Question

A fraction becomes 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 5/6. Find the fraction. ​

Answer 1

SOlUTION:

Let  $\bf \frac{x}{y}$ be the fraction.

Now,

Case 1

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

So,

The New fraction = $\frac{x+2}{y+2}$

According to the question,(ATQ)

$\implies \boxed{\bf \frac{x+2}{y+2} = \frac{9}{11}}$

⇒ $11 \times (x+2) = 9 \times (y+2)$

⇒ $11x+22 = 9y +18$

⇒ $11x-9y +4=0 ........(\red{i})$

Case 2

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

Again new fraction = $\bf \frac{x+3}{y+3}$

Now, ATQ,

$\implies \boxed{\bf \frac{x+3}{y+3} = \frac{5}{6}}$

⇒ $6 \times (x+3)=5 \times (y+3)$

⇒ $6x+18=5y+15$

⇒ $6x-5y+3=0 .........(\red{ii})$

Now,

From the equation (ii) we have,

$5y = 6x+3 \implies y = \frac{6x+3}{5}$ ......(iii)

On substituting the value of y from Eq.(iii) in Eq.(i), we get,

$11x-9 \times [\frac{6x+3}{5}]+4=0$

$\implies 55x-9 \times (6x+3)+20=0$

[ ∴ multiplying by 5 ]

$\implies 55x - 54x-27+20=0$

$\implies x-7=0$

$\implies \boxed{x=7}$

On putting x = 7 in the equation(iii), we get

$y=\frac{6 \times 7+3}{5}$

$\implies \boxed{y= \frac{45}{5}=9}$

Hence,

The required fraction is $\boxed{\frac{7}{9}}$

Answer 2

In the question there are two different parts.

First take the required fraction as p/q form

According to the first part, the fraction become 9/11, if 2 is added to both the  numerator and the denominator.

=> (p+2)/(q+2)= 9/11

cross multiplication

=> 11p + 22 = 9q +18

=> 11p - 9q + 4=0  .......(eq. i)

Now looking at the second part,

If 3 is added to both the numerator and the denominator, it becomes 5/6.

So,

=> (p+3)/(q+3) = 5/6

cross multiplication

=> 6p + 18 = 5q + 15

=> 6p - 5q + 3 = 0 .......(eq.ii)

Now,

Finding any value, may be q or p in eq.ii

=> q = (6p+3)/5

now putting the value of q in eq.i

=> 11p - 9*[ (6p+3)/5 ] + 4 = 0

=> (55p - 54p + 27 +20)/5 = 0

[taking LCM = 5]

=> 55p - 54p + 27 +20 = 0*5

=> 55p - 54p + 27 +20 = 0

=> p = 7

Now,

=> q = (6p+3)/5

=> q = (6*7 + 3)/5

=> q = 45/5

=> q = 9

So,

We got the required fraction that is p/q = 7/9