If (a + b)/c = 6/5 and (b + c)/a = 9/2, then what is the value of (a + c)/b?

Introduction / Context:
This ratio based algebra problem relates three variables a, b and c through two fractional equations. It is a typical aptitude question that checks comfort with manipulating linear equations involving ratios and expressing one required ratio in terms of the given ones.

Given Data / Assumptions:
We are given:

• (a + b)/c = 6/5
• (b + c)/a = 9/2
• a, b, c are non zero real numbers.
• We must find (a + c)/b.

Concept / Approach:
Convert the given fractional equations into linear equations by cross multiplication. Then express a and c in terms of b, or otherwise, and simplify the ratio (a + c)/b. The problem is designed so that the resulting fractions simplify nicely to a simple rational number.

Step-by-Step Solution:
From (a + b)/c = 6/5, cross multiply to get 5(a + b) = 6c. Thus c = 5(a + b)/6. From (b + c)/a = 9/2, cross multiply: 2(b + c) = 9a. Substitute c into this: 2[b + 5(a + b)/6] = 9a. Inside the brackets, b + 5(a + b)/6 = (6b + 5a + 5b)/6 = (5a + 11b)/6. So 2 * (5a + 11b)/6 = 9a. Simplify: (5a + 11b)/3 = 9a. Multiply both sides by 3: 5a + 11b = 27a. Then 11b = 22a, so b = 2a. Now compute c using c = 5(a + b)/6 = 5(a + 2a)/6 = 5*3a/6 = 5a/2. We need (a + c)/b. Substitute b = 2a and c = 5a/2. a + c = a + 5a/2 = (2a + 5a)/2 = 7a/2. So (a + c)/b = (7a/2) / (2a) = 7a/2 * 1/(2a) = 7/4.

Verification / Alternative check:
Choose a convenient value for a, say a = 2. Then b = 4, c = 5a/2 = 5. Check the given ratios: (a + b)/c = (2 + 4)/5 = 6/5 which matches. (b + c)/a = (4 + 5)/2 = 9/2 which also matches. Now (a + c)/b = (2 + 5)/4 = 7/4, confirming the result.

Why Other Options Are Wrong:
Ratios 9/5, 11/7, 7/11 and 5/4 correspond to various incorrect manipulations such as interchanging a and c or dropping a factor in cross multiplication. Because the relationships are linear, a small algebraic slip in isolating a, b or c leads to these distractors.

Common Pitfalls:
Common errors include cross multiplying incorrectly or forgetting to substitute c in terms of a and b consistently. Another issue is simplifying fractions too quickly and losing track of factors. Writing each step clearly and solving systematically for one variable in terms of another is the safest approach.

Final Answer:
The value of (a + c)/b is 7/4.