SOLUTION: 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 recipr

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: 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 recipr      Log On


   

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
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Let d = denominator
Let n = numerator.

First sentence means, d=3%2Bn, so the common fraction is highlight%28n%2F%28n%2B3%29%29.

IF 5%2F28%2Bn%2F%28n%2B3%29, THEN abs%281-%28n%2B3%29%2Fn%29. We want to find or solve for n.

The positive value of the difference is part of what we want to use, so let's just write the value in its expected positive form:
%28n%2B3%29%2Fn-1
which we could simplify
%28n%2B3%29%2Fn-n%2Fn
%28n%2B3-n%29%2Fn
3%2Fn

What should be done with that? Back to the whole second sentence and the symbolic meaning of IF,THEN...
highlight%285%2F28%2Bn%2F%28n%2B3%29=3%2Fn%29

You should expect better results continueing from that equation, so the rest is being left for you to try (but be sure to understand the path to the last highlighted equation first).

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

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%2F%28N+%2B+3%29+%2B+5%2F28+=+%28N+%2B+3%29%2FN+-+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%5E2+%2B+5N%5E2+%2B+15N+=+28%28N%5E2+%2B+6N+%2B+9%29+-+1%2828N%5E2+%2B+84N%29

28N%5E2+%2B+5N%5E2+%2B+15N+=+28N%5E2+%2B+168N+%2B+252+-+28N%5E2+-+84N

28N%5E2+%2B+5N%5E2+%2B+15N+=+28N%5E2+-+28N%5E2+%2B+168N+-+84N+%2B+252

33N%5E2+%2B+15N+=+84N+%2B+252

33N%5E2+%2B+15N+-+84N+-+252+=+0

33N%5E2+-+69N+-+252+=+0

3%2811N%5E2+-+23N+-+84%29+=+3%280%29

11N%5E2+-+23N+-+84+=+0

11N%5E2+-+44N+%2B+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%284%2F7%29

You can do the check!!

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