Multiplying Fractions: Top Times Top, Bottom Times Bottom

$\tfrac{1}{6}$

$\tfrac{1 \times 1}{2 \times 3} = \tfrac{1}{6}$.

$\tfrac{6}{35}$

$\tfrac{3 \times 2}{5 \times 7} = \tfrac{6}{35}$, and $6$ and $35$ share no factors.

Cancel before multiplying.

$\tfrac{1}{6}$

$4$ and $8$ share $4$; $3$ and $9$ share $3$. That leaves $\tfrac{1}{3} \times \tfrac{1}{2} = \tfrac{1}{6}$.

$4$

Write $6$ as $\tfrac{6}{1}$: $\tfrac{2 \times 6}{3 \times 1} = \tfrac{12}{3} = 4$.

$\tfrac{1}{2}$

The $5$s cancel, and $4$ and $8$ share $4$, leaving $\tfrac{1}{2} \times \tfrac{1}{1} = \tfrac{1}{2}$.

$\tfrac{1}{2}$

Convert: $1\tfrac{1}{4} = \tfrac{5}{4}$.

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

Is $\tfrac{7}{8}$ more or less than $1$?

Smaller.

$\tfrac{7}{8}$ is less than $1$, so taking that much of $\tfrac{9}{10}$ leaves less than $\tfrac{9}{10}$.

Working it out confirms it: $\tfrac{63}{80}$, and $\tfrac{9}{10} = \tfrac{72}{80}$.

$\tfrac{3}{5}$ of a square metre

$\tfrac{2}{3} \times \tfrac{9}{10} = \tfrac{18}{30} = \tfrac{3}{5}$.

Cancelling first is quicker: $3$ into $9$ gives $3$, and $2$ into $10$ gives $5$, leaving $\tfrac{1 \times 3}{1 \times 5} = \tfrac{3}{5}$.

Two steps, or one multiplication of three numbers.

$6$ students

Half of $30$ is $15$ students playing an instrument.

Then $\tfrac{2}{5} \times 15 = \tfrac{30}{5} = 6$ students play piano.

Read the expression using the word “of”.

Multiplying by less than $1$ shrinks the number. The answer is $\tfrac{3}{8}$.

The expression means half of three quarters, and half of something is less than the thing.

$\tfrac{3}{4} \times \tfrac{1}{2} = \tfrac{3}{8}$, which is indeed half of $\tfrac{3}{4}$.

Maya’s rule works for whole numbers bigger than $1$, which is where she learned it. It stops working the moment a factor drops below $1$.