the denominator of a fractio is an integer that is one greater than the numerator. A second fraction is the reciprocal of the integer that is one less than the numerator of the first fraction. If the product of the two fraction is 3/8 identify the first fraction.
To solve this problem, let's break it down step by step.
Step 1: Understanding the given information
From the problem statement, we know that the denominator of the first fraction is one greater than the numerator. Let's denote the numerator as "x" and the denominator as "x+1". So, the first fraction is x/(x+1).
We also know that the second fraction is the reciprocal of the integer that is one less than the numerator of the first fraction. Denoting the numerator of the first fraction as "x", the numerator of the second fraction is "x-1". Therefore, the second fraction is 1/(x-1).
Step 2: Set up the equation
The product of two fractions is obtained by multiplying their numerators and denominators. So, we can set up the equation as follows:
(x/(x+1)) * (1/(x-1)) = 3/8
Step 3: Solve the equation
To solve the equation, we'll start by cross-multiplying. That means multiplying the numerators together and the denominators together:
(x * 1) / [(x+1) * (x-1)] = 3/8
Simplifying the equation, we get:
x / (x^2 - 1) = 3/8
Now cross-multiply again:
8x = 3(x^2 - 1)
8x = 3x^2 - 3
Rearrange the equation to form a quadratic equation:
3x^2 - 8x - 3 = 0
Step 4: Solve the quadratic equation
We can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring may not be easily applicable, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -8, and c = -3. Plugging in these values, we get:
x = (-(-8) ± √((-8)^2 - 4 * 3 * -3)) / (2 * 3)
x = (8 ± √(64 + 36)) / 6
x = (8 ± √100) / 6
x = (8 ± 10) / 6
Simplifying further:
x1 = (8 + 10) / 6 = 18 / 6 = 3
x2 = (8 - 10) / 6 = -2 / 6 = -1/3
Step 5: Identify the first fraction
Since the denominator of a fraction cannot be negative, we discard x2 = -1/3 as a solution. Therefore, the first fraction is x1 = 3/(3+1), which simplifies to 3/4.
So, the first fraction is 3/4.