Question

Find the fraction equivalent to $\dfrac{{45}}{{60}}$, having denominator $240.$

Answer 1

Hint:
We will first reduce the given fraction to the simplest form by cancelling both the numerator and denominator by the common factors. Once we have the simplest form, we will multiply and divide the reduced fraction by an appropriate quantity which will give the denominator 240. This will give an equivalent fraction which has denominator 240.

Complete step by step solution:
The given fraction is $\dfrac{{45}}{{60}}$. We observe that both 45 and 60 are multiples of 5.
So, we can cancel out 45 and 60 with 5 to get a reduced form. We know that $45 = 5 \times 9$ and $60 = 5 \times 12$.
Thus, cancelling out 5 from 45 and 60 we get
$\dfrac{{45}}{{60}} = \dfrac{{5 \times 9}}{{5 \times 12}} = \dfrac{9}{{12}}$
We observe that $\dfrac{9}{{12}}$ is not in the simplest form. Both 9 and 12 have 3 as their common factor. So, we will cancel out 3 from both the numerator and the denominator.
We know that $9 = 3 \times 3$ and $12 = 3 \times 4$. Thus, cancelling out 3 from 9 and 12, we get
$\dfrac{9}{{12}} = \dfrac{{3 \times 3}}{{3 \times 4}} = \dfrac{3}{4}$.
Since 3 and 4 have no common factors, $\dfrac{3}{4}$ is in the lowest form. Now we shall find the equivalent fraction having denominator 240. We know that 240 is a multiple of 4 i.e., $240 = 4 \times 60$.
So, we have to multiply both the numerator and denominator in $\dfrac{3}{4}$ by 60. This gives us
$\dfrac{3}{4} \times \dfrac{{60}}{{60}} = \dfrac{{180}}{{240}}$. Here, the denominator is 240, which is what we require.

Hence the fraction equivalent to $\dfrac{{45}}{{60}}$, having denominator 240 is $\dfrac{{180}}{{240}}$.

Note:
We can also approach the given problem as follows:
We know that $240 = 60 \times 4$. The denominator in $\dfrac{{45}}{{60}}$ is 60.
To get 240 in the denominator, we have to multiply it by 4. So, multiplying both numerator and denominator by 4, we get
$\dfrac{{45}}{{60}} \times \dfrac{4}{4} = \dfrac{{180}}{{240}}$