The denominator of a fraction is one more than twice the numerator. If the sum of the . . .

Question : The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 16/21 , find the fraction. [CBSE D 2024] [5 Marks]

Solution : 

Let the numerator be x
and denominator be 1+2x
Required fraction = x/(1+2x)
Reciprocal of fraction = (1+2x)/x

ATQ
x/(1+2x) + (1+2x)/x=2 (16/21)
x/(1+2x) + (1+2x)/x=58/21

Let x/(1+2x) = y
and (1+2x)/x = 1/y

y+1/y=58/21

(y²+1)/y=58/21
21y²+21=58y
21y²-58y+21=0
21y²-(49+9)y+21=0
21y²-49y-9y+21=0
7y(3y-7)-3(3y-7)=0
(3y-7)(7y-3)=0

3y-7=0
3y=7
y=7/3

Or 

7y-3=0
7y=3
y=3/7

Now, 
x/(1+2x) = 7/3
3x=7(1+2x)
3x=7+14x
3x-14x=7
-11x=7
x=7/-11
x=-7/11
Rejected because x can't be negative.

x/(1+2x) = 3/7
7x=3+6x
7x-6x=3
x=3

Required Fraction = x/(1+2x)
=3/[1+2(3)]
=3/[1+6]
=3/7

Hence, the required fraction is 3/7.

Video Solution :