Question

Reduce the given fraction into the lowest terms: $\dfrac{{42}}{{78}}$.

Answer 1

Hint: We can use the Highest common factor (HCF) method to find the answer. The fraction with reduced terms can be obtained by cancelling the common factors if any from both the numerator and denominator. So finding the common factors of $42$ and $78$ we can proceed.

Useful formula:
We have $\dfrac{a}{b} = \dfrac{m}{n}$, if for some $k$, $a = mk$ and $b = nk$.

Complete step-by-step answer:
The given fraction is $\dfrac{{42}}{{78}}$.
A fraction can be expressed in many equivalent ways.
We have $\dfrac{a}{b} = \dfrac{m}{n}$, if for some $k$, $a = mk$ and $b = nk$.
That is, we can cancel the common factors if any from the numerator and denominator.
Two numbers may have more than one common factor.
So to reduce to its lowest terms, we have to cancel the highest common factor.
For that, consider the prime factorisation (means writing the numbers as multiple of its prime factors) of the two numbers.
$42 = 2 \times 3 \times 7$
And
$78 = 2 \times 3 \times 13$
So we can see the highest common factor of these two numbers is $2 \times 3 = 6$.
This gives us $6$ from the numerator and denominator of the fraction $\dfrac{{42}}{{78}}$.
We have, $\dfrac{{42}}{{78}} = \dfrac{{6 \times 7}}{{6 \times 13}}$
Cancelling $6$ we get, $\dfrac{{42}}{{78}} = \dfrac{7}{{13}}$
Therefore the reduced form of the fraction is $\dfrac{7}{{13}}$.

Note: The fraction $\dfrac{7}{{13}}$ is the lowest form of the given fraction. It has other equivalent fractions as well. In short we get an equivalent fraction by multiplying the numerator and denominator by any fixed number.
This gives $\dfrac{7}{{13}} = \dfrac{{7 \times 2}}{{13 \times 2}} = \dfrac{{7 \times 3}}{{13 \times 3}} = \dfrac{{7 \times 4}}{{13 \times 4}}$
$ \Rightarrow \dfrac{7}{{13}} = \dfrac{{14}}{{26}} = \dfrac{{21}}{{39}} = \dfrac{{28}}{{52}}$
Continuing this way we can find an infinite number of fractions which equal to the given fraction.