Find the reciprocal of the sum of the reciprocals of the fractions 8 divided by 7 and 5 divided by 6.

Introduction / Context:
This is another nested fraction exercise that asks you to find the reciprocal of a sum of reciprocals. Working with such expressions strengthens your fraction manipulation abilities and attention to arithmetic detail.

Given Data / Assumptions:
- The original fractions are 8 / 7 and 5 / 6.
- We first compute their reciprocals.
- Then we add these reciprocals together.
- Finally, we take the reciprocal of this resulting sum.

Concept / Approach:
Recall that the reciprocal of a / b is b / a when a is not zero. Here, 1 / (8 / 7) equals 7 / 8 and 1 / (5 / 6) equals 6 / 5. We then add 7 / 8 and 6 / 5, find their sum as a single fraction, and finally take the reciprocal of that sum, inverting numerator and denominator.

Step-by-Step Solution:
Step 1: Find the reciprocal of 8 / 7. This is 7 / 8.Step 2: Find the reciprocal of 5 / 6. This is 6 / 5.Step 3: Add these two fractions: 7 / 8 + 6 / 5.Step 4: The common denominator is 40. Convert to 35 / 40 and 48 / 40, so the sum is (35 + 48) / 40 = 83 / 40.Step 5: Take the reciprocal of 83 / 40, which is 40 / 83.Step 6: Therefore, the required value is 40 / 83.

Verification / Alternative check:
Consider the full expression 1 / (1 / (8 / 7) + 1 / (5 / 6)). The inner reciprocals are 7 / 8 and 6 / 5. Their sum, computed correctly, is 83 / 40. Inverting this gives 40 / 83. Any correct calculation path should reach the same final fraction, verifying the answer.

Why Other Options Are Wrong:
Option 83 / 40 is the sum of the reciprocals, not its reciprocal. Option 42 / 83 and 83 / 42 are incorrect numerators and denominators, likely coming from careless swapping. Option 7 / 8 is simply the reciprocal of 8 / 7 and ignores the 5 / 6 term entirely. Only 40 / 83 accurately represents the reciprocal of the sum.

Common Pitfalls:
Many learners stop after finding 83 / 40 and forget to take its reciprocal. Others make mistakes while finding a common denominator or add numerators but forget to maintain the correct denominator. Keeping track of the sequence of operations and carefully handling each step is essential.

Final Answer:
The reciprocal of the sum of the reciprocals of 8 divided by 7 and 5 divided by 6 is 40/83.