SOLUTION: Good day to you. My daughter is having a problem with algebra word problems, can you please help me, i am not so good at this either. The denominator of a fraction is 1 more th

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Good day to you. My daughter is having a problem with algebra word problems, can you please help me, i am not so good at this either. The denominator of a fraction is 1 more th      Log On


   

Question 683653: Good day to you. My daughter is having a problem with algebra word problems, can you please help me, i am not so good at this either.
The denominator of a fraction is 1 more than twice the numerator. If 4 is added to the numerator and subtracted from the denominator, the result is the reciprocal of the fraction. Find the original fraction.
Thank you very much!
Found 2 solutions by ReadingBoosters, Alan3354:
Answer by ReadingBoosters(3246) About Me  (Show Source):
You can put this solution on YOUR website!
x is numerator
y is denominator
...
y = 2x + 1
%28x+%2B+4%29%2F%28y-4%29+=+y%2Fx
...
Cross Multiply
x(x+4) = y(y-4)
x%5E2+%2B+4x+=+y%5E2+-+4y
...
x%5E2+%2B+4x+-+y%5E2+%2B+4y++=+0
...
x%5E2+%2B+4x+-%282x%2B1%29%5E2+%2B+4%282x%2B1%29+=+0 ==> substitution
x%5E2+%2B+4x+-%284x%5E2+%2B+4x+%2B+1%29+%2B+8x+%2B+4+=+0
x%5E2+%2B+4x+-+4x%5E2+-+4x+-+1+%2B+8x+%2B+4+=+0
...
-3x%5E2+%2B+8x+%2B+3+=+0
-1%283x%5E2+-+8x+-+3%29=0
(3x+1)(x-3)=0
3x+1=0
3x=-1
x = -1%2F3
x-3=0
x = 3
...
Using x = -1%2F3
y = 2x%2B1+=+2%28-1%2F3%29+%2B+1+=+-2%2F3+%2B+1+=+-2%2F3+%2B+3%2F3+=+1%2F3
Original fraction
highlight_green%28%28-1%2F3%29%2F%281%2F3%29%29 which is %28-1%2F3%29%283%29+=+-1
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Using x = 3
y = 2x + 1 = 2(3) + 1 = 6 +1 = 7
Original fraction
highlight_green%283%2F7%29
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Testing both originals
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Checking %28-1%2F3%29%2F%281%2F3%29 which is -1
%28-1%2B4%29%2F%281-4%29+=+3%2F-3+=+-1
it works! -1%2F1 and 1%2F-1 are reciprocals
...
checking 3%2F7
%283%2B4%29%2F%287-4%29+=+7%2F3
so
3%2F7 and 7%2F3 are reciprocals
it is correct!
...
Ultimately, there are two original fractions that qualify
%28-1%2F3%29%2F%281%2F3%29 and 3%2F7
..............................
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Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The denominator of a fraction is 1 more than twice the numerator. If 4 is added to the numerator and subtracted from the denominator, the result is the reciprocal of the fraction. Find the original fraction.
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OF = x/(2x + 1)
(x+4)/(2x-3) = 1/OF = (2x+1)/x
(x+4)/(2x-3) = (2x+1)/x
Cross multiply
x(x+4) = (2x-3)*2x+1)
x%5E2+%2B+4x+=+4x%5E2+-+4x+-+3
3x%5E2+-+8x+-+3+=+0
(3x + 1)*(x - 3) = 0
x = 3 --> OF = 3/7
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x = -1/3 --> OF = -1