Four positive integers have average 73.5. The highest is 108 and the lowest is 29. The remaining two differ by 15. Find the smaller of those two remaining integers.
Correct Answer: 71
Introduction / Context:This question combines an average-to-total conversion with constraints on two unknown integers. Setting variables for the two middle integers converts the text into a simple linear equation.
Given Data / Assumptions:
- Average of 4 integers = 73.5
- Highest = 108
- Lowest = 29
- Difference between the remaining two = 15
Concept / Approach:Let the remaining two numbers be x and x + 15. Use the total from the average and subtract the known extremes to isolate x.
Step-by-Step Solution:
Total of all four = 4 * 73.5 = 294Known extremes = 108 and 29 → sum = 137Equation: 137 + x + (x + 15) = 2942x + 152 = 294 ⇒ 2x = 142 ⇒ x = 71Verification / Alternative check:Numbers are 29, 71, 86, 108. Average = (29 + 71 + 86 + 108) / 4 = 294 / 4 = 73.5; difference between middle pair = 15, all conditions met.
Why Other Options Are Wrong:
- 73, 80, 86: do not satisfy the total and the exact difference simultaneously.
- Cannot be determined: the information is sufficient to solve uniquely.
Common Pitfalls:Forgetting to include the +15 when forming the equation for the two unknowns or miscalculating the total from the average.
Final Answer:71