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 denominator it becomes 5/6 . find the fraction

Answer 1

Answer:

The fraction will be 7/9

Solution:

Given,  

Let the numerator of the fraction be taken as x and the denominator be taken as y

From the given information,

The fraction will be 9/11 if we add 2 on both x and y

Thus,

$\begin{aligned} \frac { x + 2 } { y + 2 } & = \frac { 9 } { 11 } \\\\ 11 ( x + 2 ) & = 9 ( y + 2 ) \\\\ 11 x + 22 & = 9 y + 18 \\\\ 11 x - 9 y & = 18 - 22 \\\\ 11 x - 9 y & = - 4 \ldots ( i ) \end{aligned}$

Similarly, from the given,

The fraction will be 5/6 if we add 3 on both x and y

Thus,

$\begin{array} { c } { \frac { x + 3 } { y + 3 } = \frac { 5 } { 6 } } \\\\ { 6 ( x + 3 ) = 5 ( y + 3 ) } \\\\ { 6 x + 18 = 5 y + 15 } \\\\ { 6 x - 5 y = 15 - 18 } \\\\ { 6 x - 5 y = - 3 \ldots ( i i ) } \end{array}$

Multiply (i) by 6 and (ii) by 11, we get,

$\begin{array} { l } { 66 x - 54 y = - 24 } \\\\ { 66 x - 55 y = - 33 } \end{array}$

Subtracting these two, we get,

$y=33-24=9\\\\y=9$

Substituting this in equation (i), we get,

$\begin{array} { c } { 11 x - 9 \times 9 = - 4 } \\\\ { 11 x - 81 = - 4 } \\\\ { 11 x = - 4 + 81 } \\\\ { 11 x = 77 } \\\\ { x = 7 } \end{array}$

Thus, the fraction will be 7/9

Answer 2

Answer:

Let the numerator and denominator be x and y respectively.

Therefore, the required fraction will be x/y

According to Question now,

➳ (x + 2)/(y + 2) = 9/11

➳ 11(x + 2) = 9(y + 2)

➳ 11x + 22 = 9y + 18

➳ 22 - 18 = 9y - 11x

➳ 4 = 9y - 11x

➳ 11x = 9y - 4

➳ x = 9y - 4/11.......[Equation (i)]

Now, it is given that 3 is added to both numerator and denominator it becomes 5/6 :]

➳ (x + 3)/(y + 3) = 5/6

➳ (x + 3)6 = 5(y + 3)

➳ 6x + 18 = 5y + 15

➳ 6x - 5y + 18 - 15 = 0

➳ 6x - 5y + 3 = 0

➳ 6(9y - 4)/11 - 5y + 3 = 0

➳ 54y - 24/11 = 5y - 3

➳ 54y - 24 = 11(5y - 3)

➳ 54y - 24 = 55y - 33

➳ -24 + 33 = 55y - 54y

➳ 9 = y

Now, Putting y = 9 in equation (i) we get,

➳ x = 9y - 4/11

➳ x = 9(9) - 4/11

➳ x = 81 - 4/11

➳ x = 77/11

➳ x = 7

Therefore, the required fraction is 7/9.