Revision Library

revisionlibrary.co.uk

11+ · Mathematics

EP.M82 Number Puzzles and Logic Set E

GL ASSESSMENT GL · Calculators not allowed · about 83 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

In a chess tournament, Priya won fewer games than Zach. Zach won fewer games than Wren. Write the three players in order from most games won to fewest games won.

(Total for Question 1 is 2 marks)

Harry is 9 years old. His cousin Elsie is 6 years older than him. How old was Elsie 4 years ago?

(Total for Question 2 is 2 marks)

Mia has 6 twenty-pence coins and 3 pound coins in her purse. How much money does she have in total? Give your answer in pounds.

(Total for Question 3 is 2 marks)

Look at this sequence: 4, 9, 14, 19, ... What is the next term?

(Total for Question 4 is 1 mark)

List all the two-digit numbers that can be made by choosing two different digits from 2, 5 and 8 (no digit repeated within a number) that are even.

(Total for Question 5 is 2 marks)

A necklace is made using red beads and blue beads in the ratio 2:3. If there are 35 beads in total, how many blue beads are there?

(Total for Question 6 is 2 marks)

Three friends - Anya, Ben and Cara - are each in a different school club: Chess, Drama or Art.
1. Anya is not in Drama.
2. The Chess club member is not Cara.
3. Ben is in Art.
Work out which club each friend is in.

(Total for Question 7 is 3 marks)

Five pupils - Femi, Grace, Hugo, Isla and Jae - are standing in a single line for a fire drill, numbered position 1 (front) to position 5 (back).
1. Hugo is exactly in the middle.
2. Femi is somewhere in front of Isla.
3. Jae is at the very back.
4. Grace is directly in front of Hugo.
Work out the order of the line from front to back.

(Total for Question 8 is 3 marks)

In 7 years' time, Ayesha will be three times as old as she was 5 years ago. How old is Ayesha now?

(Total for Question 9 is 3 marks)

Compost mix A is made from topsoil and manure in the ratio 4:1. Compost mix B is made from topsoil and manure in the ratio 3:2. Ronke mixes 250 g of Mix A with 250 g of Mix B. How much manure is there in total in the new mixture?

(Total for Question 10 is 3 marks)

Anya is thinking of a two-digit number. The sum of its digits is 11. The tens digit is 3 more than the units digit. What is Anya's number?

(Total for Question 11 is 3 marks)

How many three-digit numbers greater than 500 can be made using the digits 1, 4, 5 and 9, if each digit can be used at most once per number?

(Total for Question 12 is 3 marks)

Priya's purse contains only 10p, 20p and 50p coins. She has twice as many 10p coins as 50p coins, and 3 more 20p coins than 50p coins. Altogether the coins are worth GBP 4.20. How many of each coin does she have?

(Total for Question 13 is 4 marks)

Four pupils - Kai, Lily, Mo and Nadia - each play a different musical instrument (piano, violin, flute or drums) and each has lessons on a different day (Monday, Tuesday, Wednesday or Thursday).
1. The pupil who plays drums has lessons on Wednesday.
2. Kai does not play piano or violin.
3. Lily has lessons on Monday.
4. Mo plays flute.
5. Nadia does not have lessons on Wednesday.
6. The violin player has lessons on Thursday.
Work out each pupil's instrument and lesson day.

(Total for Question 14 is 4 marks)

Three bus routes stop at the same bus stop. Route A stops every 12 minutes, Route B stops every 18 minutes, and Route C stops every 30 minutes. All three routes stop together at 09:00. What is the next time all three routes stop together?

(Total for Question 15 is 4 marks)

In a sequence, each term after the first two is the sum of the two terms directly before it. The 4th term is 15 and the 5th term is 24. Find the 1st term and the 6th term of the sequence.

(Total for Question 16 is 4 marks)

A two-digit number has tens digit X and units digit Y. When its digits are reversed, the new two-digit number is 27 more than the original number. The sum of the two digits is 9. Find the original two-digit number.

