Intermediate maths challenge practice

Intermediate maths challenge practice paper for Year 10

A free paper in the intermediate challenge's shape: 25 questions in 60 minutes, five choices each, for 135 marks. Print it, then check every answer and its working at the end.

See the answersSit a new paper online

Scholar Tests · Sample paper

Intermediate maths challenge-style paper, Year 10

25 questions · 135 marks · 60 minutes · no negative marking

Name ______________________ Date ____________

Questions 1 to 15

Questions 1 to 15, 5 marks each. Circle the letter of the best answer.

  1. 1.

    A right-angled triangle has a longest side of 39 cm and another side of 15 cm. What is its area, in square centimetres?

    1. A271
    2. B269
    3. C270
    4. D540
    5. E702
  2. 2.

    Gus, Omar and Lena are playing, and one of them has the ball.
    Gus says: "Lena does not have the ball."
    Omar says: "Lena has the ball."
    Lena says: "Omar does not have the ball."
    Exactly one of them is telling the truth. Who has the ball?

    1. AOmar
    2. BCarlos
    3. CLena
    4. DGus
    5. EWes
  3. 3.

    The mean of 9 numbers is 33. One of the numbers is taken away, and the mean of the 8 numbers left is 30. What number was taken away?

    1. A56
    2. B57
    3. C58
    4. D3
    5. E27
  4. 4.

    How many positive whole numbers divide exactly into 135?

    1. A9
    2. B8
    3. C7
    4. D6
    5. E3
  5. 5.

    What is the value of 1³ + 2³ + 3³ + 4³ + 5³?

    1. A15
    2. B125
    3. C55
    4. D225
    5. E100
  6. 6.

    66 ÷ 6 × 2 is the same as which of these?

    1. A66 × 6 ÷ 2
    2. B66 ÷ (6 × 2)
    3. C66 × 2 ÷ 6
    4. D(66 + 2) ÷ 6
    5. E6 ÷ 66 × 2
  7. 7.

    A rectangle is made from 20 square tiles, with none left over. A square counts as a rectangle, and a rectangle that is only turned on its side is still the same rectangle. How many different rectangles can be made?

    1. A4
    2. B2
    3. C6
    4. D3
    5. E5
  8. 8.

    A spinner has 5 equal parts, and one of them is blue. It is spun 80 times. About how many times would you expect it to land on blue?

    1. A20
    2. B64
    3. C40
    4. D80
    5. E16
  9. 9.

    Find the value of x.
    11x − 11 = 7x + 29

    1. A12
    2. B10
    3. C11
    4. D40
    5. E9
  10. 10.

    A tub holds 11 balls numbered 1 to 11. Balls 1 to 3 are white, 4 to 6 are black and 7 to 11 are grey. One ball is taken out without looking. What is the chance that it is not grey and its number is not odd?

    1. A3/11
    2. B6/11
    3. C5/11
    4. D4/11
    5. E8/11
  11. 11.

    Three different whole numbers multiply to 112 and add to 17. Square the smallest number, then add the largest number. What is the result?

    1. A11
    2. B57
    3. C100
    4. D12
    5. E66
  12. 12.

    How many more factors does 98 have than 39?

    1. A2
    2. B10
    3. C6
    4. D1
    5. E4
  13. 13.

    What is the last digit of 81692?

    1. A7
    2. B6
    3. C8
    4. D5
    5. E2
  14. 14.

    The equation x2 − 22x + 117 = 0 has two whole-number solutions. What is the sum of the squares of its two solutions?

    1. A22
    2. B718
    3. C367
    4. D484
    5. E250
  15. 15.

    How many numbers are in the list 26, 38, 50, …, 1046, where each number is 12 more than the one before?

    1. A86
    2. B84
    3. C87
    4. D88
    5. E85

Questions 16 to 25

Questions 16 to 25, 6 marks each. Circle the letter of the best answer.

  1. 16.

    How many three-digit whole numbers have digits that add up to 6?

    1. A10
    2. B23
    3. C22
    4. D20
    5. E21
  2. 17.

    In the drawing below, the two lines marked with arrows are parallel. What is the size of the angle marked x?

    1. A255°
    2. B123°
    3. C75°
    4. D105°
    5. E132°
  3. 18.

    11 points are marked on a circle. How many triangles have all 3 of their corners at these points?

    1. A164
    2. B166
    3. C990
    4. D165
    5. E55
  4. 19.

    The number 153! means 1 × 2 × 3 × … × 153. How many zeros are at the end of 153! when it is written out in full?

    1. A38
    2. B15
    3. C37
    4. D36
    5. E30
  5. 20.

    A competition has 16 teams in one pool. Every team plays every other team in its pool twice, once at home and once away. How many games are played altogether?

    1. A120
    2. B240
    3. C239
    4. D480
    5. E241
  6. 21.

    How many positive whole numbers divide exactly into 21 × 55 × 75?

    1. A70
    2. B14
    3. C73
    4. D25
    5. E72
  7. 22.

    If 8x = 16x − 5, what is the value of x?

    1. A21
    2. B22
    3. C5
    4. D20
    5. E19
  8. 23.

    In a Year 6 group with 34 students, 6 play in the band, 21 take art classes and 2 do both. How many do neither?

