Multiply two numbers near 100 in 5 seconds

The method in 3 moves

For two numbers $a$ and $b$ just below 100:

1. Write down what each is missing from 100: $\alpha = 100 - a$ and $\beta = 100 - b$.
2. Multiply $\alpha \times \beta$ → these are the last two digits.
3. Compute $a - \beta$ (or equivalently $b - \alpha$) → these are the first two digits.

Stick the two pieces together — done.

A magical example: $96 \times 97$

• Missing from 96: $\alpha = 4$. Missing from 97: $\beta = 3$.
• $\alpha \times \beta = 4 \times 3 = 12$ → last two digits: 12.
• $96 - 3 = 93$ (or $97 - 4 = 93$) → first two digits: 93.
• Result: 9312.

Check: $96 \times 97 = 9312$ ✓

Another: $92 \times 88$

• $\alpha = 8$, $\beta = 12$.
• $8 \times 12 = 96$ → last two digits: 96.
• $92 - 12 = 80$ → first two digits: 80.
• Result: 8096. ✓

When $\alpha \times \beta$ exceeds 99

Take $85 \times 88$:

• $\alpha = 15$, $\beta = 12$.
• $15 \times 12 = 180$ → write 80 and carry 1.
• $85 - 12 = 73$, then $+1$ carried = 74.
• Result: 7480. ✓

Why it works

$a \times b = (100 - \alpha)(100 - \beta) = 10000 - 100(\alpha + \beta) + \alpha\beta$

$= 100 \times \bigl(100 - (\alpha + \beta)\bigr) + \alpha\beta = 100 \times (a - \beta) + \alpha\beta$

The 100 factor pushes the first part two notches left, and $\alpha\beta$ tucks in on the right.

?Your turn

What is 94 × 98?

Check

?Your turn

No calculator: 89 × 91 = ?

A.8009B.8099C.8091D.8199

Check