What if you say the opposite, like $\unicode{x201C}\text{My brother is half as tall as my friend}\unicode{x201D}$ can you do exactly the same thing to calculate your brother's height from your friend's height? Yes, you can. You can multiply your friend's height by the fraction $\dfrac{1}{2}$ and get your brother's height.

If your friend's height is $140\ cm$, your brother's height would be $\dfrac{1}{2} \times 140 = 70\ cm$

So, what do you do when you multiply one fraction by another. Let us get back to our chocolate bar sharing problem. Let us say you have a chocolate bar like below, and two friends to share it with.

Let us say you give half of this bar to your first friend. We know this means you gave him a portion contain $9$ small tablets and you are left with a portion containing $9$ small tablets.

From what you are left with, you give $\dfrac{2}{3}$ to your second friend, which means you gave $2$ out of $3$ equal portion of what you had left. If you divide your remaining share into $3$ equal portions, each equal portion will contain $3$ tablets, and you gave $2$ of those portions, that means you gave $6$ small tablets to your second friend. We can also say the same thing as $\unicode{x201C}\text{You gave }\dfrac{2}{3}\text{ of } \dfrac{1}{2} \text{ of what you had}\unicode{x201D}$. Which also means you gave your friend $\dfrac{2}{3} \times \dfrac{1}{2}$ of what you had.

When you are multiplying two fractions, you are actually multiplying the numerator with the numerator and the denominator with the denominator.

In our case you have given $\dfrac{2}{3} \times \dfrac{1}{2} = \dfrac{2 \times 1}{3 \times 2} = \dfrac{2}{6}$

We can cancel the common factors in our result and get:

$\dfrac{\cancel{2} \raise{0.2em}{1}}{\cancel{6} \lower{0.2em}{3}} = \dfrac{1}{3}$.

So you gave $\dfrac{1}{3}$ of your original chocolate bar to your second friend. And you know, that one third of 18 tablets is 6 tablets, which matches our earlier result.

Tip: Although here we have done the multiplication and then we have cancelled out the common factors, we should first cancel out the common factors then we should multiply. That we our numbers will remain smaller and calculations will be easier. What we should do, is:

$\dfrac{\cancel{2} \raise{0.2em}{1}}{3} \times \dfrac{1}{\cancel {2} \lower{0.2em}{1}} = \dfrac{1}{3}$

We will take one more example of multiplication. How much is $\dfrac{15}{56} \times \dfrac{42}{55}$

As suggested earlier, we will perform the cancellations first and then do the multiplications.

$\dfrac{\cancel{15} \raise{0.2em}{3}}{56} \times \dfrac{42}{\cancel{55} \lower{0.2em}{11}}$

$= \dfrac{3}{\cancel{56} \lower{0.2em}{\cancel{8} \lower{0.2em}{4}}} \times \dfrac{\cancel{42} \raise{0.2em}{\cancel{6} \raise{0.2em}{3}}}{11}$

$= \dfrac{3}{4} \times \dfrac{3}{11}$

$= \dfrac{9}{44}$

Multiplying fraction with integer is equally simple, all you need to remember is that integers can also be written as a fraction, for example the integer $17$ can be written as $\dfrac{17}{1}$. Once you have represented the integer as fraction, the multiplication with fraction is same as we have seen before.

Before we start fraction division, let us understand what a $reciprocal$ is. Reciprocal of a number is the number we obtain by interchanging the numerator and the denominator. For example the reciprocal of the fraction $\dfrac{21}{11}$ is $\dfrac{11}{21}$. What is the reciprocal of an integer. We saw just now that, integers can also be represented as a fraction with $1$ as the denominator. So the reciprocal of the integer $15$ is same as the reciprocal of the fraction $\dfrac{15}{1}$ , which is $\dfrac{1}{15}$

Division of one number by another number is same as the multiplication with the reciprocal of the divisor.

So, $\dfrac{15}{22} \div \dfrac{3}{4}$  is same as $\dfrac{15}{22} \times \dfrac{4}{3}$. Let us check how much we get.

$\dfrac{15}{22} \div \dfrac{3}{4}$

$= \dfrac{15}{22} \times \dfrac{4}{3}$

$= \dfrac{\cancel{15} \raise{0.2em}{5}}{\cancel{22} \lower{0.2em}{11}}  \times \dfrac{\cancel{4} \raise{0.2em}{2}}{\cancel{3} \lower{0.2em}{1}}$

$= \dfrac{5}{11} \times \dfrac{2}{1}$

$= \dfrac{10}{11}$

To multiply or divide mixed fractions, you must convert them to improper fractions first and then you can multiply or divide them as needed. We will see an example for this.

How much is $3\dfrac{4}{7} \div 4\dfrac{2}{7}$ ?

$3\dfrac{4}{7} = \dfrac{3 \times 7 + 4}{7} = \dfrac{25}{7}$

$4\dfrac{2}{7} = \dfrac{4 \times 7 + 2}{7} = \dfrac{30}{7}$

$\texttip{\therefore}{therefore} 3\dfrac{4}{7} \div 4\dfrac{2}{7}$

$= \dfrac{25}{7} \div \dfrac{30}{7}$

$= \dfrac{25}{7} \times \dfrac{7}{30}$

$= \dfrac{25}{\cancel{7} \lower{0.2em}{1}} \times \dfrac{\cancel{7} \raise{0.2em}{1}}{30}$

$= \dfrac{\cancel{25} \raise{0.2em}{5}}{1} \times \dfrac{1}{\cancel{30} \raise{0.2em}{6}}$

$= \dfrac{5}{6}$

Quick Recap:

- Fraction multiplication is done by multiplying numerators with numerators and denominators with denominators.

- You should cancel common factors before multiplying.

- Multiplication of fractions with integers can be done by putting 1 as the denominator, with the integer as the numerator.

- Reciprocal of a fraction is obtained by interchanging the numerator with the denominator.

- Division is same as multiplication with the reciprocal of the divisor.

- Multiplication or division of mixed fraction can be done after converting the mixed fraction to improper fraction.