Problem 11 · 2010 AMC 8
Easy
Algebra & Patterns
ratio-and-difference
The top of one tree is 16 feet higher than the top of another tree. The heights of the two trees are in the ratio 3 : 4. In feet, how tall is the taller tree?
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Answer: B — 64 feet.
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Hint 1 of 2
Think of the trees as made of equal ‘parts’: one is 3 parts tall, the other 4. The 16-foot difference is just the one extra part. So one part = 16 ft.
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Hint 2 of 2
In a ratio, the difference between the numbers is also measured in those same parts. Find the value of one part, then scale up.
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Approach: find the value of one ratio part
- The heights are 3 parts and 4 parts. The gap between them is 4 − 3 = 1 part, and that gap is the given 16 ft — so 1 part = 16 ft.
- The taller tree is 4 parts: 4 · 16 = 64 ft.
- Why this transfers: ‘ratio + a difference (or sum)’ almost always cracks open by pricing one part. Match the given number to how many parts it represents, then multiply.
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