🔢 Arithmetic — Formulas

Essential arithmetic formulas and shortcuts for faster, more accurate calculations.

Basic Operation Formulas

Addition Properties

• Commutative: $a + b = b + a$
• Associative: $(a + b) + c = a + (b + c)$
• Identity: $a + 0 = a$
• Inverse: $a + (-a) = 0$

Example: $47 + 23 = 23 + 47 = 70$

Subtraction Properties

• Identity: $a - 0 = a$
• Inverse: $a - a = 0$
• Not commutative: $a - b \neq b - a$ (unless $a = b$)
• Not associative: $(a - b) - c \neq a - (b - c)$

Example: $47 - 23 = 24$, but $23 - 47 = -24$

Multiplication Properties

• Commutative: $a \times b = b \times a$
• Associative: $(a \times b) \times c = a \times (b \times c)$
• Distributive: $a \times (b + c) = a \times b + a \times c$
• Identity: $a \times 1 = a$
• Zero: $a \times 0 = 0$

Example: $6 \times 47 = 6 \times (40 + 7) = 240 + 42 = 282$

Division Properties

• Identity: $a \div 1 = a$
• Inverse: $a \div a = 1$ (when $a \neq 0$)
• Zero: $a \div 0$ is undefined
• Not commutative: $a \div b \neq b \div a$ (unless $a = b$)

Example: $47 \div 23 \approx 2.04$, but $23 \div 47 \approx 0.49$

Mental Math Shortcuts

Addition Shortcuts

• Adding 9: $a + 9 = a + 10 - 1$
• Adding 11: $a + 11 = a + 10 + 1$
• Adding near multiples of 10: $a + 19 = a + 20 - 1$

Examples:

• $47 + 9 = 47 + 10 - 1 = 56$
• $47 + 11 = 47 + 10 + 1 = 58$
• $47 + 19 = 47 + 20 - 1 = 66$

Subtraction Shortcuts

• Subtracting 9: $a - 9 = a - 10 + 1$
• Subtracting 11: $a - 11 = a - 10 - 1$
• Subtracting near multiples of 10: $a - 19 = a - 20 + 1$

Examples:

• $47 - 9 = 47 - 10 + 1 = 38$
• $47 - 11 = 47 - 10 - 1 = 36$
• $47 - 19 = 47 - 20 + 1 = 28$

Multiplication Shortcuts

• Multiplying by 5: $a \times 5 = a \times 10 \div 2$
• Multiplying by 25: $a \times 25 = a \times 100 \div 4$
• Multiplying by 11: $a \times 11 = a \times 10 + a$
• Multiplying by 12: $a \times 12 = a \times 10 + a \times 2$

Examples:

• $47 \times 5 = 470 \div 2 = 235$
• $47 \times 25 = 4700 \div 4 = 1175$
• $47 \times 11 = 470 + 47 = 517$
• $47 \times 12 = 470 + 94 = 564$

Division Shortcuts

• Dividing by 5: $a \div 5 = a \times 2 \div 10$
• Dividing by 25: $a \div 25 = a \times 4 \div 100$
• Dividing by 4: $a \div 4 = a \div 2 \div 2$
• Dividing by 8: $a \div 8 = a \div 2 \div 2 \div 2$

Examples:

• $47 \div 5 = 94 \div 10 = 9.4$
• $47 \div 25 = 188 \div 100 = 1.88$
• $47 \div 4 = 23.5 \div 2 = 11.75$
• $47 \div 8 = 23.5 \div 2 \div 2 = 5.875$

Powers and Roots

Perfect Squares

Squares 1-20:

• $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$
• $6^2 = 36$, $7^2 = 49$, $8^2 = 64$, $9^2 = 81$, $10^2 = 100$
• $11^2 = 121$, $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, $15^2 = 225$
• $16^2 = 256$, $17^2 = 289$, $18^2 = 324$, $19^2 = 361$, $20^2 = 400$

Squares ending in 5:

• $15^2 = 225$, $25^2 = 625$, $35^2 = 1225$, $45^2 = 2025$
• Pattern: $n5^2 = n(n+1) \times 100 + 25$

