Math - Fractions II

Addition & Subtraction (same denominator)

$\frac{a}{b} \pm \frac{c}{b} = \frac{a \pm c}{b}$

When adding/subtracting fractions with the same denominator, add/subtract the numerators.

$\frac{4}{3} + \frac{7}{3} = \frac{11}{3}$
$\frac{-12}{4} - \frac{-15}{4} = \frac{-27}{4}$

$\frac{a}{b} \pm \frac{c}{d} = \frac{(a*d)\pm(c*b)}{b*d}$

When adding/subtracting fractions with different denominators, find a common denominator. Then add/subtract the numerators

$\frac{5}{2} + \frac{7}{9} = \frac{(5*9) + (7*2)}{9*2} = \frac{59}{18}$
$\frac{9}{4} - \frac{7}{6} = \frac{(9*6) - (7*4)}{4*6} = \frac{26}{24}$

The LCD (or LCM of the denominators) finds the smallest common denominator, making calculations easier.

$\frac{10}{9} + \frac{9}{12}\\ \frac{10*4}{9*4} + \frac{9*3}{12*3}\\ \frac{40}{36} + \frac{27}{36}\\ \frac{40 + 27}{36}\\ \frac{67}{36}$

LCD(9,12) = $36$

$\frac{a}{b} * \frac{c}{d} = \frac{a * b}{c * d}$

When multiplying fractions, multiply numerators and denominators.

$\frac{5}{3} * \frac{7}{2} = \frac{35}{6}$
$\frac{3}{4} * \frac{9}{2} = \frac{27}{8}$

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} * \frac{d}{c}$

When dividing fractions, invert the divisor and multiply.

$\frac{6}{3} \div \frac{5}{3} = \frac{6}{3} * \frac{3}{5} = \frac{18}{15}$
$\frac{7}{8} \div \frac{4}{7} = \frac{7}{8} * \frac{7}{4} = \frac{49}{32}$

$\frac{ac}{bc} = \frac{a\cancel{c}}{b\cancel{c}} = \frac{a}{b}$

Cancel numbers that are common factors in numerator and denominator

$\frac{10 * 5}{2 * 5} = \frac{10 * \cancel{5}}{2 * \cancel{5}} = \frac{10}{2}$
$\frac{7 * 3}{3} = \frac{7 * \cancel{3}}{\cancel{3}} = 7$