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 numerator of the required fraction be x.
Then, its denominator `=(2x+1).`
`:." ""fraction "=(x)/((2x+1))" and its reciprocal "=((2x+1))/(x).`
`:." "(x)/((2x+1))+((2x+1))/(x)=(58)/(21)`
`implies" "21xx[x^(2)+(2x+1)^(2)]=58x(2x+1)`
`implies" "21xx[5x^(2)+4x+1]=116x^(2)+58x`
`implies" "11x^(2)-26x-21=0`
`implies" "11x^(2)-33x+7x-21=0implies11x(x-3)+7(x-3)=0`
`implies" "(x-3)(11x+7)=0impliesx-3=0" or "11x+7=0`
`implies" "x=3" or "x=(-7)/(11)`
`:." "x=3" "[because" numerator cannot be a negative fraction"].`
`:." ""required fraction "=(x)/((2x+1))=(3)/(7).`