Mental arithmetic
By the end of this chapter you will be able to
- Add and subtract large numbers in your head, by splitting or rounding
- Recall times tables to 12 × 12 and use them to work out other facts
- Double and halve numbers and multiply by 10 and 100
- Add, subtract, multiply and divide simple decimals and fractions in your head
- Use an estimate to check that a mental answer is sensible
You should already know
- Place value and the four operations (Chapter 1 of this book)
- Multiplication and division facts (Chapter 2 of this book)
Where this leads
- Measures and conversions, using mental facts (Chapter 8 of this book)
KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Perform mental calculations, including with mixed operations and large numbers (Year 6), Add and subtract mentally with increasingly large numbers (Year 5), Multiply and divide mentally, drawing on known facts, including times tables to 12 × 12 and doubling and halving (Year 5), Mental calculation with decimals and fractions (Year 6), Use estimation to check mental answers (Year 5-6).
Exercise P Do you remember?
Non-calculatorMethod. Use a number bond to 10.
- Split 6. 9 + 1 = 10, and 6 - 1 = 5.10 + 5
- Add. 10 + 5 = 15.15
- Write the answer. 15
Check. 15 - 6 = 9. ✓
Method. Add the tens first.
- Add the tens. 40 + 20 = 60.60
- Add the ones. 60 + 5 = 65.65
- Write the answer. 65
Check. 65 - 25 = 40. ✓
Method. Count back 7 from 12.
- Count back. 12 - 2 = 10, then 10 - 5 = 5.5
- Write the answer. 5
Check. 5 + 7 = 12. ✓
Method. Use a times table fact.
- Recall the fact. Six fives are thirty.6 × 5 = 30
- Write the answer. 30
Check. 5 × 6 = 30 as well. ✓
Method. Count up from 37 to 100.
- Split the count. 37 to 40 is 3; 40 to 100 is 60.3 + 60
- Add. 3 + 60 = 63.63
- Write the answer. 63
Check. 37 + 63 = 100. ✓
Adding and subtracting large numbers mentally
Rule
To add in your head, split each number into its big part and its small part. Add the big parts, then the small parts, and put them together.
To subtract, you can round the number taken away to a friendly number, subtract that, then correct the answer.
Practice guide: work out each answer in your head, then write only the number. Do not use a calculator or paper for the mental parts.
Key ideas
- Split large numbers into thousands and the rest.
- Round a number to a friendly number, then correct the answer.
- Estimate first: an answer far from the estimate has gone wrong.
Method. Split the numbers into thousands and the rest, then add.
- Split each number into thousands and the rest. 12,500 is 12,000 and 500; 8,700 is 8,000 and 700.12,000 + 500 and 8,000 + 700
- Add the thousands. The thousands add first.12,000 + 8,000 = 20,000
- Add the rest. Add the smaller parts.500 + 700 = 1200
- Put the parts together. Add the two totals.20,000 + 1200 = 21,200
- Write the answer. 21,200
Check. About 13,000 + 9,000 = 22,000, close to 21,200. ✓
Method. Subtract a friendly number, then correct the answer.
- Round the number taken away to the nearest thousand. 4,300 is close to 4,000.4,300 ≈ 4,000
- Subtract the rounded number. This is easier to work out.9,000 - 4,000 = 5,000
- Correct the answer. 4,000 is 300 less than 4,300, so take 300 away.5,000 - 300 = 4,700
- Write the answer. 4,700
Check. 4,700 + 4,300 = 9,000. ✓
Watch out. Check by adding the answer back to 4,300.
Method. Add the thousands, then the rest.
- Split each number into thousands and the rest. 45,000 is 45,000 and 0; 8,900 is 8,000 and 900.45,000 + 0 and 8,000 + 900
- Add the thousands. The thousands add first.45,000 + 8,000 = 53,000
- Add the rest. Add the smaller parts.0 + 900 = 900
- Put the parts together. Add the two totals.53,000 + 900 = 53,900
- Write the answer. 53,900
Check. About 45,000 + 9,000 = 54,000, close to 53,900. ✓
Exercise 7A Practice: adding and subtracting large numbers
Non-calculatorMethod. Split into thousands and the rest, then add.
- Split each number into thousands and the rest. 3,400 is 3,000 and 400; 2,800 is 2,000 and 800.3,000 + 400 and 2,000 + 800
- Add the thousands. The thousands add first.3,000 + 2,000 = 5,000
- Add the rest. Add the smaller parts.400 + 800 = 1200
- Put the parts together. Add the two totals.5,000 + 1200 = 6,200
- Write the answer. 6,200
Check. 6,200 - 2,800 = 3,400. ✓
Method. Split into thousands and the rest, then add.
- Split each number into thousands and the rest. 56,000 is 56,000 and 0; 7,000 is 7,000 and 0.56,000 + 0 and 7,000 + 0
- Add the thousands. The thousands add first.56,000 + 7,000 = 63,000
- Add the rest. Add the smaller parts.0 + 0 = 0
- Put the parts together. Add the two totals.63,000 + 0 = 63,000
- Write the answer. 63,000
Check. 63,000 - 7,000 = 56,000. ✓
Method. Split into thousands and the rest, then add.
