# The sum of a numerator and denominator of a fraction is 18. If the denominator increased by 2, the fraction reduces to $\frac{1}{3}$. Find the fraction.

Last updated date: 31st Mar 2023

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Answer

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Hint: Use Substitution method to find the value of x and y.

Let the numerator be x and denominator be y.

And it is given to us that the sum of numerator and denominator of a fraction is 18.

Therefore we have, x + y = 18.................(i)

And also it is given to us that the denominator increased by 2 and fraction reduces to $\frac{1}{3}$ and hence we have,

$\frac{x}{{y + 2}} = \frac{1}{3}$ ..................(ii)

Now from equation (i) , we have

y= 18 – x

Again on simplifying the equation (ii) we have

3x = y + 2

Therefore on rearranging, we have

3x – y – 2 =0...................(iii)

So on substituting the value of y on equation (iii), we have

$ \Rightarrow $ 3x –(18-x) -2=0

And hence on simplification, we have

$ \Rightarrow $4x – 20=0

$ \Rightarrow $4x = 20

Now 4 will cancel out 20 in 5 times therefore we have

$ \Rightarrow $x =5

Now we have the value of x and hence on putting the value of x in equation(i) we have,

$ \Rightarrow $5 + y=18

$ \Rightarrow $y = 18 – 5

$ \Rightarrow $y = 13

Thus the original fraction is $\frac{x}{y} = \frac{5}{{13}}$ .

Note: In this type of question we have to find the value of x and y. So in order to find the value of x and y we’ll use a substitution method after forming two equations with the help of given data and hence after substituting the value of either x or y on those equations, we’ll have our answer.

Let the numerator be x and denominator be y.

And it is given to us that the sum of numerator and denominator of a fraction is 18.

Therefore we have, x + y = 18.................(i)

And also it is given to us that the denominator increased by 2 and fraction reduces to $\frac{1}{3}$ and hence we have,

$\frac{x}{{y + 2}} = \frac{1}{3}$ ..................(ii)

Now from equation (i) , we have

y= 18 – x

Again on simplifying the equation (ii) we have

3x = y + 2

Therefore on rearranging, we have

3x – y – 2 =0...................(iii)

So on substituting the value of y on equation (iii), we have

$ \Rightarrow $ 3x –(18-x) -2=0

And hence on simplification, we have

$ \Rightarrow $4x – 20=0

$ \Rightarrow $4x = 20

Now 4 will cancel out 20 in 5 times therefore we have

$ \Rightarrow $x =5

Now we have the value of x and hence on putting the value of x in equation(i) we have,

$ \Rightarrow $5 + y=18

$ \Rightarrow $y = 18 – 5

$ \Rightarrow $y = 13

Thus the original fraction is $\frac{x}{y} = \frac{5}{{13}}$ .

Note: In this type of question we have to find the value of x and y. So in order to find the value of x and y we’ll use a substitution method after forming two equations with the help of given data and hence after substituting the value of either x or y on those equations, we’ll have our answer.

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