4.1.1 Multiplying Three-Digit Number by Two-Digit Number

We briefly touched on how to apply FOILing/LIOFing principles in Section 1.1 – chiefly concerning ourselves with two-digit number multiplication – but more recent exams have really emphasized the multiplication of three-digit numbers, starting around the third column.

The Method

When multiplying a three-digit number $abc$ by a two-digit number $ef$:

1. Treat $ef$ as a three-digit number with leading 0: $0ef$
2. Group the digits: $(a)(bc)$ and $(0)(ef)$
3. Perform FOIL/LIOF twice

$$abc \times 0ef = 100a(ef) + (bc)(ef)$$

The Rule

1. Ones and Tens digits = last two digits of $(bc) \times (ef)$
2. Carry = remaining digits from that multiplication (can be two digits!)
3. Rest of answer = $a \times (ef)$ + carry

Example: 117 × 15

Step	Calculation	Result
Units/Tens	17 × 15	255 (carry 2)
Remaining	1 × 15 + 2	17

Answer: 1755

Example: 233 × 37

Step	Calculation	Result
Units/Tens	33 × 37 = 35² − 2² = 1225 − 4	1221 (carry 12)
Remaining	2 × 37 + 12	86

Answer: 8621

Tip: Don’t be surprised if these multiplications require other tricks! In the second example, we used the “Multiplying Two Numbers Equidistant from a Third” trick (Section 1.2.10) to compute 33 × 37 = 35² − 2².

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Problem Set 4.1.1

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