Free UCAT Quantitative Reasoning guide
Most percentage errors are base errors, not calculator errors. Before entering a number, identify what represents 100%, whether the question asks for a change or a final value, and whether several percentage moves act on the same amount.
You will turn percentage language into multipliers, distinguish percentage points from relative change, and reverse a final value without relying on trial and error.
The repeatable method
A percentage is always relative to a base value; successive changes multiply rather than cancel or add directly.
Identify the original or comparison base.
Convert the percentage change to a multiplier such as 1.12 or 0.92.
For successive changes, apply each multiplier in order and round only once at the end.
Understand the method
The denominator is the decision that controls the calculation.
In ‘A is what percentage of B?’, B is the base: calculate A ÷ B × 100. In ‘increase A by 12%’, A is again the base: multiply it by 1.12. Writing the base first stops the common reversal of numerator and denominator.
When comparing two years, the original year is normally the base for percentage change. The formula is change ÷ original × 100, with the sign determined by whether the value rose or fell.
A multiplier records both direction and size in one step.
An increase of 18% is ×1.18; a decrease of 18% is ×0.82. For successive changes, apply the multipliers in order. Adding the percentages assumes both changes use the same starting base, which is usually false.
Keep full calculator precision through the chain and round only to the unit requested. An early rounded intermediate value can shift the final option when money, rates or large populations are involved.
Similar wording can require three different operations.
If a rate rises from 40% to 50%, it rises by 10 percentage points but by 25% relative to the original rate. State which measure the question requests before calculating.
For a reverse percentage, divide the final value by its multiplier. If A$96 is the value after a 20% fall, A$96 represents 80% of the original, so divide by 0.80 rather than adding 20% to 96.
Original guided practice
Work each item before reading the answer. These are original Zhou Education teaching questions, written to isolate the method rather than predict a scaled score.
Successive discounts
A revision course is advertised at A$200. An early-bird discount removes 15%, and a separate code then removes 10% from the discounted price.
What is the final price?
Worked answer
A$153Apply each discount to the amount that exists at that stage: A$200 × 0.85 × 0.90 = A$153. Subtracting 25% directly would give A$150, but that treats both discounts as though they were calculated from the original A$200.Percentage points versus relative change
The proportion of survey respondents choosing option A increased from 42% in May to 51% in June.
What was the relative percentage increase, to one decimal place?
Worked answer
21.4%The increase is 51 − 42 = 9 percentage points. For relative change, the original 42% is the base: 9 ÷ 42 × 100 = 21.428…%, which rounds to 21.4%. The 9-point change answers a different question.Reverse percentage
After a supplier reduced a fee by 20%, the new fee was A$96.
What was the fee before the reduction?
Worked answer
A$120After a 20% reduction, A$96 represents 80% of the original. Divide by the remaining multiplier: A$96 ÷ 0.80 = A$120. Adding 20% to A$96 gives A$115.20 because it incorrectly uses the smaller final fee as the base.Error correction
The percentage is divided by the comparison value rather than the original or stated whole.
Write ‘100% = …’ before using the calculator.
Two percentage movements are combined as though they act on the same amount.
Convert each movement to a multiplier and multiply in sequence.
A difference between two rates is reported without checking the requested measure.
Subtract for percentage points; divide the difference by the original for relative change.
A decrease is undone by adding the same percentage to the smaller final value.
Divide the final value by the remaining multiplier, such as 0.80.
Questions students ask
Use whichever makes the relationship clearest. Decimal multipliers are usually fastest for successive changes; simple fractions can be efficient for familiar values such as 25%, 50% or 75%.
Equal rises and falls do not usually cancel because they act on different bases. They cancel only when the combined multipliers equal 1, such as an increase by 25% followed by a decrease of 20%.
Decide whether the original must be larger or smaller than the final value. After a reduction, the original must be larger; after an increase, it must be smaller. Then reapply the percentage to verify the result.
Apply the method
Start a targeted adaptive session for this method, use the full Quantitative Reasoningbank, or take the free 16-question diagnostic to identify the skill that needs attention first.
Personal UCAT support
Zhou Education is independent and is not affiliated with or endorsed by the UCAT ANZ Consortium or Pearson VUE. These original learning materials are educational practice and do not predict a scaled UCAT result. Use the official UCAT ANZ question tutorials as the primary reference for current test format.