Question

The denominator of a fraction is $1$ less than twice its numerator. If $1$ is added to numerator and denominator respectively, the ratio of numerator to denominator is $3:5$ Find the fraction.

Answer 1

Step 1.Let the numerator of fraction be $x$.

Therefore denominator $=2x-1$

So, the fraction is $\frac{x}{2x-1}$

Step 2.According to question if $1$ is added to numerator and denominator respectively, the ratio of numerator to denominator is $3:5$

Therefore we have ,

$$\begin{gathered} (x+1):(2x-1+1)=3:5 \\ \Rightarrow (x+1):2x=3:5 \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{x+1}{2x}=\frac{3}{5} \\ \Rightarrow 5(x+1)=3x2x \\ \Rightarrow 5x+5=6x \\ \Rightarrow 5x-6x=-5 \\ \Rightarrow -x=-5 \\ \Rightarrow x=5 \end{gathered}$$

Therefore numerator of the fraction is $5$

Step 3. Now the denominator $=2x-1$

$$\begin{gathered} =2x5-1 \\ =10-1 \\ =9 \end{gathered}$$

Therefore the denominator of the fraction is $9$

Hence the fraction is $\frac{5}{9}$