Problem 19 · 2022 Math Kangaroo
Hard
Algebra & Patterns
sum-constraint
The arithmetic mean of five numbers is 24. The mean of the three smallest numbers is 19 and that of the three biggest is 28. What is the median of the five numbers?
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Answer: B — 21
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Hint 1 of 2
Write the three totals: all five, the three smallest, the three largest.
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Hint 2 of 2
The median is counted in both the bottom-three and top-three sums.
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Approach: overlap counts the median twice
- Sum of all five = 120; smallest three sum to 57; largest three sum to 84.
- 57 + 84 counts every number once except the median, which is counted twice: 57 + 84 = 120 + median.
- Median = 141 − 120 = 21.
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