4.1.2 Multiplying Three-Digit Number by Three-Digit Number

This is an extension of the previous section. The procedure is the same but requires more multiplication and bookkeeping.

The Method

For $abc \times def$, group as $(a)(bc)$ and $(d)(ef)$:

$$[100a + (bc)] \times [100d + (ef)] = 10000ad + 100[a(ef) + d(bc)] + (bc)(ef)$$

The Rule

1. Ones and Tens = last two digits of $(bc) \times (ef)$
2. Hundreds and Thousands = $a \times (ef) + d \times (bc)$ + carry
3. Remaining = $a \times d$ + carry

Example: 211 × 416

Step	Calculation	Result
Units/Tens	16 × 11	172 (carry 1)
Hundreds/Thousands	16 × 2 + 11 × 4 + 1	77
Remaining	4 × 2	8

Answer: 87772

Example: 217 × 245

Step	Calculation	Result
Units/Tens	17 × 45	765 (carry 7)
Hundreds/Thousands	45 × 2 + 17 × 2 + 7	131 (carry 1)
Remaining	2 × 2 + 1	5

Answer: 53165

Tip: Most of the time, the digits in three-digit by three-digit multiplication are pretty low, making the actual multiplication easy – the challenge is keeping track of everything in your head!

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Problem Set 4.1.2

Practice these problems. Type your answer and press Enter to check:

Basic 3×3 Multiplication

Squares of Three-Digit Numbers

More Practice