Taking a Fraction of a Number

When we say that a runner missed out on first place by a fraction of a second, we mean that he was several tenths or hundredths of a second behind the winner. In mathematics, taking a fraction of a number has a very precise meaning. What is it?

I. General method

A. Example

In a group of 120 people, three quarters are wearing pants. To find the number of people wearing pants, all we need to know is how to work out three-quarters of 120.


Three quarters is the same as three times one quarter $\left(\frac{3}{4}=3\times \frac{{\displaystyle 1}}{{\displaystyle 4}}\right)$.


If we know how to find a quarter of 120, then we know how to find three-quarters of 120, i.e. , by multiplying by 3.


As one quarter of 120 is $120÷4=\frac{120}{4}$, finding three-quarters of 120 is the same as calculating $3\times \frac{120}{4}$.


Since $\frac{120}{4}=30$, we find that $3\times \frac{120}{4}=90$.


Therefore, 90 people in this group are wearing pants.


Note: $\frac{3\times 120}{4}=\frac{360}{4}=90$. From this we can see that $3\times \frac{120}{4}=\frac{{\displaystyle 3\times 120}}{{\displaystyle 4}}$.

B. Generalization

We have just seen that three-quarters of 120 is the same as $3\times \frac{120}{4}$ and that $3\times \frac{120}{4}=\frac{{\displaystyle 3\times 120}}{{\displaystyle 4}}$. So, we can make sense of the product $\frac{3}{4}\times 120$, which means three-quarters of 120.


In more general terms, given a number n, three quarters of n will be $\frac{3}{4}\times n=3\times \frac{n}{4}=\frac{3\times n}{4}$ .


We can generalize further by replacing the fraction $\frac{3}{4}$ with any other fraction. So, if we take a fraction written as $\frac{a}{b}$, where b ≠  0, then $\frac{a}{b}$ of n is: $a\times \frac{n}{b}=\frac{{\displaystyle a\times n}}{{\displaystyle b}}$.

Special case for decimal fractions:

Let’s say that a is a natural number. If we want to take a fraction $\frac{a}{10}$, $\frac{a}{100}$, or $\frac{a}{1000}$ of a number, we simply multiply by a, then move the decimal point one, two, or three places to the left respectively in the result obtained.


For example, $\frac{37}{100}$ of 12 (thirty-seven hundredths of 12) is 4.44, since 37 × 12 = 444.


In the same way, $\frac{23}{10}$ of 5.4 (twenty-three tenths of five point four) is 12.42, since 23 × 5.4 = 124.2.