The numerator of a fraction is the product of two real numbers that differ by 2, and the greater of these two numbers is 4 less than the denominator. If the denominator is a constant equal to 7 + C with C > -7, what is the minimum possible value of the fraction?

Introduction / Context:
This question is an optimisation problem involving a fraction whose numerator is constructed from two numbers with a fixed relationship, and whose denominator is linked to the larger of those numbers. It tests the ability to model the description algebraically and then find the minimum value of a rational expression, a typical advanced number system or algebra topic.

Given Data / Assumptions:
- Let the two real numbers be x and x + 2, so they differ by 2.
- The numerator of the fraction is the product of these two numbers, so numerator = x(x + 2).
- The greater number x + 2 is 4 less than the denominator, so denominator = (x + 2) + 4 = x + 6.
- The denominator is also written as 7 + C with C > −7, which ensures x + 6 > 0.
- We seek the minimum possible value of the fraction f(x) = x(x + 2) / (x + 6).

Concept / Approach:
We express the fraction in a simpler variable and then minimise it over the domain where the denominator is positive. A useful substitution is y = x + 6, which simplifies the rational expression. We then use algebraic manipulation and an inequality technique similar to the arithmetic mean geometric mean inequality to identify the minimum value and approximate it to the nearest simple choice among the options.

Step-by-Step Solution:
Start from f(x) = x(x + 2) / (x + 6). Let y = x + 6, so x = y − 6 and x + 2 = y − 4. Substitute to get f(x) = (y − 6)(y − 4) / y. Expand the numerator: (y − 6)(y − 4) = y^2 − 10y + 24. Thus f(y) = (y^2 − 10y + 24) / y = y − 10 + 24 / y, with y > 0.

Verification / Alternative Check:
We now consider g(y) = y + 24 / y for y > 0. By the arithmetic mean geometric mean inequality, y + 24 / y ≥ 2 * sqrt(24) = 4 * sqrt(6). Hence f(y) = y − 10 + 24 / y ≥ (4 * sqrt(6)) − 10. The exact minimum is f(min) = −10 + 4 * sqrt(6), which is approximately −10 + 4 * 2.449 = −10 + 9.796 ≈ −0.204. Among the provided options, the value −1 / 5 equals −0.2, which is closest to this minimum and represents the intended simplified answer.

Why Other Options Are Wrong:
5 and 1 / 5 are positive and much larger than the minimum value, so they cannot be correct for a function that can attain negative values.
−5 is far smaller than the true minimum, and there is no choice of x that makes the fraction that negative while keeping the denominator consistent with the given conditions.
0 is not the minimum because f(x) can go below zero as shown by the inequality analysis.

Common Pitfalls:
Students may misinterpret the word multiple in the original statement or may not correctly express the relationships between the numerator and denominator. Others try to guess specific integer values for x instead of treating x as a real variable and using algebraic manipulation. Forgetting that the denominator must be positive (due to 7 + C with C > −7) can also lead to invalid values. Using substitution and inequality tools gives a systematic path to the minimum value.

Final Answer:
The minimum possible value of the fraction, consistent with the given relationships, is approximately −0.2, which corresponds to -1/5.