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Eton College — 13+ Maths Scholarship 2022

Eton College set this 13+ Maths paper in 2022. Hearthprep hosts it as an interactive past paper — 38 questions worth 100 marks in 90 minutes, each marked the moment the paper is submitted and backed by a step-by-step worked solution. Most 13+ entrance exams are sat between November and January of the year of entry.

Interactive version of the official Eton College King's 13+ Maths Scholarship Paper A from 2022. Practise with instant marking and worked solutions. Original PDF also available. Covers Maths at 13+ level.

This is the school's own past paper, reproduced here with worked solutions by Hearthprep. Most 13+ entrance exams are sat between November and January of the year of entry, so a timed attempt at this paper is worth doing early enough to leave room for the topics it exposes. Work through it on screen with instant marking, or print the PDF and sit it under exam conditions.

School
Eton College
Entry level
13+
Subject
Maths
Year
2022
Questions
38
Marks
100
Time
90 minutes
Exam board
School-specific

Questions in this paper

  1. Question 1short answer · 1 mark

    Q1(a): Fully simplify the following expression: 5y + 16 - 8y - 4

  2. Question 2short answer · 1 mark

    Q1(b): Fully simplify the following expression: (3/4)y - y/2

  3. Question 3short answer · 1 mark

    Q1(c): Fully simplify the following expression: (y * y * y * y * y) / y^3

  4. Question 4short answer · 3 marks

    Q2(a): Find the value of the following, giving your answer as a reduced, mixed fraction: 115 1/3 + 62 2/5 - 71 4/9

  5. Question 5short answer · 4 marks

    Q2(b): Find the value of the following, giving your answer as a reduced, mixed fraction: (81/98 / 3 6/7) * 154/9

  6. Question 6short answer · 3 marks

    Q2(c): Find the value of the following, giving your answer as a reduced, mixed fraction: (3 2/3 - 2)^2

  7. Question 7short answer · 3 marks

    Q3(a): Find the value of the following, giving your answer as a decimal: 0.018 * 0.0045

  8. Question 8short answer · 4 marks

    Q3(b): Find the value of the following, giving your answer as a decimal: 0.2 - (0.2)^3

  9. Question 9short answer · 3 marks

    Q3(c): Find the value of the following, giving your answer as a decimal: 0.403 / 0.062

  10. Question 10short answer · 1 mark

    Q4(a): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is odd.

  11. Question 11short answer · 1 mark

    Q4(b): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is a multiple of 5.

  12. Question 12short answer · 1 mark

    Q4(c): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is a prime number.

  13. Question 13short answer · 1 mark

    Q4(d): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is not a multiple of 3.

  14. Question 14short answer · 3 marks

    Q5(a): If p = 6, q = -2 and r = 1/2, find the value of the following expression, fully simplifying your answer: p^2 - 2q/r

  15. Question 15short answer · 3 marks

    Q5(b): If p = 6, q = -2 and r = 1/2, find the value of the following expression, fully simplifying your answer: (p^3 + q^3) / (p + q)

  16. Question 16short answer · 4 marks

    Q6: Solve the following simultaneous equations: 5x + 7y = 11 and 4x + 3y = 14. Give the values of x and y.

  17. Question 17short answer · 3 marks

    Q7: Solve the following equation for x. Give your answer as a reduced, mixed fraction: (2x + 1)/4 + (x - 1)/10 = 2

  18. Question 18short answer · 3 marks

    Q8: There are 8 more boys than girls in a football squad of 32. What is the ratio of girls to boys in the squad in simplified form?

  19. Question 19short answer · 3 marks

    Q9: Annie is 44 years old and her father is 36 years older than her. How many years ago was Annie's father three times her age?

  20. Question 20short answer · 4 marks

    Q10: Megan, Flossy and Emma are sisters. Megan is 80% taller than Emma. Flossy is 25% taller than Emma. Find the percentage by which Megan is taller than Flossy.

  21. Question 21short answer · 3 marks

    Q11(a): I visit three stores with some pocket money. I spend one fifth of my pocket money in the first shop and one quarter of what remains in the second shop. What fraction of the original amount is left over to spend in the third shop?

  22. Question 22short answer · 3 marks

    Q11(b): I am choosing a new washing machine. The standard model costs £200 and uses 17 1/2 pence of electricity per hour in operation. The energy-saving model costs £550 but only uses 3 1/2 pence of electricity per hour in operation. I operate my washing machine for twenty hours a week, on average. If I buy the energy-saving model, after how many weeks would my reduced electricity costs balance…

  23. Question 23short answer · 4 marks

    Q11(c): I run a cake stall on a Friday afternoon and spend 10 hours making 60 cakes. The ingredients for each cake cost me £1.60 and I charge £6 an hour for my labour. The sale price for a cake is 75% more than the total cost of making it. I have a special offer this week which gives 20% discount on my normal sale price. How much will a customer pay for one of my cakes this week?

