Problem 16 · 2013 AMC 8
Medium
Ratios, Rates & Proportions
lcm-for-ratios
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8th-graders to 6th-graders is 5 : 3, and the ratio of 8th-graders to 7th-graders is 8 : 5. What is the smallest number of students that could be participating in the project?
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Answer: E — 89 students.
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Hint 1 of 2
The 8th-graders appear in both ratios — they're the shared hinge. In ratio 5:3 their count is a multiple of 5; in ratio 8:5 it's a multiple of 8. Make those two pictures agree first.
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Hint 2 of 2
To merge two ratios that share a quantity, force the shared term to a common value — the smallest is the LCM. Then every group count comes out whole.
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Approach: make the shared 8th-grader count an LCM
- 8th-graders link both ratios, so their count must be a multiple of 5 (from 5:3) and of 8 (from 8:5). Smallest such count = lcm(5, 8) = 40 — pick that to keep everyone whole.
- 6th-graders: 40 × 3/5 = 24 (from 5:3).
- 7th-graders: 40 × 5/8 = 25 (from 8:5).
- Total = 40 + 24 + 25 = 89.
- Why this transfers: any "chain" of ratios sharing a common term is stitched together by setting that term to the LCM — the same trick scales recipe ratios, gear ratios, and unit conversions.
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