Question

The numerator of a fraction is 3 less than its denominator. If we add 1 to both numerator and denominator it becomes equal to \[{1}/{2}\;\].

Answer 1

Hint: For solving this question we will use assuming the terms and to find those terms by given information. In this problem assume either numerator or denominator to be an unknown quantity. Then use the condition given to solve for the unknown quantity, and thus form the original fraction.

Complete step by step answer:
According to our question the numerator of a fraction is 3 less than its denominator. And if we add 1 to both numerator and denominator it becomes equal to \[{1}/{2}\;\]. Then find the original fraction.
Here, we have to give two conditions and we will use those conditions to solve this. And both are used in it unless we can’t get the answer.
So, as our condition the numerator of a fraction is 3 less than its denominator.
\[\begin{align}
  & n=d-3 \\
 & .. \\
 & .. \\
\end{align}\]
Now, the other condition is that if we add 1 to both numerator and denominator it becomes equal to \[{2}/{3}\;\].
So, we can write it as: \[\dfrac{n+1}{d+1}={2}/{3}\;\]
Then: \[3\left( n+1 \right)=2\left( d+1 \right)\]
\[3n+3=2d+2\]
Now, if we change the sides of these then:
\[3n=2d+2-3\Rightarrow 3n=2d-1\]
As it is given that, \[n=d-3\]
So, \[3\left( d-3 \right)=2d-1\]
=\[\begin{align}
  & 3d-9=2d-1 \\
 & d=9-1=8 \\
\end{align}\]
So, the denominator is 8, and
\[n=8-3=5\]
The numerator is equal to 5.
Therefore, our fraction will be \[{5}/{8}\;\].

Note: While solving these type questions you have to remember that the equation solves with the help of each other. The most important point is that when the question is to determine how many variable values then it will provide the same number of conditions and if it will not provide then you can’t solve that exactly. But at this time we can assume the values for that special case.