Intermediate maths challenge practice

Intermediate maths challenge practice paper for Year 9

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.

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Scholar Tests · Sample paper

Intermediate maths challenge-style paper, Year 9

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.

    How many lines of symmetry does this shape have?

    1. A1
    2. B4
    3. C0
    4. D2
    5. E3
  2. 2.

    Each of these three numbers has one digit hidden by a question mark: 12?, 2?7, 3?7. The average of the three numbers is 243. What is the sum of the three hidden digits?

    1. A9
    2. B0
    3. C95
    4. D14
    5. E5
  3. 3.

    Three consecutive prime numbers add up to 131. (Consecutive primes come straight after one another, with no other prime between them.) What is the smallest of them?

    1. A43
    2. B37
    3. C44
    4. D47
    5. E41
  4. 4.

    A bag holds 17 marbles numbered 1 to 17. Marbles 1 to 6 are red, 7 to 10 are green and 11 to 17 are blue. One marble is taken out without looking. What is the chance that it is green and its number is prime?

    1. A4/17
    2. B1/4
    3. C11/17
    4. D7/17
    5. E1/17
  5. 5.

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

    1. AYara
    2. BJada
    3. CBen
    4. DMilo
    5. ELena
  6. 6.

    I have a bag of buttons. There are more than 9 and fewer than 19. When I share them out, I find they leave 1 left over when grouped in 4s, and leave 2 left over when grouped in 5s.   How many buttons are there?

    1. A17
    2. B19
    3. C12
    4. D13
    5. E18
  7. 7.

    115 ÷ 5 is the same as which of these?

    1. A(115 ÷ 3) + (115 ÷ 2)
    2. B(5 ÷ 100) + (5 ÷ 15)
    3. C(100 ÷ 5) − (15 ÷ 5)
    4. D(100 ÷ 5) + (15 ÷ 5)
    5. E(100 ÷ 5) × (15 ÷ 5)
  8. 8.

    Square tiles are joined edge to edge to make flat shapes. A shape that matches another when it is turned round or flipped over is not a new shape. How many different shapes can be made from four tiles?

    1. A7
    2. B4
    3. C2
    4. D5
    5. E12
  9. 9.

    A palindrome is a number that reads the same forwards and backwards, such as 373. How many 6-digit palindromes are there?

    1. A9000
    2. B1000
    3. C900
    4. D900000
    5. E90
  10. 10.

    Two whole numbers multiply together to make 72, and one of them is 2 times the other. What is the smaller number?

    1. A6
    2. B8
    3. C36
    4. D12
    5. E24
  11. 11.

    In the triangle below, what is the size of the angle marked x?

    1. A144°
    2. B121°
    3. C95°
    4. D265°
    5. E85°
  12. 12.

    Find the value of x.
    11x + 4 = 5x + 16

    1. A1
    2. B3
    3. C12
    4. D4
    5. E2
  13. 13.

    In a class, 36 children play in the band, 35 have a cat and 13 do both. How many play in the band but do not have a cat?

    1. A13 children
    2. B49 children
    3. C23 children
    4. D36 children
    5. E1 child
  14. 14.

    Mia rolls a die 14 times and gets: 5, 4, 5, 2, 6, 3, 2, 3, 5, 2, 1, 2, 2, 5. From these results, what is the experimental probability of rolling a number greater than 4?

    1. A5/9
    2. B5/6
    3. C5/14
    4. D3/7
    5. E1/3
  15. 15.

    What is the highest common factor of 20, 30, 120?

    1. A170
    2. B20
    3. C120
    4. D10
    5. E5

Questions 16 to 25

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

  1. 16.

    The mean of 24 numbers is 37. One of the numbers is taken away, and the mean of the 23 numbers left is 36. What number was taken away?

    1. A24
    2. B60
    3. C1
    4. D59
    5. E61
  2. 17.

    The equation x2 − 24x + 143 = 0 has two whole-number solutions. What is the sum of the squares of its two solutions?

    1. A290
    2. B576
    3. C24
    4. D862
    5. E433
  3. 18.

    If 25x = 625x − 2, what is the value of x?

    1. A8
    2. B3
    3. C2
    4. D5
    5. E4
  4. 19.

    Four consecutive prime numbers add up to 258. What is the smallest of them?

    1. A59
    2. B61
    3. C53
    4. D65
    5. E71
  5. 20.

    A right-angled triangle has a longest side of 13 cm and another side of 5 cm. What is its perimeter, in centimetres?

    1. A30
    2. B28
    3. C29
    4. D31
    5. E22
  6. 21.

    What is the value of 1³ + 2³ + 3³ + ... + 10³?

    1. A55
    2. B385
    3. C3025
    4. D1000
    5. E2025
  7. 22.

    What is the number formed by the last two digits of 71859?

