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Calculation: the four operations

Buzz buzz, it is Bea! Calculation is the toolkit for almost every maths question. Learn the methods, check your answers, and you will be quick and accurate.

1. Cheat sheet: Calculation

Everything you need on one page. Press the button to print it on one sheet of A4.

Bonds and mental maths

  • Bonds to 10, 20, 100: 64 + 36 = 100.
  • Split the second number: 47 + 38 = 47 + 30 + 8.
  • Missing number: use the inverse. □ + 27 = 63 so 63 − 27.

Times tables

  • Learn up to 12 × 12.
  • Split: 6 × 7 = 5 × 7 + 7.
  • ×5: ×10 then halve. ×4: double twice.

Column methods

  • Line up the columns. Start at the units.
  • Carry when a column makes 10 or more.
  • Subtraction: exchange 1 ten for 10 units.

Multiplying

  • Short: 243 × 6 = 1,458.
  • Long: ×ones, then ×tens (put a 0 first), then add.
  • 268 × 34 = 1,072 + 8,040 = 9,112.

Dividing

  • Short division (bus stop): 748 ÷ 4 = 187.
  • Remainders: bags needed means round UP.
  • Long division: 972 ÷ 12 = 81.

× and ÷ by 10, 100, 1,000

  • Digits move left (×) or right (÷). The point stays.
  • 9.5 × 10 = 95, 9.5 × 100 = 950.
  • 450 ÷ 100 = 4.5.

Order of operations

  • Brackets, Indices, Divide/Multiply, Add/Subtract.
  • 3 + 4 × 5 = 23. (3 + 4) × 5 = 35.
  • 92 − 36 ÷ 9 = 88.

Estimate and check

  • Round first: 398 × 21 is about 400 × 20 = 8,000.
  • Check with the inverse: if 612 + 289 = 901 then 901 − 289 = 612.
  • Multi-step: underline what is asked.

2. Mini lessons

Number bonds, mental adding and missing numbers

CALC-01CALC-03CALC-13

Number bonds are pairs that make a target, like 64 and 36 making 100. For mental adding, split the second number into tens and ones and jump along a number line. A missing-number question is solved by using the inverse operation.

Animation: 47 + 38 in jumps

405060708090+30+8

Start at 47. Add the tens first: 47 + 30 = 77. Then add the ones: 77 + 8 = 85. So 47 + 38 = 85. Split the second number into tens and ones and jump.

Worked example: Work out 47 + 38. Then find the missing number: □ + 27 = 63.

  1. 47 + 30 = 77.
  2. 77 + 8 = 85.
  3. For the missing number, subtract: 63 − 27 = 36.
  4. Check: 36 + 27 = 63.

Answer: 85, and the missing number is 36

Times tables, mental multiplying and dividing

CALC-02CALC-05

Know your tables to 12 × 12. For harder facts, split one number into easy parts. Division is the inverse of multiplication, so 56 ÷ 8 asks "what times 8 makes 56?".

5 × 7 = 351 × 7 = 7
6 × 7 = (5 × 7) + (1 × 7) = 35 + 7 = 42. Split a hard fact into easy ones.

Worked example: Work out 14 × 5 and 56 ÷ 8.

  1. 14 × 5: do × 10 then halve. 14 × 10 = 140, and 140 ÷ 2 = 70.
  2. 56 ÷ 8: 7 × 8 = 56, so 56 ÷ 8 = 7.

Answer: 70 and 7

Column addition and subtraction

CALC-04

Line the digits up by place value and start at the units. In addition you carry when a column makes 10 or more. In subtraction you exchange: take 1 ten and turn it into 10 units.

Animation: column addition with carrying (487 + 356)

+48735634811HTU

Start with the units (U) column. Units: 7 + 6 = 13. Write 3, carry the 1 ten. Tens (T): 8 + 5 + 1 = 14. Write 4, carry the 1 hundred. Hundreds (H): 4 + 3 + 1 = 8. So 487 + 356 = 843.

Worked example: Work out 503 − 276.

  1. Units: 3 is less than 6. There are no tens to exchange, so exchange a hundred: 500 becomes 4 hundreds and 10 tens, then one ten becomes 10 units. Now we have 4 hundreds, 9 tens and 13 units.
  2. Units: 13 − 6 = 7. Tens: 9 − 7 = 2. Hundreds: 4 − 2 = 2.
  3. Check by adding: 227 + 276 = 503.

Answer: 227

Short and long multiplication

CALC-06CALC-07

Short multiplication is for a one-digit multiplier. Long multiplication multiplies by the ones, then by the tens, then adds the two rows. The tens row needs a 0 placeholder because you are really multiplying by a multiple of 10.

40520380010012015800 + 100 + 120 + 15 = 1,035
Grid method: 23 × 45 = 1,035. Long multiplication does the same job in columns.

Worked example: Work out 268 × 34.