(Total for Question 17 is 4 marks)

In a group of 40 pupils, 24 attend the coding club, 19 attend the chess club, and 9 attend neither club. How many pupils attend both clubs?

(Total for Question 18 is 3 marks)

A four-digit number is formed using the digits 2, 3, 5 and 8, each digit used exactly once. There are 24 different arrangements of these four digits. How many of these 24 arrangements give a number divisible by 3? Explain your reasoning without listing all 24 numbers.

(Total for Question 19 is 4 marks)

The combined age of a father and his two children is 42. The father is 3 times as old as his elder child. The younger child is two-thirds the age of the elder child. How old is each person?

(Total for Question 20 is 4 marks)

Using the digits 1, 2, 3, 4 and 5, with each digit used at most once, how many three-digit numbers greater than 300 can be formed that are multiples of 5?

(Total for Question 21 is 4 marks)

Five friends - Ade, Bo, Cerys, Dexter and Ffion - each cycled a different distance last weekend, chosen from 3 km, 6 km, 9 km, 15 km and 21 km (one distance each, no ties).
1. Cerys cycled further than Bo but less far than Dexter.
2. The difference between Ade's distance and Ffion's distance is 12 km.
3. Bo cycled the shortest distance of the five.
4. Dexter did not cycle the furthest distance.
5. Ade cycled 9 km.
Work out how far each friend cycled, and state who cycled the furthest.

(Total for Question 22 is 5 marks)

A digital clock shows the time 08:47. A 'palindrome time' is a time where the four digits (ignoring the colon) read the same forwards and backwards, for example 12:21. How many minutes will pass before the clock next shows a palindrome time?

(Total for Question 23 is 5 marks)

In a survey of 60 pupils: 28 like apples, 25 like bananas, 26 like cherries, 10 like both apples and bananas, 11 like both bananas and cherries, 9 like both apples and cherries, and 4 like all three fruits. How many pupils like none of the three fruits, and how many like exactly one of the three fruits?

(Total for Question 24 is 5 marks)

At a chess club meeting, every member shakes hands with every other member exactly once. There are 91 handshakes in total. How many members are at the meeting? If 3 new members then join, and everyone shakes hands with every other member again (including the new arrivals), how many additional handshakes take place?

(Total for Question 25 is 5 marks)

The nth term of a sequence is given by the formula n² + 4n. Find the first three terms of the sequence, and find the value of n for which the nth term equals 140.

(Total for Question 26 is 5 marks)

Mark scheme · EP.M82 Number Puzzles and Logic Set E

Question 1

B1 correctly identifies Zach is in the middle position oe
B1 fully correct order: Wren, Zach, Priya cao
Answer: Wren, Zach, Priya

Question 2

M1 Elsie's age now = 9 + 6 = 15 oe
A1 11 cao
Answer: 11

Question 3

M1 6 x 20 + 3 x 100 = 120 + 300 = 420 (pence) oe
A1 GBP 4.20 cao
Answer: GBP 4.20

Question 4

B1 24 cao
Answer: 24

Question 5

B1 at least 2 of the 4 correct numbers listed, with no repeats or invalid digits
B1 all four correct numbers listed: 28, 52, 58, 82 cao
Answer: 28, 52, 58, 82

Question 6

M1 1 part = 35 / 5 = 7 oe
A1 21 cao
Answer: 21

Question 7

M1 places Ben in Art, as given
M1 uses clue 1 to show Anya cannot be in Drama or Art (Art is taken), so Anya must be in Chess
A1 fully correct: Anya - Chess, Ben - Art, Cara - Drama cao
Answer: Anya - Chess, Ben - Art, Cara - Drama

Question 8

M1 places Hugo in position 3 and Jae in position 5, as given
M1 places Grace in position 2, directly in front of Hugo
A1 fully correct order: Femi, Grace, Hugo, Isla, Jae cao (using clue 2 to put Femi before Isla in the remaining positions 1 and 4)
Answer: Femi, Grace, Hugo, Isla, Jae

