Admissions prep · Mathematics

Quantitative reasoning practice problems

Short original problems on percent change, ratios, rates, estimation, tables, graphs and probability, each with a fully worked answer.

38 cards5 topicsAbout 15 minutes a roundNumeracy practice for admissions tests and first-year quantitative courses

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About this deck

Quantitative reasoning questions appear in some admissions tests, scholarship applications and first-year courses in business, health and social science. They rarely need advanced math; instead they test whether you can handle percentages, ratios, rates, averages and simple probability quickly and without falling for traps. Each card in this deck is a short original problem on the front with a worked solution on the back. Topics include percent increase and decrease, reverse percentages, compound growth, ratios and unit rates, average speed, estimation, reading tables and charts, and basic probability.

Use the deck as timed practice. Cover the answer, give yourself about a minute, and write the setup before calculating; most errors come from setting up the wrong ratio, not from arithmetic. Check whether your answer is reasonable before flipping. Classic traps include adding percentage changes that apply to different bases, averaging two speeds instead of dividing total distance by total time, confusing percentage points with percent change, reading a chart with a truncated axis, and assuming a positive result on an accurate test means the condition is almost certainly present. This deck is general practice and is not linked to any specific test.

Before you start: Grade 10 math: fractions, decimals, percentages and basic algebra.

All 38 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Percent

A $80 jacket is 25% off. What is the sale price?: The discount is 0.25 x 80 = $20, so the price is 80 - 20 = $60. Faster: 80 x 0.75 = $60.
A price rises from $40 to $46. Percent increase?: (46 - 40)/40 = 6/40 = 0.15, so a 15% increase.
Enrolment falls from 250 to 215. Percent change?: (215 - 250)/250 = -35/250 = -0.14, so a 14% decrease.
A value rises 20% and then falls 20%. Net change?: 1.20 x 0.80 = 0.96, so a net 4% decrease. The two percentages apply to different bases, so they do not cancel.
After a 15% raise, pay is $23 per hour. What was it before?: Old pay x 1.15 = 23, so old pay = 23/1.15 = $20. Taking 15% off $23 would wrongly give $19.55.
What is 12% of 350?: Write 12% as 0.12, then 0.12 x 350 = 42.
18 is what percent of 72?: 18/72 = 0.25, which is 25%.
Sales tax is 13%. What is the total on a $60 item?: Multiply by 1.13 to add the tax in one step: 60 x 1.13 = $67.80.
A mark goes from 60% to 75%. Change in points and in percent?: It rose 15 percentage points. As a relative change, 15/60 = 0.25, a 25% increase.
$1000 grows at 4% compounded yearly. Value after 3 years?: 1000 x 1.04^3 = 1000 x 1.124864 = $1124.86, a bit more than the $1120 simple interest would give.

Ratios and rates

A class of 32 has boys to girls in the ratio 3:5. How many girls?: There are 3 + 5 = 8 parts, so each part is 32/8 = 4 students. Girls = 5 x 4 = 20.
A recipe uses 2 cups of flour per 3 eggs. Flour for 9 eggs?: 9/3 = 3 times the recipe, so 2 x 3 = 6 cups.
A car uses 7.5 L per 100 km. Fuel needed for 340 km?: 340/100 = 3.4 hundreds of km, and 3.4 x 7.5 = 25.5 L.
You drive 180 km in 2 h 15 min. Average speed?: 2 h 15 min = 2.25 h, so 180/2.25 = 80 km/h.
Out at 60 km/h, back the same route at 40 km/h. Average speed?: Try 120 km each way: 2 h out, 3 h back, so 240 km in 5 h = 48 km/h, not 50. Average speed is total distance over total time.
Pipe A fills a tank in 4 h, pipe B in 6 h. Time together?: Rates add: 1/4 + 1/6 = 5/12 of the tank per hour, so time = 12/5 = 2.4 h, which is 2 h 24 min.
Map scale 1:50 000. How far is 3 cm on the map?: 3 x 50 000 = 150 000 cm = 1500 m = 1.5 km.
500 g costs $4.50 and 750 g costs $6.30. Better value?: Per 100 g: 4.50/5 = $0.90 versus 6.30/7.5 = $0.84, so the 750 g package is cheaper per gram.
Convert 90 km/h to metres per second.: 90 x 1000 m / 3600 s = 25 m/s. Shortcut: divide km/h by 3.6.
A dose is 5 mg per kg of body mass. Dose for 64 kg?: Multiply dose rate by mass: 5 mg/kg x 64 kg = 320 mg.

Estimation

Estimate 49 x 21.: Round to 50 x 20 = 1000. The exact product is 1029, so the estimate is close.
Estimate 397 / 8.: Round to 400/8 = 50. The exact value is 49.625.
Is 0.3 x 0.4 bigger or smaller than 0.3?: Smaller: multiplying by a number between 0 and 1 shrinks the value. The product is 0.12.
Estimate 19% of 412.: Round to 20% of 400 = 80. The exact value is 0.19 x 412 = 78.28.
Roughly how many seconds are in a day?: 24 x 60 x 60 = 86 400, which is about 9 x 10^4.

Tables and graphs

Find the mean of 4, 7, 7, 10, 22.: The sum is 50 and there are 5 values, so 50/5 = 10.
Find the median and mode of 4, 7, 7, 10, 22.: The middle value is 7 and the most frequent value is also 7. The large value 22 pulls the mean (10) above them.
Five scores average 70; a sixth makes the average 72. Sixth score?: Totals: 5 x 70 = 350 and 6 x 72 = 432, so the sixth score is 432 - 350 = 82.
A table shows sales of 120 in 2024 and 150 in 2025. Growth rate?: (150 - 120)/120 = 30/120 = 0.25, so 25% growth.
Axis starts at 90; bars reach 92 and 96. Is one three times bigger?: No. The bars look 2 and 6 units tall only because the axis is cut off; 96/92 is about 1.043, just over 4% more.
A pie chart slice is 72 degrees. What percent is it?: 72/360 = 0.2, which is 20% of the whole.
Find the range of 3, 9, 14, 2, 11.: Largest minus smallest: 14 - 2 = 12.

Probability

Probability of a sum of 7 when rolling two fair dice?: Six of the 36 equally likely outcomes give 7 (1 and 6, 2 and 5, 3 and 4, and the reverses), so 6/36 = 1/6.
Probability of two heads in two fair coin flips?: The flips are independent, so multiply: 1/2 x 1/2 = 1/4.
Probability of at least one head in three fair flips?: Use the complement: 1 - P(no heads) = 1 - (1/2)^3 = 1 - 1/8 = 7/8.
Draw 2 marbles without replacement from 5 red, 3 blue. Both red?: 5/8 for the first, then 4/7 for the second: 5/8 x 4/7 = 20/56 = 5/14, about 0.357.
An event has probability 0.3. What are the odds in favour?: Odds compare success to failure: 0.3 to 0.7, which simplifies to 3 to 7.
A test is 90% accurate. Does a positive mean a 90% chance of illness?: Not necessarily. If the condition is rare, false positives can outnumber true ones, so you need the base rate to find the real chance.