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11+ · Numerical Reasoning and Logical Problem-Solving

EP.M43 Multi-Step Reasoning Challenge Set 2

GL ASSESSMENT GL-11PLUS-MATHS · Calculators not allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Priya is 7 years old. Her mother is exactly 5 times Priya's age. How old will Priya be when her mother is exactly 3 times Priya's age?

(Total for Question 1 is 2 marks)

A jar contains only 10p and 20p coins. There are 23 coins in total, worth £3.30 altogether. How many 20p coins are in the jar?

(Total for Question 2 is 3 marks)

Four friends - Ben, Chloe, Dan and Ella - each choose a different sandwich filling for lunch: cheese, ham, tuna or egg.
- Ben does not choose cheese or ham.
- Chloe chooses tuna.
- Dan chooses ham.
Work out which filling Ella chooses.

(Total for Question 3 is 2 marks)

Using the digits 2, 5 and 7, with each digit used at most once, how many different two-digit even numbers can be made?

(Total for Question 4 is 2 marks)

Tariq thinks of a number. He doubles it, then adds 7. He then subtracts 3 and divides the result by 2, giving a final answer of 15. What number did Tariq start with?

(Total for Question 5 is 3 marks)

A fruit drink is made by mixing orange juice and lemonade in the ratio 3:5. If 40 litres of the drink are made in total, how many more litres of lemonade than orange juice are used?

(Total for Question 6 is 3 marks)

The sum of the ages of a father and his daughter is 52. In 6 years' time, the father will be exactly 3 times as old as the daughter. How old is the daughter now?

(Total for Question 7 is 3 marks)

Freya spent half of her money on a book. She then spent £4 on a magazine, which left her with £9. How much money did Freya start with?

(Total for Question 8 is 3 marks)

A padlock code uses three different digits chosen from 1, 2, 3 and 4, with no digit repeated. How many different codes are possible if the code must end in an odd digit?

(Total for Question 9 is 3 marks)

The first term of a sequence is 5. Each term after the first is found by doubling the previous term and then subtracting 3. Find the fourth term of the sequence.

(Total for Question 10 is 3 marks)

Four pupils - Rosie, Sam, Priya and Tom - each sit one exam: Maths, English, Science or French.
- Sam sits English.
- Tom's exam is Science.
- Rosie does not sit French.
Work out which exam Rosie sits and which exam Priya sits.

(Total for Question 11 is 3 marks)

A fruit squash is made from concentrate and water mixed in the ratio 1:4. Ffion wants to make 2.5 litres of the same-strength squash. How many millilitres of concentrate does she need?

(Total for Question 12 is 2 marks)

Using the digits 1, 2, 3, 4 and 5, how many three-digit numbers can be formed that are multiples of 5, if all three digits in the number must be different?

(Total for Question 13 is 3 marks)

A number is multiplied by 4, then 8 is subtracted. The result is divided by 4, and then 5 is added, giving a final answer of 12. What was the original number?

(Total for Question 14 is 3 marks)

Sasha, Ben and Chidi have 5, 8 and 11 stickers between them, though not necessarily in that order.
- Sasha has fewer stickers than Ben.
- Chidi does not have the fewest stickers.
- Ben does not have 11 stickers.
Work out how many stickers each person has.

(Total for Question 15 is 3 marks)

After spending 1/3 of his money on a comic, and then 1/4 of what remained on sweets, Oscar has £6 left. How much money did Oscar start with?

(Total for Question 16 is 3 marks)

A three-digit number is formed using each of the digits 3, 4 and 6 exactly once. How many of the possible three-digit numbers are both greater than 400 and even?

(Total for Question 17 is 3 marks)

A gardener has 12 litres of plant feed made from concentrate and water mixed in the ratio 1:5 (concentrate:water). She wants to strengthen the mixture to a ratio of 1:2 by adding more concentrate, without adding any more water. How many litres of concentrate must she add?

(Total for Question 18 is 3 marks)

A sequence of matchstick patterns follows the rule 4, 7, 10, 13, ... with the same difference between consecutive terms. How many matchsticks are needed for the 12th pattern in the sequence?

(Total for Question 19 is 3 marks)

Four friends - Owen, Priti, Quinn and Reema - each ran a different distance in a charity run: 5 km, 8 km, 10 km and 15 km.
- Owen ran further than Priti but less far than Reema.
- Quinn ran the shortest distance of the four.
Work out how far each person ran.

