The denominator of a fraction is one more than thrice the numerator. The difference between the reciprocal of the fraction and the fraction is 15/4

if the fraction is n/(3n+1) then you want to solve

(3n+1)/n - n/(3n+1) = 15/4

To solve the problem, let's start by setting up an equation based on the given information.

Let's assume the numerator of the fraction is represented by 'x'.

According to the given information, the denominator of the fraction is one more than thrice the numerator. So, the denominator can be represented by the expression '1 + 3x'.

The fraction can, therefore, be written as x / (1 + 3x).

The reciprocal of the fraction is obtained by flipping the numerator and denominator, which gives (1 + 3x) / x.

The difference between the reciprocal of the fraction and the fraction is 15/4, so we can set up the following equation:

(1 + 3x) / x - x / (1 + 3x) = 15/4.

To simplify the equation, we can multiply through by the LCD (Least Common Denominator) of 4x(1 + 3x), which is 4x(1 + 3x).

4x(1 + 3x) * [(1 + 3x) / x] - 4x(1 + 3x) * [x / (1 + 3x)] = 15/4 * 4x(1 + 3x).

Simplifying further, we get:

4(1 + 3x)^2 - 4x^2 = 15(1 + 3x).

Expanding the squares:

4(1 + 6x + 9x^2) - 4x^2 = 15 + 45x.

Now, distribute the 4 on the left side:

4 + 24x + 36x^2 - 4x^2 = 15 + 45x.

Combine like terms:

32x + 32x^2 = 15 + 45x.

Rearranging terms:

32x^2 - 13x - 15 = 0.

Now, we can use the quadratic formula to find the values of x that satisfy this equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a.

For this equation, a = 32, b = -13, and c = -15.

Substituting these values into the quadratic formula:

x = (-(-13) ± √((-13)^2 - 4(32)(-15))) / (2(32)).

Simplifying further:

x = (13 ± √(169 + 1920)) / 64.

x = (13 ± √2089) / 64.

These are the possible values for the numerator of the fraction.

To find the corresponding denominators, we can substitute each value of x into the expression for the denominator, which is 1 + 3x.

For example, if we take the positive square root:

Denominator = 1 + 3x

= 1 + 3 * [(13 + √2089) / 64].

This will give us the corresponding value of the denominator.

Similarly, substitute the other value of x to find the second denominator.

Therefore, the numerator and denominator of the fraction can be calculated by evaluating the expressions for x and substituting it into the respective equations above.

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