If (2 + a)/a + (2 + b)/b + (2 + c)/c = 4 for non zero real numbers a, b and c, then what is the value of the expression (ab + bc + ca)/(abc)?

Introduction / Context:
This algebra question involves rational expressions in several variables and asks for a symmetric combination of a, b, and c. It focuses on manipulating fractions and recognizing a hidden expression involving the sum of reciprocals 1/a, 1/b, and 1/c.

Given Data / Assumptions:

• a, b, and c are non zero real numbers.
• (2 + a)/a + (2 + b)/b + (2 + c)/c = 4.
• We must find (ab + bc + ca)/(abc).

Concept / Approach:
The key approach is to separate each fraction into two parts and rewrite everything in terms of 1/a, 1/b, and 1/c. The expression (ab + bc + ca)/(abc) is equal to 1/a + 1/b + 1/c, so once we compute the sum of reciprocals from the given equation, we can read the desired value directly.

Step-by-Step Solution:
Step 1: Rewrite each fraction: (2 + a)/a = 2/a + a/a = 2/a + 1. Step 2: Do the same for the others: (2 + b)/b = 2/b + 1 and (2 + c)/c = 2/c + 1. Step 3: Add them: (2 + a)/a + (2 + b)/b + (2 + c)/c = (2/a + 1) + (2/b + 1) + (2/c + 1). Step 4: Combine like terms: (2/a + 2/b + 2/c) + 3 = 4. Step 5: Subtract 3 from both sides: 2(1/a + 1/b + 1/c) = 1. Step 6: Therefore 1/a + 1/b + 1/c = 1/2. Step 7: Note that (ab + bc + ca)/(abc) = 1/a + 1/b + 1/c, so the required value is 1/2.

Verification / Alternative check:
We can imagine special values that satisfy 1/a + 1/b + 1/c = 1/2 and then verify that the original equation holds. For example, if we pick any triple with that property, substituting into the transformed form 2(1/a + 1/b + 1/c) + 3 reproduces the right hand side equal to 4. This confirms the algebraic deduction.

Why Other Options Are Wrong:
Option A (2), option B (1), and option C (0) all correspond to different values of 1/a + 1/b + 1/c and would change the left side of the original equation away from 4. Option E (3/2) is far too large and would make the left side equal to 2 * 3/2 + 3 = 6, not 4.

Common Pitfalls:
Students often try to clear denominators too early and end up with a complicated expression in a, b, and c. Another common mistake is to forget that (ab + bc + ca)/(abc) simplifies to 1/a + 1/b + 1/c. Recognizing symmetric patterns in numerators and denominators makes such problems much easier.

Final Answer:

The value of the expression (ab + bc + ca)/(abc) is 1/2.