Scholarship level: problem-solving maths and the classics
Some independent schools set scholarship papers that ask for reasoning rather than routine. Prism's scholarship maths runs in six strands at two tiers, and its scholarship English works through inference from classic texts.
Every question here is written by Prism in the style of the tests. None is copied from a real paper, and a fresh one is drawn each time a child practises.
Maths strands
Number and proof
Factors, remainders and consecutive numbers: reasoning about numbers rather than just calculating with them.
Counting factorsRemaindersConsecutive numbersWhat must be true of a numberNearest square rootDivisibility without dividing
Standard tier
Question 1Aim for about 90 seconds
How many factors does 50 have? Count 1 and 50 themselves.
Show the worked answer
Answer: E, 6
How to work it out
Find the factors in pairs that multiply to 50: 1 × 50, 2 × 25, 5 × 10.
That is 6 factors.
Check: 50 = 5^2 × 2, and (2 + 1) × (1 + 1) = 6.
Why the other options are wrong
A (3): This counts the prime factors with repeats, not the factors of the number.
B (5): This misses one factor. Pairing the factors (1 × 50, 5 × 10, …) finds them all.
C (4): This leaves out 1 and 50. The question says to count them.
D (2): This counts only the prime factors 5 and 2. Every number that divides 50 exactly is a factor.
Ratio, rate and proportion
Sharing in a ratio, speed and average speed, and inverse proportion.
Ratio from a differenceSpeed and average speedInverse proportionBest buyMap scalesWhich mix is stronger
Standard tier
Question 2Aim for about 90 seconds
Isla and Layla share some money in the ratio 5 : 6. Layla receives £3 more than Isla. How much money do they share altogether?
Show the worked answer
Answer: B, £33.00
How to work it out
The difference between Isla’s and Layla’s shares is 6 − 5 = 1 parts, and that is £3.
So one part is £3 ÷ 1 = £3.
The total is 5 + 6 = 11 parts: 11 × £3 = £33.
Why the other options are wrong
A (£18.00): This is only Layla’s share.
C (£3.00): This is only the difference between the two shares.
D (£34.00): A near miss: check each step of the working again.
E (£36.00): This counts one part too many.
Algebra and sequences
Finding the rule of a sequence, two unknowns at once, and unpicking a chain of operations.
Rule of a sequenceTwo unknownsI think of a numberWhich rule fits the patternUndoing a formulaWhat must be true of a and b
Standard tier
Question 3Aim for about 90 seconds
Here is a sequence: 6, 9, 12, 15, … What is the 30th term?
Show the worked answer
Answer: C, 93
How to work it out
The terms go up by 3 each time, so the rule is 3n + 3 (check: 3 × 1 + 3 = 6).
The 30th term is 3 × 30 + 3 = 93.
Why the other options are wrong
A (90): This uses the rule 3n and forgets to add 3.
B (99): A near miss: check each step of the working again.
D (96): This adds 3 to the first term 30 times instead of 29 times.
E (87): A near miss: check each step of the working again.
Shape and space
Compound shapes, angles in polygons, and cubes built from cubes.
Compound shapesAngles in polygonsPainted cubesTiles on a floor
Standard tier
Question 4Aim for about 90 seconds
A rectangle 19 cm by 12 cm has a rectangular hole 6 cm by 2 cm cut out of it. What is the area of the shape that is left?
Show the worked answer
Answer: C, 216 cm²
How to work it out
Area of the whole rectangle: 19 × 12 = 228 cm².
Area of the hole: 6 × 2 = 12 cm².
Area left: 228 − 12 = 216 cm².
Why the other options are wrong
A (46 cm²): This works with perimeters, not areas.
B (240 cm²): This adds the hole instead of subtracting it.
D (228 cm²): This is the whole rectangle; the hole has not been taken away.
E (130 cm²): This subtracts the lengths before multiplying. Subtract the areas, not the sides.
Logic and counting
Handshakes and fixtures, arranging digits, and being certain without luck.
Handshakes and fixturesArranging digitsBeing certainWhich statement must be trueFind the error in the workingWhat else do you need to know
Standard tier
Question 5Aim for about 90 seconds
10 people meet, and every person shakes hands exactly once with every other person. How many handshakes are there?
Show the worked answer
Answer: C, 45
How to work it out
Each of the 10 people shakes hands with 9 others: 10 × 9 = 90.
But that counts every handshake twice (once for each person), so halve it: 90 ÷ 2 = 45.
Why the other options are wrong
A (10): This is the number of people.
B (90): This counts every handshake twice, once for each of the two people.
D (9): This is how many handshakes one person makes, not the total.
E (55): This lets each person shake hands with themselves.
Fractions, decimals and percentages
Working backwards from a percentage, fractions of what is left, and repeated changes.
Reverse percentagesFractions of what is leftTwo percentage changesWhich percentage is largerA fraction between two fractionsLargest of fractions, decimals and percentages
Standard tier
Question 6Aim for about 90 seconds
The price of a tablet rises by 25% to £80. What was the price before the rise?
Show the worked answer
Answer: A, £64.00
How to work it out
The new price is 125% of the old price.
So 1% of the old price is £80 ÷ 125 = £0.64, and 100% is £0.64 × 100 = £64.
Check: 25% of £64 is £16, and £64 + £16 = £80.
Why the other options are wrong
B (£60.00): This takes 25% of the NEW price. The percentage was of the old price, so divide by 125 and multiply by 100.
C (£80.00): This is the current price.
D (£100.00): This changes the price by 25% again in the same direction.
E (£20.00): This is 25% of the current price.
Classics comprehension
Extracts from 6 classic texts, each with its own questions:
The ChurchyardThe Window-SeatThe Door in the WallThe FireMarley’s GhostThe Visitor’s Stick
Scholarship practice opens at the standard
Prism starts with a free assessment across maths, English, verbal and non-verbal reasoning, then builds a weekly plan for your target schools. Parents follow progress free.