Long Multiplication Step-by-Step (2-Digit & 3-Digit)
[40ร30 = 1200] + [40ร6 = 240] + [7ร30 = 210] + [7ร6 = 42] = 1,692
โข 2. โ ๏ธ Place Placeholder 0: Drop 0 in Row 2, then multiply by Tens
โข 3. Add Rows: Sum Row 1 + Row 2 = Final Product!
| Problem Type | Step-by-Step Vertical Stack | Answer & Proof Check |
|---|---|---|
| 1. 2-Digit ร 2-Digit 38 ร 24 | โข Row 1 (38 ร 4): 8ร4=32 (carry 3), 3ร4=12+3=15 โ 152 โข Row 2 (38 ร 20): Place 0! 8ร2=16 (carry 1), 3ร2=6+1=7 โ 760 โข Sum: 152 + 760 = 912 | = 912 Check: 912 รท 24 = 38 โ |
| 2. 3-Digit ร 2-Digit 415 ร 32 | โข Row 1 (415 ร 2): 5ร2=10 (carry 1), 1ร2=2+1=3, 4ร2=8 โ 830 โข Row 2 (415 ร 30): Place 0! 5ร3=15 (carry 1), 1ร3=3+1=4, 4ร3=12 โ 12,450 โข Sum: 830 + 12,450 = 13,280 | = 13,280 13,280 รท 32 = 415 โ |
| 3. Zero in Middle 604 ร 57 | โข Row 1 (604 ร 7): 4ร7=28 (carry 2), 0ร7=0+2=2, 6ร7=42 โ 4,228 โข Row 2 (604 ร 50): Place 0! 4ร5=20 (carry 2), 0ร5=0+2=2, 6ร5=30 โ 30,200 โข Sum: 4,228 + 30,200 = 34,428 | = 34,428 34,428 รท 57 = 604 โ |
Multi-Digit Long Multiplication: Partial Products & Standard Algorithm
Why Multi-Digit Multiplication Works (The Area Model)
Multiplying 47 ร 36 seems intimidating until you split both numbers into tens and ones: (40 + 7) ร (30 + 6).
This creates 4 simple partial products:
โข 40 ร 30 = 1,200 ย |ย โข 40 ร 6 = 240
โข 7 ร 30 = 210 ย |ย โข 7 ร 6 = 42
Adding them all together: 1,200 + 240 + 210 + 42 = 1,692!
How to Execute Vertical Long Multiplication
0 in the ones place! Then multiply the top number by the bottom tens digit.Interactive Step-by-Step Multiplication Stack
2-Digit ร 2-DigitMultiply 47 by 6. First, 7 ร 6 = 42 (write 2 down, carry 4). Next, 4 ร 6 = 24, plus the carried 4 = 28. Row 1 partial product is 282.
The 3 Mistakes That Cost Students Marks in Multiplication
When multiplying by the tens digit, you MUST write 0 in the ones place on Row 2. Forgetting it makes 47 ร 30 become 47 ร 3, giving the wrong answer 423 instead of 1,692.
Always multiply first, then add the carried digit. E.g., in $4 \times 6 + 4$, calculate $4 \times 6 = 24$, THEN add $4 = 28$.
Cross out carried numbers from Row 1 before multiplying Row 2 so you don't accidentally add them twice!
3 Step-by-Step Worked Examples
Master the exact derivation across 2-digit and 3-digit multiplication:
0 in ones place! 8 ร 2 = 16 (write 6, carry 1). 3 ร 2 = 6 + 1 = 7 โ 760.0 in ones place! 5 ร 3 = 15 (write 5, carry 1). 1 ร 3 = 3 + 1 = 4. 4 ร 3 = 12 โ 12,450.0 in ones place! 4 ร 5 = 20 (write 0, carry 2). 0 ร 5 = 0 + 2 = 2. 6 ร 5 = 30 โ 30,200.๐ฏ Active Recall Quiz (Test Your Long Multiplication Mastery)
Test your multi-digit multiplication skills with 3 quick questions. Instant feedback provided:
Frequently Asked Questions
When you multiply by the second digit, you are actually multiplying by the TENS place (e.g., in 47 ร 36, multiplying by 3 means multiplying by 30). The placeholder zero shifts the entire row one place to the left to preserve the true place value.
Both methods calculate the exact same mathematical partial products. The Box (Area) Method writes out all products in a visual 2x2 grid (e.g. 40ร30, 40ร6, 7ร30, 7ร6), while the Standard Algorithm combines them vertically into 2 rows with carrying.
Always multiply the top number by the bottom digit FIRST, and only add the carried digit AFTER multiplying (never add before multiplying!). Cross out old carried numbers when moving to the next row.
You can verify your multiplication answer using division (Product รท Bottom Number = Top Number) or by using the Area/Box Method to add the individual partial products.
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