Question

a fraction becomes 9 by 11 if 2 is added to both the numerator and the denominator if 3 is added to both the numerator and denominator it becomes 5 by 6 find the fraction​

Answer 1

Answer :-

Fraction is 7/9.

Explanation

Let the fraction be x/y

If 2 is added to both numerator and denominator fraction becomes 9/11

⇒ (x + 2)/(y + 2) = 9/11

⇒ 11(x + 2) = 9(y + 2)

⇒ 11x + 22 = 9y + 18

⇒ 11x - 9y = 18 - 22

⇒ 11x - 9y = - 4 ---eq(1)

If 3 is added to both numerator and denominator fraction becomes 5/6

⇒ (x + 3)/(y + 3) = 5/6

⇒ 6(x + 3) = 5(y + 3)

⇒ 6x + 18 = 5y + 15

⇒ 6x - 5y = 15 - 18

⇒ 6x - 5y = - 3

⇒ 6x = - 3 + 5y

⇒ x = (-3 + 5y)/6

Substitute x = (-3 + 5y)/6 in eq(1)

$\sf 11x - 9y = - 4 \\ \\ \\ \sf 11 \bigg( \dfrac{ - 3 + 5y}{6} \bigg) - 9y = - 4 \\ \\ \\ \sf \dfrac{11( - 3 + 5y)}{6} - 9y = - 4 \\ \\ \\ \sf \dfrac{ - 33 + 55y}{6} - 9y = - 4 \\ \\ \\ \sf \dfrac{ - 33 + 55y - 54y}{6} = - 4 \\ \\ \\ \sf - 33 + y = - 4(6) \\ \\ \\ \sf - 33 + y = - 24 \\ \\ \\ \sf y = - 24 + 33 \\ \\ \\ \sf y = 9$

Substitute y = 9 in x = (-3 + 5y)/6

⇒ x = (-3 + 5y)/6

⇒ x ={-3 + 5(9)}/6

⇒ x = (-3 + 45)/6

⇒ x = 42/6

⇒ x = 7

Therefore fraction = x/y = 7/9.

Answer 2

Solution:

Let Numerator of the fraction be x.

And denominator of the fraction be y.

$\sf{So,\;fraction=\dfrac{x}{y}}$

According to question,

Case-1

When 2 is added to both numerator and denominator, then

$\sf{\implies \dfrac{x+2}{y+2}=\dfrac{9}{11}}$

=> 11x + 22 = 9y + 18

=> 11x = 9y - 4           ............(1)

Case-2

When 3 is added to both numerator and denominator, then

$\sf{\implies \dfrac{x+3}{y+3}=\dfrac{5}{6}}$

=> 6x + 18 = 5y + 15

=> 6x = 5y - 3

$\sf{\implies x = \dfrac{5y-3}{6}\;\;\;\;.......(2)}$

Put the value of x in equation (1), we get

$\sf{\implies 11\bigg(\dfrac{5y-3}{6}\bigg)=9y-4}$

=> 11(5y - 3) = 54y - 24

=> 55y - 33 = 54y - 24

=> y = 9

Put the value of y in equation 2, we get

$\sf{\implies x = \dfrac{5y-3}{6}}$

$\sf{\implies \dfrac{5\times 9-3}{6}=\dfrac{42}{6}=7}$

=> x = 7

$\sf{\implies So,\;required\;fraction=\dfrac{7}{9}}$