Question

Question 3 (v)
Form the pair of linear equations for the following problems and find their solution by substitution method:
A fraction becomes $\frac{9}{11}$, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes $\frac{5}{6}$. Find the fraction.

Answer 1

Let Numerator = x
Denominator = y
Fraction will be $\frac{x}{y}$

A fraction becomes $\frac{9}{11}$, if 2 is added to both the numerator and the denominator.

$\Rightarrow$ $\frac{(x+2)}{(y+2)}=\frac{9}{11}$

By cross multiplication, we get;
11x + 22 = 9y + 18
Subtracting 22 from both sides, we get
11x = 9y - 4
Dividing by 11, we get,
$x=\frac{9y-4}{11}$ … (i)

Given that 3 is added to both the numerator and the denominator it becomes $\frac{5}{6}$.

$\frac{(x+3)}{(y+3)}=\frac{5}{6}$ … (ii)

By Cross multiplication, we get
6x + 18 = 5y + 15
Substituting the value of x, we get;
$\frac{6(9y-4)}{11}+18=5y+15$
Subtract 18 from both sides, we get;
$\frac{6(9y-4)}{11}=5y-3$
$54y-24=55y-33$
-y = -9
y = 9

Putting this value of y in equation (i), we get,
$x$ = $\frac{9y-4}{11}$ … (i)
$x=\frac{(81-4)}{11}$
$x=\frac{77}{11}$
$x$ = 7

Hence, our fraction is $\frac{x}{y}$ = $\frac{7}{9}$.