Question 753794

The denominator of a positive common fraction is 3 more than its numerator.  If 5/28 is added to this fraction, the result is the same as the positive difference between the reciprocal of the original fraction and 1.  What is the common fraction?

Here is the equation I come up with but I have stalled...
(x/x+3)+ 5/28 = (x+3/x)-1


Let numerator be N
Then denominator is N + 3
{{{N/(N + 3) + 5/28 = (N + 3)/N - 1}}} (You’re correct up to this point)


28N(N) + 5N(N + 3) = 28(N + 3)(N + 3) – 1(28N)(N + 3) ----- Multiplying by LCD, 28(N)(N + 3)


{{{28N^2 + 5N^2 + 15N = 28(N^2 + 6N + 9) - 1(28N^2 + 84N)}}}


{{{28N^2 + 5N^2 + 15N = 28N^2 + 168N + 252 - 28N^2 - 84N}}}


{{{28N^2 + 5N^2 + 15N = 28N^2 - 28N^2 + 168N - 84N + 252}}}


{{{33N^2 + 15N = 84N + 252}}}


{{{33N^2 + 15N - 84N - 252 = 0}}}


{{{33N^2 - 69N - 252 = 0}}}


{{{3(11N^2 - 23N - 84) = 3(0)}}}


{{{11N^2 - 23N - 84 = 0}}}


{{{11N^2 - 44N + 21N - 84 = 0}}}


11N(N - 4) + 21(N - 4) = 0


(N - 4)(11N + 21) = 0


N – 4 = 0   or   11N + 21 = 0 


N, or numerator = 4   OR  11N = - 21 (ignore)


Since N or numerator = 4, then denominator = 4 + 3, or 7


Common fraction is: {{{highlight_green(4/7)}}}


You can do the check!!


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