Addition & Subtraction
Left-to-Right Addition
add
Add the largest place value first, carry forward.
1
Start with the hundreds (or largest digit).
2
Add tens, running total in your head.
3
Add ones last.
Example — 348 + 276
300+200 = 500500+40+70 = 610
610+8+6 = 624
Complement to 10/100
add
Find what each number needs to reach the next round number.
1
Round one number up to a nice multiple of 10 or 100.
2
Subtract the same amount from the other number.
3
Add the adjusted pair.
Example — 67 + 38
67 + 38 → 67 + 40 − 2 = 105
Example — 293 + 148
300 + 148 − 7 = 441
Subtraction by Addition
subtract
Count up from the smaller number instead of subtracting down.
1
Start at the smaller number.
2
Count up to the next round number, note the jump.
3
Continue to the larger number; sum all jumps.
Example — 1003 − 867
867 → 900: +33900 → 1000: +100
1000 → 1003: +3
Total: 136
Double & Halve Trick
add
Double one number, halve the other to get an easier sum.
1
Double the smaller, halve the larger (for even numbers).
2
If the result is simpler, use it directly.
Example — 16 + 34
16→32, 34→17 → 32+17 = 49✓(same as 16+34=50? No — better: 16+34 directly = 50)
Best for × not +. See Multiplication section.
Multiplication
Multiply by 5
×
÷2 then ×10 is always faster.
n × 5 = (n ÷ 2) × 10
Examples
48 × 5 = 24 × 10 = 24073 × 5 = 36.5 × 10 = 365
Multiply by 9
×
×10 then subtract the number once.
n × 9 = (n × 10) − n
Examples
47 × 9 = 470 − 47 = 42383 × 9 = 830 − 83 = 747
Multiply by 11
×
For 2-digit numbers: outer digits stay, middle is their sum.
1
Write the first digit, then the sum of both digits, then the last digit.
2
If the middle sum ≥ 10, carry 1 to the first digit.
Examples
53 × 11 = 5_(5+3)_3 = 58378 × 11 = 7_(7+8)_8 = 7_15_8 → 858
Multiply by 25
×
÷4 then ×100 (25 = 100/4).
n × 25 = (n ÷ 4) × 100
Examples
36 × 25 = 9 × 100 = 90052 × 25 = 13 × 100 = 1300
Double & Halve
×
Keep halving one factor and doubling the other until easy.
1
Halve the trickier number (if even).
2
Double the other number.
3
Repeat until one factor is 1 or trivially easy.
Example — 32 × 15
32×15 → 16×30 → 8×60 → 4×120 → 2×240 → 1×480
FOIL for 2-Digit × 2-Digit
×
Split both numbers and use the distributive property.
1
Split: (a+b)(c+d) = ac + ad + bc + bd.
2
Choose splits to give round numbers.
Example — 37 × 43
= (40−3)(40+3)= 40² − 3² = 1600 − 9 = 1591
Example — 23 × 47
= 23×(50−3) = 1150 − 69 = 1081
Squaring Numbers Ending in 5
×
Multiply the leading digit by the next integer, append 25.
(n5)² = n×(n+1) | 25
Examples
35² = 3×4 | 25 = 122575² = 7×8 | 25 = 5625
95² = 9×10 | 25 = 9025
Near-Square Shortcut
×
a² − b² = (a+b)(a−b) — works backwards too.
x × y = ((x+y)/2)² − ((x−y)/2)²
Example — 18 × 22
avg = 20, diff = 2= 20² − 2² = 400 − 4 = 396
Example — 31 × 29
= 30² − 1² = 900 − 1 = 899
Division
Divisibility Rules
÷
Instantly know if a number divides evenly.
| Divisor | Rule |
|---|---|
| 2 | Last digit is even |
| 3 | Digit sum divisible by 3 |
| 4 | Last 2 digits divisible by 4 |
| 5 | Ends in 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 7 | Double last digit, subtract from rest; repeat |
| 8 | Last 3 digits divisible by 8 |
| 9 | Digit sum divisible by 9 |
| 11 | Alternating digit sum divisible by 11 |
Divide by 5
÷
Multiply by 2, then divide by 10.
n ÷ 5 = (n × 2) ÷ 10
Examples
340 ÷ 5 = 680 ÷ 10 = 6885 ÷ 5 = 170 ÷ 10 = 17
Divide by 25
÷
Multiply by 4, then divide by 100.
n ÷ 25 = (n × 4) ÷ 100
Examples
375 ÷ 25 = 1500 ÷ 100 = 15225 ÷ 25 = 900 ÷ 100 = 9
Long Division Shortcut
÷
Break the dividend into parts that divide cleanly.