Question 9

M1 sets up the equation x + 7 = 3(x - 5), where x is Ayesha's age now
M1 correctly expands and rearranges: x + 7 = 3x - 15, so 22 = 2x oe
A1 11 cao
Answer: 11

Question 10

M1 manure in mix A = 1/5 x 250 = 50 g oe
M1 manure in mix B = 2/5 x 250 = 100 g oe
A1 150 g cao
Answer: 150 g

Question 11

M1 sets up tens + units = 11 and tens = units + 3 oe
M1 substitutes to solve: (units + 3) + units = 11, so 2 x units = 8, units = 4 oe
A1 74 cao
Answer: 74

Question 12

M1 identifies that the hundreds digit must be 5 or 9 for the number to exceed 500 (2 choices) oe
M1 correctly counts 6 arrangements for each choice of hundreds digit (3 x 2 for the remaining two positions) oe
A1 12 cao
Answer: 12

Question 13

M1 lets the number of 50p coins be x, so 10p coins = 2x and 20p coins = x + 3
M1 forms the value equation 10(2x) + 20(x + 3) + 50x = 420, i.e. 90x + 60 = 420 oe
M1 solves to x = 4 oe
A1 8 ten-pence coins, 7 twenty-pence coins and 4 fifty-pence coins cao
Answer: 8 ten-pence coins, 7 twenty-pence coins, 4 fifty-pence coins

Question 14

B1 Kai - drums, Wednesday cao (deduced: Kai cannot play piano or violin, and flute is Mo's, so Kai must play drums; the drums player has lessons on Wednesday)
B1 Lily - piano, Monday cao (Lily's day is given as Monday; since the violin player has Thursday, Lily cannot play violin, so she plays piano)
B1 Nadia - violin, Thursday cao (the remaining instrument, confirmed by clue 6 and consistent with clue 5)
B1 Mo - flute, Tuesday cao (deduced by elimination, the only day left once Monday, Wednesday and Thursday are taken)
Answer: Kai - drums, Wednesday; Lily - piano, Monday; Mo - flute, Tuesday; Nadia - violin, Thursday

Question 15

M1 recognises this is a lowest common multiple problem and finds prime factors: 12 = 2² x 3, 18 = 2 x 3², 30 = 2 x 3 x 5 oe
M1 LCM = 2² x 3² x 5 = 180 oe
M1 converts 180 minutes to 3 hours oe
A1 12:00 cao
Answer: 12:00

Question 16

M1 term 3 = term 5 - term 4 = 24 - 15 = 9 oe
M1 term 2 = term 4 - term 3 = 15 - 9 = 6, then term 1 = term 3 - term 2 = 9 - 6 = 3 oe
A1 1st term = 3 cao
A1 6th term = term 5 + term 4 = 24 + 15 = 39 cao
Answer: 1st term = 3, 6th term = 39

Question 17

M1 sets up (10Y + X) - (10X + Y) = 27, which simplifies to 9(Y - X) = 27, so Y - X = 3 oe
M1 uses the digit sum X + Y = 9 oe
M1 solves simultaneously: Y = X + 3 substituted into X + Y = 9 gives 2X + 3 = 9, so X = 3 oe
A1 36 cao
Answer: 36

Question 18

M1 pupils attending at least one club = 40 - 9 = 31 oe
M1 forms the equation 24 + 19 - both = 31 oe
A1 12 cao
Answer: 12

Question 19

M1 finds the digit sum = 2 + 3 + 5 + 8 = 18 oe
M1 recognises that a number is divisible by 3 exactly when its digit sum is divisible by 3, and rearranging the digits does not change this digit sum
A1 18 is divisible by 3 (18 = 3 x 6), so this digit sum condition is met regardless of the order of the digits
A1 24 (all of the arrangements) cao
Answer: 24 (all of the arrangements); the digit sum 2+3+5+8 = 18 is divisible by 3, and rearranging digits never changes the digit sum, so every arrangement is divisible by 3

