Dividing Fractions: Keep, Change, Flip — and Why

How many quarters fit into a half?

$2$

$\tfrac{1}{2} \times \tfrac{4}{1} = \tfrac{4}{2} = 2$.

Two quarters make a half, so the count checks out.

$\tfrac{3}{2}$, which is $1\tfrac{1}{2}$

$\tfrac{3}{5} \times \tfrac{5}{2} = \tfrac{15}{10} = \tfrac{3}{2}$.

$\tfrac{1}{6}$

$\tfrac{5}{6} \times \tfrac{1}{5} = \tfrac{5}{30} = \tfrac{1}{6}$.

$12$

$\tfrac{4}{1} \times \tfrac{3}{1} = 12$.

Each whole holds three thirds, and there are four wholes.

$\tfrac{1}{2}$

$\tfrac{7}{10} \times \tfrac{5}{7} = \tfrac{35}{70} = \tfrac{1}{2}$.

The divisor is bigger than $1$, so a shrinking answer is expected.

Convert the mixed number first.

$6$

$2\tfrac{1}{4} = \tfrac{9}{4}$, so $\tfrac{9}{4} \times \tfrac{8}{3} = \tfrac{72}{12} = 6$.

Bigger.

The divisor $\tfrac{1}{5}$ is less than $1$, so the answer grows. Working it out gives $\tfrac{10}{3}$, which is $3\tfrac{1}{3}$.

$6$ glasses

$\tfrac{3}{4} \div \tfrac{1}{8} = \tfrac{3}{4} \times \tfrac{8}{1} = \tfrac{24}{4} = 6$.

This is the “how big is each group” kind of division.

$\tfrac{3}{4}$ of a metre each

$3 \div 4 = \tfrac{3}{1} \times \tfrac{1}{4} = \tfrac{3}{4}$.

Here the divisor is bigger than $1$, so the answer shrinks — which is right, since sharing gives each person less than the whole.

Which operation did she actually perform?

She multiplied without flipping. The answer is $\tfrac{1}{2}$.

$\tfrac{8}{25}$ is $\tfrac{2}{5} \times \tfrac{4}{5}$, so she changed the sign but forgot the flip.

Correctly: $\tfrac{2}{5} \times \tfrac{5}{4} = \tfrac{10}{20} = \tfrac{1}{2}$.

A check: $\tfrac{2}{5}$ is exactly half of $\tfrac{4}{5}$, so only half of a $\tfrac{4}{5}$ piece fits inside it. The answer $\tfrac{1}{2}$ matches that reading.