  1. Multiply by 4: 268 × 4 = 1,072.
  2. Multiply by 30: 268 × 30 = 8,040 (write the 0 first, then 268 × 3 = 804).
  3. Add the rows: 1,072 + 8,040 = 9,112.
  4. Estimate: 300 × 30 = 9,000, so 9,112 is sensible.

Answer: 9,112

Short and long division, remainders

CALC-08CALC-09

Short division (the bus stop) shares one digit at a time. Long division does the same job with the working written out. When a division leaves a remainder, the story decides whether to round up, round down or give the remainder.

114 full bags of 6 (84 sweets)The 15th bag holds the 1 sweet left over85 ÷ 6 = 14 remainder 1
Remainders in real life: 85 sweets in bags of 6 need 15 bags.

Worked example: Work out 972 ÷ 12. Then: 85 sweets are put in bags of 6. How many bags are needed?

  1. Long division: 12 into 97 goes 8 times (8 × 12 = 96). Remainder 1.
  2. Bring down the 2 to make 12. 12 ÷ 12 = 1. So 972 ÷ 12 = 81. Check: 81 × 12 = 972.
  3. For the sweets: 85 ÷ 6 = 14 remainder 1.
  4. The 1 left over still needs a bag, so round UP to 15 bags.

Answer: 81, and 15 bags

Multiplying and dividing by 10, 100 and 1,000

CALC-10

When you multiply by 10, every digit moves one place to the left. By 100 it moves two places, by 1,000 three places. Dividing moves digits to the right. The decimal point stays still.

Animation: multiplying by 10 and 100

HTUth9509.59.5 × 10 = 959.5 × 100 = 950

The columns are H (hundreds), T (tens), U (units), then tenths (t) and hundredths (h). Start with 9.5. × 10: every digit moves ONE column to the left. 9.5 becomes 95. × 100: every digit moves TWO columns left and a 0 fills the gap. 9.5 becomes 950. The decimal point never moves, only the digits. To divide, move them right.

Worked example: Work out 4.7 × 100 and 450 ÷ 1,000.

  1. 4.7 × 100: move each digit two places left: 4.7 becomes 470 (a 0 fills the units).
  2. 450 ÷ 1,000: move each digit three places right: 450 becomes 0.45.

Answer: 470 and 0.45

Order of operations

CALC-11

A calculation is not always done from left to right. Do brackets first, then powers, then multiply and divide, then add and subtract. When the operations are on the same level, go left to right.

BBrackets firstIIndices (powers)DMDivide and Multiply, left to rightASAdd and Subtract, left to rightWork from the top row to the bottom row
The order of operations (BIDMAS).

Worked example: Work out 6 + 4 × 5 − 2.

  1. Multiply first: 4 × 5 = 20.
  2. Now the sum is 6 + 20 − 2.
  3. Left to right: 6 + 20 = 26, then 26 − 2 = 24.

Answer: 24

Estimating and checking with inverses

CALC-12

Round the numbers to get a quick estimate. Then check the exact answer with the inverse operation: addition is checked by subtracting, and multiplication by dividing.

380390400387
387 is nearer to 390 than 380, so it rounds to 390.

Worked example: Estimate 612 + 289, then work it out and check it.

  1. Round: 612 is about 600, 289 is about 300. 600 + 300 = 900.
  2. Exact: 612 + 289 = 901.
  3. Check with the inverse: 901 − 289 = 612. It matches.

Answer: 901 (estimate 900)

Multi-step word problems

CALC-14

Read the question, underline what it asks, and draw a bar model. Do one step at a time and write what each answer means. The last step must answer the question.

£10 paid6 pens: £7.20? change£10 − £7.20 = £2.80
A bar model shows the whole and its parts.

Worked example: Pens cost £1.20 each. Mo buys 6 and pays with a £10 note. How much change does he get?

  1. Cost of the pens: 6 × £1.20 = £7.20.
  2. Change: £10 − £7.20 = £2.80.
  3. The question asks for the change, not the cost.

Answer: £2.80

3. Worked examples from easy to hard

Level 1 is Year 3, one step. Level 5 is a challenge with 3 to 5 steps. Try each one yourself before you read the steps.

Level 1One step

What must be added to 64 to make 100?

  1. The number bond: 100 − 64 = 36.
  2. Check: 64 + 36 = 100.

Answer: 36

Level 2One or two steps

A box holds 12 eggs. Hana buys 7 boxes and 5 eggs break. How many eggs are not broken?

  1. Eggs bought: 12 × 7 = 84.
  2. Take away the broken eggs: 84 − 5 = 79.

Answer: 79

Level 3Two steps

A school buys 24 packs of 36 pencils and gives out 500 of them. How many pencils are left?

  1. Pencils bought: 24 × 36 = 864 (24 × 30 = 720 and 24 × 6 = 144, then 720 + 144 = 864).
  2. Take away: 864 − 500 = 364.

Answer: 364

Level 4Exam standard

Priya buys 4 comics at £1.85 each and a poster for £3.20. She pays with a £20 note. How much change does she get?

