Multiplying 3 Digits By 2 Digits

Multiplying 3 digits by 2 digits is one of the most important milestones in elementary arithmetic, because it combines everything a student has learned about place value, multiplication facts, and regrouping into a single, reliable procedure. Once you can confidently solve problems like 324 × 56, you tap into the ability to handle money calculations, area problems, unit conversions, and the multi-step multiplication that appears in algebra and science. This guide walks you through the standard algorithm step by step, explains why it works, highlights the mistakes that trip most learners up, and gives you practice problems with full solutions Most people skip this — try not to..

Why Multiplying 3 Digits by 2 Digits Matters

Up to this point, most students have multiplied by single digits or worked with smaller factors. Which means multiplying a three-digit number by a two-digit number is different in one crucial way: you must produce two partial products and then add them together. That single change is where nearly all errors happen Small thing, real impact..

Quick note before moving on.

Understanding this skill also builds number sense. When you know that 324 × 56 is roughly 300 × 60, you can instantly judge whether an answer is reasonable. That habit of estimating before computing is what separates strong math students from students who guess and hope Not complicated — just consistent..

The Foundation: Place Value Does the Heavy Lifting

Before touching the algorithm, make sure you can break a two-digit number apart. The number 56 is not just "five and six." It is 50 + 6. Similarly, 324 is 300 + 20 + 4 And it works..

This matters because multiplying 324 × 56 really means:

• 324 × 6 (the ones digit)
• 324 × 50 (the tens digit, which is 5 tens, or 50)

If you're write the second partial product in the standard algorithm, you shift it one place to the left. That shift is not a trick or a rule to memorize blindly — it is simply a shortcut for multiplying by 50 instead of by 5.

Step-by-Step: The Standard Algorithm

Here is the complete procedure for multiplying any three-digit number by any two-digit number.

1. Write the problem vertically, aligning the digits by place value. The three-digit number goes on top, the two-digit number below it, and the ones digits line up.
2. Multiply by the ones digit first. Multiply the top number by the ones digit of the bottom number, working right to left. Carry any tens just as you would in single-digit multiplication.
3. Write the first partial product directly beneath the line.
4. Place a zero (or leave a blank space) in the ones column of the second row. This accounts for the fact that you are now multiplying by tens, not ones.
5. Multiply by the tens digit. Multiply the top number by the tens digit of the bottom number, again working right to left.
6. Write the second partial product, shifted one place to the left.
7. Add the two partial products to get the final answer.
8. Check with an estimate to confirm the answer is reasonable.

Worked Example 1: 324 × 56

Let's apply the steps That alone is useful..

• Step 1: Multiply 324 by the ones digit, 6.
  • 6 × 4 = 24 → write 4, carry 2
  • 6 × 2 = 12, plus 2 = 14 → write 4, carry 1
  • 6 × 3 = 18, plus 1 = 19 → write 19
  • First partial product: 1,944
• Step 2: Multiply 324 by the tens digit, 5, and shift one place left.
  • 5 × 4 = 20 → write 0, carry 2
  • 5 × 2 = 10, plus 2 = 12 → write 2, carry 1
  • 5 × 3 = 15, plus 1 = 16 → write 16
  • Second partial product: 16,200
• Step 3: Add them.
  • 1,944 + 16,200 = 18,144

Estimate check: 300 × 60 = 18,000. Our answer of 18,144 is very close, so it is almost certainly correct.

Worked Example 2: 638 × 47

• Multiply by 7: 7 × 8 = 56 (write 6, carry 5); 7 × 3 = 21 + 5 = 26 (write 6, carry 2); 7 × 6 = 42 + 2 = 44. First partial product: 4,466
• Multiply by 4 (tens), shifted left: 4 × 8 = 32 (write 2, carry 3); 4 × 3 = 12 + 3 = 15 (write 5, carry 1); 4 × 6 = 24 + 1 = 25. Second partial product: 25,520
• Add: 4,466 + 25,520 = 29,986

Estimate check: 600 × 50 = 30,000. The answer 29,986 is right on target It's one of those things that adds up..

Worked Example 3: When a Zero Appears — 205 × 34

Zeros are a common source of confusion, so handle them carefully.

• Multiply by 4: 4 × 5 = 20 (write 0, carry 2); 4 × 0 = 0 + 2 = 2 (write 2); 4 × 2 = 8. First partial product: 820
• Multiply by 3 (tens), shifted left: 3 × 5 = 15 (write 5, carry 1); 3 × 0 = 0 + 1 = 1 (write 1); 3 × 2 = 6. Second partial product: 6,150
• Add: 820 + 6,150 = 6,970

Notice that multiplying by zero gives zero in that column — but you still must add the carried digit. Skipping that step is one of the most frequent errors.

The Five Most Common Mistakes

• Forgetting the shift. Writing the second partial product directly under the first, without moving it one place left, produces an answer that is far too small.
• Dropping the zero. If you write the zero placeholder, remember it is part of the