The denominator of a fraction is one more than thrice the numerator.the difference is between the reciprocal of the fraction and the fraction is 15/4.find the fraction.
n(3n + 1)/n - n/(3n + 1)
To find the fraction described in the problem, we need to set up an equation based on the given information.
Let's assume the numerator of the fraction is represented by the variable 'x'. According to the problem, the denominator of the fraction is one more than thrice the numerator.
So, the denominator can be written as 3x + 1.
The problem further states that the difference between the reciprocal of the fraction and the fraction is 15/4.
The reciprocal of a fraction is obtained by flipping the numerator and the denominator of the fraction.
So, the reciprocal of the fraction can be written as (3x + 1)/x.
Now, the difference between the reciprocal and the original fraction is given as 15/4. We can set up the equation:
(3x + 1)/x - x/(3x + 1) = 15/4
To solve this equation, we can start by clearing the fractions:
(4)((3x + 1)/x) - (4)(x/(3x + 1)) = 15
Now, we simplify:
(12x + 4)/x - 4x/(3x + 1) = 15
Next, we can remove the denominators by multiplying both sides of the equation by x(3x + 1):
(x(3x + 1))((12x + 4)/x) - (x(3x + 1))(4x/(3x + 1)) = (15)(x(3x + 1))
Simplifying further, we get:
(12x + 4)(3x + 1) - 4x(3x + 1) = 15x(3x + 1)
Now, expand and simplify:
36x^2 + 12x + 12x + 4 - 12x^2 - 4x = 45x^2 + 15x
Combining like terms:
36x^2 + 24x + 4 = 45x^2 + 15x
Bringing all terms to one side, we get:
45x^2 + 15x - 36x^2 - 24x - 4 = 0
Combine like terms again:
9x^2 - 9x - 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 9, b = -9, and c = -4.
Substituting these values into the formula, we get:
x = (-(-9) ± √((-9)^2 - 4(9)(-4))) / (2(9))
Simplifying further:
x = (9 ± √(81 + 144)) / 18
x = (9 ± √225) / 18
Now, we have two possible solutions for x:
x = (9 + 15) / 18 = 24 / 18 = 4/3
or
x = (9 - 15) / 18 = -6 / 18 = -1/3
Since the denominator of a fraction cannot be negative, we discard the solution x = -1/3.
Therefore, the fraction is 4/3.
put the words into math ...
fraction = n/(3n+1)
so now do the math
(3n+1)/n - n/(3n+1) = 15/4
now just find n and thus the fraction.