Long multiplication and long division are the written column methods for multiplying and dividing numbers too large for mental arithmetic. At KS3 you must carry out these methods without a calculator on non-calculator papers. Practising the layout carefully is the fastest way to avoid lost marks.

What is long multiplication and when do you use it?

Long multiplication is the method for multiplying two numbers when at least one has two or more digits. It breaks the calculation into a series of simpler multiplications, recorded in rows, then adds the rows together.

You use long multiplication when both factors are at least two digits — for example, 347 × 26. Mental methods become unreliable for calculations like these, so a written layout is essential.

How do you carry out long multiplication step by step?

Worked example: Calculate 347 × 26.

Step 1: Write the numbers one above the other, aligned on the right.

    3 4 7
  ×   2 6
  -------

Step 2: Multiply 347 by the units digit of 26, which is 6.

  • 7 × 6 = 42. Write 2, carry 4.
  • 4 × 6 = 24, plus carry 4 = 28. Write 8, carry 2.
  • 3 × 6 = 18, plus carry 2 = 20. Write 20.

Row 1: 2 0 8 2

Step 3: Multiply 347 by the tens digit of 26, which is 2 (representing 20). Write a placeholder 0 in the units column before starting.

  • 7 × 2 = 14. Write 4, carry 1.
  • 4 × 2 = 8, plus carry 1 = 9. Write 9.
  • 3 × 2 = 6. Write 6.

Row 2: 6 9 4 0 (with the placeholder 0)

Step 4: Add the two rows.

    2 0 8 2
+   6 9 4 0
-----------
    9 0 2 2

Answer: 347 × 26 = 9 022

Quick check: 350 × 26 ≈ 350 × 25 = 8 750. Our answer of 9 022 is close to this estimate. ✓

What is long division and when do you use it?

Long division is the method for dividing a number (the dividend) by another number (the divisor) when the divisor is larger than 12 or when you need to find a decimal remainder. It works digit by digit from left to right.

The layout is sometimes called the "bus stop" method when used for short division, but the full long division layout is needed for two-digit divisors.

How do you carry out long division step by step?

Worked example: Calculate 924 ÷ 12.

Step 1: Set up the division frame: 12 ) 924

Step 2: Work digit by digit from left to right.

  • How many times does 12 go into 9? 0 times (9 < 12). Bring the next digit alongside: consider 92.
  • How many times does 12 go into 92? 12 × 7 = 84, 12 × 8 = 96 (too large). 7 times. Write 7 above.
  • Subtract: 92 − 84 = 8. Bring down the 4 to make 84.
  • How many times does 12 go into 84? 12 × 7 = 84. 7 times exactly. Write 7.
  • Subtract: 84 − 84 = 0.

Answer: 924 ÷ 12 = 77

Check: 77 × 12 = 77 × 10 + 77 × 2 = 770 + 154 = 924. ✓

How do you handle a remainder in long division?

Worked example: Calculate 950 ÷ 12.

Following the same steps:

  • 12 into 95: 7 times (7 × 12 = 84). Remainder: 95 − 84 = 11. Bring down 0 → 110.
  • 12 into 110: 9 times (9 × 12 = 108). Remainder: 110 − 108 = 2.

Answer: 950 ÷ 12 = 79 remainder 2, or as a fraction: 79 and 2/12 = 79 and 1/6.

If a decimal answer is needed, add a decimal point and zeros after the dividend:

  • Bring down 0 → 20. 12 into 20 = 1 remainder 8. Bring down 0 → 80.
  • 12 into 80 = 6 remainder 8. The remainder 8 repeats → 79.1̄6̄ (a recurring decimal).

What layout errors cost marks in exams?

Common error How to fix it
Not writing the placeholder 0 in long multiplication Always add the 0 before starting the second row
Misaligning columns Use squared paper or draw your own columns
Forgetting to carry Write carry digits small above the next column
Forgetting to bring down in long division Circle each digit as you bring it down
Not checking the answer Do a quick reverse calculation (multiply back)

Presentation is worth marks on non-calculator papers. An illegible layout makes it impossible for an examiner to award method marks.

How do you multiply or divide decimals using these methods?

For multiplication: multiply as if the decimal points are not there, then count the total number of decimal places in both numbers and insert the decimal point that many places from the right.

Example: 3.47 × 2.6. Ignore decimals: 347 × 26 = 9 022. Total decimal places = 2 + 1 = 3. Insert point 3 from the right: 9.022.

For division: shift the decimal point in the divisor to make it a whole number, and shift the decimal point in the dividend by the same amount.

Example: 9.24 ÷ 1.2. Multiply both by 10: 92.4 ÷ 12 = 7.7. Answer: 7.7.

Frequently asked questions

How do I know which row goes first in long multiplication?

Always multiply by the units digit first (bottom-right digit), then by the tens digit, then hundreds, and so on. The first row needs no placeholder; the second row has one zero at the end; the third row has two zeros, and so on. This ensures each row represents the correct power of ten.

What if there is a zero in the middle of one of the numbers?

Zero times anything is zero, but you still need to record the result in the correct column. For example, multiplying by 206: do the 6-row, then the 0-row (which gives a row of zeros — you can skip writing it and just add a second placeholder), then the 2-row with two placeholders. In long division, a zero in the dividend simply means "bring down 0" — proceed exactly as normal.

Is the grid method the same as long multiplication?

The grid method (also called the box method) splits each number into its tens and units, multiplies all four combinations in a grid, then sums them. It gives the same answer as long multiplication and is a valid method. The long multiplication column layout is faster for large numbers and is the expected method from Year 7 onwards.

Do I need to know long division for GCSE?

Yes — non-calculator GCSE papers can ask for division of large numbers or division that produces decimals. Long division is the reliable method. Practising it regularly at KS3 means you will not panic in a GCSE exam when you cannot reach for a calculator.

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