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11+ · GL ASSESSMENT

EP.M23 Multi-Step Reasoning Challenge

GL ASSESSMENT GL-11PLUS-MATHS · Calculators not allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Amina thinks of a number. She adds 12 to it, then multiplies the result by 3, giving 90. What number did Amina start with?

(Total for Question 1 is 2 marks)

A fruit drink is made by mixing orange juice and water in the ratio 2:5. If 400 ml of orange juice is used, how much water is needed?

(Total for Question 2 is 2 marks)

Priya is 4 years older than her brother Kofi. In 6 years' time, the sum of their ages will be 36. How old is Priya now?

(Total for Question 3 is 3 marks)

A rectangular lawn is 3 times as long as it is wide. Its perimeter is 48 m. What is the area of the lawn?

(Total for Question 4 is 3 marks)

The first four terms of a sequence are 5, 11, 21, 35. The differences between consecutive terms increase in a regular pattern. If this pattern continues, what is the 6th term of the sequence?

(Total for Question 5 is 3 marks)

A three-digit number has digits that add up to 15. The hundreds digit is twice the units digit, and the tens digit is 3 more than the units digit. What is the number?

(Total for Question 6 is 3 marks)

In a school choir, the ratio of sopranos to altos is 3:2, and the ratio of altos to tenors is 4:5. There are 30 tenors in the choir. How many sopranos are there?

(Total for Question 7 is 3 marks)

A number is multiplied by 5, then 8 is subtracted. This gives the same result as when the original number is multiplied by 3 and then increased by 20. What is the original number?

A) 6
B) 10
C) 14
D) 16
E) 28

(Total for Question 8 is 2 marks)

Six years ago, Tomasz was three times as old as his sister Ola. In four years' time, he will be twice as old as her. How old is Tomasz now?

(Total for Question 9 is 4 marks)

A square garden has the same area as a rectangular garden measuring 12 m by 3 m. What is the perimeter of the square garden?

(Total for Question 10 is 3 marks)

A sequence of patterns is made from dots. Pattern 1 has 4 dots, Pattern 2 has 9 dots, and Pattern 3 has 16 dots. If this pattern continues, how many dots will Pattern 7 have?

(Total for Question 11 is 2 marks)

A number between 100 and 200 is a multiple of both 6 and 8. It is also one more than a multiple of 7. What is the number?

A) 96
B) 120
C) 144
D) 168
E) 192

(Total for Question 12 is 3 marks)

Leila, Ben and Aisha share a sum of money in the ratio 5:3:4. Ben receives £18 less than Leila. How much money did they share in total?

(Total for Question 13 is 3 marks)

In a class, everyone who plays chess also plays draughts. Sara plays draughts but does not play chess. Tom plays chess. Which statement must be true?

A) Sara plays chess
B) Tom plays draughts
C) Everyone who plays draughts also plays chess
D) Sara and Tom play the same games
E) No one plays both chess and draughts

(Total for Question 14 is 2 marks)

A garden is L-shaped, formed from a 10 m by 8 m rectangle with a 4 m by 3 m rectangular corner cut away. What is the perimeter of the L-shaped garden?

(Total for Question 15 is 3 marks)

The four-digit number formed by the digits 3, an unknown digit, 7 and 4, in that order, is exactly divisible by 9. What is the unknown digit?

(Total for Question 16 is 2 marks)

Jug A contains orange squash and water in the ratio 1:4. Jug B contains 600 ml of the same squash and 900 ml of water. When Jug A and Jug B are combined, the mixture contains 800 ml of squash in total. How much water was in Jug A before the jugs were combined?

(Total for Question 17 is 3 marks)

The sum of the ages of a mother and her twin daughters is 57. In 5 years' time, the mother will be twice as old as one of the twins. How old is each twin now, and how old is the mother?

(Total for Question 18 is 4 marks)

A cyclist travels from Norbridge to Elmstead, a distance of 60 km, at an average speed of 15 km/h. She rests for 45 minutes, then continues to Fenwick, a further 40 km, arriving at 15:15 having started her journey at 10:00. What was her average speed, in km/h, for the second part of the journey (Elmstead to Fenwick)?

(Total for Question 19 is 4 marks)

Three consecutive even numbers have a sum equal to twice the largest number, plus 18. What is the smallest of the three numbers?

(Total for Question 20 is 3 marks)

A rectangle has a perimeter of 50 cm. Its length is 5 cm more than its width. If the width is increased by 2 cm and the length is decreased by 3 cm, the new rectangle is a square. What is the area of the original rectangle?

(Total for Question 21 is 4 marks)

In a sequence, each term after the first two is found by adding the two previous terms. The first term is 4 and the fourth term is 30. What is the second term?