Question 20

M1 sets up elder child = e, father = 3e, younger child = (2/3)e
M1 forms the equation e + 3e + (2/3)e = 42, i.e. (14/3)e = 42 oe
M1 solves to e = 9 oe
A1 father = 27, elder child = 9, younger child = 6 cao
Answer: Father = 27, elder child = 9, younger child = 6

Question 21

M1 identifies the units digit must be 5, since 5 is the only available digit that is a multiple of 5
M1 identifies the hundreds digit must be 3 or 4 for the number to exceed 300 (2 choices, since 5 is already used) oe
M1 the tens digit is then chosen from the remaining 3 digits (3 choices) oe
A1 6 cao
Answer: 6

Question 22

B1 Ade = 9 km cao (given directly)
B1 Bo = 3 km cao (the shortest of the five distances)
M1 Ffion = Ade + 12 = 21 km cao (rejecting Ade - 12 = -3, which is not a valid distance)
M1 Cerys = 6 km and Dexter = 15 km, using Bo < Cerys < Dexter on the remaining values {6, 15}
A1 fully correct: Ade 9 km, Bo 3 km, Cerys 6 km, Dexter 15 km, Ffion 21 km, and Ffion cycled the furthest cao
Answer: Ade = 9 km, Bo = 3 km, Cerys = 6 km, Dexter = 15 km, Ffion = 21 km; Ffion cycled the furthest

Question 23

M1 recognises that a palindrome time HH:MM needs the first digit of the hour to equal the second digit of the minutes, and the second digit of the hour to equal the first digit of the minutes oe
M1 shows hour 08 would need minutes '80' (invalid, minutes cannot exceed 59) oe
M1 shows hour 09 would need a minutes tens digit of 9 (invalid, minutes tens digit cannot exceed 5) oe
A1 identifies hour 10 gives valid minutes '01', so the next palindrome time is 10:01 cao
A1 74 minutes cao
Answer: 10:01 (74 minutes after 08:47)

Question 24

M1 finds the number liking exactly two fruits: apples-and-bananas only = 10 - 4 = 6, bananas-and-cherries only = 11 - 4 = 7, apples-and-cherries only = 9 - 4 = 5
M1 finds the number liking exactly one fruit: apples only = 28 - 6 - 5 - 4 = 13, bananas only = 25 - 6 - 7 - 4 = 8, cherries only = 26 - 5 - 7 - 4 = 10 oe
A1 exactly one fruit = 13 + 8 + 10 = 31 cao
M1 total liking at least one fruit = 31 + 6 + 7 + 5 + 4 = 53 oe
A1 none = 60 - 53 = 7 cao
Answer: 31 pupils like exactly one fruit; 7 pupils like none of the three fruits

Question 25

M1 sets up n(n-1)/2 = 91, i.e. n(n-1) = 182 oe
M1 solves (e.g. by testing values or factorising n² - n - 182 = 0) to find n = 14, rejecting the negative root n = -13 oe
A1 14 members cao
M1 total handshakes with 17 members = 17 x 16 / 2 = 136 oe
A1 additional handshakes = 136 - 91 = 45 cao
Answer: 14 members originally; 45 additional handshakes when 3 more join

Question 26

M1 substitutes n = 1, 2, 3 into n² + 4n oe
A1 first three terms: 5, 12, 21 cao
M1 sets up the equation n² + 4n = 140, i.e. n² + 4n - 140 = 0 oe
dM1 factorises to (n - 10)(n + 14) = 0, or solves using the quadratic formula, giving n = 10 or n = -14 oe
A1 n = 10 cao (rejecting the negative solution as n must be a positive whole number)
Answer: First three terms: 5, 12, 21. n = 10

Number Puzzles and Logic Set E - Worksheets, Questions and Revision