UCAT Quantitative Reasoning study guide
Choose the right data, set up the calculation and check the units. Start with a short skill drill, then practice managing the full section.
Know the section
Standard Quantitative Reasoning has 36 questions in 26 minutes, after two minutes of instructions. Choose one of five answers. Questions often share a table, chart or other numerical information; some stand alone. A basic on-screen calculator is available.
Getting the calculation right starts with understanding what is being asked. A fast calculation using the wrong year, group or unit still produces the wrong answer.
Official format and scoringMethods and worked examples
These original examples introduce the method. Open a topic, work out the answer, then compare your reasoning before taking its drill.
Percentages and ratios
Name the base before calculating. An increase of 15% means multiplying the original amount by 1.15; a 15% reduction means multiplying by 0.85. Reverse a percentage by dividing by its multiplier. For a ratio, divide the total by the sum of the parts before scaling to the share requested. Keep percentage points separate from percentage change.
Common mistake: Adding two percentage changes as though both use the same base. A 20% decrease followed by a 20% increase does not restore the original amount.
Practice this skillTry this example
A supplier reduces a listed price by 15%, then adds tax of 20% to the discounted price. The final bill is £122.40. What was the listed price?
Show answer and method
£120
- The two multipliers are 0.85 and 1.20. Together they give 1.02.
- Let the listed price be P. Then P × 1.02 = £122.40, so P = £122.40 ÷ 1.02 = £120.
- Check: £120 becomes £102 after the discount, and £102 × 1.20 = £122.40. The net increase is 2%, not 5%.
Time, distance and speed
Write the units beside each quantity. Convert minutes to hours before combining times with a speed in km/h. Average speed is total distance divided by total time, rather than the average of two speeds. Distinguish travel time from elapsed time when a journey includes a stop.
Common mistake: Using a simple average when the quantities have different weights, or treating 1 hour 30 minutes as 1.30 hours.
Practice this skillTry this example
A bus travels 36 km at an average speed of 48 km/h. It returns along the same route at 72 km/h. Ignoring stops, what is its average speed for the whole journey?
Show answer and method
57.6 km/h
- The outward journey takes 36 ÷ 48 = 0.75 hours; the return takes 36 ÷ 72 = 0.5 hours.
- Total distance is 72 km and total time is 1.25 hours. The average is 72 ÷ 1.25 = 57.6 km/h.
- The bus spends longer at the slower speed. Averaging 48 and 72 to get 60 km/h gives the wrong weighting.
Averages and statistics
Convert percentages to counts before combining groups of different sizes. For a median, order the values first; for a mean, use the total divided by the number of observations. The mode is the most frequent value and the range is the largest minus the smallest.
Common mistake: Averaging two percentages without checking the group sizes, or confusing the mean with the middle observation.
Practice this skillTry this example
Department A has 80 staff, of whom 25% completed training. Department B has 120 staff, of whom 60% completed training. What percentage of all staff completed training?
Show answer and method
46%
- Department A contributes 80 × 0.25 = 20 trained staff. Department B contributes 120 × 0.60 = 72.
- There are 92 trained staff out of 200: 92 ÷ 200 × 100 = 46%.
- The simple mean of 25% and 60% is 42.5%, but the larger department must have more weight.
Rates and costs
Separate fixed charges from charges per item, hour or unit. Write a short expression for each option before comparing costs. For tiered pricing, apply each rate only to the quantity in that tier. Check whether tax, discounts and minimum charges apply.
Common mistake: Multiplying every unit by the final tier’s rate, or omitting a fixed charge because the variable rate looks cheaper.
Practice this skillTry this example
Delivery A charges £12 plus £0.80 per kilogram. Delivery B charges £4 plus £1.20 per kilogram. How much cheaper is A for a 35 kg parcel?
Show answer and method
£6
- Delivery A costs £12 + (35 × £0.80) = £40.
- Delivery B costs £4 + (35 × £1.20) = £46. The difference is £6.
- Comparing the rates alone would give £14, which ignores A’s additional £8 fixed charge.
Geometry and measurement
Match units before using a formula. Convert all relevant lengths, then calculate the required perimeter, area or volume. When using a scale drawing, a length factor applies once to lengths, twice to areas and three times to volumes.
Common mistake: Applying a linear scale factor directly to an area, or confusing square centimeters with square meters.
Practice this skillTry this example
A rectangular garden measures 8 cm by 5 cm on a plan drawn at a scale of 1:250. What is its actual area in square meters?
Show answer and method
250 m²
- The actual lengths are 8 × 250 = 2,000 cm and 5 × 250 = 1,250 cm.
