Arithmetic

Arithmetic Formula Reference

Percentages

OperationFormula
Percent of a numberResult=Number×%100\text{Result} = \text{Number} \times \frac{\%}{100}
What percent%=PartWhole×100\% = \frac{\text{Part}}{\text{Whole}} \times 100
Percent change% Change=New−OldOld×100\% \text{ Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100
Percent to decimalDivide by 100: 45%=0.4545\% = 0.45
Decimal to percentMultiply by 100: 0.72=72%0.72 = 72\%

Fractions

OperationFormula
Addition (unlike)ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
Subtraction (unlike)ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}
Multiplicationab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
Divisionab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
SimplifyDivide numerator and denominator by GCD
Mixed to improperabc=ac+bca\frac{b}{c} = \frac{ac + b}{c}

Common Fraction/Decimal/Percent Equivalents

FractionDecimalPercent
12\frac{1}{2}0.550%
13\frac{1}{3}0.333…33.3%
14\frac{1}{4}0.2525%
15\frac{1}{5}0.220%
18\frac{1}{8}0.12512.5%
110\frac{1}{10}0.110%
23\frac{2}{3}0.667…66.7%
34\frac{3}{4}0.7575%
38\frac{3}{8}0.37537.5%

Ratios and Proportions

ConceptFormula
Ratioa:b=aba : b = \frac{a}{b}
Proportionab=cd\frac{a}{b} = \frac{c}{d}
Cross multiplyad=bcad = bc
Scale factorScale=NewOriginal\text{Scale} = \frac{\text{New}}{\text{Original}}

Order of Operations (PEMDAS)

  1. Parentheses (innermost first)
  2. Exponents
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)

Rounding Rules

  • If the digit to the right is 0–4: round down
  • If the digit to the right is 5–9: round up