Question

Find the equivalent fraction of $\dfrac{{54}}{{72}}$, with the following condition: numerator 6

Answer 1

Hint:
We will first reduce the given fraction to the simplest form by canceling 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 numerator 6. This will give an equivalent fraction which has a denominator 6.

Complete step by step solution:
The given fraction is $\dfrac{{54}}{{72}}$.
We observe that both 54 and 72 are even numbers. So, they will have 2 as a common factor.
We know that $54 = 2 \times 27$ and $72 = 2 \times 36$.
Cancelling out 2 from both 54 and 72, we get
$\dfrac{{54}}{{72}} = \dfrac{{2 \times 27}}{{2 \times 36}} = \dfrac{{27}}{{36}}$
This is not in the simplest form. The reason is that both 27 and 36 have a common factor, which is 3. So, we have to cancel out 3 from 27 and 36.
We know that $27 = 3 \times 9$and $36 = 3 \times 12$. Cancelling out 3 from 27 and 36, we get
$\dfrac{{27}}{{36}} = \dfrac{{3 \times 9}}{{3 \times 12}} = \dfrac{9}{{12}}$
Again, this is not in the simplest form since 9 and 12 have 3 as their common factor. Hence, we have to cancel out 3 from 9 and 12.
We know that $9 = 3 \times 3$ and $12 = 3 \times 4$.
Cancelling out 3 from 9 and 12, we have
 $\dfrac{9}{{12}} = \dfrac{{3 \times 3}}{{3 \times 4}} = \dfrac{3}{4}$
Since 3 and 4 have no common factors, we have obtained the simplest form. Now, we must find an equivalent fraction with numerator 6. We know that $6 = 3 \times 2$.
So, to get 6 in the numerator, we have to multiply the numerator by 2. We should also multiply the denominator by 2. Therefore, we get
$\dfrac{3}{4} = \dfrac{{3 \times 2}}{{4 \times 2}} = \dfrac{6}{8}$.

Hence, the fraction equivalent to $\dfrac{{54}}{{72}}$ with numerator 6 is $\dfrac{6}{8}$.

Note:
We can also approach the given problem as follows:
We know that $54 = 9 \times 6$ and $72 = 9 \times 8$.
The numerator in $\dfrac{{54}}{{72}}$ is 54.
To get 6 in the numerator, we have to cancel out 9 from both 54 and 72, since both 54 and 72 are multiples of 9. Therefore,
\[\dfrac{{54}}{{72}} = \dfrac{{9 \times 6}}{{9 \times 8}} = \dfrac{6}{8}\]
Hence, the fraction equivalent to $\dfrac{{54}}{{72}}$ with numerator 6 is $\dfrac{6}{8}$.