The denominator of a fraction is one more than thrice the numerator.The difference between the reciprocal of the fraction and the fraction is 3 3/4. Find the fraction. Show your workings,pls I need it now
(3x+1)/x - x/(3x+1) = 15/4
solve for x
x / (3x +1)
(3x+1) /x - 15/4 = x / (3x+1)
To find the fraction, let's first assign variables to represent the numerator and denominator. Let's call the numerator "x" and the denominator "y".
According to the problem, the denominator is one more than three times the numerator. So, we can write the equation:
y = 3x + 1 ...(Equation 1)
Now, let's consider the second statement. The difference between the reciprocal of the fraction and the fraction is 3 3/4, which can be written as 15/4. The reciprocal of a fraction can be found by inverting the numerator and denominator. So, the reciprocal of the fraction is "y/x" (since the fraction is x/y). So, we can write another equation:
(y/x) - (x/y) = 15/4
Now, let's simplify the second equation by getting rid of the fractions. We can do this by multiplying both sides of the equation by (4xy) to eliminate the denominators:
(4xy) * [(y/x) - (x/y)] = (4xy) * (15/4)
4y^2 - 4x^2 = 15xy
Now, let's simplify the equation:
4y^2 - 4x^2 - 15xy = 0 ...(Equation 2)
We have obtained two equations:
y = 3x + 1 ...(Equation 1)
4y^2 - 4x^2 - 15xy = 0 ...(Equation 2)
Now, we can solve these two equations simultaneously to find the values of x and y, which will give us the numerator and denominator of the fraction.
To solve the equations, we can substitute Equation 1 into Equation 2, replacing "y" with "3x + 1":
4(3x + 1)^2 - 4x^2 - 15x(3x + 1) = 0
Next, simplify and expand that expression:
4(9x^2 + 6x + 1) - 4x^2 - 45x - 15x = 0
36x^2 + 24x + 4 - 4x^2 - 45x - 15x = 0
(36x^2 - 4x^2) + (24x - 45x - 15x) + 4 = 0
32x^2 - 36x + 4 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, the equation can be factored as:
4(8x^2 - 9x + 1) = 0
Setting each factor equal to zero:
8x^2 - 9x + 1 = 0
Now, we can solve for "x" by factoring or using the quadratic formula. Factoring does not work for this equation, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 8, b = -9, and c = 1. Plugging these values into the quadratic formula:
x = (-(-9) ± √((-9)^2 - 4(8)(1))) / (2(8))
x = (9 ± √(81 - 32)) / 16
x = (9 ± √49) / 16
Now, we have two possible solutions for "x":
x1 = (9 + 7) / 16 = 16 / 16 = 1
x2 = (9 - 7) / 16 = 2 / 16 = 1/8
So, the possible values for "x" are 1 and 1/8.
Now, we can substitute these values back into Equation 1 to find the corresponding values for "y":
For x = 1:
y = 3(1) + 1 = 4
For x = 1/8:
y = 3(1/8) + 1 = 3/8 + 8/8 = 11/8
Therefore, the possible fractions are:
1/4 and 1/8
Both of these fractions satisfy the given conditions: the denominator is one more than three times the numerator, and the difference between their reciprocal and the fraction is 3 3/4.