- Split each number into thousands and the rest. 1,200,000 is 1,200,000 and 0; 350,000 is 350,000 and 0.1,200,000 + 0 and 350,000 + 0
- Add the thousands. The thousands add first.1,200,000 + 350,000 = 1,550,000
- Add the rest. Add the smaller parts.0 + 0 = 0
- Put the parts together. Add the two totals.1,550,000 + 0 = 1,550,000
- Write the answer. 1,550,000
Check. 1,550,000 - 350,000 = 1,200,000. ✓
Method. Split into thousands and the rest, then add.
- Split each number into thousands and the rest. 45,000 is 45,000 and 0; 8,900 is 8,000 and 900.45,000 + 0 and 8,000 + 900
- Add the thousands. The thousands add first.45,000 + 8,000 = 53,000
- Add the rest. Add the smaller parts.0 + 900 = 900
- Put the parts together. Add the two totals.53,000 + 900 = 53,900
- Write the answer. 53,900
Check. 53,900 - 8,900 = 45,000. ✓
Method. Split into thousands and the rest, then add.
- Split each number into thousands and the rest. 6,700 is 6,000 and 700; 1,500 is 1,000 and 500.6,000 + 700 and 1,000 + 500
- Add the thousands. The thousands add first.6,000 + 1,000 = 7,000
- Add the rest. Add the smaller parts.700 + 500 = 1200
- Put the parts together. Add the two totals.7,000 + 1200 = 8,200
- Write the answer. 8,200
Check. 8,200 - 1,500 = 6,700. ✓
Method. Split into thousands and the rest, then add.
- Split each number into thousands and the rest. 23,000 is 23,000 and 0; 9,900 is 9,000 and 900.23,000 + 0 and 9,000 + 900
- Add the thousands. The thousands add first.23,000 + 9,000 = 32,000
- Add the rest. Add the smaller parts.0 + 900 = 900
- Put the parts together. Add the two totals.32,000 + 900 = 32,900
- Write the answer. 32,900
Check. 32,900 - 9,900 = 23,000. ✓
Method. Round the number taken away, subtract, then correct.
- Round the number taken away to the nearest thousand. 4,300 is close to 4,000.4,300 ≈ 4,000
- Subtract the rounded number. This is easier to work out.9,000 - 4,000 = 5,000
- Correct the answer. 4,000 is 300 less than 4,300, so take 300 away.5,000 - 300 = 4,700
- Write the answer. 4,700
Check. 4,700 + 4,300 = 9,000. ✓
Method. Round the number taken away, subtract, then correct.
- Round the number taken away to the nearest thousand. 9,500 is close to 10,000.9,500 ≈ 10,000
- Subtract the rounded number. This is easier to work out.72,000 - 10,000 = 62,000
- Correct the answer. 10,000 is 500 more than 9,500, so add 500 back.62,000 + 500 = 62,500
- Write the answer. 62,500
Check. 62,500 + 9,500 = 72,000. ✓
Method. Round the number taken away, subtract, then correct.
- Round the number taken away to the nearest thousand. 85,000 is close to 85,000.85,000 ≈ 85,000
- Subtract the rounded number. This is easier to work out.640,000 - 85,000 = 555,000
- Correct the answer. 85,000 is 0 more than 85,000, so add 0 back.555,000 + 0 = 555,000
- Write the answer. 555,000
Check. 555,000 + 85,000 = 640,000. ✓
Method. Round the number taken away, subtract, then correct.
- Round the number taken away to the nearest thousand. 1,250,000 is close to 1,250,000.1,250,000 ≈ 1,250,000
- Subtract the rounded number. This is easier to work out.3,000,000 - 1,250,000 = 1,750,000
- Correct the answer. 1,250,000 is 0 more than 1,250,000, so add 0 back.1,750,000 + 0 = 1,750,000
- Write the answer. 1,750,000
Check. 1,750,000 + 1,250,000 = 3,000,000. ✓
Method. Round the number taken away, subtract, then correct.
- Round the number taken away to the nearest thousand. 2,600 is close to 3,000.2,600 ≈ 3,000
- Subtract the rounded number. This is easier to work out.8,100 - 3,000 = 5,100
- Correct the answer. 3,000 is 400 more than 2,600, so add 400 back.5,100 + 400 = 5,500
- Write the answer. 5,500
Check. 5,500 + 2,600 = 8,100. ✓
Method. Add the two amounts.
- Split each number into thousands and the rest. 12,500 is 12,000 and 500; 8,700 is 8,000 and 700.12,000 + 500 and 8,000 + 700
- Add the thousands. The thousands add first.12,000 + 8,000 = 20,000
- Add the rest. Add the smaller parts.500 + 700 = 1200
- Put the parts together. Add the two totals.20,000 + 1200 = 21,200
- Write the answer. £21,200
Check. About £13,000 + £9,000 = £22,000; £21,200 is close. ✓
Method. Add the two distances.
