If a proper fraction in lowest terms is subtracted from its reciprocal, the difference is 77/18. What is the proper fraction.

x - 1/x = 77/18

multiply by the common denominator 18x

18x (x - 1/x) = 77/18 * 18x

distribute

18x^2 - 18 = 77x

subtract 77 x from each side

18x^2 - 77x-18=0

factor

(2 x - 9) (9 x + 2) = 0

using the zero property

2x-9 = 0 9x+2 = 0

2x=9 9x=-2

x = 9/2 x = - 2/9

x cannot = 9/2 because this is not a proper fraction in the lowest terms

so x = - 2/9