Mental Math – Complete Reference Guide

Multiplying by 11

Two-Digit Numbers

Split digits, add middle: $ab \times 11$

$32 \times 11 = 3(3+2)2 = 352$

$47 \times 11 = 4(11)7 = 517$ (carry the 1)

$63 \times 11 = 6(9)3 = 693$

Three-Digit Numbers

$abc \times 11 = a(a+b)(b+c)c$

$234 \times 11 = 2(5)(7)4 = 2574$

Carry when sums exceed 9

Pattern: Each inner digit = sum of neighbors

Multiply by 5, 25, 50

Multiply by 5

$n \times 5 = \frac{n \times 10}{2}$

$48 \times 5 = 480 \div 2 = 240$

$137 \times 5 = 1370 \div 2 = 685$

Multiply by 25

$n \times 25 = \frac{n \times 100}{4}$

$48 \times 25 = 4800 \div 4 = 1200$

$36 \times 25 = 3600 \div 4 = 900$

Multiply by 50

$n \times 50 = \frac{n \times 100}{2}$

$68 \times 50 = 6800 \div 2 = 3400$

Key: Use division by 2 or 4 after scaling up

Numbers Near 100

Both Below 100

$98 \times 96$: Deficits: $-2, -4$

Left: $98-4 = 94$ (or $96-2$)

Right: $(-2) \times (-4) = 08$

Result: $9408$

Both Above 100

$103 \times 107$: Excesses: $+3, +7$

Left: $103+7 = 110$

Right: $3 \times 7 = 21$

Result: $11021$

One Above, One Below

$97 \times 104$: $(-3)(+4)$

Left: $97+4 = 101$; Right: $-12$

Result: $10100 - 12 = 10088$

Vedic Multiplication

Vertically & Crosswise

For $ab \times cd$:

Units: $b \times d$

Middle: $a \times d + b \times c$

Hundreds: $a \times c$

Example: 23 × 41

Units: $3 \times 1 = 3$

Middle: $2 \times 1 + 3 \times 4 = 14$

Hundreds: $2 \times 4 = 8$

Result: $8|14|3 = 943$

Practice: Works for any 2-digit × 2-digit

Squaring Numbers Ending in 5

The Rule

$(n5)^2 = n(n+1) | 25$

First part: $n \times (n+1)$

Last part: always 25

Examples

$25^2: 2 \times 3 = 6 \to 625$

$35^2: 3 \times 4 = 12 \to 1225$

$65^2: 6 \times 7 = 42 \to 4225$

$85^2: 8 \times 9 = 72 \to 7225$

$115^2: 11 \times 12 = 132 \to 13225$

Why: $(10n+5)^2 = 100n(n+1)+25$

Squaring Near 50 & 100

Near 50

$(50+d)^2 = (25+d)|d^2$

$53^2: (25+3)|(09) = 2809$

$48^2: (25-2)|04 = 2304$

$56^2: (25+6)|36 = 3136$

Near 100

$(100+d)^2 = (100+2d)|d^2$

$103^2: (106)|09 = 10609$

$97^2: (94)|09 = 9409$

$108^2: (116)|64 = 11664$

Pad: Right side must be 2 digits (or 4 for 1000)

Difference of Squares

Formula

$a^2 - b^2 = (a+b)(a-b)$

For Any Square

$n^2 = (n+d)(n-d) + d^2$

Choose $d$ to make calculation easy

Example: 47²

$47^2 = (47+3)(47-3) + 9$

$= 50 \times 44 + 9$

$= 2200 + 9 = 2209$

Example: 67²

$67^2 = 70 \times 64 + 9 = 4489$

Strategy: Round to nearest 10, compute, add $d^2$

Duplex Method for Squares

Duplex Rules

Single digit $a$: $D(a) = a^2$

Two digits $ab$: $D(ab) = 2ab$

Three $abc$: $D = 2ac + b^2$

To Square abc

Units: $D(c) = c^2$

Tens: $D(bc) = 2bc$

Hundreds: $D(abc) = 2ac + b^2$

Thousands: $D(ab) = 2ab$

Ten-thousands: $D(a) = a^2$

Example: 123²

$9 | 6 | (6+4) | 4 | 1 = 15129$

Divisibility Rules

Quick Tests

2: Last digit even

3: Digit sum divisible by 3

4: Last 2 digits ÷ 4

5: Ends in 0 or 5

6: Divisible by 2 AND 3

8: Last 3 digits ÷ 8

9: Digit sum divisible by 9

10: Ends in 0

Tricky Ones

7: Double last, subtract from rest

11: Alternating sum = 0 or ±11

Ex: $1364 \div 11$: $1-3+6-4 = 0$ ✓

Dividing by 5, 25, 50

Divide by 5

$n \div 5 = n \times 2 \div 10$

$235 \div 5 = 470 \div 10 = 47$

$845 \div 5 = 1690 \div 10 = 169$

Divide by 25

$n \div 25 = n \times 4 \div 100$

$650 \div 25 = 2600 \div 100 = 26$

Divide by 50

$n \div 50 = n \times 2 \div 100$

$350 \div 50 = 700 \div 100 = 7$

Divide by 125

$n \div 125 = n \times 8 \div 1000$

Key: Multiply then shift decimal

Percentages

Key Equivalents

$50\% = \frac{1}{2}$, $25\% = \frac{1}{4}$, $20\% = \frac{1}{5}$

$10\% = \frac{1}{10}$, $5\% = \frac{1}{20}$, $1\% = \frac{1}{100}$

$33.\overline{3}\% = \frac{1}{3}$, $12.5\% = \frac{1}{8}$

Breakdown Method

15% of 80:

$10\%$ of $80 = 8$

$5\%$ of $80 = 4$

$15\%$ of $80 = 12$

Flip Trick

$x\%$ of $y = y\%$ of $x$

$8\%$ of $50 = 50\%$ of $8 = 4$

Always: Find 10% first, then scale

Percent Increase/Decrease

Multipliers

Increase 20%: multiply by 1.2

Decrease 15%: multiply by 0.85

Increase $x\%$: multiply by $(1 + \frac{x}{100})$

Quick Calculations

$\$80$ + 15% = $80 \times 1.15$

$= 80 + 8 + 4 = \$92$

Finding Original

After 20% off, price is $\$64$

Original = $64 \div 0.8 = \$80$

Successive Changes

Up 10% then down 10% ≠ same!