- Add the thousands. 6,000 + 3,000 = 9,000.9,000
- Add the rest. 450 + 800 = 1,250.1,250
- Put together. 9,000 + 1,250 = 10,250.10,250
- Write the answer. 10,250 km
Check. 10,250 - 3,800 = 6,450. ✓
Method. Round 1,850 to 2,000, subtract, then correct.
- Round. 1,850 is about 2,000.2,000
- Subtract. 5,000 - 2,000 = 3,000.3,000
- Correct. 2,000 is 150 more than 1,850, so add 150.3,150
- Write the answer. 3,150
Check. 3,150 + 1,850 = 5,000. ✓
Method. Round 375,000 to 400,000, then correct by 25,000.
- Round. 375,000 is about 400,000.400,000
- Subtract. 1,000,000 - 400,000 = 600,000.600,000
- Correct. 400,000 is 25,000 more, so add 25,000.625,000
- Write the answer. 625,000
Check. 625,000 + 375,000 = 1,000,000. ✓
Method. Check the subtraction by adding back.
- Add. 7,000 + 6,000 = 13,000.13,000
- Check. 13,000 - 6,000 = 7,000, the same as the start.7,000
- Write the answer. Yes, because subtracting 6,000 undoes adding 6,000, so the answer goes back to 7,000.
Check. The check works. ✓
Method. Use the number just below 50,000.
- Add 50,000 and 49,999. That is 50,000 + 50,000 - 1.100,000 - 1
- Write the answer. 100,000 - 1 = 99,999.99,999
- Write the answer. 99,999
Check. 99,999 is one less than 100,000. ✓
Method. Count on from 1,999,998 to 2,000,000.
- Count on. From 1,999,998 to 2,000,000 is 2 steps.2 steps
- Write the answer. 2
Check. 1,999,998 + 2 = 2,000,000. ✓
Times tables to 12 × 12
Rule
Learn the times tables to 12 × 12, then use them to work out others. For 12 × 9, use 10 × 9 = 90 and 2 × 9 = 18, which gives 108.
Multiplication can be done in either order: 9 × 12 = 12 × 9.
Key ideas
- Know the times tables to 12 × 12 by heart.
- Split a bigger number to multiply it in parts.
- Swap the numbers: 9 × 12 is the same as 12 × 9.
Method. Split 12 into 10 and 2, then multiply each part by 9.
- Split the bigger number into 10 and the rest. 12 is 10 + 2.12 = 10 + 2
- Multiply each part. 10 × 9 = 90 and 2 × 9 = 18.90 + 18
- Add the parts. Add the two products.90 + 18 = 108
- Write the answer. 108
Check. 9 × 12 = 12 × 9. ✓
Watch out. Do not use 12 × 9 = 90 + 9: the 12 splits into 10 and 2, not 10 and 9.
Method. Use a known fact from the times table.
- Use the known fact. 7 × 8 is in the times table.7 × 8 = 56
- Check by swapping the numbers. 8 × 7 gives the same answer.8 × 7 = 56
- Write the answer. 56
Check. 8 × 7 = 56. ✓
Method. Split 12 into 10 and 2, or use a known fact.
- Use the known fact. 6 × 12 is in the times table.6 × 12 = 72
- Check by swapping the numbers. 12 × 6 gives the same answer.12 × 6 = 72
- Write the answer. 72
Check. 12 × 6 = 6 × 12. ✓
Exercise 7B Practice: times tables to 12 × 12
Non-calculatorMethod. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 7 × 8 is in the times table.7 × 8 = 56
- Check by swapping the numbers. 8 × 7 gives the same answer.8 × 7 = 56
- Write the answer. 56
Check. 8 × 7 = 56 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 6 × 9 is in the times table.6 × 9 = 54
- Check by swapping the numbers. 9 × 6 gives the same answer.9 × 6 = 54
- Write the answer. 54
Check. 9 × 6 = 54 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Split the bigger number into 10 and the rest. 12 is 10 + 2.12 = 10 + 2
- Multiply each part. 10 × 11 = 110 and 2 × 11 = 22.110 + 22
- Add the parts. Add the two products.110 + 22 = 132
- Write the answer. 132
Check. 11 × 12 = 132 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 9 × 7 is in the times table.9 × 7 = 63
- Check by swapping the numbers. 7 × 9 gives the same answer.7 × 9 = 63
- Write the answer. 63
Check. 7 × 9 = 63 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 8 × 8 is in the times table.8 × 8 = 64
- Check by swapping the numbers. 8 × 8 gives the same answer.8 × 8 = 64
- Write the answer. 64
Check. 8 × 8 = 64 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 4 × 12 is in the times table.4 × 12 = 48
- Check by swapping the numbers. 12 × 4 gives the same answer.12 × 4 = 48
- Write the answer. 48
Check. 12 × 4 = 48 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 3 × 11 is in the times table.3 × 11 = 33
- Check by swapping the numbers. 11 × 3 gives the same answer.11 × 3 = 33
- Write the answer. 33
Check. 11 × 3 = 33 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 5 × 9 is in the times table.5 × 9 = 45
- Check by swapping the numbers. 9 × 5 gives the same answer.9 × 5 = 45
- Write the answer. 45
Check. 9 × 5 = 45 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Split the bigger number into 10 and the rest. 12 is 10 + 2.12 = 10 + 2
- Multiply each part. 10 × 12 = 120 and 2 × 12 = 24.120 + 24
- Add the parts. Add the two products.120 + 24 = 144
- Write the answer. 144
Check. 12 × 12 = 144 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 6 × 7 is in the times table.6 × 7 = 42
- Check by swapping the numbers. 7 × 6 gives the same answer.7 × 6 = 42
- Write the answer. 42
Check. 7 × 6 = 42 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Use the known fact. 8 × 9 is in the times table.8 × 9 = 72
- Check by swapping the numbers. 9 × 8 gives the same answer.9 × 8 = 72
- Write the answer. 72
Check. 9 × 8 = 72 as well. ✓
Method. Use a known fact, or split one number into 10 and the rest.
