Question

A fraction becomes $\dfrac{9}{{11}}$ if 2 is added to both numerator and denominator. If 3 is added to both numerator and denominator it becomes $\dfrac{5}{6}$ to find the fraction.

Answer 1

Hint: The system of two equations with two unknown values, the solution can be obtained by using the below steps. Here, the list of steps is provided to solve the linear equation. They are
Simplify the given equation by expanding the parenthesis
Solve one of the equations for either x or y
Substitute the step 2 solution in the other equation
Now solve the new equation obtained using elementary arithmetic operations
Finally, solve the equation to find the value of the second variable

Complete step-by-step answer:
Let the two fractions be $x$ and $y$.
Now as per the first statement equation becomes: -
\[
   \Rightarrow \dfrac{{x + 2}}{{y + 2}} = \dfrac{9}{{11}} \ldots \ldots \ldots \ldots \ldots \ldots (1) \\
   \Rightarrow 11\left( {x + 2} \right) = 9\left( {y + 2} \right) \\
   \Rightarrow 11x + 22 = 9y + 18 \\
   \Rightarrow 11x - 9y = 18 - 22 \\
   \Rightarrow 11x - 9y = - 4 \\
   \Rightarrow 9y - 11x = 4 \ldots \ldots \ldots \ldots \ldots \ldots (1) \\
 \]
and as per the second statement equation becomes: -
$
   \Rightarrow \dfrac{{x + 3}}{{y + 3}} = \dfrac{5}{6} \ldots \ldots \ldots \ldots \ldots \ldots (2) \\
   \Rightarrow 6\left( {x + 3} \right) = 5\left( {y + 3} \right) \\
   \Rightarrow 6x + 18 = 5y + 15 \\
   \Rightarrow 6x - 5y = 15 - 18 \\
   \Rightarrow 6x - 5y = - 3 \\
   \Rightarrow 5y - 6x = 3 \ldots \ldots \ldots \ldots \ldots \ldots (2) \\
 $
Now we will use elimination method:-
Multiply equation (1) with 5
\[ \Rightarrow 45y - 55x = 20 \ldots \ldots \ldots \ldots \ldots \ldots (3)\]
Multiply equation (2) with 9
$ \Rightarrow 45y - 54x = 27 \ldots \ldots \ldots \ldots \ldots \ldots (4)$
Now subtracting equation (3) from equation (4)
\[ \Rightarrow x = 7 \ldots \ldots \ldots \ldots \ldots \ldots (5)\]
Now we put the value of x in equation (1) and find the value of y
\[
   \Rightarrow 9y - 11x = 4 \ldots \ldots \ldots \ldots \ldots \ldots (1) \\
   \Rightarrow 9y - 11\left( 7 \right) = 4 \\
   \Rightarrow 9y - 77 = 4 \\
   \Rightarrow 9y = 4 + 77 \\
   \Rightarrow y = \dfrac{{81}}{9} \\
   \Rightarrow y = 9 \ldots \ldots \ldots \ldots \ldots \ldots (6) \\
 \]
By using equation (5) and (6) the fraction is $\dfrac{7}{9}$.

Note: To check whether the obtained solution is correct or not, substitute the values of x and y in any of the given equations.