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ExplainerMathGrades 3–5

Long Multiplication

What Is This?

Long multiplication is a written method for multiplying large numbers — numbers too big to calculate in your head. It works by breaking the multiplication into smaller, manageable steps and adding the results together. Once you understand the method, you can multiply any two whole numbers, no matter how large.

Why Does It Matter?

Long multiplication is the foundation for multiplying decimals, working with area and volume, scaling recipes, and solving rate problems. Every multiplication algorithm used by computers and calculators is built on the same principles you use when you do long multiplication by hand.

Key Vocabulary

Term Meaning
Factor A number being multiplied (e.g. in 34 × 6, both 34 and 6 are factors)
Product The result of a multiplication
Partial product One of the intermediate results when multiplying by each digit separately
Carrying (regrouping) Moving a digit to the next column when a result is 10 or more
Placeholder zero A zero written in the ones column of the second partial product row to hold the place value

Method 1 — Short Multiplication (2-digit × 1-digit)

Before tackling long multiplication, make sure you are comfortable multiplying a 2-digit number by a single digit.

Example: 47 × 6

    4 7
  ×   6
  -----
  • Ones: 7 × 6 = 42 → write 2, carry 4
  • Tens: 4 × 6 = 24, plus 4 carried = 28 → write 28
    4 7
  ×   6
  -----
  2 8 2

Answer: 282 ✓

Method 2 — Long Multiplication (2-digit × 2-digit)

Example: 34 × 27

Step 1: Multiply 34 by the ones digit of 27 (which is 7).

    3 4
  × 2 7
  -----
  2 3 8   ← 34 × 7
  • Ones: 4 × 7 = 28 → write 8, carry 2
  • Tens: 3 × 7 = 21, plus 2 carried = 23 → write 23

Step 2: Multiply 34 by the tens digit of 27 (which is 2). Write a placeholder zero in the ones column first, because this row represents tens.

    3 4
  × 2 7
  -----
  2 3 8   ← 34 × 7
  6 8 0   ← 34 × 20 (note the placeholder 0)
  • Ones placeholder: 0
  • Ones: 4 × 2 = 8
  • Tens: 3 × 2 = 6

Step 3: Add the partial products.

    3 4
  × 2 7
  -----
  2 3 8
+ 6 8 0
  -----
  9 1 8

Answer: 918 ✓

Method 3 — Long Multiplication (3-digit × 2-digit)

Example: 243 × 35

Step 1: Multiply 243 × 5 (ones digit).

  • Ones: 3 × 5 = 15 → write 5, carry 1
  • Tens: 4 × 5 = 20 + 1 = 21 → write 1, carry 2
  • Hundreds: 2 × 5 = 10 + 2 = 12 → write 12

First partial product: 1215

Step 2: Multiply 243 × 30 (tens digit). Write placeholder zero first.

  • Ones: 0 (placeholder)
  • Ones: 3 × 3 = 9
  • Tens: 4 × 3 = 12 → write 2, carry 1
  • Hundreds: 2 × 3 = 6 + 1 = 7

Second partial product: 7290

Step 3: Add.

    2 4 3
  ×   3 5
  -------
  1 2 1 5   ← 243 × 5
+ 7 2 9 0   ← 243 × 30
  -------
  8 5 0 5

Answer: 8505 ✓

Method 4 — Long Multiplication (3-digit × 3-digit)

Example: 312 × 124

Step 1: 312 × 4 = 1248 Step 2: 312 × 20 → placeholder 0, then 312 × 2 = 624 → 6240 Step 3: 312 × 100 → two placeholder zeros, then 312 × 1 = 312 → 31200

Step 4: Add all three partial products.

    3 1 2
  × 1 2 4
  -------
    1 2 4 8   ← 312 × 4
    6 2 4 0   ← 312 × 20
+ 3 1 2 0 0   ← 312 × 100
  ---------
  3 8 6 8 8

Answer: 38688 ✓

The Placeholder Zero Rule

Every time you move to the next digit of the multiplier, add one more placeholder zero at the right of that row.

Multiplier digit position Placeholder zeros to add
Ones 0
Tens 1
Hundreds 2
Thousands 3

This is the most common mistake in long multiplication — forgetting the placeholder zero shifts every digit one column to the left, which is what multiplying by a tens digit requires.

Estimating to Check

Before calculating, round each factor to the nearest 10 or 100 and multiply mentally. If your exact answer is far from the estimate, recheck.

Example: 34 × 27

  • Estimate: 30 × 30 = 900
  • Exact: 918 ✓ (close to 900)

Example: 243 × 35

  • Estimate: 240 × 35 ≈ 200 × 35 = 7000 or 240 × 40 = 9600 → middle estimate ≈ 8500
  • Exact: 8505 ✓

Worked Examples

Worked Example 1: 56 × 43

    5 6
  × 4 3
  -----
  1 6 8   ← 56 × 3
+ 2 2 4 0   ← 56 × 40
  -----
  2 4 0 8
  • 56 × 3: ones 6×3=18 write 8 carry 1; tens 5×3=15+1=16. → 168
  • 56 × 40: placeholder 0, then 56×4: ones 6×4=24 write 4 carry 2; tens 5×4=20+2=22. → 2240
  • 168 + 2240 = 2408

Worked Example 2: 125 × 48

    1 2 5
  ×   4 8
  -------
  1 0 0 0   ← 125 × 8
+ 5 0 0 0   ← 125 × 40
  -------
  6 0 0 0
  • 125 × 8: ones 5×8=40 write 0 carry 4; tens 2×8=16+4=20 write 0 carry 2; hundreds 1×8=8+2=10. → 1000
  • 125 × 40: placeholder 0, then 125×4: 5×4=20 write 0 carry 2; 2×4=8+2=10 write 0 carry 1; 1×4=4+1=5. → 5000
  • 1000 + 5000 = 6000

Common Mistakes

Mistake 1: Forgetting the placeholder zero When multiplying by the tens digit, you must write a 0 in the ones column of that row first. Without it, you are multiplying by 2 instead of 20.

Mistake 2: Carrying errors Always write the carry digit above the next column in small writing and add it in before you move on. Forgetting a carry is the most frequent arithmetic error.

Mistake 3: Adding partial products incorrectly After all partial products are written, take your time adding them column by column. This is a separate addition step — treat it carefully.

Mistake 4: Misaligning columns Keep digits in their correct columns throughout. Use squared paper or draw vertical lines to help keep hundreds, tens, and ones separate.

Key Takeaways

  • Long multiplication breaks a large multiplication into partial products, one for each digit of the multiplier
  • Always multiply by the ones digit first, then the tens digit, then hundreds, and so on
  • Add a placeholder zero for each step as you move left through the multiplier digits
  • Carry digits carefully and write them clearly above the next column
  • Add all partial products to get the final answer
  • Always estimate first so you can catch major errors

Practice and Resources

Ready to practise? Try our Grade 3-5 Long Multiplication Worksheet with problems from 2-digit × 1-digit up to 3-digit × 3-digit. Test yourself with the Grade 3-5 Math Practice Test.

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