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Foundations · Arithmetic

The four operations and mental maths tricks

Column addition and subtraction, long multiplication and division, and the shortcuts that make mental arithmetic fast.

First, the rules

Written methods that always work, shortcuts that often do

Column methods never fail; the tricks are faster when the numbers cooperate. Know both, and know which one the question wants.

Written methods

★ Column add / subtract
Example 1
Line up the units. Work right to left. Add: a column over 9 carries 1 left. Subtract: a column that would go below 0 borrows 10 from the left.
★ Long multiplication (area model)
Example 2
Split both numbers into tens and units, multiply every pair, add the four partial products. 23 × 47 = 20·40 + 20·7 + 3·40 + 3·7 = 800 + 140 + 120 + 21 = 1 081.
★ Long division
Example 3
Divide, multiply, subtract, bring down — repeat. 858 ÷ 6: 6 into 8 goes 1 (r 2), 6 into 25 goes 4 (r 1), 6 into 18 goes 3 → 143.

Mental tricks

× 10, 100, 1000
Shift the digits left: 37 × 100 = 3 700. ÷ 10 shifts right: 37 ÷ 10 = 3.7.
× 5 and × 25
Practice
× 5: × 10 then halve (46 × 5 = 460 ÷ 2 = 230). × 25: × 100 then ÷ 4 (36 × 25 = 3600 ÷ 4 = 900).
× 11
Put the digit sum in the middle: 52 × 11 = 5 [5+2] 2 = 572; carry if the sum passes 9 (78 × 11 = 7 [15] 8 → 858).
× 9, × 99
× 10 minus the number: 43 × 9 = 430 − 43 = 387. × 99: × 100 minus the number.
Doubling and halving
Example 2
16 × 35 = 8 × 70 = 4 × 140 = 560. Keep halving one and doubling the other until it is easy.
Squares ending in 5
tens digit × (tens digit + 1), then write 25: 65² = 6·7 = 42 → 4 225.
Compensation
Example 1
298 + 467 = 300 + 467 − 2 = 765. 1 003 − 648 = 1 000 − 648 + 3 = 355.

See it

Long multiplication as an area

The product of two numbers is the area of a rectangle. Split each side into tens and units and the four smaller rectangles are the four partial products you add in column multiplication.

Step by step

Worked examples

1

Carrying and borrowing

column methods

Work out (a) 4 768 + 2 859 and (b) 5 003 − 1 647.

Ask yourself: Start at the units column. What happens when a column passes 9, or cannot be taken away?

Show the solution
  1. (a) Units 8 + 9 = 17: write 7, carry 1. Tens 6 + 5 + 1 = 12: write 2, carry 1. Hundreds 7 + 8 + 1 = 16: write 6, carry 1. Thousands 4 + 2 + 1 = 7. Result 7 627.
  2. (b) Units 3 − 7 cannot: borrow — but the tens and hundreds are 0, so borrow from the thousands: 5 003 becomes 4 thousands, 9 hundreds, 9 tens, 13 units. Then 13 − 7 = 6, 9 − 4 = 5, 9 − 6 = 3, 4 − 1 = 3: 3 356.
  3. Mental check of (b) by compensation: 5 000 − 1 647 = 3 353, plus the 3 we left out = 3 356. ✓

Answer 7 627 · 3 356

2

One product, three ways

long multiplication · tricks

Work out 24 × 35 (a) by the area model, (b) by doubling and halving, (c) by × 5 then × 7.

Ask yourself: Which pieces of 24 and 35 are easy to multiply?

Show the solution
  1. (a) 20·30 + 20·5 + 4·30 + 4·5 = 600 + 100 + 120 + 20 = 840.
  2. (b) 24 × 35 = 12 × 70 = 6 × 140 = 840.
  3. (c) 35 = 5 × 7: 24 × 5 = 120, then 120 × 7 = 840.

Answer 840

3

Long division with a remainder

division

Work out 1 379 ÷ 8. Give the quotient and remainder, then as a decimal to one place.

Ask yourself: Divide, multiply, subtract, bring down — one digit at a time.

Show the solution
  1. 8 into 13 goes 1, remainder 5. Bring down 7: 8 into 57 goes 7 (56), remainder 1. Bring down 9: 8 into 19 goes 2 (16), remainder 3.
  2. Quotient 172, remainder 3. Check: 172 × 8 + 3 = 1 376 + 3 = 1 379 ✓.
  3. Decimal: bring down a 0: 8 into 30 goes 3 (24), remainder 6 — and 6 ≥ 4 would round the next digit, so 172.375 → 172.4.

Answer 172 r 3 ≈ 172.4

Practice: mental and written arithmetic

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Tricks for × 5, × 11, × 25, × 99 and squares ending in 5, plus column addition, long multiplication and division. Your best score is kept in this browser only.

About this lesson

The four operations and mental maths tricks: column addition and subtraction with carrying and borrowing, long multiplication through the area model, long division, and the shortcuts worth knowing — doubling and halving, multiplying by 5, 11, 25 and 99, squaring numbers ending in 5, compensation. An area-model multiplier you can move, worked examples and practice. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.