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

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Cecilia -Let a/b = fractionLet b/a = reciprocalNow create your equation:b/a - a/b = 77/18Get a common denominator for the right side:(b^2 - a^2) / (a*b) = 77/18Now you have two equations ... set the numerators equal to each other and the denominators equal to each other:b^2 - a^2 = 77a*b =18Just solve these two equations using substitution by letting a = 18*b and substitute this into the first equation to solve for b.Hope this helps

Corrections - (1) Get a common denominator for the "left" side of the equation(2) If a*b = 18, then let a = 18 ÷ b.  Now substitute this into the first equation to solve for b,  Once you know b, then a = 18 ÷ b.