- Split the bigger number into 10 and the rest. 11 is 10 + 1.11 = 10 + 1
- Multiply each part. 10 × 6 = 60 and 1 × 6 = 6.60 + 6
- Add the parts. Add the two products.60 + 6 = 66
- Write the answer. 66
Check. 6 × 11 = 66 as well. ✓
Doubling, halving and multiplying by 10 and 100
Rule
Doubling means multiply by 2; halving means divide by 2. Double the number, or halve it, one part at a time.
Multiplying by 10 moves digits one place left; by 100, two places left.
Key ideas
- Double: add the number to itself, or double each part.
- Halve: halve each part, then add the halves.
- Multiply by 100 by adding two zeros.
Method. Double each part of the number.
- Split the number. 3,450 is 3,000 + 400 + 50.3,000 + 400 + 50
- Double each part. 6,000 + 800 + 100.6,000 + 800 + 100
- Add. 6,000 + 800 + 100 = 6,900.6,900
- Write the answer. 6,900
Check. 6,900 ÷ 2 = 3,450. ✓
Method. Halve each part of the number.
- Split the number. 764 is 700 + 60 + 4.700 + 60 + 4
- Halve each part. 350 + 30 + 2.350 + 30 + 2
- Add. 350 + 30 + 2 = 382.382
- Write the answer. 382
Check. 382 × 2 = 764. ✓
Method. Multiply by 100 by moving the digits two places left.
- Move the digits. 46 moves two places left.46 → 4,600
- Check. 4,600 ÷ 100 = 46.4,600
- Write the answer. 4,600
Check. 46 with two zeros is 4,600. ✓
Exercise 7C Practice: doubling, halving and multiplying by 10 and 100
Non-calculatorMethod. Double each part of the number.
- Split and double. Double each part of 2,450.2 × 2,450
- Add the parts. Put the doubled parts together.4,900
- Write the answer. 4,900
Check. 4,900 ÷ 2 = 2,450. ✓
Method. Double each part of the number.
- Split and double. Double each part of 6,780.2 × 6,780
- Add the parts. Put the doubled parts together.13,560
- Write the answer. 13,560
Check. 13,560 ÷ 2 = 6,780. ✓
Method. Double each part of the number.
- Split and double. Double each part of 34,500.2 × 34,500
- Add the parts. Put the doubled parts together.69,000
- Write the answer. 69,000
Check. 69,000 ÷ 2 = 34,500. ✓
Method. Double each part of the number.
- Split and double. Double each part of 1,250.2 × 1,250
- Add the parts. Put the doubled parts together.2,500
- Write the answer. 2,500
Check. 2,500 ÷ 2 = 1,250. ✓
Method. Double each part of the number.
- Split and double. Double each part of 480.2 × 480
- Add the parts. Put the doubled parts together.960
- Write the answer. 960
Check. 960 ÷ 2 = 480. ✓
Method. Double each part of the number.
- Split and double. Double each part of 9,900.2 × 9,900
- Add the parts. Put the doubled parts together.19,800
- Write the answer. 19,800
Check. 19,800 ÷ 2 = 9,900. ✓
Method. Halve each part of the number.
- Split and halve. Halve each part of 1,840.1,840 ÷ 2
- Add the halves. Put the halves together.920
- Write the answer. 920
Check. 920 × 2 = 1,840. ✓
Method. Halve each part of the number.
- Split and halve. Halve each part of 6,300.6,300 ÷ 2
- Add the halves. Put the halves together.3,150
- Write the answer. 3,150
Check. 3,150 × 2 = 6,300. ✓
Method. Halve each part of the number.
- Split and halve. Halve each part of 25,000.25,000 ÷ 2
- Add the halves. Put the halves together.12,500
- Write the answer. 12,500
Check. 12,500 × 2 = 25,000. ✓
Method. Halve each part of the number.
