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# The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio $2:3$. Determine the fraction.

Last updated date: 18th Jun 2024
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Hint: First we will assume that the numerator be $x$ and denominator be $y$. Then we will use the given conditions to form linear equations in two variables and then using the substitution method so that we will get the required values.

We are given that the sum of the numerator and denominator of a fraction is 4 more than twice the numerator and if the numerator and denominator are increased by 3, they are in the ratio $2:3$.
Let us assume that the numerator be $x$ and the denominator be $y$.
Using the given conditions, we get
$\Rightarrow x + y = 2x + 4$
Subtracting the above equation by $2x$ on both sides, we get
$\Rightarrow x + y - 2x = 2x + 4 - 2x \\ \Rightarrow y - x = 4 \\$
Rewriting the above equation, we get
$\Rightarrow x - y = - 4$
Adding the above equation by $y$ on both sides, we get
$\Rightarrow x - y + y = - 4 + y \\ \Rightarrow x = - 4 + y{\text{ ......eq.(1)}} \\$
Also using the numerator and denominator are increased by 3, they are in the ratio $2:3$, we get
$\Rightarrow \dfrac{{x + 3}}{{y + 3}} = \dfrac{2}{3}$
Cross-multiplying the above equation, we get
$\Rightarrow 3\left( {x + 3} \right) = 2\left( {y + 3} \right) \\ \Rightarrow 3x + 9 = 2y + 6 \\$
Subtracting the above equation by $2y + 9$ on both sides, we get
$\Rightarrow 3x + 9 - 2y - 9 = 2y + 6 - 2y - 9 \\ \Rightarrow 3x - 2y = - 3{\text{ ......eq.(2)}} \\$

Using the equation (1) in the above equation, we get
$\Rightarrow 3\left( { - 4 + y} \right) - 2y = - 3 \\ \Rightarrow - 12 + 3y - 2y = - 3 \\ \Rightarrow - 12 + y = - 3 \\$
Adding the above equation by 12 on both sides, we get
$\Rightarrow - 12 + y + 12 = - 3 + 12 \\ \Rightarrow y = 9 \\$
Substituting the above value in the equation (1), we get
$\Rightarrow x = - 4 + 9 \\ \Rightarrow x = 5 \\$
Thus, the numerator is 5 and the denominator is 9.
Therefore, $\dfrac{5}{9}$ is the required fraction.

Note: We need to know that numerator is the top number in a fraction and denominator is the bottom number. Do not confuse with that or else the answer will be wrong. We can assume any variables for numerator and denominator, it is not necessary to use only $x$ and $y$. We can also solve this problem using elimination methods, this is up to the students.