QUESTION

# The denominator of a rational number is greater than its numerator by 7. If the numerator is increased by 17 and the denominator is decreased by 6, the new number becomes 2. Find the original number.

Hint: Here, take the rational number as $\dfrac{x}{y}$, with $x$ as the numerator and $y$ as the denominator. Given that $y=x+7$ and $\dfrac{x+17}{y-6}=2$. Now, we have to solve these two equations and find the values of $x$ and $y$ to get the rational number $\dfrac{x}{y}$.

Here, let us take the rational number as $\dfrac{x}{y}$ where $x$ is the numerator and $y$ is the denominator.
Now, we have to find the value of $\dfrac{x}{y}$.
Given that the denominator of the rational number is greater than its numerator by 7.
Hence, we will get the equation:
$y=x+7$
By taking $x$ to the left side $x$ becomes $-x$, our equation becomes:
$y-x=7$
It is also given that if the numerator is increased by 17 and denominator is decreased by 6, the new number becomes 2. Hence, we obtain the equation:
$\dfrac{x+17}{y-6}=2$
Now, by cross multiplication we get:
\begin{align} & x+17=2(y-6) \\ & x+17=2\times y-2\times 6 \\ & x+17=2y-12 \\ \end{align}
In the next step take variables to one side and constants to the other, then 17 becomes -17 and $2y$ becomes $-2y$. Therefore, we will get the equation:
\begin{align} & x-2y=-12-17 \\ & x-2y=-29 \\ \end{align}
Hence, the two equations are:
$-x+y=7$ …… (1)
$x-2y=-29$ …… (2)
Now, to obtain the value of $y$, add equation (1) and equation (2) we get:
$-y=-22$
By multiplying by -1 we get:
$y=22$
Substitute the value of $y=22$ in equation (1) we obtain:
$-x+22=7$
Now take 22 to the right side, 22 becomes -22. Hence we obtain:
\begin{align} & -x=7-22 \\ & -x=-15 \\ \end{align}
Now, multiplying the equation by -1 we get:
$x=15$
Hence, we got $x=15$ and $y=22$.
Therefore, the rational number is $\dfrac{x}{y}=\dfrac{15}{22}$

Note: Here, instead of finding $y$ from equation (1) and equation (2) you can also find $x$. For that you have to multiply equation (1) by 2 to make the coefficients of $y$ equal. Then adding newly obtained equation and equation (2) will give the value of $x$. Then substituting that value in any one equation will give the value of $y$.