Find the reciprocal of the sum of the reciprocals of the fractions 3 divided by 8 and 8 divided by 11.

Introduction / Context:
This question tests manipulation of complex fractions, specifically reciprocals of sums of reciprocals. It involves nested operations and is a good check of fraction arithmetic skills.

Given Data / Assumptions:
- The given fractions are 3 / 8 and 8 / 11.
- We first take the reciprocals of these fractions.
- Then we add those reciprocals together.
- Finally, we take the reciprocal of that sum.

Concept / Approach:
The reciprocal of a fraction a / b is b / a, provided a is not zero. Here we compute 1 / (3 / 8) and 1 / (8 / 11), add them, and then take the reciprocal of the result. Careful simplification is required to avoid errors in the numerator and denominator.

Step-by-Step Solution:
Step 1: Compute the reciprocal of 3 / 8. This is 8 / 3.Step 2: Compute the reciprocal of 8 / 11. This is 11 / 8.Step 3: Add the two reciprocals: 8 / 3 + 11 / 8.Step 4: Find a common denominator of 24. Convert to 64 / 24 and 33 / 24, so the sum is (64 + 33) / 24 = 97 / 24.Step 5: Now take the reciprocal of 97 / 24, which is 24 / 97.Step 6: Therefore, the required value is 24 / 97.

Verification / Alternative check:
An alternative is to keep track of the structure: required value = 1 / (1 / (3 / 8) + 1 / (8 / 11)) = 1 / (8 / 3 + 11 / 8). Evaluating this systematically as above always gives 24 / 97. Any computational variation that is correct in arithmetic will agree with this final fraction, confirming the solution.

Why Other Options Are Wrong:
Option 97 / 24 is the sum of the reciprocals, not the reciprocal of that sum. Option 88 / 97 or 97 / 88 result from incorrect numerator or denominator additions. Option 11 / 8 is simply the reciprocal of 8 / 11 and ignores the presence of 3 / 8. Only 24 / 97 correctly represents the reciprocal of the sum.

Common Pitfalls:
It is easy to get confused by the nested reciprocals and either stop after computing the sum 97 / 24 or mistakenly invert only part of the expression. Another common issue is adding 8 / 3 and 11 / 8 without finding the correct common denominator, leading to wrong numerators or denominators. Writing each step clearly helps avoid these issues.

Final Answer:
The reciprocal of the sum of the reciprocals of 3 divided by 8 and 8 divided by 11 is 24/97.