(Total for Question 20 is 4 marks)

Two sisters, Nadia and Zara, together have £45. After Nadia gives Zara £6, Zara then has exactly twice as much money as Nadia. How much money did Nadia have originally?

(Total for Question 21 is 3 marks)

In a sequence, the first term is 2 and the second term is 3. Every term after that is found by adding the two previous terms together and then subtracting 1. Find the sixth term of the sequence.

(Total for Question 22 is 3 marks)

A box contains 5 different coloured hats: red, blue, green, yellow and purple. Three pupils sitting in a row each take one hat, so that no colour is used twice. How many different arrangements are possible in which the red hat is given to one of the three pupils?

(Total for Question 23 is 4 marks)

A fruit bowl contains only apples and oranges.

(a)There are 3 more apples than oranges, and 27 pieces of fruit in total. How many oranges are there?(2)

(b)Each apple weighs 120 g and each orange weighs 150 g. What is the total weight of the fruit in the bowl, in kilograms?(3)

(Total for Question 24 is 5 marks)

At a school fete, three stalls - Cakes, Books and Games - each raised a different amount of money: £15, £30 and £45, though not necessarily in that order.
- The Games stall raised exactly twice as much as the Cakes stall.
- The Books stall did not raise the least amount.
Work out how much money each stall raised.

(Total for Question 25 is 4 marks)

Mrs Whitfield needs to give a customer exactly £1 in change, using only 20p, 10p and 5p coins. She must use at least one of each type of coin. In how many different ways can she make the £1?

(Total for Question 26 is 4 marks)

Mark scheme · EP.M43 Multi-Step Reasoning Challenge Set 2

Question 1

M1 forms an equation such as 35+t=3(7+t), where t is the number of years from now (oe)
A1 14 cao
Answer: Priya will be 14 years old (her mother will be 42)

Question 2

M1 forms x+y=23 and 10x+20y=330, where x is the number of 10p coins and y the number of 20p coins (oe)
M1 divides the value equation by 10 to get x+2y=33, then subtracts x+y=23 to get y=10 (oe)
A1 10 twenty pence coins cao
Answer: 10 twenty pence coins (and 13 ten pence coins)

Question 3

B1 identifies Ben's filling as egg, the only one left once cheese and ham are ruled out (oe)
B1 Ella's filling is cheese cao
Answer: Ella chooses cheese

Question 4

M1 recognises that only the digit 2 can be the units digit of an even number from this set (oe)
A1 2 cao (the numbers 52 and 72)
Answer: 2 (the numbers 52 and 72)

Question 5

M1 forms the equation (2x+7-3)/2=15, where x is the starting number (oe)
M1 simplifies to 2x+4=30 (oe)
A1 13 cao
Answer: 13

Question 6

M1 finds the total number of parts = 3+5=8 (oe)
M1 finds 1 part = 40/8=5 litres, so orange juice=15 litres and lemonade=25 litres (oe)
A1 10 litres cao
Answer: 10 litres

Question 7

M1 sets up f+d=52, so f=52-d (oe)
M1 forms (52-d)+6=3(d+6) (oe)
A1 10 cao
Answer: The daughter is 10 now (the father is 42)

Question 8

M1 adds back the £4 spent on the magazine: 9+4=13 (oe)
M1 recognises £13 is half of the original amount, so doubles it: 13x2=26 (oe)
A1 £26 cao
Answer: £26

Question 9

M1 identifies 2 choices for the units digit, 1 or 3 (oe)
M1 finds the remaining two digits can be arranged in the first two positions in 3x2=6 ways (oe)
A1 12 cao
Answer: 12

Question 10

M1 finds the second term: 2x5-3=7 (oe)
M1 finds the third term: 2x7-3=11 (oe)
A1 19 cao, from 2x11-3
Answer: 19

Question 11

M1 recognises the two remaining exams for Rosie and Priya are Maths and French (oe)
A1 Rosie sits Maths cao
A1 Priya sits French cao
Answer: Rosie sits Maths, Priya sits French

Question 12

M1 recognises concentrate makes up 1/5 of the total mixture (oe)
A1 500 ml cao
Answer: 500 ml

Question 13

M1 recognises the units digit must be 5, the only multiple of 5 available (oe)
M1 finds the hundreds digit has 4 choices and the tens digit has 3 remaining choices (oe)
A1 12 cao
Answer: 12

