Question

Divide the sum of -13/5 and 12/7 by the product of -31/7 and -1/2

Answer 1

Let's compute the two factors separately: as for "the sum of -13/5 and 12/7", we have to rewrite the two fractions so that they have the same denominator. Since the least common multiple of 5 and 7 is 35, we will change both fractions so that they have denominator 35:

$\frac{-13}{5} = \frac{-13}{5}\frac{7}{7} = \frac{-91}{35} \qquad \qquad \frac{12}{7} = \frac{12}{7}\frac{5}{5} = \frac{60}{35}$

So now we can sum them:

$\frac{-13}{5}+\frac{12}{7} = \frac{-91}{35}+\frac{60}{35} = \frac{-91+60}{35} = \frac{-31}{35}$

Multiplication is easier, since you just have to multiply numerators and denominators with each others:

$\frac{-31}{7} \frac{-1}{2} = \frac{-31 \cdot (-1)}{7\cdot 2} = \frac{31}{14}$

Finally, we must divide the two fractions. Dividing by a fraction is the same this as multiplying by the inverse of that fraction, i.e. we have to switch numerator and denominator:

$\frac{-31}{35} \div \frac{31}{14} = \frac{-31}{35} \cdot \frac{14}{31} = -\frac{2}{5}$