Problem 22 · 2018 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Four positive numbers are given. Take three of them, find their mean, then add the fourth number. This can be done in four ways, giving the results 17, 21, 23 and 29. Which is the biggest of the four numbers?
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Answer: C — 21
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Hint 1 of 2
Each result is “mean of three” plus the one left out; adding the four results relates them to the total.
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Hint 2 of 2
The largest result corresponds to leaving out (adding back) the largest number.
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Approach: add all four results to find the total, then back out each number
- Let the total of the four numbers be T. Each result is \(\frac{T-x}{3}+x = \frac{T+2x}{3}\), where x is the added number.
- Summing the four results: \(\frac{4T + 2T}{3} = 2T\), and \(17+21+23+29 = 90\), so \(2T = 90\), giving \(T = 45\).
- The biggest result 29 comes from the biggest x: \(\frac{45+2x}{3}=29\Rightarrow x = 21\).
- So the biggest number is 21.
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