Mental arithmetic
Mental arithmetic gives you one sum, like 87 − 54 or □ + 49 = 79, to work out in your head. Here’s how to pick the quickest way every time, and spot the traps in the wrong answers.
Chapter 1
What are mental arithmetic questions?
One sum, worked out in your head. Some are plain sums, like 87 − 54. Some have a box where a number is missing, like this one: something add 49 makes 79.
The sum. Adding, taking away, times or dividing. Harder ones have decimals or units.
The box. When there is one, it’s the number you’re finding.
Look at the numbers before you start. A number close to 100, two numbers close together, a box to fill: each one points to a quicker way than columns in your head.
□ + 49 = 79
The options
- Option 1: 30
- Option 2: 128
- Option 3: 29
- Option 4: 20
A Maths question from Rush, level 2.
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
□ + 49 = 79
The options
- Option 1: 30
- Option 2: 128
- Option 3: 29
- Option 4: 20
A Maths question from Rush, level 2.
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Step 1
Undo the box.
Something add 49 makes 79. Taking away undoes adding, so the box is 79 − 49. The box must be less than 79, so 128 is out already.
- □ + 49 = 79
- □ = 79 − 49
Take away undoes add.
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Step 2
Round, then adjust.
49 is nearly 50, and 79 − 50 = 29 is easy. But that took away 1 too many, so give it back: 30.
Back 50 is one too far, so forward 1.
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Step 3
Put it back in.
30 + 49 = 79, so it fits. A fast check: 0 + 9 ends in 9, and so does 79.
30+49=79
30 fits in the box.
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Adding the numbers you can see. 128 is 79 + 49. Three in four of Rush’s box sums offer this answer. | Undo it: the box is less than 79. |
| Not giving the 1 back. 29 is 79 − 50, but you only had to take 49. | Rounded up? Add the extra back on. |
| Being a ten out. 20 takes away 59. Most of Rush’s adding and taking away questions offer an answer a ten or a hundred out. | Answers differ only in the tens? Redo the tens: 7 − 4 = 3. |
So it’s 30. The box is 79 − 49, and 30 + 49 makes 79 again.
Remember: Undo it. Round it. Put it back.
Chapter 3
Adding and taking away.
Every kind of Add & Subtract question: what to look for, the quick way, and the trap.
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No shortcut? Tens, then ones
Keep the first number whole. Take the tens, then the ones.
87 − 54: take 50 to get 37, then 4 to get 33. Trap: 43, which takes only 4 tens.
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Adding 99 or 299? Add the round number
Add a little too much, then take the extra back.
465 + 299 is 465 + 300 = 765, take back 1: 764. Trap: 765, from forgetting the 1.
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Close together? Count up
Close together, or zeros to cross: count up from the smaller number, through round numbers.
700 − 337: 3 to 340, 60 to 400, 300 to 700. That’s 363, with no exchanging. Trap: 437, from taking the smaller digit from the bigger.
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A missing digit? Columns, from the right
Ones first, and remember the carry.
4□9 + 454 = 903. 9 + 4 = 13: write 3, carry 1. Then □ + 5 + 1 ends in 0, so □ is 4. Trap: 5, from forgetting the carry.
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Decimals or units? Make them match
Line up the points. Change to the smaller unit: 5 km is 5000 m.
0.08 + 6.9: write 6.9 as 6.90, so it’s 6.98. Trap: 7.7, from reading 0.08 as 0.8.
-
- Double 76 = 140 + 12
- = 152
Doubles and halves? Split them too
Double or halve the tens, then the ones, then add.
Double 76 is 140 + 12 = 152. Half of 76 is 35 + 3 = 38. And 38 + 39 is double 38, plus 1. Trap: 142, from losing the ten in 12.
Learn by heart: pairs that make 100
The ones make 10 and the tens make 9. When both end in 0, just the tens make 10, like 40 + 60. Trap: 73 for 100 − 37, which makes 110.
Chapter 4
Times and dividing.
Every kind of Multiply & Divide question, set out the same way.
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- 7 × 8 = 56
- 8 × 7 = 56
- 56 ÷ 8 = 7
- 56 ÷ 7 = 8
Tables: know them both ways
Stuck? Build from an easier fact: × 9 is × 10 take one lot, and × 4 is double, then double again.
7 × 8 = 56 gives 56 ÷ 8 = 7 and 56 ÷ 7 = 8. Trap: the facts next door, like 48 (6 × 8) or 63 (7 × 9).
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H T O t 4.5 4 . 5 × 10 4 5 × 100 4 5 0 × or ÷ by 10, 100, 1000? Move the digits
× 10 moves each digit one place left, × 100 two. ÷ moves them right.
4.5 × 10 = 45, not 4.50. Trap: 1900 for 19 × 10, a place too far.
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- 56 × 5 is 560 ÷ 2 = 280
- 36 × 99 = 3600 − 36
× 5, × 25 or × 99? Go via 10 or 100
× 5 is × 10 halved, × 25 is × 100 quartered, × 99 is × 100 take one lot.
36 × 99 is 3600 take away one 36, which is 3564. Trap: 3599, from taking away 1 instead of 36.
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- □ × 9 = 225
- 180 ÷ 9 = 20, 45 ÷ 9 = 5
- □ = 25
Big numbers? Split them
Multiply the tens and ones apart: 47 × 6 = 240 + 42. Divide in easy chunks.
