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 FACTS TO KNOW BY HEART
Column methods: line up the digits by place value and start with the ones.
Order of operations (BIDMAS): Brackets, Indices, then Division and Multiplication (left to right), then Addition and Subtraction (left to right).
Inverses: + undoes −, and × undoes ÷. Use them to check your answers.
Remainders: decide whether to round up (how many coaches?) or round down (how many full boxes?).
Level 1 – SATs core: written methods and mental strategies
Matches KS2 SATs Paper 1 (arithmetic). Use the column method and short division, and spot quick mental shortcuts.
✏️ WORKED EXAMPLE: Work out 4738 + 2596 using the column method.
Step 1: Ones: 8 + 6 = 14, write 4 carry 1.
Step 2: Tens: 3 + 9 + 1 = 13, write 3 carry 1.
Step 3: Hundreds: 7 + 5 + 1 = 13, write 3 carry 1.
Step 4: Thousands: 4 + 2 + 1 = 7. Answer: 7334
Common mistake: forgetting to add the carried digits, which gives 6224.
Your turn
Q1.1 Work out 365 + 248
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Ones: 5 + 8 = 13, write 3 carry 1. Tens: 6 + 4 + 1 = 11, write 1 carry 1. Hundreds: 3 + 2 + 1 = 6. Answer: 613
Q1.2 Work out 802 − 357
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Ones: 2 − 7 can’t be done, so exchange. 0 tens can’t give, so exchange 1 hundred for 10 tens, then 1 ten for 10 ones. 12 − 7 = 5. Tens: 9 − 5 = 4. Hundreds: 7 − 3 = 4. Answer: 445
Q1.3 Work out 4516 + 3879
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Ones: 6 + 9 = 15, write 5 carry 1. Tens: 1 + 7 + 1 = 9. Hundreds: 5 + 8 = 13, write 3 carry 1. Thousands: 4 + 3 + 1 = 8. Answer: 8395
Q1.4 Work out 7003 − 2468
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Ones: 3 − 8 can’t be done, so exchange. 0 tens and 0 hundreds can’t give, so exchange 1 thousand for 10 hundreds, then 1 hundred for 10 tens, then 1 ten for 10 ones. 13 − 8 = 5. Tens: 9 − 6 = 3. Hundreds: 9 − 4 = 5. Thousands: 6 − 2 = 4. Answer: 4535
Q1.5 Work out 238 × 4
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Ones: 8 × 4 = 32, write 2 carry 3. Tens: 3 × 4 = 12, add the 3 to make 15, write 5 carry 1. Hundreds: 2 × 4 = 8, add the 1 to make 9, write 9. Answer: 952
Q1.6 Work out 1346 × 7
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Ones: 6 × 7 = 42, write 2 carry 4. Tens: 4 × 7 = 28, add the 4 to make 32, write 2 carry 3. Hundreds: 3 × 7 = 21, add the 3 to make 24, write 4 carry 2. Thousands: 1 × 7 = 7, add the 2 to make 9, write 9. Answer: 9422
Q1.7 Work out 852 ÷ 6
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6 into 8 goes 1 remainder 2, carry the 2. 6 into 25 goes 4 remainder 1, carry the 1. 6 into 12 goes 2 exactly. Answer: 142
Q1.8 Work out 4536 ÷ 8
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8 into 4 won’t go, so carry the 4. 8 into 45 goes 5 remainder 5, carry the 5. 8 into 53 goes 6 remainder 5, carry the 5. 8 into 56 goes 7 exactly. Answer: 567
Q1.9 Work out 499 + 386 in your head.
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499 is 1 less than 500. 500 + 386 = 886, then take away 1: 885
Q1.10 Work out 25 × 36 in your head.
Show answer
36 = 4 × 9 and 25 × 4 = 100, so 25 × 36 = 100 × 9 = 900
Level 2 – SATs stretch: long multiplication, long division and BIDMAS
Matches the harder questions in SATs Paper 1. Do multiplication and division before addition and subtraction.
✏️ WORKED EXAMPLE: Work out 2507 × 36 using long multiplication.
Step 1: multiply by the 6 ones: 2507 × 6 = 15042. Write the small carries under the row.
Step 2: multiply by the 3 tens: put a 0 in the ones column first, then 2507 × 3 = 7521, so the row is 75210.
Step 3: add the two rows: 15042 + 75210 = 90252. Answer: 90252
Common mistake: forgetting the placeholder 0 in Step 2, which gives 15042 + 7521 = 22563.