- Split and halve. Halve each part of 740.740 ÷ 2
- Add the halves. Put the halves together.370
- Write the answer. 370
Check. 370 × 2 = 740. ✓
Method. Halve each part of the number.
- Split and halve. Halve each part of 9,700.9,700 ÷ 2
- Add the halves. Put the halves together.4,850
- Write the answer. 4,850
Check. 4,850 × 2 = 9,700. ✓
Method. Two zeros on the end.
- Add two zeros. 37 × 100 = 3,700.3,700
- Write the answer. 3,700
Check. 3,700 ÷ 100 = 37. ✓
Method. One zero on the end.
- Add one zero. 540 × 10 = 5,400.5,400
- Write the answer. 5,400
Check. 5,400 ÷ 10 = 540. ✓
Method. Double 3,450.
- Double. 3,450 × 2 = 6,900.6,900
- Write the answer. 6,900
Check. 6,900 ÷ 2 = 3,450. ✓
Method. Multiply by 10.
- Multiply by 10. 480 × 10 = 4,800.4,800
- Write the answer. 4,800
Check. 4,800 ÷ 10 = 480. ✓
Method. Multiply by 100.
- Multiply by 100. 25 × 100 = 2,500.2,500
- Write the answer. 2,500
Check. 2,500 ÷ 25 = 100. ✓
Method. Halve twice.
- First halve. 1,000 ÷ 2 = 500.500
- Halve again. 500 ÷ 2 = 250.250
- Write the answer. 250
Check. Halving twice is the same as dividing by 4: 1,000 ÷ 4 = 250. ✓
Method. Think about the two steps as opposites.
- Double. 2,450 × 2 = 4,900.4,900
- Halve. 4,900 ÷ 2 = 2,450.2,450
- Write the answer. Doubling multiplies by 2 and halving divides by 2, so the two steps undo each other.
Check. 4,900 ÷ 2 = 2,450. ✓
Mental calculation with decimals and fractions
Rule
Add and subtract decimals by lining up the decimal points, or by thinking of money in pounds and pence.
To find a fraction of an amount, divide by the bottom number, then multiply by the top number.
Key ideas
- Line up the decimal points when adding or subtracting.
- Think of money: £3.50 is 350 pence.
- A fraction of an amount: divide by the bottom, multiply by the top.
Method. Add the whole numbers, then add the tenths and hundredths.
- Add the whole numbers. 3 + 2 = 5.5
- Add the parts. 0.5 + 0.75 = 1.25.1.25
- Put together. 5 + 1.25 = 6.25.6.25
- Write the answer. 6.25
Check. £3.50 + £2.75 = 350p + 275p = 625p = £6.25. ✓
Method. Split the 2.5 into 2 and 0.5, then multiply each part by 4.
- Split. 2.5 = 2 + 0.5.2 + 0.5
- Multiply each part. 2 × 4 = 8 and 0.5 × 4 = 2.8 + 2
- Add. 8 + 2 = 10.10
- Write the answer. 10
Check. 2.5 is a quarter of 10 times 4. ✓
Method. Find one fifth, then take two fifths.
- Divide the amount by the bottom number. 45 ÷ 5 = 9.45 ÷ 5 = 9
- Multiply by the top number. Take 2 of those parts.9 × 2 = 18
- Write the answer. 18
Check. 9 × 2 = 18 and 45 ÷ 5 = 9. ✓
Exercise 7D Practice: decimals and fractions in your head
Non-calculatorMethod. Add the tenths: 6 + 8 = 14 tenths.
- Add the tenths. 6 tenths + 8 tenths = 14 tenths.14 tenths
- Write it as a decimal. 14 tenths is 1 whole and 4 tenths.1.4
- Write the answer. 1.4
Check. 1.4 - 0.8 = 0.6. ✓
Method. Subtract 2 from 4.2, then add back the 0.1 that was taken too much.
- Subtract 2. 4.2 - 2 = 2.2.2.2
- Add 0.1. 1.9 is 0.1 more than 2, so add 0.1 back.2.3
- Write the answer. 2.3
Check. 2.3 + 1.9 = 4.2. ✓
Method. Think of £3.50 + £2.75 in pence.
- Change to pence. 350p + 275p = 625p.625p
- Change back to pounds. 625p is £6.25.6.25
- Write the answer. 6.25
Check. 625 - 275 = 350. ✓
Method. Count up from 3.6 to 10.
- Count up. 3.6 + 0.4 = 4, then 4 + 6 = 10.0.4 + 6 = 6.4
- Write the answer. 6.4
Check. 3.6 + 6.4 = 10. ✓
Method. Half of 18 is 9.
- Halve 18. 18 ÷ 2 = 9.9
- Write the answer. 9
Check. 9 + 9 = 18. ✓
Method. Think of 1.2 as 1 and 0.2.
- Split. 1 × 5 = 5 and 0.2 × 5 = 1.5 + 1
- Add. 5 + 1 = 6.6
- Write the answer. 6
Check. 6 ÷ 5 = 1.2. ✓
Method. A quarter of 8 is 2.
