🔢 Fractions, Decimals, Percents — Formulas

Essential formulas and shortcuts for working with rational numbers in MATHCOUNTS.

Fraction Operations

Basic Operations

Addition: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ Subtraction: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$ Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Mixed Numbers

Convert to improper: $a\frac{b}{c} = \frac{ac + b}{c}$ Convert to mixed: $\frac{a}{b} = \left\lfloor \frac{a}{b} \right\rfloor + \frac{a \bmod b}{b}$

Examples:

• $2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$
• $\frac{11}{4} = 2 + \frac{3}{4} = 2\frac{3}{4}$

Simplifying Fractions

GCF method: $\frac{a}{b} = \frac{a \div \text{GCF}(a,b)}{b \div \text{GCF}(a,b)}$ Prime factorization: Cancel common prime factors

Example: Simplify $\frac{24}{36}$

• GCF(24, 36) = 12
• $\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}$

Decimal Operations

Addition and Subtraction

Method: Line up decimal points, then add/subtract Example: $3.47 + 2.15 = 5.62$

Multiplication

Method: Multiply as whole numbers, count decimal places Formula: $(a \times 10^m) \times (b \times 10^n) = ab \times 10^{m+n}$

Example: $3.4 \times 2.5 = 8.5$ (1 + 1 = 2 decimal places)

Division

Method: Move decimal point to make divisor whole number Formula: $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$

Example: $3.6 \div 0.4 = 36 \div 4 = 9$

Percent Formulas

Basic Percent Calculations

Percent of number: $a%$ of $b = \frac{a}{100} \times b$ What percent: $\frac{a}{b} \times 100%$ Percent change: $\frac{\text{new} - \text{original}}{\text{original}} \times 100%$

Examples:

• $25%$ of 80 = $0.25 \times 80 = 20$
• What percent is 15 of 60? $\frac{15}{60} \times 100% = 25%$
• Increase from 80 to 100: $\frac{100-80}{80} \times 100% = 25%$

Compound Percent

Simple interest: $I = P \times r \times t$ Compound interest: $A = P(1 + r)^t$

Where:

• $P$ = principal
• $r$ = rate (as decimal)
• $t$ = time
• $A$ = amount
• $I$ = interest

Conversion Formulas

Fraction to Decimal

Method 1: $\frac{a}{b} = a \div b$ Method 2: Use equivalent fractions with powers of 10

Examples:

• $\frac{3}{4} = 3 \div 4 = 0.75$
• $\frac{3}{4} = \frac{75}{100} = 0.75$

Decimal to Fraction

Method: $0.abc = \frac{abc}{1000}$, then simplify

Example: $0.75 = \frac{75}{100} = \frac{3}{4}$

Fraction to Percent

Method: $\frac{a}{b} = \frac{a}{b} \times 100%$

Example: $\frac{3}{4} = \frac{3}{4} \times 100% = 75%$

Percent to Fraction

Method: $a% = \frac{a}{100}$, then simplify

Example: $75% = \frac{75}{100} = \frac{3}{4}$

Decimal to Percent

Method: $0.abc = 0.abc \times 100% = abc%$

Example: $0.75 = 0.75 \times 100% = 75%$

Percent to Decimal

Method: $a% = a \div 100$

Example: $75% = 75 \div 100 = 0.75$

Common Fraction-Decimal-Percent Equivalents

Mental Math Shortcuts

Percent Shortcuts

10% of $n$: $n \div 10$ 5% of $n$: $10%$ of $n \div 2$ 1% of $n$: $n \div 100$ 20% of $n$: $10%$ of $n \times 2$ 25% of $n$: $n \div 4$ 50% of $n$: $n \div 2$ 75% of $n$: $n \div 4 \times 3$

Examples:

• 10% of 80 = 8
• 5% of 80 = 4
• 20% of 80 = 16
• 25% of 80 = 20

Fraction Shortcuts

Half: $\frac{1}{2}$ of $n = n \div 2$ Quarter: $\frac{1}{4}$ of $n = n \div 4$ Third: $\frac{1}{3}$ of $n = n \div 3$ Fifth: $\frac{1}{5}$ of $n = n \div 5$

Examples:

• $\frac{1}{2}$ of 60 = 30
• $\frac{1}{4}$ of 60 = 15
• $\frac{1}{3}$ of 60 = 20
• $\frac{1}{5}$ of 60 = 12

Word Problem Formulas

Part-Whole Problems

Find part: Part = Percent × Whole Find whole: Whole = Part ÷ Percent Find percent: Percent = Part ÷ Whole × 100%

Examples:

• 25% of 80 = 0.25 × 80 = 20
• 20 is what percent of 80? 20 ÷ 80 × 100% = 25%
• 20 is 25% of what number? 20 ÷ 0.25 = 80

Mixture Problems

Weighted average: $\frac{a_1 \times p_1 + a_2 \times p_2 + … + a_n \times p_n}{a_1 + a_2 + … + a_n}$

Example: Mix 2 liters of 20% solution with 3 liters of 80% solution

• $\frac{2 \times 0.20 + 3 \times 0.80}{2 + 3} = \frac{0.4 + 2.4}{5} = \frac{2.8}{5} = 0.56 = 56%$

Rate Problems

Distance: $d = rt$ Rate: $r = \frac{d}{t}$ Time: $t = \frac{d}{r}$

With percents: $d = r(1 + p%)t$ where $p%$ is percent change

Scientific Notation

Basic Format

Standard form: $a \times 10^n$ where $1 \leq a < 10$

Examples:

• $3,400 = 3.4 \times 10^3$
• $0.0007 = 7 \times 10^{-4}$

Operations

Multiplication: $(a \times 10^m) \times (b \times 10^n) = ab \times 10^{m+n}$ Division: $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$ Addition/Subtraction: Convert to same power of 10

Examples:

• $(2 \times 10^3) \times (3 \times 10^4) = 6 \times 10^7$
• $\frac{6 \times 10^8}{2 \times 10^3} = 3 \times 10^5$

Rounding Rules

Standard Rounding

0-4: Round down 5-9: Round up Ties: Round to even (banker’s rounding)

Significant Figures

Count digits from left, starting with first non-zero digit Round to specified number of significant figures

Examples:

• 3.14159 to 3 sig figs = 3.14
• 0.0001234 to 2 sig figs = 0.00012

Previous: Problem Types
Back to: Fractions, Decimals, Percents