Multiplying Fractions

Multiplying Fractions – 3 Simple Steps

To multiply fractions –

Multiply the top numbers (numerators).
Multiply the bottom numbers (denominators).
Simplify the product fraction (if possible).

To get a better understanding of what they mean, let’s use these steps to solve a couple of examples.

Then, we’ll look at some special cases. And also something important that will make multiplications easier for you.

Example

Simplify -

(a) \hspace{0.3cm} \frac{1}{2} \times \frac{3}{5}

(b) \hspace{0.3cm} \frac{2}{5} \times \frac{3}{8}

Solution (a)

Step 1 & 2. We multiply the top numbers together and the bottom numbers together.

\begin{align*} \frac{1}{2} \times \frac{3}{5} \hspace{0.2em} &= \hspace{0.2em} \frac{1 \times 3}{2 \times 5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3}{10} \end{align*}

Step 3. We can’t simplify the product any further. It’s already in its lowest terms So, that’s our answer.

Solution (b)

\begin{align*} \frac{2}{5} \times \frac{3}{8} \hspace{0.2em} &= \hspace{0.2em} \frac{2 \times 3}{5 \times 8} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{6}{40} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3}{20} \end{align*}

This time, the resulting fraction could be simplified further. And so in the last step, we simplified it by canceling out the common factor ( $\hspace{0.2em} 2 \hspace{0.2em}$ ) from the top and bottom numbers.

Multiplying Fractions and Whole Numbers

When multiplying fractions with whole numbers, write the whole numbers as fractions (with $\hspace{0.2em} 1 \hspace{0.2em}$ as their denominators). Then proceed as you would, to multiply any other group of factions.

Remember – a denominator of $\hspace{0.2em} 1 \hspace{0.2em}$ makes no difference. You can add it or remove it as you see fit.

Example

Simplify -

(a) \hspace{0.3cm} \frac{4}{11} \times 2

(b) \hspace{0.3cm} 10 \times \frac{3}{5}

Solution (a)

We rewrite $\hspace{0.2em} 2 \hspace{0.2em}$ as a fraction – $\hspace{0.2em} 2 \hspace{0.2em}$ over $\hspace{0.2em} 1 \hspace{0.2em}$ – and proceed to get the required product.

\begin{align*} \frac{4}{11} \times {\color{Red} 2} \hspace{0.2em} &= \hspace{0.2em} \frac{4}{11} \times {\color{Red} \frac{2}{1}} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{4 \times 2}{11 \times 1} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{8}{11} \end{align*}

That's it.

Solution (b)

\begin{align*} \frac{3}{5} \times {\color{Red} 10} \hspace{0.2em} &= \hspace{0.2em} \frac{4}{11} \times {\color{Red} \frac{10}{1}} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3 \times 10}{5 \times 1} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{30}{5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{6}{1} \hspace{0.25em} = \hspace{0.25em} 6 \end{align*}

See how I dropped the denominator in the last step? I repeat – a denominator of $\hspace{0.2em} 1 \hspace{0.2em}$ makes no difference to the number – $\hspace{0.2em} 6 \hspace{0.2em}$ over $\hspace{0.2em} 1 \hspace{0.2em}$ is the same as $\hspace{0.2em} 6 \hspace{0.2em}$ .

How to Multiply Fractions and Mixed Numbers

When you have a multiplication involving mixed numbers, convert the mixed numbers into improper (top-heavy) fractions.

Example

Simplify -

(a) \hspace{0.3cm} \frac{5}{9} \times 1 \frac{2}{10}

(b) \hspace{0.3cm} 1 \frac{1}{3} \times \frac{6}{16}

Solution (a)

In the first step, we convert the mixed number into an improper fraction. And then it’s business as usual.

\begin{align*} \frac{5}{9} \times {\color{Red} 1 \frac{2}{10}} \hspace{0.2em} &= \hspace{0.2em} \frac{5}{9} \times {\color{Red} \frac{12}{10}} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{5 \times 12}{9 \times 10} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{60}{90} \hspace{0.25em}=\hspace{0.25em} \frac{2}{3} \end{align*}

Again, don’t forget to simplify your answer. In the next section, we’ll look at a better way of doing it – canceling common factors during multiplication.