- Find a quarter of 8. 8 ÷ 4 = 2.2
- Write the answer. 2
Check. 2 × 4 = 8, so 0.25 × 8 = 2. ✓
Method. Divide by the bottom number, then multiply by the top number.
- Divide the amount by the bottom number. 36 ÷ 3 = 12.36 ÷ 3 = 12
- Multiply by the top number. Take 1 of those parts.12 × 1 = 12
- Write the answer. 12
Check. One 3th of 36 is 12, and 1 of those is 12. ✓
Method. Divide by the bottom number, then multiply by the top number.
- Divide the amount by the bottom number. 20 ÷ 4 = 5.20 ÷ 4 = 5
- Multiply by the top number. Take 3 of those parts.5 × 3 = 15
- Write the answer. 15
Check. One 4th of 20 is 5, and 3 of those is 15. ✓
Method. Divide by the bottom number, then multiply by the top number.
- Divide the amount by the bottom number. 45 ÷ 5 = 9.45 ÷ 5 = 9
- Multiply by the top number. Take 2 of those parts.9 × 2 = 18
- Write the answer. 18
Check. One 5th of 45 is 9, and 2 of those is 18. ✓
Method. Divide by the bottom number, then multiply by the top number.
- Divide the amount by the bottom number. 80 ÷ 8 = 10.80 ÷ 8 = 10
- Multiply by the top number. Take 5 of those parts.10 × 5 = 50
- Write the answer. 50
Check. One 8th of 80 is 10, and 5 of those is 50. ✓
Method. Add the two prices.
- Add. £3.50 + £2.75 = £6.25.£6.25
- Write the answer. £6.25
Check. £6.25 - £2.75 = £3.50. ✓
Method. Find one fifth of 45, then take two.
- Divide the amount by the bottom number. 45 ÷ 5 = 9.45 ÷ 5 = 9
- Multiply by the top number. Take 2 of those parts.9 × 2 = 18
- Write the answer. 18
Check. 18 + 27 = 45. ✓
Method. Subtract 0.9 from 2.4.
- Subtract. 2.4 - 0.9 = 1.5.1.5
- Write the answer. 1.5 m
Check. 1.5 + 0.9 = 2.4. ✓
Method. Think of 0.5 as half.
- Half of 30. 30 ÷ 2 = 15.15
- Compare. 0.5 × 30 = 15 as well.15
- Write the answer. Yes, because multiplying by 0.5 is the same as halving, and half of 30 is 15.
Check. 15 + 15 = 30. ✓
Method. Find 3/4 of 60, then add 0.5.
- Find 3/4 of 60. 60 ÷ 4 = 15, then 15 × 3 = 45.45
- Add 0.5. 45 + 0.5 = 45.5.45.5
- Write the answer. 45.5
Check. 45.5 - 0.5 = 45. ✓
Timed set: mixed mental questions
Rule
Timings in this book are a practice guide only. They are not a rule of the official test. Work steadily: check each answer with a quick estimate, then move on.
Key ideas
- Work out each question in your head before writing the answer.
- Estimate first to see whether the answer is sensible.
- If you are stuck, split the numbers and try again.
Method. Multiply first, then add.
- Multiply. 8 × 7 = 56.56
- Add. 56 + 9 = 65.65
- Write the answer. 65
Check. About 8 × 7 = 56, plus 9 is about 65. ✓
Method. Think of 25 as a quarter of 100.
- Use the fact. 25 is one quarter of 100, so 25 × 4 is 100.25 × 4
- Check. Four lots of 25 make 100.25 + 25 + 25 + 25 = 100
- Write the answer. 100
Check. 100 ÷ 4 = 25. ✓
Method. Subtract 3 and then add back the 0.1 taken too much.
- Subtract 3. 6.4 - 3 = 3.4.3.4
- Add 0.1. 2.9 is 0.1 more than 3, so add 0.1 back.3.5
- Write the answer. 3.5
Check. 3.5 + 2.9 = 6.4. ✓
Exercise 7E Timed set: 20 mental questions
Non-calculatorMethod. Add the thousands, then the rest.
- Work it out in your head. Add the thousands, then the rest.8,500
- Write the answer. 8,500
Check. Estimate to check the size of 8,500. ✓
Method. Split 15 into 10 and 5.
- Work it out in your head. Split 15 into 10 and 5.90
- Write the answer. 90
Check. Estimate to check the size of 90. ✓
Method. Subtract the thousands, then take away 250.
- Work it out in your head. Subtract the thousands, then take away 250.750
- Write the answer. 750
Check. Estimate to check the size of 750. ✓
Method. Split 12 into 10 and 2.
- Work it out in your head. Split 12 into 10 and 2.108
- Write the answer. 108
Check. Estimate to check the size of 108. ✓
Method. Add the tenths.
- Work it out in your head. Add the tenths.1.3
- Write the answer. 1.3
Check. Estimate to check the size of 1.3. ✓
Method. Think of the times table fact.
- Work it out in your head. Think of the times table fact.8
- Write the answer. 8
Check. Estimate to check the size of 8. ✓
Method. Multiply by 10 moves the digit left.
