Q.14 Calculate the harmonic mean of the numbers 4, 5, and 10 collected in one experiment.
- 6.83
- 5.45
- 7
- 6
The harmonic mean of 4, 5, and 10 is 5.45, calculated using the reciprocal formula for three values. This matches option 2 precisely after step-by-step computation.
Calculation Steps
Harmonic mean (HM) formula for n numbers: HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ).
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Reciprocals: 1/4 = 0.25, 1/5 = 0.2, 1/10 = 0.1.
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Sum: 0.25 + 0.2 + 0.1 = 0.55.
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HM = 3 / 0.55 ≈ 5.4545 (rounds to 5.45).
Option Analysis
6.83: Wrong; exceeds arithmetic mean (6.33), impossible for HM.
5.45: Correct; exact match from 3/0.55.
7: Wrong; too high, confuses with arithmetic mean.
6: Wrong; approximate but ignores precise reciprocal sum.Harmonic mean of 4, 5, and 10 is essential for statistics in competitive exams like GATE Life Sciences, especially for experimental data rates. Use HM = 3 / (1/4 + 1/5 + 1/10) = 5.45 for accurate averages when dealing with rates or speeds.
Formula Breakdown
The harmonic mean formula prioritizes smaller values, ideal for rates: HM = n / Σ(1/xᵢ). For 4, 5, 10: sum of reciprocals (0.55), then 3/0.55 = 5.45.
Step Calculation Value Reciprocals 1/4 + 1/5 + 1/10 0.55 Divide by n=3 3 / 0.55 5.45 Exam Relevance
In GATE PYQs, harmonic mean of 4, 5, and 10 tests reciprocal understanding over arithmetic mean (6.33). Option 5.45 is correct; avoid traps like 6 or 7.
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