Question

A fraction becomes $\dfrac{1}{3}$ when 1 is subtracted from the numerator and it becomes $\dfrac{1}{4}$ when 8 is added to its denominator. Find the fraction

Answer 1

- Hint: Here, consider the fraction as $\dfrac{x}{y}$, where $x$ is the numerator and $y$is the denominator. While subtracting 1 from the numerator we get the equation $\dfrac{x-1}{y}=\dfrac{1}{3}$. Next by adding 8 to the denominator we get the equation $\dfrac{x}{y+8}=\dfrac{1}{4}$. Now solve for $x$ and $y$.

Complete step-by-step solution -

Let $\dfrac{x}{y}$ be the fraction where $x$ is called the numerator and $y$ is called the denominator.
Here, it is given that the fraction $\dfrac{x}{y}$ becomes $\dfrac{1}{3}$ when 1 is subtracted from the numerator $x$. Therefore we will get the equation:
$\dfrac{x-1}{y}=\dfrac{1}{3}$
Now, by cross multiplying we get:
$\begin{align}
  & 3(x-1)=1\times y \\
 & 3x-3=y \\
\end{align}$
In the next step, take $y$ to the left side then $y$ becomes $-y$ and -3 is taken to the right side then -3 becomes 3. Hence we obtain:
$3x-y=3$
It is also given that the fraction $\dfrac{x}{y}$ becomes $\dfrac{1}{4}$ when 8 is added to its denominator. Therefore we will get the equation:
$\dfrac{x}{y+8}=\dfrac{1}{4}$
 Now, by cross multiplication we obtain the equation:
$\begin{align}
  & 4\times x=1(y+8) \\
 & 4x=y+8 \\
\end{align}$
In the next step, take $y$ to the right side then $y$ becomes $-y$. Hence, we get the equation:
$4x-y=8$
 Therefore, the two equations are:
$3x-y=3$ …….. (1)
$4x-y=8$ …….. (2)
Now we have to find the value of $x$ and $y$.
 To find $x$subtract equation (2) from equation (1) then we get:
$-x=-5$
 In the next step, multiplying the equation with -1 we obtain:
$x=5$
 Hence the value of $x$ is 5.
To find $y$, first make the coefficients of $x$in equation (1) and equation (2) equal. For that multiply equation (1) by the coefficient of $x$ in equation (2) and multiply equation (2) by the coefficient of $x$ in equation (1).
First we can multiply equation (1) by 4 we get:
$\begin{align}
  & 4(3x-y)=4\times 3 \\
 & 4\times 3x-4\times y=12 \\
 & 12x-4y=12 \\
\end{align}$
 In the next step, multiply equation (2) by 3 we obtain:
$\begin{align}
  & 3(4x-y)=3\times 8 \\
 & 3\times 4x-3\times y=24 \\
 & 12x-3y=24 \\
\end{align}$
We got the equations as:
$12x-4y=12$ .….. (3)
$12x-3y=24$ …… (4)
Now, to obtain $x$ subtract equation (4) from equation (3), we get:
$-y=-12$
 Now, by multiplying the above equation by -1 we get:
$y=12$
Therefore, the value of $y$ is 12
Hence, the fraction, $\dfrac{x}{y}=\dfrac{5}{12}$.

Note: After getting the values of $x$ and $y$ you can verify the values by substituting into any one of the equations $\dfrac{x-1}{y}=\dfrac{1}{3}$ or $\dfrac{x}{y+8}=\dfrac{1}{4}$. If it does not satisfy the given conditions then the values might be wrong.