All the kangaroos on AKSF Island are either grey or green. Every time when the clock strikes midnight, some of the kangaroos change their colors. Yesterday, there were 6 grey kangaroos for every 5 green ones. Today, there are 3 grey ones for every 4 green ones. What is the smallest possible number of kangaroos that changed the color overnight?
Question
All the kangaroos on AKSF Island are either grey or green. Every time when the clock strikes midnight, some of the kangaroos change their colors. Yesterday, there were 6 grey kangaroos for every 5 green ones. Today, there are 3 grey ones for every 4 green ones. What is the smallest possible number of kangaroos that changed the color overnight?
Solution
To solve this problem, we need to find the least common multiple (LCM) of the numbers of kangaroos from both days.
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Yesterday, the ratio of grey to green kangaroos was 6:5. This means for every 6 grey kangaroos, there were 5 green ones. So, the total number of kangaroos could be any multiple of 11 (6+5).
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Today, the ratio is 3:4. This means for every 3 grey kangaroos, there are 4 green ones. So, the total number of kangaroos could be any multiple of 7 (3+4).
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We need to find the smallest number that is a multiple of both 11 and 7. This is the least common multiple (LCM) of 11 and 7. The LCM of 11 and 7 is 77.
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So, the smallest possible number of kangaroos on the island is 77.
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Yesterday, there were (6/11)*77 = 42 grey kangaroos and (5/11)*77 = 35 green kangaroos.
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Today, there are (3/7)*77 = 33 grey kangaroos and (4/7)*77 = 44 green kangaroos.
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Therefore, the smallest possible number of kangaroos that changed color overnight is the difference in the number of grey kangaroos plus the difference in the number of green kangaroos, which is (42-33) + (44-35) = 9 + 9 = 18.
So, at least 18 kangaroos must have changed their color overnight.
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