In an aptitude based algebra question, real numbers a and b satisfy a^2 + b^2 = 80 and ab = 32. Using these conditions, find the exact value of the ratio (a − b) / (a + b).

Introduction / Context:
This algebra question from aptitude practice tests your ability to use standard identities involving a^2 + b^2 and ab in order to simplify a ratio. Rather than solving directly for a and b, you can use relationships between sums, differences, and products of two real numbers. Understanding these identities is very useful in many competitive examinations where speed and accuracy are important.

Given Data / Assumptions:

• a and b are real numbers.
• a^2 + b^2 = 80.
• ab = 32.
• We must find the exact value of (a − b) / (a + b).
• a and b are not equal, because otherwise ab would be 40, not 32.

Concept / Approach:
The key idea is to use formulas for (a − b)^2 and (a + b)^2 in terms of a^2 + b^2 and ab. Once we know (a − b)^2 and (a + b)^2, we can obtain the magnitude of the ratio (a − b)/(a + b). Because the equation is symmetric in a and b, the sign of the ratio depends on which variable is considered first, but the magnitude is fixed and matches the exact fractional option provided.

Step-by-Step Solution:
Step 1: Use the identity (a − b)^2 = a^2 + b^2 − 2ab.Step 2: Substitute known values: (a − b)^2 = 80 − 2 × 32 = 80 − 64 = 16.Step 3: Therefore |a − b| = √16 = 4.Step 4: Use the identity (a + b)^2 = a^2 + b^2 + 2ab.Step 5: Substitute again: (a + b)^2 = 80 + 64 = 144.Step 6: Hence |a + b| = √144 = 12.Step 7: The magnitude of the ratio is |(a − b)/(a + b)| = 4/12 = 1/3.Step 8: Since the problem and options focus on the numerical value and all listed options are positive, we take the value as 1/3.

Verification / Alternative check:
We can construct an example pair that matches the conditions. Suppose a and b are roots of t^2 − st + p = 0 with s = a + b and p = ab = 32. Then a^2 + b^2 = s^2 − 2p = 80, so s^2 = 80 + 64 = 144 and s = ±12. Choosing s = 12 gives one valid pair. In that case (a − b)^2 = 16, so |a − b| = 4 and the ratio magnitude is 4/12 = 1/3, which is consistent with our earlier result.

Why Other Options Are Wrong:
The decimal options 0.333, 0.335, and 0.339 are approximations, but only 1/3 is the exact simplified fraction. The value 2/3 would require the difference between a and b to be 8 for the same sum 12, which conflicts with the given values of a^2 + b^2 and ab. Therefore, 1/3 is the only exact and correct ratio.

Common Pitfalls:
A common mistake is to attempt to solve directly for a and b using simultaneous equations, which is slower and more error prone. Another error is to confuse the identities and use a^2 + b^2 = (a + b)^2 − 2ab incorrectly, or to forget to take square roots correctly. Keeping the identities clear and working systematically with squares first leads to the correct value of the ratio.

Final Answer:
The exact value of the ratio (a − b) / (a + b) is 1/3.