For: Year 5 and Year 6 | Exams: KS2 SATs and 11+ | Calculator: not allowed
Jump to: Level 1 – SATs core · Level 2 – SATs stretch · Level 3 – 11+ standard · Level 4 – 11+ selective
📌 KEY STRATEGIES TO REMEMBER
Work systematically: list possibilities in an order (biggest first, or smallest first) so you never miss one or count one twice.
Work backwards: start from the answer and undo each step in reverse order (+ becomes −, × becomes ÷).
Trial and improvement: make a sensible guess, check it, and use the result to make a better guess.
Check every clue: odd + odd = even, odd + even = odd. Test your final answer against every condition in the question.
Level 1 – SATs core: “find the number”, number patterns and odd and even
Matches the short reasoning questions in KS2 SATs Papers 2 and 3.
✏️ WORKED EXAMPLE: I think of a number. I multiply it by 4, then add 7. My answer is 43. What was my number?
Step 1: work backwards from 43, doing the opposite operations in reverse order.
Step 2: the last thing done was “add 7”, so subtract 7: 43 − 7 = 36
Step 3: undo “multiply by 4” by dividing: 36 ÷ 4 = 9. Check: 9 × 4 + 7 = 43 ✓
Common mistake: undoing the steps in the wrong order (43 ÷ 4 first). Always undo the last step first.
Your turn
Q1.1 I think of a number and add 15. The answer is 42. What was my number?
Show answer
Undo “add 15”: 42 − 15 = 27
Q1.2 I think of a number, double it and then subtract 6. The answer is 30. What was my number?
Show answer
Work backwards: 30 + 6 = 36, then 36 ÷ 2 = 18
Q1.3 Here is a number pattern: 3, 7, 11, 15, …
What are the next two numbers?
Show answer
The rule is “add 4”: 15 + 4 = 19, 19 + 4 = 23. 19 and 23
Q1.4 Sam says: “When you add two odd numbers, the answer is always odd.” Is Sam right? Explain.
Show answer
Try one: 3 + 5 = 8, which is even. Two odd numbers always make an even total. No, Sam is wrong.
Q1.5 Two numbers add up to 20. Their difference is 6. What are the two numbers?
Show answer
Take the difference away: 20 − 6 = 14, then 14 ÷ 2 = 7. The other is 7 + 6 = 13. 7 and 13
Q1.6 Lily buys 2 pencils at 8p each and some rubbers at 5p each. She spends 31p altogether. How many rubbers does she buy?
Show answer
Pencils cost 2 × 8 = 16p. Rubbers cost 31 − 16 = 15p, and 15 ÷ 5 = 3 rubbers
Q1.7 Three consecutive whole numbers add up to 36. What are they?
Show answer
The middle number is 36 ÷ 3 = 12. 11, 12 and 13
Q1.8 Here is a number pattern: 5, 11, 17, 23, …
What is the 10th number in the pattern?
Show answer
The rule is “add 6”. From the 1st to the 10th number is 9 steps: 5 + 9 × 6 = 5 + 54 = 59
Q1.9 Max adds together five odd numbers. He says the answer is 40. Explain why Max must have made a mistake.
Show answer
Two odds make an even, so four odds make an even. Adding a fifth odd number makes the total odd. 40 is even, so the total of five odd numbers cannot be 40
Q1.10 Ella has three times as many stickers as Jo. Together they have 48 stickers. How many stickers does Ella have?
Show answer
Jo has 1 part and Ella 3 parts, so 4 parts = 48. 1 part = 12. Ella has 3 × 12 = 36 stickers
Level 2 – SATs stretch: trial and improvement, listing systematically and simple logic
Matches the harder multi-step questions in SATs Papers 2 and 3. Be organised so you do not miss any answers.
✏️ WORKED EXAMPLE: How many different ways can you make exactly 20p using only 2p, 5p and 10p coins?
Step 1: work systematically. Start with the most 10p coins and work down.
Step 2: two 10p coins: 10 + 10. One 10p: 10 + 5 + 5, or 10 + 2 + 2 + 2 + 2 + 2. No 10p: 5 + 5 + 5 + 5, or 5 + 5 + five 2p coins, or ten 2p coins.
(One or three 5p coins leave an odd amount, which 2p coins cannot make.)
Step 3: count the list: 6 ways
Common mistake: writing down ways at random and missing some, or counting 10 + 5 + 5 and 5 + 10 + 5 as different ways.
Your turn
Q2.1 Which whole number, multiplied by itself, gives 169?
Show answer
Try and improve: 12 × 12 = 144 (too small), 13 × 13 = 169 ✓. 13
Q2.2 Ann, Ben and Cal stand in a line. How many different orders can they stand in?
Show answer
List them: ABC, ACB, BAC, BCA, CAB, CBA. That is 6 orders
Q2.3 Two whole numbers multiply to make 48 and add up to 14. What are they?
