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The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, then they are in the ratio 2:3. What is the fraction?

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59

39

23

13

answer is A.

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Detailed Solution

Given that, sum of the numerator and denominator of a fraction is 4 more than twice the numerator.
Let the numerator be x and the denominator be y.
So, the fraction is equal to

x y

.
According to the problem, the sum of the numerator and denominator of a fraction is 4 more than twice the numerator. Then,

x + y = 2 x + 4 ⇒ x − 2 x + y = 4 ⇒ − x + y = 4 ....... (1)

According to the problem, if the numerator and denominator are increased by 3, then they are in the ratio 2 : 3. Then,

x + 3 y + 3 = 23 ⇒ 3(x + 3) = 2 y + 3 ​ ​ [​ ​ cross multiply the terms ​ ​] ​ ​ ⇒ 3 x + 9 = 2 y + 6 ⇒ 3 x − 2 y = 6 − 9 ⇒ 3 x − 2 y = − 3 ...... 2

From equation 1, we get

y = 4 + x ... (3)

Substituting the value of

y = 4 + x

in second equation and solving it, we get,

3 x − 2 y = − 3 ⇒ 3 x − 2 4 + x = − 3 ⇒ 3 x − 8 − 2 x = − 3 ⇒ x = − 3 + 8 ⇒ x = 5

Putting the value of x = 5 in equation 3 to find the value of y, we get,

y = 4 + x ⇒ y = 4 + 5 ⇒ y = 9

So, the numerator and denominator are 5 and 9.
Now,

F r a c t i o n = Numerator Denominator ⇒ F r a c t i o n = x y ⇒ F r a c t i o n = 59 .

Hence, the fraction is

59

.
Thus, option 1 is correct.

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