Question 14

M1 undoes the final '+5' step: 12-5=7 (oe)
M1 undoes the '/4' and '-8' steps: 7x4=28, then 28+8=36 (oe)
A1 9 cao, from 36/4
Answer: 9

Question 15

M1 uses 'Ben does not have 11' to show Ben has 5 or 8 stickers (oe)
M1 tests Ben=5 and rejects it, since Sasha would then need fewer than 5, which is impossible, so Ben=8 (oe)
A1 Sasha=5, Ben=8, Chidi=11, all cao
Answer: Sasha has 5 stickers, Ben has 8 stickers, Chidi has 11 stickers

Question 16

M1 recognises the £6 remaining is 3/4 of the amount he had before buying sweets, so finds 6 divided by 3/4 = £8 (oe)
M1 recognises £8 is 2/3 of the original amount, so finds 8 divided by 2/3 = £12 (oe)
A1 £12 cao
Answer: £12

Question 17

M1 lists all 6 permutations of 3, 4 and 6: 346, 364, 436, 463, 634, 643 (oe)
M1 identifies which are greater than 400 (first digit 4 or 6): 436, 463, 634, 643 (oe)
A1 2 cao, since only 436 and 634 also end in an even digit
Answer: 2 (the numbers 436 and 634)

Question 18

M1 finds the current amounts: concentrate=2 litres, water=10 litres, from the ratio 1:5 of 12 litres (oe)
M1 forms the equation (2+x):10=1:2, where x is the concentrate added (oe)
A1 3 litres cao
Answer: 3 litres

Question 19

M1 finds the common difference is 3 (oe)
M1 forms the nth term rule 3n+1, or builds the sequence up to the 12th term systematically (oe)
A1 37 cao
Answer: 37

Question 20

B1 Quinn ran 5 km, the shortest distance, cao
M1 recognises the remaining three distances must be assigned in increasing order Priti < Owen < Reema (oe)
A1 Priti=8 km and Owen=10 km cao
A1 Reema=15 km cao
Answer: Quinn 5 km, Priti 8 km, Owen 10 km, Reema 15 km

Question 21

M1 forms n+z=45 and z+6=2(n-6), where n and z are Nadia's and Zara's original amounts (oe)
M1 substitutes z=45-n and solves to give n=21 (oe)
A1 £21 cao
Answer: £21

Question 22

M1 finds the third and fourth terms: 3+2-1=4, then 4+3-1=6 (oe)
M1 finds the fifth term: 6+4-1=9 (oe)
A1 14 cao, from 9+6-1
Answer: 14

Question 23

M1 finds the total number of ways to give 3 different-coloured hats to 3 pupils with no restriction: 5x4x3=60 (oe)
M1 chooses which of the 3 pupils receives the red hat: 3 ways (oe)
M1 finds the remaining 2 pupils receive 2 of the remaining 4 colours, in order: 4x3=12 ways (oe)
A1 36 cao, from 3x12
Answer: 36

Question 24

(a) M1 forms 2o+3=27, where o is the number of oranges (oe)
(a) A1 12 cao
(a) Answer: 12 oranges
(b) M1 finds the apples' weight: 15x120=1800 g (ft from a)
(b) M1 finds the oranges' weight: 12x150=1800 g (ft from a)
(b) A1 3.6 kg cao
(b) Answer: 3.6 kg

Question 25

M1 tests each pair of the three values for a doubling relationship and finds only 15 and 30 work, since 30=2x15 (oe)
A1 Cakes=£15 and Games=£30 cao
B1 Books must be the remaining value, £45, cao
B1 checks Books (£45) is not the least amount, since the least is Cakes (£15), so the clue is satisfied (oe)
Answer: Cakes £15, Games £30, Books £45

Question 26

M1 recognises at most 4 twenty pence coins can be used, since at least one 10p and one 5p coin must remain (oe)
M1 for each number of 20p coins used (1, 2, 3 or 4), systematically counts the valid combinations of 10p and 5p coins that make up the rest, e.g. with one 20p coin there are 7 ways (oe)
A1 finds 7, 5, 3 and 1 ways respectively for 1, 2, 3 and 4 twenty pence coins, all cao
A1 16 cao, from 7+5+3+1
Answer: 16 ways

Multi-Step Reasoning Challenge Set 2 - Worksheets, Questions and Revision