Mental math shortcuts guide
These are the methods worth learning once so you stop recalculating them from scratch. Each one applies directly to a practice mode on this site — links are included so you can drill a trick right after reading it.
Squares
Numbers ending in 5
For any number ending in 5, take the digit(s) before the 5, multiply that by itself plus one, then append 25.
Example: 65² — the leading part is 6, so compute 6 × 7 = 42, then append 25: 65² = 4225.
Numbers close to 50
For numbers near 50, use (50 ± x)² = 2500 ± 100x + x², where x is the distance from 50.
Example: 48² — x = 2 (48 is 2 below 50), so 2500 − 200 + 4 = 2304.
Numbers close to 100
Same idea, anchored at 100: (100 ± x)² = 10000 ± 200x + x².
Example: 97² — x = 3, so 10000 − 600 + 9 = 9409.
General case: break into tens and units
For anything else, split the number as (a + b) and expand using (a + b)² = a² + 2ab + b².
Example: 43² = (40 + 3)² = 1600 + 240 + 9 = 1849.
Cubes
Memorize 1³ through 10³
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 — knowing these cold makes every other cube faster to estimate or verify.
Binomial expansion for larger numbers
Split the number into tens and units, then expand using (a + b)³ = a³ + 3a²b + 3ab² + b³.
Example: 12³ = (10 + 2)³ = 1000 + 600 + 120 + 8 = 1728.
Multiplication
Multiply by 11
Add the two digits of the other number and insert the sum between them (carrying if the sum is 10 or more).
Example: 23 × 11 — 2 + 3 = 5, insert between: 253.
Multiply by 9, 99, or other "one less than a round number"
Multiply by the round number instead, then subtract the original number once.
Example: 47 × 9 = 47 × 10 − 47 = 470 − 47 = 423.
Multiply by 5, 25, or 125
These are all fractions of powers of ten: 5 = 10 ÷ 2, 25 = 100 ÷ 4, 125 = 1000 ÷ 8. Multiply by the power of ten, then divide.
Example: 24 × 25 = (24 × 100) ÷ 4 = 2400 ÷ 4 = 600.
Both numbers close to 100 (or another round base)
Write each number as 100 plus or minus a small deviation, then use (100+x)(100+y) = 10000 + 100(x+y) + xy.
Example: 103 × 107 → x=3, y=7 → 10000 + 1000 + 21 = 11021.
Same tens digit, units that add to 10
For numbers like 42 × 48 (same tens digit, units summing to 10): multiply the tens digit by itself-plus-one for the leading part, then multiply the units directly for the trailing two digits.
Example: 42 × 48 → tens digit 4, so 4 × 5 = 20; units 2 × 8 = 16 → 2016.
Vedic cross-multiplication (Urdhva Tiryagbhyam)
For two-digit numbers generally: multiply the left digits, cross-multiply and add for the middle, multiply the right digits for the end, then carry as needed.
Example: 23 × 14 → left: 2×1=2, middle: (2×4)+(3×1)=11, right: 3×4=12 → combine with carries: 322.