Problem:
The problem states that the sum of four times a fraction and seven times its reciprocal is 16. We need to find the value of the fraction.

Solution:
Let's assume the fraction to be x.

Step 1: Translate the problem into an equation:
According to the problem, the sum of four times the fraction and seven times its reciprocal is 16. This can be written as:

4x + 7(1/x) = 16

Step 2: Simplify the equation:
To simplify the equation, we need to eliminate the denominator. We can do this by multiplying the entire equation by x:

4x^2 + 7 = 16x

Step 3: Rearrange the equation:
Rearrange the equation to bring all the terms on one side:

4x^2 - 16x + 7 = 0

Step 4: Solve the quadratic equation:
To solve the quadratic equation, we can either factorize it or use the quadratic formula. In this case, the equation cannot be easily factorized, so we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our equation, a = 4, b = -16, and c = 7. Substituting these values into the quadratic formula, we get:

x = (-(-16) ± √((-16)^2 - 4(4)(7))) / (2(4))
= (16 ± √(256 - 112)) / 8
= (16 ± √144) / 8
= (16 ± 12) / 8

This gives us two possible solutions for x:

x1 = (16 + 12) / 8 = 28 / 8 = 7/2
x2 = (16 - 12) / 8 = 4 / 8 = 1/2

Step 5: Check the solutions:
We need to check if the solutions satisfy the given equation. Let's substitute x = 7/2 and x = 1/2 into the original equation:

For x = 7/2:
4(7/2) + 7(1/(7/2)) = 16
14 + 7(2/7) = 16
14 + 2 = 16
16 = 16

For x = 1/2:
4(1/2) + 7(1/(1/2)) = 16
2 + 7(2) = 16
2 + 14 = 16
16 = 16

Both solutions satisfy the equation. However, the problem asks for the value of the fraction, which is x. Therefore, the correct solution is x = 7/2, which is option B.