If ab = 45 and bc = 60 for nonzero real numbers a, b, and c, find the value of: (c^2 - a^2) / (c^2 + a^2). Choose the correct simplified value.

Introduction:
This question tests algebraic manipulation using given products. Even though a, b, and c are not individually known, ab and bc allow you to express a and c in terms of b. Since the expression involves a^2 and c^2, representing both as something divided by b^2 helps cancel b. The key insight is that the expression (c^2 - a^2)/(c^2 + a^2) depends only on the ratio of c^2 and a^2, and b will cancel completely.

Given Data / Assumptions:

• ab = 45
• bc = 60
• b ≠ 0 (so we can divide by b)
• We need (c^2 - a^2) / (c^2 + a^2)

Concept / Approach:
From ab = 45, write a = 45/b. From bc = 60, write c = 60/b. Then compute a^2 and c^2. Substitute into the expression. Because both a^2 and c^2 contain 1/b^2, it cancels out, leaving a simple fraction involving 45^2 and 60^2.

Step-by-Step Solution:
a = 45/b and c = 60/b a^2 = 45^2 / b^2 = 2025 / b^2 c^2 = 60^2 / b^2 = 3600 / b^2 (c^2 - a^2) / (c^2 + a^2) = (3600/b^2 - 2025/b^2) / (3600/b^2 + 2025/b^2) = (3600 - 2025) / (3600 + 2025) = 1575 / 5625 Divide numerator and denominator by 225: 1575/225 = 7 and 5625/225 = 25 So value = 7/25

Verification / Alternative check:
Pick a convenient b, say b = 15. Then a = 45/15 = 3 and c = 60/15 = 4. Expression becomes (16 - 9)/(16 + 9) = 7/25, confirming the result.

Why Other Options Are Wrong:
7/5 and 7 are too large (the fraction must be less than 1 in magnitude because |c^2 - a^2| < c^2 + a^2 for positive squares). 7/125 is too small. 5/7 is a different ratio and does not match substitution.

Common Pitfalls:
Trying to solve for a, b, c uniquely (not possible), forgetting to square, or failing to cancel b^2 correctly when combining fractions.

Final Answer:
The value is 7/25.