If Andy drove 84 miles in 1 hour 45 minutes, how many miles can he drive in 5 hours?
Convert 1 hour 45 minutes to a decimal: 45 minutes is 3/4 of an hour, so the time is 1.75 hours.
Find Andy's speed: 84 ÷ 1.75 = 48 miles per hour.
At a constant rate of 48 mph, in 5 hours he travels 48 × 5 = 240 miles.
The correct answer is C.
A collection of quarters, dimes, and nickels is worth $5.00. If the ratio of quarters to dimes to nickels is 2:4:7, how many quarters are there?
Unlike a simple count ratio, here each coin type has a different dollar value, so let the number of quarters, dimes, and nickels be 2k, 4k, and 7k respectively, and write the total value in dollars:
0.25(2k) + 0.10(4k) + 0.05(7k) = 5.00
0.5k + 0.4k + 0.35k = 5.00 → 1.25k = 5.00 → k = 4
The number of quarters is 2k = 2(4) = 8. The correct answer is C.
If 5x/3 = (x + 14)/2, what is the value of x?
This is a proportion — two fractions set equal — so cross-multiply to clear both denominators at once:
2(5x) = 3(x + 14)
10x = 3x + 42 → 7x = 42 → x = 6
The correct answer is C.
A trail mix contains raisin, peanut, and chocolate. The ratio of raisin to peanut is 2:3 and the ratio of peanut to chocolate is 5:8. What is the ratio of raisin to chocolate?
To combine the two ratios, the "peanut" value must match in both. Currently peanut is 3 in the first ratio and 5 in the second — find a common value using the LCM of 3 and 5, which is 15.
Scale raisin:peanut (2:3) by 5: 2:3 → 10:15. Scale peanut:chocolate (5:8) by 3: 5:8 → 15:24.
Now all three ingredients share the same peanut value: raisin:peanut:chocolate = 10:15:24.
The ratio of raisin to chocolate is 10:24, which simplifies (dividing both by 2) to 5:12.
The correct answer is C.
13 down, 7 to go — keep practicing.
Explore 2,500+ Free PSAT/NMSQT Math ProblemsThe ratio of 1 3/4 to 2 1/2 is equal to the ratio of 14 to what number?
Convert both mixed numbers to improper fractions: 1(3/4) = 7/4 and 2(1/2) = 5/2.
Set up the proportion, letting n be the unknown number: (7/4) / (5/2) = 14/n.
Simplify the left side by multiplying by the reciprocal: (7/4) × (2/5) = 14/20 = 7/10.
So 7/10 = 14/n. Cross-multiply: 7n = 140 → n = 20.
The correct answer is B.
The sum of two numbers is 14 and the ratio of the two numbers is −3. What is the product of the two numbers?
Let the two numbers be a and b, with a = −3b (their ratio is −3) and a + b = 14 (their sum).
Substitute: −3b + b = 14 → −2b = 14 → b = −7. Then a = −3(−7) = 21.
The product is a × b = 21 × (−7) = −147.
The correct answer is D.
If 2(x − y) = 3y, what is the ratio x/y?
Distribute the left side: 2x − 2y = 3y.
Get all the x-terms on one side and y-terms on the other: 2x = 3y + 2y = 5y.
Divide both sides by 2y to isolate the ratio: x/y = 5/2.
The correct answer is C.
The ratio of length to width of a rectangular garden is 6:7. If the perimeter of the rectangle is 78 meters, what is the area of the garden in square meters?
Let the length be 6k and the width be 7k, matching the given ratio. The perimeter is 2(L + W) = 2(6k + 7k) = 26k.
Set the perimeter equal to 78: 26k = 78 → k = 3.
So the length is 6(3) = 18 meters and the width is 7(3) = 21 meters.
Area: 18 × 21 = 378 square meters. The correct answer is D.
A car travels 218.5 miles on 9.5 gallons of gas. What is the car's gas mileage?
Gas mileage is simply a unit rate: miles traveled divided by gallons used.
218.5 ÷ 9.5 = 23 miles per gallon.
The correct answer is C.
At a grocery store, 20 fl oz of brand A vitamin water is sold for $0.95. What is the unit price of the vitamin water per ounce, to the nearest cent?
Unit price is total cost divided by total quantity: 0.95 ÷ 20 = 0.0475 dollars per ounce.
Rounding $0.0475 to the nearest cent means looking at the digit right after the hundredths place — the thousandths digit is 7, which rounds the hundredths digit up.
$0.0475 → $0.05
The correct answer is B.
The density of aluminum is 2.7 grams per cm³. How many grams does 12 cm³ of aluminum weigh?
Density is a rate — grams per cubic centimeter. To find total weight, multiply the rate by the volume:
2.7 g/cm³ × 12 cm³ = 32.4 grams.
The correct answer is C.
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