$1.1 \times 0.9 = 0.99$ (1% loss)

Percent changes don't simply add/subtract!

Estimating Square Roots

Linear Approximation

$$\sqrt{n} \approx s + \frac{n - s^2}{2s}$$

$s$ = nearest perfect square root

Example: √27

Nearest: $\sqrt{25} = 5$

$\sqrt{27} \approx 5 + \frac{2}{10} = 5.2$

Actual: $5.196...$

Example: √50

$\sqrt{49} = 7$

$\sqrt{50} \approx 7 + \frac{1}{14} \approx 7.07$

Perfect Squares to Know

$1, 4, 9, 16, 25, 36, 49, 64, 81, 100$

$121, 144, 169, 196, 225, 256...$

Better: $\sqrt{n} \approx s + \frac{n-s^2}{2s+1}$

Cube Roots & Powers

Perfect Cubes

$1, 8, 27, 64, 125, 216, 343, 512, 729$

Last Digit Pattern

$n^3$ ends in same digit as $n$ for:

$1, 4, 5, 6, 9, 0$

$2 \leftrightarrow 8$, $3 \leftrightarrow 7$ swap

Estimating ∛n

$$\sqrt[3]{n} \approx c + \frac{n - c^3}{3c^2}$$

$\sqrt[3]{30} \approx 3 + \frac{3}{27} \approx 3.11$

Powers of 2

$2, 4, 8, 16, 32, 64, 128, 256, 512, 1024$

Last digit of $\sqrt[3]{n}$ tells you last digit of answer!

Fraction ↔ Decimal

Memorize These

$\frac{1}{2} = 0.5$, $\frac{1}{4} = 0.25$, $\frac{3}{4} = 0.75$

$\frac{1}{5} = 0.2$, $\frac{2}{5} = 0.4$, $\frac{3}{5} = 0.6$

$\frac{1}{8} = 0.125$, $\frac{3}{8} = 0.375$

$\frac{1}{3} = 0.\overline{3}$, $\frac{2}{3} = 0.\overline{6}$

$\frac{1}{6} = 0.1\overline{6}$, $\frac{5}{6} = 0.8\overline{3}$

$\frac{1}{7} = 0.\overline{142857}$

$\frac{1}{9} = 0.\overline{1}$, $\frac{1}{11} = 0.\overline{09}$

Quick Division

$\frac{a}{b}$: Find equivalent with denominator 10, 100...

$\frac{7}{25} = \frac{28}{100} = 0.28$

$\frac{1}{7}$ pattern: 142857 cycles (memorize!)

General Strategies

Left to Right

Add/subtract big parts first

$347 + 286 = 500 + 130 + 3 = 633$

Compensation

$299 + 47 = 300 + 46 = 346$

$98 \times 7 = 100 \times 7 - 14 = 686$

Factoring

$36 \times 25 = 9 \times 4 \times 25 = 9 \times 100$

Anchor Numbers

Use 10, 50, 100 as reference points

$17 + 38 = (17+3) + 35 = 55$

Practice: Speed comes from pattern recognition!

Multiply by 9 & 99

Multiply by 9

$n \times 9 = n \times 10 - n$

$47 \times 9 = 470 - 47 = 423$

Multiply by 99

$n \times 99 = n \times 100 - n$

$34 \times 99 = 3400 - 34 = 3366$

Multiply by 101

$n \times 101 = n \times 100 + n$

$34 \times 101 = 3400 + 34 = 3434$

Pattern: $(10^k - 1)$ and $(10^k + 1)$ are easy!

Same Tens, Units Sum to 10

The Rule

When tens digit same, units sum to 10:

$a \times (a+1)$ | product of units

Examples

$23 \times 27$: $2 \times 3 = 6$, $3 \times 7 = 21$

Result: $621$

$34 \times 36$: $3 \times 4 = 12$, $4 \times 6 = 24$

Result: $1224$

$71 \times 79$: $7 \times 8 = 56$, $1 \times 9 = 09$

Result: $5609$

Why: $(10a+b)(10a+(10-b)) = 100a(a+1)+b(10-b)$

Quick Estimation

Rounding Strategy

Round to 1 significant figure

$487 \times 23 \approx 500 \times 20 = 10000$

Actual: $11201$

Balancing Errors

If rounding one up, round other down

$48 \times 52 \approx 50 \times 50 = 2500$

Actual: $2496$

Order of Magnitude

$347 \times 891 \approx 300 \times 900 = 270000$

Ballpark: between 200k and 400k

Goal: Quick sanity check, not exact answer

Common Calculations

Tip Calculation

15% tip: 10% + half of that

$\$47$ → $\$4.70 + \$2.35 = \$7.05$

20% tip: double 10%

Time Calculations

Minutes to hours: divide by 60

135 min = 2h 15m

Unit Conversions

km to mi: multiply by 0.6

kg to lb: multiply by 2.2

°C to °F: $\times 1.8 + 32$

Build mental reference points for everyday math!

MENTAL MATH Complete Reference