- Convert to meters: 20 m and 12.5 m. The area is 20 × 12.5 = 250 m².
- The plan’s 40 cm² cannot simply be multiplied by 250: the scale applies in both dimensions.
Tables and charts
Read the title, headings, units and footnotes before selecting data. Identify the exact group and period requested. Check whether the question compares a difference, a ratio or a percentage change; the largest absolute change need not be the largest percentage change.
Common mistake: Choosing the biggest visible number without making the requested comparison, or reading a chart value in thousands as a single unit.
Practice this skillTry this example
A sales table uses units of £000. North rises from 120 in Year 1 to 156 in Year 2. South rises from 200 to 248. Which region has the larger percentage increase?
Show answer and method
North: 30%, compared with South’s 24%
- North increases by 36 on a base of 120: 36 ÷ 120 × 100 = 30%.
- South increases by 48 on a base of 200: 48 ÷ 200 × 100 = 24%.
- South has the larger cash increase (£48,000 rather than £36,000), but North has the larger percentage increase. The shared £000 unit cancels in each ratio.
Keep useful relationships ready
Percentage change
(new − original) ÷ original × 100
Use the original value as the denominator.
One ratio part
total ÷ sum of ratio parts
Multiply one part by the number of parts requested.
Speed
distance ÷ time
Match the time unit to the required speed unit.
Weighted mean
sum of (group mean × group count) ÷ total count
A larger group contributes more observations.
Area and scale
length factor squared for area
A length doubled in both directions gives four times the area.
Range and median
largest − smallest; ordered middle
For an even count, average the two central values.
Use the calculator after setting up the problem
- Identify the output. Write or think “percentage of all staff,” “cost per month” or the other quantity requested before entering numbers.
- Estimate the scale. A rough result catches a misplaced decimal or an answer in the wrong unit. Estimation can sometimes eliminate options, but close options still need precise work.
- Keep intermediate precision. Avoid rounding a converted currency amount or a rate too early. Round to the requested precision at the end.
- Check the display. Clear an old calculation and confirm decimal points and operations. Mental arithmetic may be quicker for a simple fraction; use the calculator when it reduces effort or error.
Try the official calculator and navigation on a desktop before test day. Our practice tool supports learning, but familiarity with the official interface needs its own rehearsal.
Official calculator and test toolsMove from accurate working to timed practice
Twenty-six minutes averages about 43 seconds per question, including time spent understanding the shared data. Use that as a planning average, not a rule to abandon every item at 43 seconds.
- Learn one skill at a time. Review errors in a short drill. Separate mistakes in reading, method, arithmetic and time management.
- Take advantage of shared information. Once you understand a chart’s units and layout, reuse that understanding for its next question. Recheck the specific row, group and time period each time.
- Move on deliberately. If an item needs lengthy working and you have no clear approach, choose your best answer, flag it and return if time remains. The official test has no deduction for a wrong answer.
- Review coverage as well as accuracy. A strong percentage among attempted items can hide unanswered questions. Inspect all planned questions and compare attempts using the same timing.
Select any applicable extra-time practice setting before starting. Official access arrangements require approval; practice settings do not provide that approval or reproduce every permitted arrangement.
Official timing optionsCommon questions
How long is UCAT Quantitative Reasoning?
The current standard section has 36 questions in 26 minutes, following a separate two-minute instruction period. Check the official UCAT timings for approved extra-time arrangements.
Can I use a calculator?
A basic on-screen calculator is available for Quantitative Reasoning and Decision Making. Practice using the official tutorial’s calculator as well as our practice tool; keyboard behavior and layout can differ.
What math should I practice?
Focus on percentages, ratios, rates, unit conversions, tables, charts and summary statistics. The main challenge is identifying the relevant data and the right calculation quickly. Advanced scientific calculator functions are not needed.
Do practice marks convert to a UCAT score?
No. These original questions provide raw practice results and explanations. They are not equated to the official test, so a practice percentage cannot be converted reliably to a 300–900 section score.
Does a UCAT pass include the other sections?
One UCAT pass covers the section tests, full assessment and drill replays across Verbal Reasoning, Decision Making, Quantitative Reasoning and Situational Judgement. Available free drill attempts are shown before you start.
Practice the section, then the full assessment
Use a targeted drill to work on an error pattern, or take a timed section with explanations after finishing. One UCAT pass includes all four sections and the full assessment.
Use the free official practice tests tooOriginal independent preparation. Not affiliated with or endorsed by the UCAT Consortium. Raw practice marks are not official scores or an admissions prediction.