Example: $25^2 = 2 \times 3 \times 100 + 25 = 600 + 25 = 625$

Perfect Cubes

Cubes 1-10:

• $1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$
• $6^3 = 216$, $7^3 = 343$, $8^3 = 512$, $9^3 = 729$, $10^3 = 1000$

Square Root Estimation

For perfect squares: Use memorized values For other numbers: Use approximation

• $\sqrt{n} \approx \frac{a + b}{2}$ where $a^2 < n < b^2$

Example: $\sqrt{50} \approx \frac{7 + 8}{2} = 7.5$ (actual: 7.07)

Divisibility Rules

Basic Rules

• 2: Last digit is even (0, 2, 4, 6, 8)
• 3: Sum of digits is divisible by 3
• 4: Last two digits form a number divisible by 4
• 5: Last digit is 0 or 5
• 6: Divisible by both 2 and 3
• 8: Last three digits form a number divisible by 8
• 9: Sum of digits is divisible by 9
• 10: Last digit is 0
• 11: Alternating sum of digits is divisible by 11

Examples

• 2: 1,234 is even (ends in 4)
• 3: $1 + 2 + 3 + 4 = 10$, not divisible by 3
• 4: Last two digits 34, not divisible by 4
• 5: Doesn’t end in 0 or 5
• 6: Not divisible by 2 or 3
• 9: $1 + 2 + 3 + 4 = 10$, not divisible by 9
• 10: Doesn’t end in 0
• 11: $1 - 2 + 3 - 4 = -2$, not divisible by 11

Number Properties

Even and Odd Properties

• Even + Even = Even
• Odd + Odd = Even
• Even + Odd = Odd
• Even × Even = Even
• Odd × Odd = Odd
• Even × Odd = Even

Prime Properties

• 2 is the only even prime
• All other primes are odd
• 1 is neither prime nor composite
• Every composite number has at least one prime factor

GCD and LCM

• GCD(a, b): Greatest common divisor
• LCM(a, b): Least common multiple
• Relationship: $a \times b = \text{GCD}(a, b) \times \text{LCM}(a, b)$

Example: GCD(12, 18) = 6, LCM(12, 18) = 36

• $12 \times 18 = 216 = 6 \times 36$ ✓

Estimation Formulas

Front-End Estimation

Method: Use first digit of each number Example: $347 + 256 \approx 300 + 200 = 500$

Compatible Numbers

Method: Use numbers that are easy to work with Example: $47 \times 12 \approx 50 \times 10 = 500$

Benchmark Estimation

Method: Use known reference points Example: $0.47 \approx 0.5 = \frac{1}{2}$

Range Estimation

Method: Find upper and lower bounds Example: $23 \times 47$ is between $20 \times 40 = 800$ and $30 \times 50 = 1500$

Order of Operations

PEMDAS/BODMAS

1. Parentheses/Brackets: $( )$, $[ ]$, ${ }$
2. Exponents/Orders: $a^n$, $\sqrt[n]{a}$
3. Multiplication/Division: $\times$, $\div$ (left to right)
4. Addition/Subtraction: $+$, $-$ (left to right)

Examples

• $2 + 3 \times 4 = 2 + 12 = 14$
• $(2 + 3) \times 4 = 5 \times 4 = 20$
• $2 + 3^2 \times 4 = 2 + 9 \times 4 = 2 + 36 = 38$

Common Conversion Factors

Length

• 1 foot = 12 inches
• 1 yard = 3 feet
• 1 mile = 5,280 feet
• 1 meter = 100 centimeters
• 1 kilometer = 1,000 meters

Area

• 1 square foot = 144 square inches
• 1 square yard = 9 square feet
• 1 acre = 43,560 square feet

Volume

• 1 cubic foot = 1,728 cubic inches
• 1 gallon = 4 quarts
• 1 quart = 2 pints
• 1 pint = 2 cups

Time

• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 week = 7 days
• 1 year = 365 days (366 in leap year)

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