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A fraction becomes $\frac{9}{11}$ , if $2$ is added to both the numerator and the denominator. If $3$ is added to both the numerator and the denominator it becomes $\frac{5}{6}$ . Find the fraction.

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\frac{7}{9}

\frac{5}{7}

\frac{9}{11}

None of these

answer is A.

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Detailed Solution

Given a fraction.
Let

\frac{x}{y}

be the fraction, where

x

and

y

are positive integers.

⇒ x + 2 y + 2 = 911

……....(1)

\Rightarrow \frac{x + 3}{y + 3} = \frac{5}{6}

………...(2)
From equation (1) we get,

⇒ 11 x + 22 = 9 y + 18

⇒ 11 x − 9 y = − 4

..............(3)
From equation (2) we get,

⇒ 6 x + 18 = 5 y + 15

⇒ 6 x − 5 y = − 3

……....(4)
From equation (4) will get,

y = 3 + 6 x 5

Substituting

y = 3 + 6 x 5

in equation (3) and then multiply it by 5 throughout,

⇒ 11 x − 9 3 + 6 x 5 = − 4 ⇒ 55 x − 27 − 54 x = − 20 ⇒ 55 x − 54 x = − 20 + 27 ⇒ x = 7

Substituting

x = 7

in equation (4),

⇒ 6 × 7 − 5 y = − 3 ⇒ 42 − 5 y = − 3 ⇒ − 5 y = − 3 − 42 ⇒ − 5 y = − 45 ⇒ y = 9

Thus, the fraction

\frac{x}{y} = \frac{7}{9}

.
Therefore, option (1) is correct.

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