- Work it out in your head. Multiply by 10 moves the digit left.32
- Write the answer. 32
Check. Estimate to check the size of 32. ✓
Method. Halve 90.
- Work it out in your head. Halve 90.45
- Write the answer. 45
Check. Estimate to check the size of 45. ✓
Method. Multiply 7 by 6, then add two zeros.
- Work it out in your head. Multiply 7 by 6, then add two zeros.4,200
- Write the answer. 4,200
Check. Estimate to check the size of 4,200. ✓
Method. Divide by 100 moves the digits two places right.
- Work it out in your head. Divide by 100 moves the digits two places right.56
- Write the answer. 56
Check. Estimate to check the size of 56. ✓
Method. Round 99 to 100, then take away 2.
- Work it out in your head. Round 99 to 100, then take away 2.198
- Write the answer. 198
Check. Estimate to check the size of 198. ✓
Method. Count on from 634 to 1,000.
- Work it out in your head. Count on from 634 to 1,000.366
- Write the answer. 366
Check. Estimate to check the size of 366. ✓
Method. Use the 11 times table.
- Work it out in your head. Use the 11 times table.77
- Write the answer. 77
Check. Estimate to check the size of 77. ✓
Method. Double 12.5.
- Work it out in your head. Double 12.5.25
- Write the answer. 25
Check. Estimate to check the size of 25. ✓
Method. Find a third of 15, then double it.
- Work it out in your head. Find a third of 15, then double it.10
- Write the answer. 10
Check. Estimate to check the size of 10. ✓
Method. Subtract the whole numbers and the decimals.
- Work it out in your head. Subtract the whole numbers and the decimals.2.2
- Write the answer. 2.2
Check. Estimate to check the size of 2.2. ✓
Method. Quarter of 1,000 is 250.
- Work it out in your head. Quarter of 1,000 is 250.1,000
- Write the answer. 1,000
Check. Estimate to check the size of 1,000. ✓
Method. Square 6, then add 4.
- Work it out in your head. Square 6, then add 4.40
- Write the answer. 40
Check. Estimate to check the size of 40. ✓
Method. Add first, then subtract.
- Work it out in your head. Add first, then subtract.4,500
- Write the answer. 4,500
Check. Estimate to check the size of 4,500. ✓
Method. Multiply by 10 first, then divide by 4.
- Work it out in your head. Multiply by 10 first, then divide by 4.25
- Write the answer. 25
Check. Estimate to check the size of 25. ✓
Chapter summary
| Skill | Method |
|---|---|
| Add large numbers | Split into thousands and the rest: 12,500 + 8,700 = 21,200. |
| Subtract large numbers | Round the number taken away, subtract, then correct. |
| Times tables | Know 12 × 12 by heart; split 12 into 10 and 2 for 12 × 9 = 108. |
| Doubling and halving | Double or halve each part; 3,450 doubled is 6,900. |
| Powers of 10 | × 10 moves digits one place left; × 100 two places left. |
| Decimals | Line up the points, or think in pence: 3.5 + 2.75 = 6.25. |
| Fraction of amount | Divide by the bottom, multiply by the top: 2/5 of 45 = 18. |
Can you…?
- I can add and subtract large numbers in my head by splitting or rounding
- I know the times tables to 12 × 12 and can use them for other facts
- I can double, halve and multiply by 10 and 100 in my head
- I can add, subtract, multiply and divide simple decimals in my head
- I can find a fraction of an amount in my head
- I can estimate to check that my answer is sensible
Key terms
- mental method
- a way of working out a calculation in your head
- round
- to change a number to a friendly number close to it
- double
- multiply by 2
- halve
- divide by 2
- estimate
- a sensible rounded answer used to check a calculation
Exercise 7Z Check-up
Non-calculatorMethod. Add the thousands, then the rest.
- Split each number into thousands and the rest. 6,300 is 6,000 and 300; 2,850 is 2,000 and 850.6,000 + 300 and 2,000 + 850
- Add the thousands. The thousands add first.6,000 + 2,000 = 8,000
- Add the rest. Add the smaller parts.300 + 850 = 1150
- Put the parts together. Add the two totals.8,000 + 1150 = 9,150
- Write the answer. 9,150
Check. About 6,000 + 3,000 = 9,000, close to 9,150. ✓
Method. Subtract a friendly number, then correct.
- Round the number taken away to the nearest thousand. 3,400 is close to 3,000.3,400 ≈ 3,000
- Subtract the rounded number. This is easier to work out.8,000 - 3,000 = 5,000
- Correct the answer. 3,000 is 400 less than 3,400, so take 400 away.5,000 - 400 = 4,600
- Write the answer. 4,600
Check. 4,600 + 3,400 = 8,000. ✓
Method. Split 12 into 10 and 2.
- Split the bigger number into 10 and the rest. 12 is 10 + 2.12 = 10 + 2
- Multiply each part. 10 × 7 = 70 and 2 × 7 = 14.70 + 14
- Add the parts. Add the two products.70 + 14 = 84
- Write the answer. 84
Check. 7 × 12 = 84. ✓
Method. Use the 9 times table.