Your turn
Q2.1 Work out 436 × 23
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Multiply by the ones: 436 × 3 = 1308. Multiply by the tens: put a 0 in the ones column first, then 436 × 2 = 872, so the row is 8720. Add the rows: 1308 + 8720 = 10028. Answer: 10028
Q2.2 Work out 1265 × 24
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Multiply by the ones: 1265 × 4 = 5060. Multiply by the tens: put a 0 in the ones column first, then 1265 × 2 = 2530, so the row is 25300. Add the rows: 5060 + 25300 = 30360. Answer: 30360
Q2.3 Work out 4183 × 47
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Multiply by the ones: 4183 × 7 = 29281. Multiply by the tens: put a 0 in the ones column first, then 4183 × 4 = 16732, so the row is 167320. Add the rows: 29281 + 167320 = 196601. Answer: 196601
Q2.4 Work out 7605 ÷ 15
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15 into 7 won’t go, so use 76. 15 into 76 goes 5 (5 × 15 = 75). 76 − 75 = 1. Bring down the 0 to make 10. 15 into 10 won’t go, write 0. Bring down the 5 to make 105. 15 into 105 goes 7 (7 × 15 = 105). 105 − 105 = 0. Check: 507 × 15 = 7605 ✓ Answer: 507
Q2.5 Work out 9085 ÷ 23
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23 into 9 won’t go, so use 90. 23 into 90 goes 3 (3 × 23 = 69). 90 − 69 = 21. Bring down the 8 to make 218. 23 into 218 goes 9 (9 × 23 = 207). 218 − 207 = 11. Bring down the 5 to make 115. 23 into 115 goes 5 (5 × 23 = 115). 115 − 115 = 0. Check: 395 × 23 = 9085 ✓ Answer: 395
Q2.6 Work out 943 ÷ 21. Give your answer with a remainder.
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21 into 9 won’t go, so use 94. 21 into 94 goes 4 (4 × 21 = 84). 94 − 84 = 10. Bring down the 3 to make 103. 21 into 103 goes 4 (4 × 21 = 84). 103 − 84 = 19. Check: 44 × 21 + 19 = 943 ✓ Answer: 44 remainder 19
Q2.7 Work out 5 + 4 × 3
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Multiply before you add: 4 × 3 = 12, then 5 + 12 = 17
Q2.8 Work out 30 − 18 ÷ 3
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Divide before you subtract: 18 ÷ 3 = 6, then 30 − 6 = 24
Q2.9 Work out 6 × 7 − 24 ÷ 8
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Do the multiplication and division first: 6 × 7 = 42 and 24 ÷ 8 = 3. Then 42 − 3 = 39
Q2.10 Work out 3² + 4 × 5 − 2
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Indices first: 3² = 9. Then multiply: 4 × 5 = 20. Then 9 + 20 − 2 = 27
Level 3 – 11+ standard: word problems, remainders and missing digits
Typical of grammar-school 11+ papers. Read carefully: should a remainder make you round up or round down?
✏️ WORKED EXAMPLE: Amir and Bella share £84. Bella gets £12 more than Amir. How much does Bella get?
Step 1: draw a bar for each person. Bella’s bar is the same as Amir’s plus an extra £12.
Step 2: take away the extra: £84 − £12 = £72. This is two equal bars.
Step 3: one bar: £72 ÷ 2 = £36. This is Amir’s share.
Step 4: Bella gets £36 + £12 = £48
Check: £36 + £48 = £84 ✓
Common mistake: halving £84 to get £42 and then adding £12. That makes the total £96, not £84.
Your turn
Q3.1 Work out (15 − 7) × 6
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Brackets first: 15 − 7 = 8. Then 8 × 6 = 48
Q3.2 Work out 72 ÷ (4 + 5) − 3
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Brackets first: 4 + 5 = 9. Then 72 ÷ 9 = 8, and 8 − 3 = 5
Q3.3 230 pupils are going on a school trip. Each coach holds 52 pupils. How many coaches are needed?
Show answer
230 ÷ 52 = 4 remainder 22. Four coaches leave 22 pupils behind, so one more coach is needed: 5 coaches
Q3.4 A farmer has 200 eggs. She packs them into boxes of 6. How many boxes can she fill completely?
Show answer
200 ÷ 6 = 33 remainder 2. The 2 left-over eggs cannot fill a box, so the answer is 33 boxes
Q3.5 Each box stands for a single digit. Find the missing digits: 3□ × □ = 296
Show answer
A number in the 30s times one digit makes 296, so the digit must be 8 or 9 (30 × 7 is only 210 and 39 × 7 is only 273). 296 ÷ 9 is not a whole number, but 296 ÷ 8 = 37. So 37 × 8 = 296
Q3.6 Adult tickets cost £14 and child tickets cost £9. Mr Patel buys 3 adult tickets and 5 child tickets. He pays with a £100 note. How much change does he get?
