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Fractions

A fraction $\frac{a}{b}$ means $a$ divided by $b$ . For addition and subtraction the fractions need a common denominator, for multiplication we multiply numerator by numerator and denominator by denominator, and for division we multiply by the reciprocal.

\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}

addition (common denominator)

\frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd}

multiplication

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b}\cdot\frac{d}{c}

division: multiply by the reciprocal

Symbols

$a,\ c$	numerators
$b,\ d$	denominators

Example

$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .

$\frac{2}{3}\cdot\frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ .

Never add the denominators directly, as in

\frac12+\frac13 \neq \frac25

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Part of Foundations of Mathematics: Numbers, fractions and percent.