Question

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

Answer 1

$\sf \large \underline{ \underline{ \: Solution : \: \: \: }}$

Let , the fraction be x/y

By the given condition ,

When 2 Is added to the both numerator and denominator , the fraction will become 9/11

$\sf \hookrightarrow \frac{x + 2}{y + 2} = \frac{9}{11} \\ \\\sf \hookrightarrow 11x + 22 = 9y + 18 \\ \\\sf \hookrightarrow 11x - 9y = - 4\: - - - - \: eq(i)$

When 3 Is added to both numerator and denominator , the fraction will become 5/6

$\sf \hookrightarrow \frac{x + 3}{y +3 } = \frac{5}{6} \\ \\\sf \hookrightarrow 6x + 18 = 5y + 15 \\ \\\sf \hookrightarrow 6x - 5y = - 3 \: - - - - \: eq(i)$

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

$\star \: \: \sf 66x - 54y = - 24 \: - - - - \: eq(iii) \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: and \\ \star \: \: \sf 66x - 55y = - 33 \: - - - - \: eq(iv)$

Subtract eq (iii) from eq (iv) , we obtain

$\sf \hookrightarrow 66x - 54y - (66x - 55y) = - 24 - (-33) \\ \\\sf \hookrightarrow - 54y + 55y = 9 \\ \\\sf \hookrightarrow y = 9$

Put the value of y = 9 in eq(i) , we obtain

$\sf \hookrightarrow 11x - 9(9) = - 4 \\ \\\sf \hookrightarrow 11x = - 4 + 81 \\ \\\sf \hookrightarrow 11x = 77 \\ \\\sf \hookrightarrow x = 7$

Therefore , the fraction is 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.