(Total for Question 22 is 3 marks)

Mark scheme · EP.M23 Multi-Step Reasoning Challenge

Question 1

M1 reverses the operations: 90 / 3 = 30
A1 18, from 30 - 12
Answer: 18

Question 2

M1 finds 1 part = 400 / 2 = 200 ml
A1 1000 ml (1 litre) of water, from 5 x 200
Answer: 1000 ml (1 litre)

Question 3

M1 sets Kofi's age now as k and Priya's as k + 4
M1 forms (k + 6) + (k + 4 + 6) = 36 and simplifies to 2k + 16 = 36
A1 14, from k = 10 so Priya = k + 4
Answer: Priya is 14 now

Question 4

M1 forms 2(w + 3w) = 48, i.e. 8w = 48
M1 w = 6, so length = 18
A1 108 m², from 6 x 18
Answer: 108 m²

Question 5

M1 finds the differences 6, 10, 14 and spots they increase by 4 each time
M1 extends the differences to 18 and 22
A1 75, from 35 + 18 = 53 then 53 + 22 = 75
Answer: 75

Question 6

M1 lets the units digit be u and forms hundreds = 2u, tens = u + 3
M1 forms 2u + (u + 3) + u = 15, i.e. 4u + 3 = 15, giving u = 3
A1 663, from hundreds = 6, tens = 6, units = 3
Answer: 663

Question 7

M1 uses altos:tenors = 4:5 with 30 tenors to find altos = 24
M1 uses sopranos:altos = 3:2 with 24 altos
A1 36 sopranos, from 24 / 2 x 3
Answer: 36 sopranos

Question 8

M1 forms 5x - 8 = 3x + 20 and collects terms to 2x = 28
A1 C) 14
Answer: C) 14

Question 9

M1 forms T - 6 = 3(O - 6), i.e. T = 3O - 12
M1 forms T + 4 = 2(O + 4), i.e. T = 2O + 4
M1 solves simultaneously: 3O - 12 = 2O + 4, giving O = 16
A1 36, from T = 2 x 16 + 4
Answer: Tomasz is 36 now

Question 10

M1 finds the rectangle's area = 12 x 3 = 36 m²
M1 finds the square's side length = √36 = 6 m
A1 24 m, from 4 x 6
Answer: 24 m

Question 11

M1 spots that the dot totals are square numbers of (pattern number + 1), i.e. 2², 3², 4²
A1 64, from 8²
Answer: 64

Question 12

M1 lists multiples of the lowest common multiple of 6 and 8 (24) between 100 and 200: 120, 144, 168, 192
M1 tests each against 'one more than a multiple of 7'
A1 B) 120
Answer: B) 120

Question 13

M1 finds Leila's share minus Ben's share = 2 parts = £18, so 1 part = £9
M1 finds the total number of parts = 5 + 3 + 4 = 12
A1 £108, from 12 x £9
Answer: £108

Question 14

M1 applies the rule 'chess implies draughts' to Tom, who plays chess
A1 B) Tom plays draughts
Answer: B) Tom plays draughts

Question 15

M1 recognises that the two edges removed by the notch (4 m and 3 m) are replaced by two new edges of the same total length
M1 finds the perimeter of the original rectangle: 2 x (10 + 8) = 36 m
A1 36 m
Answer: 36 m

Question 16

M1 uses the rule that a number is divisible by 9 when its digit sum is a multiple of 9, forming 3 + d + 7 + 4 = 14 + d
A1 4, since 14 + 4 = 18, a multiple of 9
Answer: 4

Question 17

M1 finds the squash in Jug A = 800 - 600 = 200 ml
M1 uses the ratio 1:4 to scale 200 ml of squash to the matching amount of water
A1 800 ml, from 200 x 4
Answer: 800 ml

Question 18

M1 lets each twin's age now be D and the mother's be M, forming M + 2D = 57
M1 forms M + 5 = 2(D + 5), i.e. M = 2D + 5
M1 solves simultaneously: 2D + 5 + 2D = 57, giving D = 13
A1 twins are 13 each, mother is 31, from M = 2 x 13 + 5
Answer: Each twin is 13; the mother is 31

Question 19

M1 finds the time for the first leg: 60 / 15 = 4 hours, arriving at Elmstead at 14:00
M1 adds the 45-minute rest to find the departure time from Elmstead: 14:45
M1 finds the travel time for the second leg: 15:15 - 14:45 = 30 minutes = 0.5 hours
A1 80 km/h, from 40 / 0.5
Answer: 80 km/h

Question 20

M1 lets the numbers be n, n + 2, n + 4 and forms their sum 3n + 6
M1 forms 3n + 6 = 2(n + 4) + 18 and simplifies to n = 20
A1 20
Answer: 20

Question 21

M1 lets the width be w and the length be w + 5
M1 uses the perimeter to form 2(w + w + 5) = 50, giving w = 10 and length = 15
M1 checks the square condition: (w + 2) = (length - 3), i.e. 12 = 12
A1 150 cm², from 10 x 15
Answer: 150 cm²

Question 22

M1 lets the second term be x and forms the third term as 4 + x
M1 forms the fourth term as x + (4 + x) = 2x + 4, and sets 2x + 4 = 30
A1 13, from x = 13
Answer: 13

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