The average of the five numbers 26, 29, n, 35 and 43 lies between 25 and 35. If n is an integer greater than this average, what is the value of n?

Introduction / Context:
This aptitude question tests your understanding of numerical averages with an unknown term and basic inequality reasoning. You must express the average in terms of n, apply the condition about the range of the average, and also use the condition that n itself is greater than this average. Finally, you have to choose the only option that satisfies all the conditions.

Given Data / Assumptions:
- The five numbers are 26, 29, n, 35 and 43.
- Their average lies between 25 and 35.
- n is an integer and n is greater than the average of these five numbers.
- You must pick n from the options 30, 31, 33 and 34.

Concept / Approach:
The average of k numbers is the sum of the numbers divided by k. Here, the average is a simple linear expression in n. Once we write the average, we can compare it with the boundary values 25 and 35. Separately, the condition that n is greater than the average eliminates some possible values. Finally, we test the remaining candidate values from the options.

Step-by-Step Solution:
Step 1: Write the sum of the five numbers: 26 + 29 + n + 35 + 43. Step 2: Compute the constant part: 26 + 29 + 35 + 43 = 133, so the sum is 133 + n. Step 3: The average is therefore (133 + n) / 5. Step 4: The condition that the average lies between 25 and 35 gives 25 < (133 + n) / 5 < 35, which implies a wide acceptable range of n, including all the options. Step 5: Now use the condition n > average. That is n > (133 + n) / 5. Step 6: Multiply both sides by 5: 5n > 133 + n, so 4n > 133, which gives n > 33.25. Step 7: Since n is an integer, n must be at least 34. Step 8: Among the options 30, 31, 33 and 34, only 34 satisfies n > 33.25 and also keeps the average between 25 and 35.

Verification / Alternative check:
For n = 34, the average is (26 + 29 + 34 + 35 + 43) / 5 = 167 / 5 = 33.4. This value is clearly between 25 and 35, and n = 34 is greater than 33.4, so both conditions are satisfied. No other listed option satisfies n greater than the average.

Why Other Options Are Wrong:
- Option 30: For n = 30, the average is (163) / 5 = 32.6, which is greater than 30, so n is not greater than the average.
- Option 31: For n = 31, the average is 164 / 5 = 32.8, again larger than 31, so the condition n greater than average fails.
- Option 33: For n = 33, the average is 166 / 5 = 33.2, which is still greater than 33, so the inequality is again not satisfied.

Common Pitfalls:
Many learners only apply the condition that the average lies between 25 and 35 and forget to check the extra condition that n must be greater than this average. Another common mistake is to treat “lies between 25 and 35” as giving a unique value of n instead of a range. You must always use every condition in the problem.

Final Answer:
Therefore, the only integer value of n that satisfies all the given conditions is 34.