□ × 9 = 225: divide, so 225 ÷ 9 = 20 + 5 = 25. Trap: 216, from taking 9 away.
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- 960too small
- 2496too small
- 4416ends in 6
- 4421ends in 1
Too big to work out? Size, then last digit
Estimate the size, then find the last digit.
192 × 23 is about 200 × 23 = 4600, and 2 × 3 ends in 6: 4416. Traps: 2496 ends in 6 but is far too small, and 4421 is the right size but ends in 1.
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- 45 ÷ 8 = 5 r 5
- 6 tents: everyone in
A story? Find the sum, then decide
“Altogether” adds. “Left” and “how many more” take away. Equal groups? Multiply when the total is missing, and divide when you know it.
A tent sleeps 8. How many tents for 45 campers? 45 ÷ 8 = 5 r 5, so 6: everyone needs a bed. Trap: 5 r 5, which isn’t a number of tents.
In the exam: a decimal × a whole number
Multiply the whole number part and the decimal part apart, then add.
3.4 × 6: 3 × 6 = 18 and 0.4 × 6 = 2.4, so 18 + 2.4 = 20.4. Check the size: 3.4 is between 3 and 4, so the answer is between 18 and 24. Traps: 18.24, from writing 0.4 × 6 as 0.24, and 204, which loses the point.
- 3 × 6 = 18
- 0.4 × 6 = 2.4
- 18 + 2.4 = 20.4
Chapter 5: In the exam
Two parts from a whole.
The exam puts most sums in words (sum and total mean add, difference means take away, product means times), and often takes two parts from a whole.
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Step 1
Find the whole and the parts.
The whole is 900. Two parts are sold: 345 and 278. What’s left is the last part.
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Step 2
Add the parts.
278 is nearly 280. 345 + 280 = 625, take back 2: 623 sold.
- 345 + 280 = 625
- 625 − 2 = 623
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Step 3
Take them from the whole.
Count up from 623: 7 to 630, 70 to 700, 200 to 900. That’s 277.
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Stopping after one step. 555 is 900 − 345, and 622 is 900 − 278. | Two days of selling: two parts to take away. |
| Adding everything. 1523 is 900 + 345 + 278. Selling leaves fewer than 900. | Ask: more or less than the whole? |
| Being a hundred out. 377 forgets the hundred exchanged when taking away in columns. | Count up. There’s nothing to exchange. |
So it’s 277. Check: 277 + 345 + 278 = 900.
Remember: Add the parts. Take from the whole.
Four more ways it’s asked
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- 36 × 14 = 504
- 36 × 15 = 504 + 36 = 540
Use the fact it gives you
Change the fact a little. Don’t start again.
36 × 15 is one more 36 than 36 × 14: 540. Trap: 505, from adding 1 instead of 36.
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- 600 ÷ 3 = 200
- 18 ÷ 3 = 6
- 618 ÷ 3 = 206
A zero in the answer
An empty column still needs its 0.
618 ÷ 3 = 200 + 6 = 206. Trap: 26, which drops the 0. 26 × 3 is only 78.
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- 47 × 6 = 282
- 47 × 30 = 1410
- 282 + 1410 = 1692
Long multiplication
One row for the ones, one for the tens. The tens row starts with a 0.
47 × 36: 47 × 6 = 282, then 47 × 30 = 1410. Add the rows: 1692. Trap: 423, from leaving out the 0, so the second row is only 141.
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- 43 ÷ 16 = 2 r 11
- 112 ÷ 16 = 7
- 432 ÷ 16 = 27
Long division
Write the 16s down the side first: 16, 32, 48, 64, 80, 96, 112.
432 ÷ 16: 16 goes into 43 twice (32), with 11 left. Bring down the 2 to make 112, which is 7 lots of 16. So it’s 27. Check: 27 × 16 = 432. Trap: 2 r 11, which stops before the last digit.
Chapter 6
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Undo the box.
- Round, then adjust.
- Put it back in.
Remember: Undo it. Round it. Put it back.
For parents.
More on mental arithmetic, and what parents ask us.
More on mental arithmetic
Written methods and longer calculations
Rush practises the mental routes, and the estimate-and-check way of choosing between close answers to a big multiplication. Papers also ask for long multiplication and long division worked out in full. Chapter 5 shows one of each. Practise the written methods on paper. Rush drills the times tables and estimating that they rely on.
Questions parents ask
How can my child get faster at mental maths for the 11+?
Start with the times tables up to 12 × 12, known without stopping to think, because every other method depends on them. Then learn the pairs that make 10 and 100 (such as 37 and 63), and practise doubling and halving.
Should my child use column methods in the 11+?
Use whichever method your child is most accurate with. Counting up, or rounding and adjusting, is quick when the numbers are close together or near a round number. Column methods written on paper, as taught at school, are safest for awkward numbers, and they are the way to do long multiplication and division. Don't switch methods close to the test.
Is BODMAS in the 11+?
Yes. BODMAS is the order to work a calculation: brackets, then powers, then division and multiplication, then addition and subtraction. Division and multiplication are done left to right as they come, and so are addition and subtraction. Rush practises it in Order & Algebra, covered on the algebra page.
Are calculators allowed in the 11+?
No. Every calculation is done in the head or on paper.
Rush: 11+ Practice & KS2 SATsPractise mental arithmetic in Rush.
- Add & Subtract and Multiply & Divide as 100-second games, easy at first and harder as your child improves
- Every miss kept with the right answer and why
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