    1. A49
    2. B44
    3. C13
    4. D42
    5. E43
  8. 23.

    Three different whole numbers multiply to 108 and add to 22. Square the smallest number, then add the largest number. What is the result?

    1. A169
    2. B10
    3. C93
    4. D13
    5. E145
  9. 24.

    What is the total of all the numbers in the list 15, 26, 37, …, 708, where each number is 11 more than the one before?

    1. A23137
    2. B23135
    3. C22428
    4. D23136
    5. E23855
  10. 25.

    How many positive whole numbers divide exactly into 25 × 31 × 55?

    1. A25
    2. B73
    3. C72
    4. D70
    5. E14

Answers and working

Questions 1 to 15

  1. 1.A (1). Only the folds that make both halves match count, so this shape has 1 line of symmetry.
  2. 2.D (14). The total is 3 × 243 = 729. The visible parts make 634, so the hidden digits are worth 95. Whatever way they fit, the digits add to 14.
  3. 3.E (41). The primes 41, 43, 47 come one after another and add up to 131. The smallest is 41.
  4. 4.E (1/17). Of the green ones, 7 to 10, the ones whose number is prime: 7. That is 1 of the 17: 1/17.
  5. 5.D (Milo). Try each child as the one with the ball and count the true statements. Only when Milo has it is exactly one statement true.
  6. 6.A (17). Checking every number from 10 to 18 against all 2 clues, only 17 fits every one.
  7. 7.D ((100 ÷ 5) + (15 ÷ 5)). 115 = 100 + 15, so 115 ÷ 5 = (100 ÷ 5) + (15 ÷ 5), which is 23.
  8. 8.D (5). Listing every shape, with turned and flipped copies counted as the rules say, gives 5 shapes.
  9. 9.C (900). The back half copies the front half, so only the first three digits are chosen. The first cannot be 0, giving 9 ways, and each other free digit has 10: 9 × 10 × 10 = 900.
  10. 10.A (6). Call the smaller number n. The larger is 2n, so the two multiply to 2 × n × n = 72. That makes n × n = 72 ÷ 2 = 36, so n = 6 and the larger number is 12.
  11. 11.E (85°). 180° − 36° − 59° = 85°.
  12. 12.E (2). Take 5x from both sides: 6x + 4 = 16. Take off 4: 6x = 12. Divide by 6: x = 2.
  13. 13.C (23 children). The 36 who play in the band include the 13 who do both, so 36 − 13 play in the band only. So the answer is 23 children.
  14. 14.C (5/14). 5 of the 14 rolls were a number greater than 4, and 5/14 in simplest form is 5/14. So the answer is 5/14.
  15. 15.D (10). 20 = 2 × 2 × 5, 30 = 2 × 3 × 5, 120 = 2 × 2 × 2 × 3 × 5. The primes they all share multiply to 10.

Questions 16 to 25

  1. 16.B (60). The 24 numbers add up to 24 × 37 = 888. The 23 left add up to 23 × 36 = 828. The number taken away is the difference, 60.
  2. 17.A (290). The solutions add to 24 and multiply to 143. The sum of the squares is 242 − 2 × 143 = 290.
  3. 18.E (4). 25 = 52 and 625 = 54, so 2x = 4(x − 2). That gives 8 = 2x, so x = 4.
  4. 19.A (59). The primes 59, 61, 67, 71 come one after another and add up to 258. The smallest is 59.
  5. 20.A (30). By Pythagoras, the third side is √(132 − 52) = 12 cm. The perimeter is 5 + 12 + 13, which is 30.
  6. 21.C (3025). The numbers 1 to 10 add to 55, and the cubes add to 55² = 3025.
  7. 22.E (43). The last two digits of the powers of 7 repeat in a cycle. Find the cycle, then find where the power 1859 falls in it: the answer is 43.
  8. 23.D (13). Only 1, 9 and 12 multiply to 108 and add to 22. Smallest squared is 1; adding the largest, 12, gives 13: 13.
  9. 24.D (23136). There are 64 numbers. Pair the first with the last, the second with the second last, and so on: each pair adds to 723. So the total is 64 × 723 ÷ 2 = 23136.
  10. 25.C (72). 300000 = 25 × 31 × 55. A divisor takes each prime to any power from 0 up, so there are 6 × 2 × 6 = 72 divisors.

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 9

    25 questions · 60 minutes

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

Questions parents ask

Is this paper right for Year 9?
Yes, for a strong Year 9. The intermediate challenge is set for Year 11 and below, so this paper starts on the junior challenge's hardest kinds of question before moving to the intermediate's.
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 10 paperYear 11 paper

The other maths challenges

Primary Maths ChallengeJunior Maths Challenge