Mental Math Tricks Explained: The Algebra Behind Squaring, ×11 and Near-100 Shortcuts
Most "mental math tricks" pages hand you a rule — multiply the tens digit by the next one up, stick 25 on the end — and leave you to memorize it like a magic spell. Forget one digit of the rule under exam pressure and you have nothing to fall back on. But nearly every shortcut for squaring, multiplying near a round number, or multiplying by 11 comes from exactly two pieces of algebra: the distributive law and the difference of squares. Once you see the proof, you can rebuild any of these tricks from scratch instead of hoping you remembered it correctly.
The two identities behind almost every trick
Every shortcut below is one of these two facts, applied to a specific shape of number:
- The distributive law (FOIL): (a + b)(c + d) = ac + ad + bc + bd. This is just how ordinary long multiplication works, split into pieces small enough to do in your head.
- The difference of squares: (m − d)(m + d) = m² − d². Any pair of numbers that sit the same distance from a convenient middle value collapses to one square minus another.
Everything else — squaring, the ×11 rule, multiplying numbers near 100 — is one of these two identities wearing a different number's clothes.
Squaring numbers ending in 5
Write a two-digit number ending in 5 as 10n + 5, where n is the tens digit. Squaring it with the distributive law:
(10n + 5)² = 100n² + 100n + 25 = 100n(n + 1) + 25
That's the whole trick: multiply the tens digit by the next integer up, then attach 25. For 65², n = 6, so 6 × 7 = 42, giving 4,225. The proof also shows why the trick scales to bigger numbers: 205² treats n as 20, so 20 × 21 = 420, giving 42,025.
Numbers the same distance from a round middle (difference of squares)
When two numbers sit equally far above and below some convenient value m, write them as (m − d) and (m + d). Their product is m² − d² — one square minus a much smaller one. For 47 × 53, the midpoint is 50 and the distance is 3: 50² − 3² = 2,500 − 9 = 2,491. This only works when both factors are equidistant from the same middle number; if they aren't, this specific shortcut doesn't apply and you need the general method below.
Multiplying numbers close to a round base (the complements method)
For two numbers both a little under a base like 100, write them as (100 − a) and (100 − b). Expanding with the distributive law:
(100 − a)(100 − b) = 100(100 − a − b) + ab
So: subtract each number's "deficiency" from the base to get the leading digits, then multiply the two deficiencies together for the trailing two digits (carrying if that product runs over 99). For 97 × 96, the deficiencies are 3 and 4: leading part is 100 − 3 − 4 = 93, trailing part is 3 × 4 = 12, giving 9,312. The same identity works above a base too — useful for numbers just over 100, where both "deficiencies" are negative and you add instead of subtract.
The ×11 shortcut
For a two-digit number with digits a and b (value 10a + b), multiplying by 11 distributes as 11(10a + b) = 100a + 10(a + b) + b — in other words, drop the sum of the digits between them. For 32 × 11: digits 3 and 2 sum to 5, giving 352. When the digit sum is 10 or more you carry: for 68 × 11, 6 + 8 = 14, so you write down 4 and carry the 1 into the leading digit, giving 6+1=7, then 4, then 8: 748.
The ×9 shortcut
Multiplying by 9 is multiplying by (10 − 1), which distributes as 10x − x: multiply by 10, then subtract the original number once. For 47 × 9: 470 − 47 = 423. The same idea extends to ×99 (×100 − x once) or ×8 (×10 − x twice).
Percentages from 10% and 5% building blocks
Finding 10% of a number just moves the decimal point one place; 5% is half of that. Any "nice" percentage can be built from sums of these two blocks instead of long multiplication. For 35% of 840: 10% is 84, so 30% is 84 × 3 = 252, and 5% is half of 84 = 42. Add them: 252 + 42 = 294.
Why the digit-sum divisibility rules work
The rules for 3, 9, and 11 all come from how powers of 10 behave in modular arithmetic, not from coincidence. Because 10 ≡ 1 (mod 9), every power of 10 is also ≡ 1 (mod 9), so a number is congruent to the plain sum of its digits mod 9 — which is why a digit sum divisible by 9 (or by 3, since 3 divides 9) means the original number is too. For 11, the relevant fact is 10 ≡ −1 (mod 11), so powers of 10 alternate between 1 and −1, and a number is congruent to the alternating sum of its digits mod 11. That's why 594 (9 − 5 + 4, read right to left, giving an alternating sum of 5 − 9 + 4 = 0) is divisible by 11, while a plain digit sum wouldn't tell you that.
Which trick fits which numbers
| Situation | Trick | Core identity |
|---|---|---|
| Squaring a number ending in 5 | tens digit × (tens digit + 1), then attach 25 | Distributive law |
| Two numbers equally spaced around a round middle | middle² − distance² | Difference of squares |
| Two numbers both near the same base (e.g. 100) | subtract deficiencies for the lead, multiply them for the tail | Distributive law |
| Multiplying a two-digit number by 11 | insert the digit sum between the digits, carry if ≥10 | Distributive law |
| Multiplying by 9 | ×10 minus the original number once | Distributive law |
| Awkward percentage of a number | build it from 10% and 5% blocks | Linearity of multiplication |
| Checking divisibility by 3 or 9 | sum the digits | 10 ≡ 1 (mod 9) |
| Checking divisibility by 11 | alternate-sum the digits | 10 ≡ −1 (mod 11) |
Recognizing which situation you're in — not memorizing eight separate rules — is the actual skill. Once a number's shape (ends in 5, sits near 100, is being multiplied by 11) matches a row in the table above, the identity behind it tells you what to do even if you've forgotten the exact wording of the "trick." Try applying these to real problems with PassDrill's mental math practice questions, which cover these same shortcuts plus the casting-out-nines check and complements-from-100 method.
Source: identities are elementary algebra (distributive law, difference of squares) verifiable in any algebra reference; divisibility-rule proofs cross-checked against Art of Problem Solving Wiki, "Divisibility rules/Rule for 11 proof" (artofproblemsolving.com); near-100 multiplication method cross-checked against VedicMaths.org's base-multiplication chapter.