Show answer
Factor pairs of 48: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8. Only 6 + 8 = 14. 6 and 8
Q2.4 An ice-cream van sells vanilla, chocolate, strawberry and mint. Priya chooses two different flavours. How many different pairs could she choose? (Vanilla and mint is the same pair as mint and vanilla.)
Show answer
List with V first: VC, VS, VM. Then C: CS, CM. Then S: SM. That is 6 pairs
Q2.5 Two consecutive whole numbers multiply to make 182. What are they?
Show answer
10 × 11 = 110 (too small), 13 × 14 = 182 ✓. 13 and 14
Q2.6 A pen and a book cost £3.20 together. The book costs 80p more than the pen. How much does the book cost?
Show answer
Take off the extra: 320p − 80p = 240p. The pen is 240 ÷ 2 = 120p, so the book is 120 + 80 = 200p = £2.00
Q2.7 I am a two-digit odd number. I am a multiple of 7 and my digits add up to 10. What number am I?
Show answer
Multiples of 7 with digit sum 10: 28 and 91. 28 is even, so the answer is 91
Q2.8 Asha, Bilal and Chloe each do a different sport: football, tennis or swimming.
Asha does not play football or tennis. Bilal does not play football.
Which sport does each child do?
Show answer
Asha must swim. Bilal does not play football, so he plays tennis. That leaves football for Chloe. Asha swimming, Bilal tennis, Chloe football
Q2.9 A farm has some hens and some sheep. There are 7 animals altogether and they have 22 legs. How many of each animal are there?
Show answer
If all 7 were hens: 14 legs. Each sheep adds 2 more legs: 22 − 14 = 8, 8 ÷ 2 = 4 sheep. 3 hens and 4 sheep
Q2.10 How many different ways can you make exactly 10p using only 1p, 2p and 5p coins?
Show answer
Two 5p: 1 way. One 5p, then 5p from 2p and 1p: 2+2+1, 2+1+1+1, five 1p, so 3 ways. No 5p, 10p from 2p and 1p with 0 to 5 twos: 6 ways. Total: 1 + 3 + 6 = 10 ways
Level 3 – 11+ standard: working backwards, logic clues, magic squares and counting with conditions
Typical of grammar-school 11+ papers. Write down what you know, then use each clue in turn.
✏️ WORKED EXAMPLE: Mia had some marbles. She gave half of them to her brother, then gave 3 to her friend. Then she gave half of what was left to her sister. Mia now has 4 marbles. How many did she start with?
Step 1: start at the end. Before giving half to her sister she had 4 × 2 = 8
Step 2: before giving 3 to her friend she had 8 + 3 = 11
Step 3: before giving half to her brother she had 11 × 2 = 22 marbles. Check: 22 → 11 → 8 → 4 ✓
Common mistake: halving 4 instead of doubling it. When you work backwards, every operation is reversed.
Your turn
Q3.1 A number goes into this function machine. The number that comes out is 41. What number went in?
Show answer
Work backwards: 41 − 5 = 36, 36 ÷ 3 = 12, 12 + 8 = 20
Q3.2 Ann, Ben, Cara and Dev each own a different pet: a cat, a dog, a goldfish or a parrot.
• Ann’s pet has four legs, but it is not the dog.
• Ben’s pet has fewer legs than Dev’s pet.
• Cara’s pet has more legs than Ben’s pet.
• Cara does not own the parrot.
Who owns which pet?
Show answer
Ann has the cat. Ben’s pet has fewer legs than both Cara’s and Dev’s pets, so it must be the goldfish. Cara has more legs than Ben and is not the parrot, so Cara has the dog. Ann cat, Ben goldfish, Cara dog, Dev parrot
Q3.3 Complete this magic square using each of the numbers 2 to 10 once. Every row, column and diagonal must have the same total.
| 5 | ||
| 8 | ||
| 2 |
Show answer
The nine numbers add up to 54, so each line totals 54 ÷ 3 = 18. The centre of a 3 by 3 magic square is always one third of the line total: 6. Then fill in each line with one gap:
5, 10, 3 / 4, 6, 8 / 9, 2, 7
Q3.4 You have four cards: 1, 2, 3 and 4. You use three of them to make a three-digit number. The number must be even. How many different even numbers can you make?
Show answer
The last digit must be 2 or 4 (2 choices). Then 3 choices for the first digit and 2 for the middle: 2 × 3 × 2 = 12 numbers
Q3.5 Raj has exactly 5 coins with a total of 50p. Every coin is a 5p, 10p or 20p coin. Find both possible sets of coins.
Show answer
With no 20p: five 10p coins = 50p ✓. With one 20p: 4 coins make 30p, so two 10p + two 5p ✓. Two 20p leaves 3 coins for 10p, which is too much. Five 10p coins, or one 20p, two 10p and two 5p
Q3.6 I think of a number. I halve it, add 4, then multiply by 3. My answer is 84. What was my number?