  24. Question 24short answer · 3 marks

    Q12(a): A triangle ABC is made up of four smaller triangles. EBF and DFC are isosceles triangles. Angle EFD is x degrees and is acute. Calculate the size of angle EAD in terms of x. (refer to original PDF for figure)

  25. Question 25short answer · 3 marks

    Q12(b)(i): A shape is formed by joining an isosceles triangle ABE and a parallelogram BCDE. ABC is a straight line and angle BCD is ((90 - 5y)/4) degrees. Find angle BAE in terms of y, fully simplifying your answer. (refer to original PDF for figure)

  26. Question 26short answer · 4 marks

    Q12(b)(ii): Using the same shape from Q12(b), the line AF intersects BE at X as shown. It is given that DC = 17 cm and the ratio of the lengths AX and XF is 3:2. Given that the area of triangle ABE is 78 cm^2, calculate the area of the parallelogram BCDE. (refer to original PDF for figure)

  27. Question 27short answer · 2 marks

    Q13(a)(i): A triangle OAB is cut out of a circle with centre O. OA and OB are radii of length 35 cm and the chord AB is of length 56 cm. Find the area of the triangle OAB. (refer to original PDF for figure)

  28. Question 28short answer · 2 marks

    Q13(a)(ii): Using 22/7 as an approximation for pi, calculate an estimate for the shaded area (the circle minus the triangle OAB), to the nearest 10 square centimetres. The circle has centre O and radius 35 cm. (refer to original PDF for figure)

  29. Question 29short answer · 2 marks

    Q13(b)(i): Pentagon ABCDE is irregular but all five sides are each 34 cm long. Diagonal AC is 66 cm and is parallel to side ED. What is the value of x, where the area of triangle ABC is 33*sqrt(x) cm^2 and x is a positive whole number less than 100?

  30. Question 30short answer · 3 marks

    Q13(b)(ii): Pentagon ABCDE has all five sides of length 34 cm. Diagonal AC = 66 cm and AC is parallel to ED. What is the area of quadrilateral ACDE in square centimetres? Show that it is a whole number.

  31. Question 31short answer · 1 mark

    Q13(b)(iii): By making a suitable estimate for sqrt(x) from part (i), find the area of pentagon ABCDE correct to the nearest 10 square centimetres.

  32. Question 32short answer · 2 marks

    Q14(a): If m = (7 + n)/2 and n = (1 + m)/2, find the value of (m + n)/2.

  33. Question 33short answer · 2 marks

    Q14(b)(i): x, y and z are positive numbers such that x * y = 6, y * z = 27, and z * x = 2. Find the value of (xyz)^2.

  34. Question 34short answer · 3 marks

    Q14(b)(ii): x, y and z are positive numbers such that x * y = 6, y * z = 27, and z * x = 2. Find the values of x, y and z.

  35. Question 35short answer · 3 marks

    Q14(c): Find the values of u, v and w given: u + v = -3, v + w = -2, w + u = 6.

  36. Question 36short answer · 4 marks

    Q15(a): X is a three-digit whole number. The sum of its digits is 12. If the second and third digits (the tens digit and the units digit) are switched, the resulting number is 45 more than X. Find all possible values for X. [Hint: let the letters a, b and c represent the three digits of X and start by finding two equations involving a, b and c.]

  37. Question 37short answer · 2 marks

    Q15(b)(i): The numbers 1327231 and 394493 are both palindromic: they read the same when the order of their digits is reversed. Find the largest five-digit palindromic whole number which is divisible by 3 but not divisible by 9.

  38. Question 38short answer · 4 marks

    Q15(b)(ii): Of all possible five-digit palindromic whole numbers divisible by 15, Y is the largest and Z is the smallest. Find the difference between Y and Z.

Frequently asked questions

What is in the Eton College — 13+ Maths Scholarship 2022?

38 questions worth 100 marks in total, written for 90 minutes. It covers Maths at 13+ level. Eton College set it as a 13+ Maths paper.

How long is the Eton College 13+ Maths paper?

This paper is timed at 90 minutes for 100 marks — roughly 54 seconds a mark. Hearthprep runs the same clock on screen, so your child sees how far through the time they are as they answer.

Can my child answer this paper online?

Yes. Every question on this page can be answered in the browser and is marked the moment the paper is submitted — multiple choice, short answers and written responses alike. Answers are matched leniently, so a correct value written a different way (units, fractions, word numbers) still scores.

Are worked solutions included?

Every question has a step-by-step worked solution and an AI tutor that will talk through it after the paper is submitted. This paper is free — the whole thing, with marking and solutions, needs no account.

Where does this Eton College paper come from?

This is the school's own past paper, reproduced here with worked solutions by Hearthprep. Use it to time a full mock attempt, then drill whichever topics your child drops marks on.

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