Question

In an unknown fraction the denominator is one more than twice the numerator. When 2 1/10 is added to this fraction, the result is equal to the reciprocal of the unknown fraction.

Find the unknown fraction.
(Hint; the the numerator of the fraction to be the x)

Answer 1

using their hint:
let the numerator of original fraction be x
then the denominator is 2x+1

so the original fraction is x/(2x+1)

condition stated:

x/(2x+1) + 2 1/10 = (2x+1)/x
x(2x+1) + 21/10 = (2x+1)/x
multiply each term by 10x(2x+1) , the LCD

10x^2 + 21x(2x+1) = 10(2x+1)^2
10x^2 + 42x^2 + 21x = 40x^2 + 40x + 10
12x^2 - 19x - 10 = 0
(x-2)(12x + 5) = 0
x = 2 or x = -5/12

if x = 2, the original fraction was 2/5
check:
2/5 + 21/10 = 5/2, that works

could it work for x = -5/12 ?
the fraction would be
(-5/12) / (-10/12 + 1)
= -5/2

what if we add 21/10 to -5/2 ??
-5/2 + 21/10 = -2/5

so the original fraction could have been
2/5 or -5/2

(I verified that either one will work)