1.3.1 Extending Foiling

You can extend the method of FOILing to quickly multiply two three-digit numbers in the form $cba \times dba$.

The general objective is you treat the digits of $ba$ as one number, so after foiling you would get:

$$cba \times dba =$$

• Ones/Tens: $(ba)^2$
• Hundreds/Thousands: $(c + d) \times (ba)$
• Rest of Answer: $c \times d$

Let’s look at a problem to practice this extension:

412 × 612 =

• Ones/Tens: $12^2 = 144$ (Write 44, carry 1)
• Hundreds/Thousands: $(4 + 6) \times 12 + 1 = 121$ (Write 21, carry 1)
• Rest of Answer: $4 \times 6 + 1 = 25$

Answer: 252144

By treating the last two digits as a single entity, you reduce the three-digit multiplication to a two-digit problem. The last two digits need not be the same in the two numbers (usually I do see this as the case though) in order to apply this method, let’s look at an example of this:

211 × 808 =

• Ones/Tens: $08 \times 11 = 88$
• Hundreds/Thousands: $08 \times 2 + 11 \times 8 = 16 + 88 = 104$ (Write 04, carry 1)
• Rest of Answer: $2 \times 8 + 1 = 17$

Answer: 170488

The method works the best when the last two digits don’t exceed 20 (after that the multiplication become cumbersome). Another good area where this approach is great for is squaring three-digit numbers:

606² = 606 × 606

• Ones/Tens: $06 \times 06 = 36$
• Hundreds/Thousands: $06 \times 6 + 6 \times 06 = 2 \times 6 \times 6 = 72$
• Rest of Answer: $6 \times 6 = 36$

Answer: 367236

In order to use this procedure for squaring, it would be beneficial to have squares of two-digit numbers memorized. Take for example this problem:

431² = 431 × 431

• Ones/Tens: $31 \times 31 = 961$ (Write 61, carry 9)
• Hundreds/Thousands: $31 \times 4 + 4 \times 31 + 9 = 2 \times 4 \times 31 + 9 = 257$ (Write 57, carry 2)
• Rest of Answer: $4 \times 4 + 2 = 18$

Answer: 185761

If you didn’t have $31^2$ memorized, you would have to calculate it in order to do the first step in the process (very time consuming). However, if you have it memorized you would not have to do the extra steps, thus saving time.

Problem Set 1.3.1

Here are some practice problems to help with understanding FOILing three-digit numbers.