  1. Comics: 4 × £1.85 = £7.40.
  2. Add the poster: £7.40 + £3.20 = £10.60.
  3. Change: £20 − £10.60 = £9.40.
  4. Many children stop at £10.60, but that is the total spent, not the change.

Answer: £9.40

Level 5Challenge

A baker makes 8 trays of 24 rolls. She packs them in bags of 5 and sells all the full bags at £2.50 each. The ingredients cost £48. How much profit does she make?

  1. Rolls: 8 × 24 = 192.
  2. Full bags: 192 ÷ 5 = 38 remainder 2, so 38 full bags (the 2 spare rolls are not sold).
  3. Money from sales: 38 × £2.50 = £95.
  4. Profit: £95 − £48 = £47.

Answer: £47

4. Common mistakes

Lots of children make these slips. Spot them early and you will win marks.

Always working left to right

ORDER_OPS_LEFT_TO_RIGHT

The mistake: Doing the sum strictly from left to right.

Why it is wrong: Multiplying and dividing come before adding and subtracting, so the order changes the answer.

How to avoid it: Look for × and ÷ first and do those. Put a ring around them before you start.

✗ Wrong: 92 − 36 ÷ 9: 92 − 36 = 56, 56 ÷ 9 = 6.2...
✓ Right: 36 ÷ 9 = 4 first, then 92 − 4 = 88.

Forgetting the 0 in long multiplication

PV_LONGMULT_NO_PLACEHOLDER

The mistake: Leaving out the 0 on the tens row.

Why it is wrong: The 3 in 34 means 30, so that row is ×30 and ends in 0. Without the 0 the row is ten times too small.

How to avoid it: Write the 0 in the units column before you start the second row.

✗ Wrong: 268 × 34: 1,072 + 804 = 1,876.
✓ Right: 1,072 + 8,040 = 9,112.

Taking the smaller digit from the bigger

PV_SMALLER_FROM_LARGER

The mistake: In every column, subtracting the smaller digit from the larger.

Why it is wrong: The order of the numbers matters. You must take the bottom digit from the top digit, exchanging when needed.

How to avoid it: Estimate first, then exchange 1 ten for 10 units when the top digit is smaller.

✗ Wrong: 52 − 27: 7 − 2 = 5 and 5 − 2 = 3, so 35.
✓ Right: Exchange: 12 − 7 = 5, 4 − 2 = 2, so 25. Check: 25 + 27 = 52.

"Add a zero" to multiply by 10

PV_ADD_A_ZERO

The mistake: Believing that multiplying by 10 means adding a zero to the end.

Why it is wrong: The digits move one place left. Adding a zero to a decimal does not change its value at all.

How to avoid it: Use a place value chart. The decimal point stays put and the digits move.

✗ Wrong: 9.5 × 10 = 9.50.
✓ Right: 9.5 × 10 = 95.

"Multiplying always makes bigger"

MULT_MAKES_BIGGER

The mistake: Expecting multiplication to give a bigger answer and division a smaller one.

Why it is wrong: Multiplying by a number less than 1 makes things smaller. Dividing by a number less than 1 makes things bigger.

How to avoid it: Think of "of": 6 × 0.5 is half of 6.

✗ Wrong: 6 × 0.5 = 12 (bigger than 6).
✓ Right: 6 × 0.5 = 3 (half of 6).

Stopping part-way

TECH_STOPPED_AT_STEP_1

The mistake: Giving an answer from the middle of a multi-step problem.

Why it is wrong: The middle answer is not what was asked.

How to avoid it: Before you answer, re-read the question and underline what it asks for.

✗ Wrong: Total spent is £10.60, so the change is £10.60.
✓ Right: Change = £20 − £10.60 = £9.40.

5. Quick check

Five questions, easy to hard. Have a go on paper, then open the answer.

  1. What must be added to 68 to make 100?

    Show answer

    32

    100 − 68 = 32. Check: 68 + 32 = 100.

  2. Work out 12 × 6.

    Show answer

    72

    10 × 6 = 60 and 2 × 6 = 12, so 60 + 12 = 72.

  3. Work out 508 − 279.

    Show answer

    229

    Exchange across the zero. Units: 18 − 9 = 9. Tens: 9 − 7 = 2. Hundreds: 4 − 2 = 2. Check: 229 + 279 = 508.

  4. Work out 9 + 3 × 6 − 5.

    Show answer

    22

    3 × 6 = 18 first. Then 9 + 18 − 5 = 22.

  5. A coach seats 48 children. How many coaches are needed for 250 children?

    Show answer

    6 coaches

    250 ÷ 48 = 5 remainder 10. The 10 left-over children still need a coach, so round up to 6.

6. For grown-ups

  • Short, daily practice beats long sessions. Five minutes on one times table fact family, said aloud, is worth more than a worksheet.
  • Ask "Is that sensible?" after every answer. Rounding to estimate is the habit that catches most slips in the exam.
  • For long multiplication and division, let your child write the layout neatly on squared paper. Most errors are misaligned columns.