Show answer
Backwards: 84 ÷ 3 = 28, 28 − 4 = 24, 24 × 2 = 48
Q3.7 Complete this magic square. Every row, column and diagonal has the same total.
| 11 | ||
| 10 | ||
| 7 |
Show answer
In a 3 by 3 magic square the centre is one third of the line total, so the total is 3 × 10 = 30. Diagonal: 30 − 7 − 10 = 13 top right. Top row: 30 − 11 − 13 = 6. Left column: 30 − 11 − 7 = 12. Then fill the remaining gaps:
11, 6, 13 / 12, 10, 8 / 7, 14, 9
Q3.8 I am a number between 40 and 80. I am a multiple of 6. I am 2 less than a multiple of 5. My digits add up to 12. What number am I?
Show answer
2 less than a multiple of 5 means I end in 3 or 8. Multiples of 6 are even, so I end in 8: 48 or 78. Digit sums: 12 and 15. 48
Q3.9 A café lunch deal is one sandwich (cheese, ham or egg) and one drink (juice, milk, water or cola). The café does not allow a ham sandwich with milk. How many different lunch deals are there?
Show answer
Without the rule: 3 × 4 = 12. Take away the one banned pair: 12 − 1 = 11
Q3.10 Zara spent one third of her money on a book. Then she spent one quarter of what was left on lunch. She had £18 left. How much money did she start with?
Show answer
Backwards: £18 is three quarters of what was left after the book, so that was £24. £24 is two thirds of the start, so she started with £36
Level 4 – 11+ selective: ages, handshakes, digit puzzles and consecutive numbers
The style of selective independent-school papers. Several clues must all be true at once, so check your answer against every one.
✏️ WORKED EXAMPLE: Dad is three times as old as Zak. In 12 years’ time, Dad will be twice as old as Zak. How old are they now?
Step 1: call Zak’s age now z. Then Dad is 3z.
Step 2: in 12 years: Dad is 3z + 12 and Zak is z + 12. So 3z + 12 = 2(z + 12) = 2z + 24
Step 3: take 2z and 12 from both sides: z = 12. Zak is 12 and Dad is 36. Check: in 12 years they are 24 and 48 ✓
Common mistake: adding 12 years to only one person. Both people get 12 years older.
Your turn
Q4.1 Two numbers add up to 23 and their difference is 7. What do you get when you multiply them?
Show answer
23 − 7 = 16, 16 ÷ 2 = 8, and 8 + 7 = 15. The numbers are 15 and 8, so the product is 120
Q4.2 Five consecutive whole numbers add up to 115. What is the largest of the five numbers?
Show answer
The middle number is 115 ÷ 5 = 23. The numbers are 21, 22, 23, 24, 25. Largest: 25
Q4.3 Eight friends meet. Each person shakes hands once with every other person. How many handshakes are there altogether?
Show answer
The first shakes 7 hands, the next 6 new hands, and so on: 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28
Q4.4 In 4 years’ time, Kai will be twice as old as he was 5 years ago. How old is Kai now?
Show answer
Call his age k: k + 4 = 2(k − 5) = 2k − 10, so k = 14. Check: 18 is twice 9 ✓. 14
Q4.5 Leo writes all the whole numbers from 1 to 100. How many times does he write the digit 7?
Show answer
7 in the units: 7, 17, … 97 is 10 times. 7 in the tens: 70 to 79 is 10 times (77 is counted in both lists because it has two 7s). Total: 20
Q4.6 How many triangles of any size are there in this diagram?
Show answer
Every triangle uses the top corner and two of the five points on the bottom line. Pairs of points: 4 + 3 + 2 + 1 = 10 triangles
Q4.7 Three different whole numbers multiply to make 60 and add up to 12. What are they?
Show answer
Try sets with product 60 and check the sum, for example: 1, 5, 12 (18); 2, 5, 6 (13); 2, 3, 10 (15); 3, 4, 5 (12 ✓). 3, 4 and 5
Q4.8 A two-digit number has a tens digit that is 3 more than its units digit. When you reverse the digits and add the new number to the original, you get 121. What is the original number?
Show answer
A number plus its reverse is 11 × (sum of digits), so the digits add to 121 ÷ 11 = 11. Digits 7 and 4 (7 − 4 = 3). 74
Q4.9 Three consecutive whole numbers multiply to make 2730. What is their sum?
Show answer
2730 ends in 0, so one number is a multiple of 5 or 10. 13 × 14 × 15 = 182 × 15 = 2730 ✓. Sum: 13 + 14 + 15 = 42
Q4.10 How many whole numbers from 1 to 100 are multiples of 3 or multiples of 5, but not multiples of both?
Show answer
Multiples of 3: 33. Multiples of 5: 20. Multiples of both (of 15): 6. Take the 6 away from each list: 27 + 14 = 41