Problem 20 · 2016 Math Kangaroo
Stretch
Algebra & Patterns
substitution
In an enclosure there are 2016 kangaroos. Each of them is either red or grey, and there is at least one red and at least one grey kangaroo amongst them. For each kangaroo K we calculate the fraction obtained, if you take the number of kangaroos of the other colour divided by the kangaroos of the own colour (including K itself). Determine the sum of these 2016 fractions.
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Answer: A — 2016
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Hint 1 of 2
Group the kangaroos by colour: say r red and g grey, with r + g = 2016.
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Hint 2 of 2
Add the fractions colour by colour and watch the counts cancel.
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Approach: sum by colour
- Each red kangaroo contributes g/r and there are r of them, totalling g.
- Each grey kangaroo contributes r/g and there are g of them, totalling r.
- The grand total is g + r = 2016.
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