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.

Explanation

Let, the fraction be `(x)/(y)` where numerator is x, then denominator =2x+1
Accroding to give statement
`(x)/(2x+1)+(2x+1)/(x)=2(16)/(21)" "……(i)`
Now, let `(x)/(2x+1)=a` than from (1)
`a+(1)/(a)=(58)/(21)" "implies" "(a^(2)+1)/(a)=(58)/(21)`
`implies21a^(2)+21=58a`
`implies21a^(2)-58a+21=0`
`implies21a^(2)-58a+21=0`
`implies21a^(2)-49a-9a+21=0`
`implies7a(3a-7)-3(3a-7)=0`
`implies(7a-3)(3a-7)=0`
`impliesa=(3)/(7)ora=(7)/(3)`
when `a=(3)/(7)implies(x)/(2x+1)=(3)/(7)`
`implies7x=6x+3`
`impliesx=3`
when `a=(7)/(3)implies(x)/(2x+1)=(7)/(3)`
`implies3x=14x+7`
`implies-11x=7`
`impliesx=(-7)/(11)`
`impliesx=3" "(x=(-7)/(11)"is not a natural number")`
Hence, required fraction is `(x)/(2x+1)=(3)/(7)`