Show answer
Adults: 3 × £14 = £42. Children: 5 × £9 = £45. Total: £87. Change: £100 − £87 = £13
Q3.7 Each box stands for a single digit. Find the missing digits: 5□3 − □87 = 286
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Use the inverse: 286 + □87 must equal 5□3. Ones: 6 + 7 = 13, so carry 1. Tens: 8 + 8 + 1 = 17, so the tens digit of the top number is 7, carry 1. Hundreds: 2 + □ + 1 = 5, so □ = 2. The calculation is 573 − 287 = 286
Q3.8 Work out 4 × (18 − 3 × 4) + 10 ÷ 2
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Inside the brackets, multiply first: 3 × 4 = 12, so 18 − 12 = 6. Then 4 × 6 = 24 and 10 ÷ 2 = 5. Finally 24 + 5 = 29
Q3.9 A baker makes 365 rolls. He puts them into packs of 8 and sells each full pack for £2. Rolls that do not fill a pack are not sold. How much money does he get if he sells every full pack?
Show answer
365 ÷ 8 = 45 remainder 5, so there are 45 full packs. 45 × £2 = £90
Q3.10 Mia thinks of a number. She multiplies it by 6 and then subtracts 18. Her answer is 84. What number did Mia think of?
Show answer
Work backwards using inverses: 84 + 18 = 102, then 102 ÷ 6 = 17. Check: 17 × 6 − 18 = 84 ✓
Level 4 – 11+ selective: brackets, digit puzzles and efficient methods
The style of selective independent-school papers. Look for clever shortcuts and use inverses to check.
✏️ WORKED EXAMPLE: Add one pair of brackets to make this calculation correct: 3 + 5 × 6 − 2 = 23
Step 1: without brackets, 3 + 5 × 6 − 2 = 3 + 30 − 2 = 31. That is too big.
Step 2: 23 − 3 = 20, so we need the rest to make 20. 5 × 4 = 20 and 6 − 2 = 4.
Step 3: put the brackets round 6 − 2: 3 + 5 × (6 − 2) = 23
Check: 6 − 2 = 4, 5 × 4 = 20, 3 + 20 = 23 ✓
Common mistake: working from left to right as if there were brackets round 3 + 5, which gives 8 × 6 − 2 = 46.
Your turn
Q4.1 Work out 25 × 47 × 4. Find a quick way.
Show answer
Change the order: 25 × 4 = 100, then 100 × 47 = 4700
Q4.2 Work out 99 × 46 in your head.
Show answer
99 × 46 is 100 × 46 take away 1 × 46: 4600 − 46 = 4554
Q4.3 Estimate 612 × 49 by rounding 612 to the nearest hundred and 49 to the nearest ten. Is the exact answer more or less than your estimate?
Show answer
Estimate: 600 × 50 = 30000. Exact: 612 × 49 = 612 × 50 − 612 = 30600 − 612 = 29988, so the exact answer is less than the estimate.
Q4.4 Leo says that 3456 ÷ 12 = 278. Use multiplication to check his answer. If he is wrong, give the correct answer.
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Check: 278 × 12 = 3336, not 3456, so Leo is wrong. 3456 − 3336 = 120 = 10 × 12, so the answer is 278 + 10 = 288. Check: 288 × 12 = 3456 ✓
Q4.5 Add one pair of brackets to make this correct: 8 + 12 ÷ 4 − 2 = 14
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Without brackets the answer is 8 + 3 − 2 = 9. We need 12 ÷ something = 6, so divide by 2 = 4 − 2: 8 + 12 ÷ (4 − 2) = 14
Q4.6 Add one pair of brackets to make this correct: 2 × 9 − 3 × 4 + 1 = 49
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49 − 1 = 48, and 48 = 2 × 6 × 4. 9 − 3 = 6, so: 2 × (9 − 3) × 4 + 1 = 49
Q4.7 Add one pair of brackets to make this correct: 7 + 3 × 4 − 2 + 5 = 28
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28 − 7 = 21 = 3 × 7, and 4 − 2 + 5 = 7. So: 7 + 3 × (4 − 2 + 5) = 28
Q4.8 Use each of the digits 2, 4, 6 and 8 exactly once to make a calculation □□ × □□. What is the largest answer you can make?
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Put the two biggest digits in the tens places: 8_ × 6_. Then try both ways: 84 × 62 = 5208 and 82 × 64 = 5248. The largest is 82 × 64 = 5248
Q4.9 In this calculation each letter stands for a different digit: AB × 3 = CAB. (AB is a two-digit number and CAB is a three-digit number.) Find AB.
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CAB is C hundreds plus AB. So 3 lots of AB = C hundreds + 1 lot of AB, which means 2 × AB = 100 × C. So AB is a multiple of 50, and the only two-digit multiple of 50 is 50. AB = 50 gives 50 × 3 = 150 ✓ (A = 5, B = 0, C = 1). AB = 50
Q4.10 When a number is divided by 7, the answer is 23 remainder 5. What is the answer when the same number is divided by 6? Give any remainder.
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The number is 23 × 7 + 5 = 166. Then 166 ÷ 6 = 27 remainder 4, so the answer is 27 remainder 4