Problem 10 · 2018 AMC 8
Medium
Fractions, Decimals & Percents
fraction-to-decimalevaluate-formula
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
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Answer: C — 12/7.
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Hint 1 of 2
The scary name is just a recipe. Read the definition backwards from the answer: it's the "reciprocal of the average of the reciprocals" — so peel it off in the forward order: flip, average, flip.
Still stuck? Show hint 2 →
Hint 2 of 2
The technique for any unfamiliar definition: execute it as literal instructions, one verb at a time, and don't skip the last step — here the final "reciprocal" is exactly the step people forget.
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Approach: follow the definition
- Flip each number: the reciprocals of 1, 2, 4 are 1, 1/2, 1/4, which sum to 7/4.
- Average the three reciprocals: (7/4) ÷ 3 = 7/12.
- Now do the final flip the name demands — the reciprocal of 7/12 is 12/7. Sanity check: the harmonic mean always lands below the ordinary average (here (1+2+4)/3 ≈ 2.33) and pulls toward the small numbers, and 12/7 ≈ 1.71 does exactly that.
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