    1. A7 students
    2. B5 students
    3. C13 students
    4. D9 students
    5. E28 students
  9. 24.

    What is the highest common factor of 45, 81?

    1. A3
    2. B126
    3. C405
    4. D45
    5. E9
  10. 25.

    A town's streets form a grid of square blocks. Ana walks from one corner of a rectangle 3 blocks wide and 5 blocks high to the opposite corner, always walking east or north along the streets. The corner 1 blocks east and 4 blocks north of the start is closed, so no route may pass through it. How many different routes can she take?

    1. A56
    2. B15
    3. C41
    4. D8
    5. E40

Answers and working

Questions 1 to 15

  1. 1.C (270). By Pythagoras, the third side is √(392 − 152) = 36 cm. The area is half of 15 × 36, which is 270.
  2. 2.A (Omar). Try each child as the one with the ball and count the true statements. Only when Omar has it is exactly one statement true.
  3. 3.B (57). The 9 numbers add up to 9 × 33 = 297. The 8 left add up to 8 × 30 = 240. The number taken away is the difference, 57.
  4. 4.B (8). 135 = 33 × 51. A divisor takes each prime to any power from 0 up, so there are 4 × 2 = 8 divisors.
  5. 5.D (225). The numbers 1 to 5 add to 15, and the cubes add to 15² = 225.
  6. 6.C (66 × 2 ÷ 6). 66 ÷ 6 × 2 = 22, and so does 66 × 2 ÷ 6.
  7. 7.D (3). Listing the rectangles with the smaller side first, 3 of them have an area of 20.
  8. 8.E (16). The chance of blue is 1/5 each spin, and 1/5 of 80 is 16. So the answer is 16.
  9. 9.B (10). Take 7x from both sides: 4x − 11 = 29. Add 11: 4x = 40. Divide by 4: x = 10.
  10. 10.A (3/11). Leaving out the grey ones and every one whose number is odd leaves 3 of the 11: 3/11.
  11. 11.D (12). Only 2, 7 and 8 multiply to 112 and add to 17. Smallest squared is 4; adding the largest, 8, gives 12: 12.
  12. 12.A (2). 98 has 6 factors and 39 has 4, so 98 has 2 more.
  13. 13.B (6). The last digit of the powers of 8 repeat in a cycle. Find the cycle, then find where the power 1692 falls in it: the answer is 6.
  14. 14.E (250). The solutions add to 22 and multiply to 117. The sum of the squares is 222 − 2 × 117 = 250.
  15. 15.A (86). From 26 to 1046 is 1020, which is 85 steps of 12. The list has one more number than steps: 86.

Questions 16 to 25

  1. 16.E (21). Fix the first digit, then count the ways the other digits can make up the rest of 6. Adding the cases gives 21.
  2. 17.C (75°). The top triangle's angle at E is 180° − 57° − 48°, and x is vertically opposite it: 75°.
  3. 18.D (165). Any 3 of the points make exactly one triangle, so count the ways to choose 3 of the 11: 165.
  4. 19.C (37). Each zero at the end comes from a 10 = 2 × 5, and there are far more 2s than 5s, so count the 5s. There are 30 multiples of 5, 6 multiples of 25, 1 multiples of 125, so 153! ends in 37 zeros.
  5. 20.B (240). In one pool there are 16 × 15 ÷ 2 = 120 pairs of teams, each meeting twice. That is 240 games.
  6. 21.E (72). 105043750 = 21 × 55 × 75. A divisor takes each prime to any power from 0 up, so there are 2 × 6 × 6 = 72 divisors.
  7. 22.D (20). 8 = 23 and 16 = 24, so 3x = 4(x − 5). That gives 20 = 1x, so x = 20.
  8. 23.D (9 students). 6 + 21 − 2 = 25 do at least one, and the other 9 of the 34 do neither. So the answer is 9 students.
  9. 24.E (9). 45 = 3 × 3 × 5, 81 = 3 × 3 × 3 × 3. The primes they all share multiply to 9.
  10. 25.C (41). Count routes corner by corner: each corner's count is the sum of the corner to its west and the corner to its south, with the closed corner set to 0. That gives 41.

Written by Scholar Tests in the intermediate maths challenge's published shape. Not a past paper; we are not connected with the UK Mathematics Trust.

Sit a new paper online

Every online paper is new, timed like the real thing, and marked the moment it ends, 5 marks for each of the first 15 and 6 for the rest.

  • Year 10

    25 questions · 60 minutes

    Number theory, algebra, counting and geometry, the hardest last. No calculator.

Questions parents ask

Is this paper right for Year 10?
Yes. It is the intermediate challenge at its own level: number theory, algebra, counting and geometry, the hardest ten last.
How is it marked?
Questions 1 to 15 are worth 5 marks each and 16 to 25 are worth 6, so the paper is out of 135. Our marking takes nothing off for a wrong answer; check the rules printed on the real paper, as some have taken marks off late in the paper.
Can my child use a calculator?
No. The challenge allows no calculator or measuring instruments, only rough paper. Practise the same way.
Are these real past papers?
No. We write our own questions in the challenge's published shape. Past papers belong to the UK Mathematics Trust, and we are not connected with it.

Papers for other years

Year 9 paperYear 11 paper

The other maths challenges

Primary Maths ChallengeJunior Maths Challenge