Question

If the numerator of a fraction is multiplied by 2 and its denominator is increased by 1, it becomes 1. However, if the numerator is increased by 4 and the denominator is multiplied by 2, the fraction becomes $ \dfrac{1}{2} $ . Find the fraction.

Answer 1

Hint: We can solve this question by letting the fraction in the form of the numerator over the denominator in terms of two variables. Then according to the given two conditions we can find two equations and solve for the numerator and denominator.

Complete step-by-step answer:

Let fraction is  $ \dfrac{x}{y} $  where x is numerator of fraction and y is denominator of fraction.

According to the first condition, if the numerator is multiplied by 2 and the denominator is increased by 1 then the fraction becomes 1.

So we can write it as

 $ \Rightarrow \dfrac{2x}{y+1}=1 $  

 $ \Rightarrow 2x=y+1 $  

 $ \Rightarrow 2x-1=y $  ……………………………………………………(i)

According to the second condition, if the numerator is increased by 4 and the denominator is multiplied by 2, then the fraction becomes  $ \dfrac{1}{2} $ .

So we can write it as

$ \Rightarrow \dfrac{x+4}{2y}=\dfrac{1}{2} $ 

$ \Rightarrow x+4=y $  

$ \Rightarrow x-y=-4 $  …………………………………………………..(ii)

Now from equation (i) we can substitute value of y in equation (ii)

$ \Rightarrow x-(2x-1)=-4 $  

$ \Rightarrow x-2x+1=-4 $ 

$ \Rightarrow -x=-4-1 $ 

$ \Rightarrow -x=-5 $ 

$ \Rightarrow x=5 $ 

On substituting value of x in equation (i)

$ \Rightarrow y=2\times 5-1=9 $  

Hence the required fraction is  $ \dfrac{5}{9} $ .

Note: In general to solve n number of variables we need n number of equations. So in the given question we have two conditions from which we can get two equations. That’s why we let fraction in terms of two variables.Also fraction is always in the form of  $ \dfrac{\text{Numerator}}{\text{Denominator}} $  that’s why we said x is numerator and y is denominator.And we need to give a final answer in the form of fraction so after getting the value of x and y we should write it in the form of fraction.