Question

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

Answer 1

let the fraction is x/y.

Then by the given question,

(x+2)/(y+2)=9/11

or, 11x+22=9y+18

or, 11x-9y=18-22

or, 11x-9y=-4

or, 11x=9y-4

or, x=(9y-4)/11.

and, (x+3)/(y+3)=5/6

or, 6x+18=5y+15

or, 6x-5y=15-18

or, 6(9y-4)/11-5y=-3

or, (54y-24-55y)11=-3

or, -y-24=-3×11

or, y+24=33

or, y=33-24=9

then, x=(9×9-4)/11

or, x=(81-4)/11.

or, x=77/11=7

∴, x=7,y=9 and the

Answer 2

Given :

• A fraction becomes 9/11 if 2 is added to both numerator and denominator.
• If 3 is added to numerator and denominator both fraction becomes 5/6.

To Find :

• The fraction.

Solution :

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Fraction = $\bold{\dfrac{x}{y}}$

Case 1 :

When 2 is added to the numerator and denominator of the fraction, it becomes 9/11.

Numerator = (x+2)

Denominator = (y+2)

Equation :

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

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

$\implies$ $\sf{11x+22=9y+18}$

$\implies$ $\sf{11x-9y=18-22}$

$\implies$ $\sf{11x-9y=-4}$

$\implies$ $\sf{11x=-4+9y}$

$\large{\sf{x=\dfrac{-4+9y}{11}\:\:\:(1)}}$

Case 2 :

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

Numerator = (x+3)

Denominator = (y+3)

Equation :

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

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

$\implies$ $\sf{6x+18=5y+15}$

$\implies$ $\sf{6x-5y=15-18}$

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

$\implies$ $\sf{6\:\Big(\dfrac{-4+9y}{11}\Big)-5y=-3}$$\bold{\Big[From\:equation\:(1)\:x\:=\:\dfrac{-4+9y}{11}\Big]}$

$\implies$ $\sf{\dfrac{-24+54y}{11}-5y\:=\:-3}$

$\implies$ $\sf{\dfrac{-24+54y-55y}{11}\:=\:-3}$

$\implies$ $\sf{-24+54y-55y=11\:\times\:-3}$

$\implies$ $\sf{-24+54y-55y=-33}$

$\implies$ $\sf{54y-55y=-33+24}$

$\implies$ $\sf{-y=-9}$

$\implies$ $\sf{y=9}$

Substitute, y = 9 in equation (1),

$\implies$ $\sf{x=\dfrac{-4+9y}{11}}$

$\implies$ $\sf{x=\dfrac{-4+9(9)}{11}}$

$\implies$ $\sf{x=\dfrac{-4+81}{11}}$

$\implies$ $\sf{x=\dfrac{77}{11}}$

$\implies$ $\sf{x=7}$

$\large{\boxed{\bold{Numerator\:of\:the\:fraction\:=\:x\:=\:7}}}$

$\large{\boxed{\bold{Denominator\:of\:the\:fraction\:=\:y\:=\:9}}}$

$\large{\boxed{\bold{\red{Fraction\:=\:\dfrac{x}{y}\:=\:\dfrac{7}{9}}}}}$