- Recall. 9 × 9 = 81.9
- Write the answer. 9
Check. 9 × 9 = 81. ✓
Method. Double each part.
- Split and double. 4,000 + 600 + 50 doubled is 8,000 + 1,200 + 100.9,300
- Write the answer. 9,300
Check. 9,300 ÷ 2 = 4,650. ✓
Method. Think of 0.7 × 4 as seven tenths times 4.
- Multiply. 7 × 4 = 28, so 0.7 × 4 = 2.8.2.8
- Write the answer. 2.8
Check. 2.8 ÷ 4 = 0.7. ✓
Method. Find one fifth, then take three.
- Divide the amount by the bottom number. 60 ÷ 5 = 12.60 ÷ 5 = 12
- Multiply by the top number. Take 3 of those parts.12 × 3 = 36
- Write the answer. 36
Check. 12 × 5 = 60. ✓
Method. Add the two numbers.
- Add. 480 + 250 = 730.730
- Write the answer. 730
Check. 730 - 250 = 480. ✓
Method. Round 39 to 40, then take away 5 from 200.
- Estimate. 40 × 5 = 200.200
- Correct. 200 - 5 = 195, because 39 is 1 less than 40.195
- Write the answer. 195
Check. 195 is close to 200. ✓
Method. Split one of the numbers into parts you know.
- Split 18. 18 = 10 + 8.10 + 8
- Multiply each part. 5 × 10 = 50 and 5 × 8 = 40.50 + 40
- Add. 50 + 40 = 90.90
- Write the answer. Split 18 into 10 and 8, then 5 × 10 = 50 and 5 × 8 = 40, so the total is 90.
Check. 5 × 18 = 90, and 90 ÷ 5 = 18. ✓
Total 19 marks. Allow about 19 minutes.
Exercise 7X Practice test questions
Non-calculatorMethod. Add the thousands, then the rest.
- Add. 5,000 + 1,000 = 6,000 and 400 + 800 = 1,200.7,200
- Write the answer. 7,200
Check. About 5,000 + 2,000 = 7,000, close to 7,200. ✓
Method. Round 2,600 to 3,000, subtract, then correct.
- Round the number taken away to the nearest thousand. 2,600 is close to 3,000.2,600 ≈ 3,000
- Subtract the rounded number. This is easier to work out.7,000 - 3,000 = 4,000
- Correct the answer. 3,000 is 400 more than 2,600, so add 400 back.4,000 + 400 = 4,400
- Write the answer. 4,400
Check. 4,400 + 2,600 = 7,000. ✓
Method. Use the times table fact.
- Split the bigger number into 10 and the rest. 11 is 10 + 1.11 = 10 + 1
- Multiply each part. 10 × 8 = 80 and 1 × 8 = 8.80 + 8
- Add the parts. Add the two products.80 + 8 = 88
- Write the answer. 88
Check. 8 × 11 = 88. ✓
Method. Split 1.5 into 1 and 0.5.
- Split. 1 × 6 = 6 and 0.5 × 6 = 3.6 + 3
- Add. 6 + 3 = 9.9
- Write the answer. 9
Check. 9 ÷ 6 = 1.5. ✓
Method. Find one eighth of 40, then take three eighths.
- Divide the amount by the bottom number. 40 ÷ 8 = 5.40 ÷ 8 = 5
- Multiply by the top number. Take 3 of those parts.5 × 3 = 15
- Write the answer. 15
Check. 15 × 8 = 120 and 40 ÷ 8 = 5, so 3 × 5 = 15. ✓
Method. Subtract the number in Key Stage 2 from the total.
- Subtract. 2,450 - 1,380. Round 1,380 to 1,400, then take away 1,400 to get 1,050, and add 20 back.1,070
- Write the answer. 1,070
Check. 1,070 + 1,380 = 2,450. ✓
Method. Double first, then subtract.
- Double. 3,000 × 2 = 6,000.6,000
- Subtract. 6,000 - 1,500 = 4,500.4,500
- Write the answer. 4,500
Check. 4,500 + 1,500 = 6,000. ✓
Method. Split 10.5 into 10 and 0.5.
- Split. 10 × 4 = 40 and 0.5 × 4 = 2.40 + 2
- Add. 40 + 2 = 42.42
- Write the answer. 42
Check. 42 ÷ 4 = 10.5. ✓
Method. Add 1,000, then take away 100.
- Add 1,000. 9,350 + 1,000 = 10,350.10,350
- Take away 100. 10,350 - 100 = 10,250.10,250
- Write the answer. 10,250
Check. 10,250 + 100 - 1,000 = 9,350. ✓
Method. Split 20 into 2 and 10.
- Split. 20 = 2 × 10.2 × 10
- Multiply. 25 × 10 = 250, then × 2 = 500.500
- Write the answer. Yes, because 25 × 20 = 500, found by doubling 250.
Check. 500 ÷ 25 = 20. ✓