Question

Which of the following is a reducible fraction?
A) \[\dfrac{105}{112}\]
B) \[\dfrac{104}{121}\]
C) \[\dfrac{77}{72}\]
D) \[\dfrac{46}{63}\]

Answer 1

Hint: In the given question, we have been asked to find the reducible fraction among the given fractions. Reducible fraction is a fraction that has some common factor and can be converted into simplest form by cancelling out that common factor. We need to reduce the fraction by dividing both the numerator and denominator by a common factor.

Complete step by step solution:
We have the following fraction;
Taking\[\dfrac{105}{112}\],
Prime factorization of both the numbers of the given fraction is,
\[\Rightarrow 105=3\times 5\times 7\]
\[\Rightarrow 112=2\times 2\times 2\times 2\times 7\]
Common factor= 7
Dividing the numerator and denominator by the common factor i.e. 7;
\[\Rightarrow \dfrac{\dfrac{105}{7}}{\dfrac{112}{7}}=\dfrac{15}{16}\]
Hence, the reduced form of \[\dfrac{105}{112}\] is\[\dfrac{15}{16}\].
Now,
Taking \[\dfrac{104}{121}\]
Prime factorization of both the numbers of the given fraction is,
\[\Rightarrow 104=2\times 2\times 2\times 13\]
\[\Rightarrow 121=11\times 11\]
No common factor, cannot be reducible.
Taking \[\dfrac{77}{72}\]
Prime factorization of both the numbers of the given fraction is,
\[\Rightarrow 77=7\times 11\]
\[\Rightarrow 72=2\times 2\times 2\times 3\times 3\]
No common factor, cannot be reducible.
Taking \[\dfrac{46}{63}\]
Prime factorization of both the numbers of the given fraction is,
\[\Rightarrow 46=2\times 23\]
\[\Rightarrow 63=3\times 3\times 7\]
No common factor, cannot be reducible.

Hence, the option (a) is the correct answer.

Note:
In order to solve these types of questions, students need to know about the concept of common factors and reducible fraction. They need to know the way of finding common factors. They should remember that the numerator and denominator of the given fraction should be divided by a same common factor, to get the reducible form.