The numerator of a fraction is four less than the denominator The sum of the fraction and its reciprocal is 5/2 Find the numerator and denominator if each is a positive integer
denominator --- x
numerator ---- x-4
(x/x-4) + (x-4)/x = 5/2
times 2x(x-4)
2x^2 + 2(x-4)^2 = 5x(x-4)
2x^2 + 2x^2 - 16x + 32 = 5x^2 - 20x
-x^2 + 4x + 32 = 0
x^2 -4x - 32 = 0
(x-8)(x+4) = 0
x = 8 or x = -4 , but x was positive
so x = 8
so original fraction was 4/8 or 1/2
check:
1/2 + 2/1 = 2 1/2 = 5/2
the denominator of a fraction is 1 more than the numerator.the sum of the fraction and its reciprocal is 2 1/30.find the fraction
Let's assume the denominator of the fraction is x.
According to the given information, the numerator would be x - 4.
The fraction itself can be written as (x - 4)/x.
The reciprocal of the fraction would be x/(x - 4).
Now, the sum of the fraction and its reciprocal is given as 5/2:
(x - 4)/x + x/(x - 4) = 5/2
To solve this equation, we can first cross-multiply to eliminate the denominators:
2(x - 4)(x - 4) + 2x*x = 5x(x - 4)
Simplifying further:
2(x^2 - 8x + 16) + 2x^2 = 5x^2 - 20x
Expanding and rearranging the terms:
2x^2 - 16x + 32 + 2x^2 = 5x^2 - 20x
4x^2 + 20x - 5x^2 + 16x - 32 = 0
Combining like terms:
-x^2 + 36x - 32 = 0
Solving this quadratic equation can be quite complex. However, we are given that the numerator and denominator are positive integers.
To find the positive integer solutions, we can use the fact that the product of the solutions of a quadratic equation in the form ax^2 + bx + c = 0 is equal to c/a.
The product of the solutions in this case is equal to -32/-1 = 32.
The positive integer pairs that multiply to 32 are:
1 × 32
2 × 16
4 × 8
Out of these pairs, we need to find a pair that can be used as the numerator and denominator.
Let's try each pair:
1. If we select 1 as the numerator and 32 as the denominator, the fraction is 1/32. However, the sum of 1/32 and its reciprocal is much smaller than 5/2.
2. If we select 2 as the numerator and 16 as the denominator, the fraction is 2/16 or 1/8. The sum of 1/8 and its reciprocal is 5/4, which does not match 5/2.
3. If we select 4 as the numerator and 8 as the denominator, the fraction is 4/8 or 1/2. The sum of 1/2 and its reciprocal is indeed 5/2, which matches the given condition.
Therefore, the numerator and denominator are 4 and 8 respectively.
To solve this problem, let's assume the numerator of the fraction is represented by the variable 'x' and the denominator is represented by the variable 'y' (both positive integers).
From the given information, we can write two equations:
1) The numerator is four less than the denominator: x = y - 4
2) The sum of the fraction and its reciprocal is 5/2: x/y + y/x = 5/2
Using these equations, we can proceed to solve the problem:
From equation 1, we can substitute x = y - 4 into equation 2:
(y - 4) / y + y / (y - 4) = 5/2
Now, let's find a common denominator (2y(y - 4)) to simplify the equation:
[(y - 4) * (y - 4)] / (y * (y - 4)) + [y * y] / (y * (y - 4)) = 5/2
Simplifying further:
(y^2 - 8y + 16) + y^2 = (5/2) * (2y * (y - 4))
2y^2 - 8y + 16 + 2y^2 = 5y^2 - 20y
4y^2 - 20y + 16 = 5y^2 - 20y
Rearranging the equation to form a quadratic equation on one side:
5y^2 - 4y^2 - 20y + 20y - 16 = 0
y^2 - 16 = 0
Using the quadratic formula, we can solve for y:
y = (-(-16) ± sqrt((-16)^2 - 4 * 1 * (-16))) / (2 * 1)
y = (16 ± sqrt(256 + 64)) / 2
y = (16 ± sqrt(320)) / 2
y = (16 ± 8√5) / 2
y = 8 ± 4√5
Since y is a positive integer, y = 8 + 4√5
Now, substituting this value of y into equation 1 to find x:
x = y - 4
x = (8 + 4√5) - 4
x = 4 + 4√5
Hence, the numerator (x) is equal to 4 + 4√5, and the denominator (y) is equal to 8 + 4√5.