1
Find the largest multiple of the divisor ≤ dividend.
2
Subtract, then divide the remainder.
Example — 672 ÷ 8
640 ÷ 8 = 8032 ÷ 8 = 4
Total: 84
Estimation
Round & Adjust
est
Round to the nearest easy number, then correct the error.
1
Round each number to the nearest 10, 100, or power of 2.
2
Calculate with the rounded numbers.
3
Add or subtract the rounding error.
Example — 198 × 52
≈ 200 × 52 = 10400Error: −2 × 52 = −104
Result: 10296
Percent Tricks
est
Flip the percent and number for easier calculation.
1
x% of y = y% of x — pick the easier direction.
2
Build from 10%: find 10%, then scale.
Shortcuts
10% of n = n ÷ 105% = half of 10%
1% = n ÷ 100
15% = 10% + 5%
20% = 10% × 2
25% = n ÷ 4
33% ≈ n ÷ 3
75% = n − (n÷4)
Example — 8% of 75
= 75% of 8 = 6 → 6
Order of Magnitude
est
Estimate by powers of 10 — get the scale right first.
1
Count digits or express as a × 10ⁿ.
2
Multiply the coefficients, add the exponents.
3
Refine if needed.
Example — 3200 × 4700
3.2×10³ × 4.7×10³≈ 3×5 × 10⁶ = 15,000,000
(exact: 15,040,000)
Fractions & Decimals
Fraction ↔ Decimal Reference
frac
Core conversions to memorize.
| Fraction | Decimal | % |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 2/3 | 0.666… | 66.7% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1666… | 16.7% |
| 1/7 | 0.142857… | 14.3% |
| 1/8 | 0.125 | 12.5% |
| 1/9 | 0.111… | 11.1% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.0833… | 8.3% |
Cross-Multiply to Compare
frac
Which fraction is bigger without finding common denominator.
a/b vs c/d → compare a×d vs b×c
Example — 3/7 vs 5/11
3×11 = 337×5 = 35
33 < 35 → 3/7 < 5/11
Simplify Before Multiplying
frac
Cancel common factors across numerators and denominators first.
1
Look for any numerator and any denominator sharing a factor.
2
Cancel (cross-cancel) before multiplying.
Example — (14/9) × (3/7)
14 and 7 share 7 → 2/9 × 3/13 and 9 share 3 → 2/3 × 1/1 = 2/3
Powers & Roots
Squares 1–30
pow
Essential squares to memorize for fast mental math.
| n | n² |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| n | n² |
|---|---|
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
| 21 | 441 |
| 22 | 484 |
| 23 | 529 |
| 24 | 576 |
| 25 | 625 |
| 26 | 676 |
| 27 | 729 |
| 28 | 784 |
| 29 | 841 |
| 30 | 900 |
Powers of 2
pow
Critical for CS, binary, and bit manipulation.
| 2ⁿ | Value |
|---|---|
| 2⁰ | 1 |
| 2¹ | 2 |
| 2² | 4 |
| 2³ | 8 |
| 2⁴ | 16 |
| 2⁵ | 32 |
| 2⁶ | 64 |
| 2⁷ | 128 |
| 2⁸ | 256 |
| 2⁹ | 512 |
| 2¹⁰ | 1,024 ≈ 1K |
| 2²⁰ | 1,048,576 ≈ 1M |
| 2³⁰ | ≈ 1B |
Estimate Square Roots
pow
Find the root between two known perfect squares, then interpolate.
1
Find the two perfect squares that bracket the number.
2
Linear interpolate: root ≈ lower + (n − lower²) / (upper² − lower²).
Example — √50
7² = 49, 8² = 6450 − 49 = 1, range = 15
√50 ≈ 7 + 1/15 ≈ 7.07
Quick Reference
Multiplication Table 1–12 (rows × columns)