The numerator of a fraction is 8 less than the denominator.when 6 is added to the numerator and 10 is added to the denominator,the value of the fraction is unchanged.find the original fraction.
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d-8/d = d-2/d+10
(d-8)(d+10) = d(d-2)
d^2 + 2d - 80 = d^2 - 2d
4d = 80
d = 20
check:
12/20 = 18/30
To find the original fraction, let's assign variables to the numerator and denominator. Let x represent the denominator of the fraction.
According to the problem, the numerator is 8 less than the denominator, so the numerator can be represented as x - 8.
The original fraction is then (x - 8)/x.
Next, we are given that when 6 is added to the numerator and 10 is added to the denominator, the value of the fraction remains unchanged.
So, we can set up the equation:
(x - 8 + 6) / (x + 10) = (x - 8) / x
Now, let's solve this equation to find the value of x.
Cross-multiply:
x - 8 + 6 = (x - 8)(x + 10)
Simplify:
x - 2 = x^2 + 2x - 80
Move all terms to one side of the equation:
0 = x^2 + 2x - x - 2 - 80
Combine like terms:
0 = x^2 + x - 82
Now, we have a quadratic equation. We can factor it or use the quadratic formula to solve for x. In this case, the equation does not factor easily, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation: x^2 + x - 82 = 0, a = 1, b = 1, c = -82.
Plugging in the values:
x = (-1 ± √(1^2 - 4(1)(-82))) / 2(1)
x = (-1 ± √(1 + 328)) / 2
x = (-1 ± √329) / 2
The solutions to this equation are approximately:
x ≈ 8.64
x ≈ -9.63
Since the denominator of a fraction cannot be negative, we discard the negative solution.
Therefore, the original fraction is (x - 8)/x = (8.64 - 8)/8.64 = 0.64/8.64.
Simplifying the fraction, we get 8/107 as the original fraction.