Solution (b)

\begin{align*} {\color{Red} 1 \frac{1}{3}} \times \frac{6}{16} \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{4}{3}} \times \frac{6}{16} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{5 \times 12}{9 \times 10} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{24}{48} \hspace{0.25em}=\hspace{0.25em} \frac{1}{2} \end{align*}

Canceling Out Common Factors Early

In all our previous examples, we waited till the end to get rid of the common factors and simplify the answer. But often, there’s a much better way.

When multiplying fractions, you can split any of the numbers into smaller factors in any step. And cancel out the common factors (between the top and bottom numbers). You can cancel them out in one step or multiple steps, it’s all good.

Example

Simplify -

\frac{25}{48} \times \frac{32}{35}

\frac{33}{40} \times \frac{25}{22}

Solution (a)

Here, instead of multiplying the top and bottom numbers right away, we split them into smaller factors and cancel out the factors common between the top and bottom parts. You might be familiar with the process if you know how to simplify fractions.

\begin{align*} \frac{25}{48} \times \frac{32}{35} \hspace{0.2em} &= \hspace{0.2em} \frac{\cancel{5} \times 5}{6 \times \cancel{8}} \times \frac{4 \times \cancel{8}}{\cancel{5} \times 7} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{5 \times 4}{6 \times 7} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{20}{42} \hspace{0.25em}=\hspace{0.25em} \frac{10}{21} \end{align*}

See how splitting and canceling make things easier by making the numbers smaller. Try solving it like we solved the earlier examples and you’ll see the difference.

Also, note how there was still a common factor of 2 left that we canceled in the last step to simplify our answer. That’s okay. There’s no compulsion to remove all common factors in one go.

Besides, with time and practice, you will get better at spotting common factors and canceling them early and easily. So don’t feel overwhelmed by the idea.

Solution (b)

Same story. In the first step, we factorize the numbers and cancel the common factors.

\begin{align*} \frac{33}{40} \times \frac{25}{22} \hspace{0.2em} &= \hspace{0.2em} \frac{\cancel{11} \times 3}{8 \times \cancel{5}} \times \frac{\cancel{5} \times 5}{\cancel{11} \times 2} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3 \times 5}{8 \times 2} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{15}{16} \end{align*}

"Of" Means Multiply

Often in the study of fractions, you will come across phrases like “one-half of his salary” or “a third of the students in the class”. Just remember, when doing calculations, “of” is the same as “times.”

Example

Simplify -

(a) \hspace{0.3cm} \frac{3}{4} \text{\,\,\,of\,\,\,} 6

(b) \hspace{0.3cm} \frac{1}{2} \text{\,\,\,of\,\,\,} \frac{2}{15}

Solution (a)

We start by replacing “of” with the multiplication symbol. Also, because we have a whole number (6), we rewrite it as a fraction (6 over 1). I’m sure by now you already know that.

\begin{align*} \frac{3}{4} {\color{Red} \text{\,\,\,of\,\,\,}} 6 \hspace{0.2em} &= \hspace{0.2em} \frac{3}{4} {\color{Red} \, \times \,} 6 \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3}{4} \times \frac{6}{1} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{18}{4} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{9}{2} \hspace{0.25em} = \hspace{0.25em} 4\frac{1}{2} \end{align*}

Solution (b)

\begin{align*} \frac{1}{2} {\color{Red} \text{\,\,\,of\,\,\,}} \frac{12}{15} \hspace{0.2em} &= \hspace{0.2em} \frac{1}{2} {\color{Red} \, \times \,} \frac{12}{15} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{1}{2} \times \frac{\cancel{3} \times \cancel{2} \times 2}{\cancel{2} \times 5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{2}{5} \end{align*}

And that brings us to the end of this tutorial on multiplying fractions. Until next time.