How to multiply numbers close to 100
97 × 103 looks like a calculator problem. It isn't — both numbers are close to a round, easy base, and that's exactly the situation this shortcut is built for.
The rule
Write each number as 100 plus or minus a small deviation. Then:
In practice: adjust one number by the other's deviation to get the leading part, then multiply the two deviations for the trailing two digits.
Why it actually works
This is just the distributive law applied to a convenient base. If a = 100 + x and b = 100 + y, expanding (100+x)(100+y) gives 10000 + 100x + 100y + xy — which regroups into 10000 + 100(x+y) + xy. The 10000 and the 100(x+y) term combine into one leading number once you cross-add, and xy is small enough to just tack onto the end.
Worked examples
97 × 96 — deviations: −3 and −4. Cross-add: 97 − 4 = 93. Tail: (−3)×(−4) = 12. Combine → 9312.
103 × 104 — deviations: +3 and +4. Cross-add: 103 + 4 = 107. Tail: 3×4 = 12. Combine → 10712.
98 × 105 — deviations: −2 and +5. Cross-add: 98 + 5 = 103. Tail: (−2)×5 = −10. Combine: 103 hundreds minus 10 → 10290.
That last example is the one case worth slowing down for: when the deviations have opposite signs, the tail can be negative, which means subtracting instead of just appending. It's still fast — just don't on autopilot glue the numbers together without checking the sign first.
Try it interactively
Pick a pair and watch the deviations, cross-add, and tail get computed step by step.
When this shortcut is worth reaching for
It's fastest when both numbers are within roughly 10–15 of 100. Much further out, the deviations get large enough that squaring or multiplying them isn't obviously faster than just doing the multiplication directly — at that point it's fine to fall back to a different method, like breaking one number into tens and units.
Practice it for real
Read through a few examples until the pattern clicks, then drill it under time pressure with random pairs — that's the point where it stops being a rule you're consciously applying and starts being instant recall.
Practice arithmetic → · See the rest of the shortcuts guide →