A car travels equal distances at speeds of 40 km/hr, 60 km/hr and 80 km/hr on three successive stretches. Find

Worked Examples · Example 8

Q. A car travels equal distances at speeds of 40 km/hr, 60 km/hr and 80 km/hr on three successive stretches. Find the average speed for the whole journey, and verify your answer is consistent with the AM \geq GM \geq HM relationship.

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Harmonic mean:

\frac{1}{40}+\frac{1}{60}+\frac{1}{80} = \frac{6}{240}+\frac{4}{240}+\frac{3}{240} = \frac{13}{240}

HM = \frac{n}{\Sigma(1/x)} = \frac{3}{13/240} = \frac{3\times240}{13} = \frac{720}{13} = 55.38\ \text{km/hr}

Independent cross-check — verify against the AM ≥ GM ≥ HM relationship:

AM = \frac{40+60+80}{3} = \frac{180}{3} = 60

$GM = \sqrt[3]{40\times60\times80} = \sqrt[3]{192{,}000} \approx 57.69$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CA Foundation 2025Set jan-20251 markMCQ
$Q. Find the Harmonic Mean of 2, 4 & 6. (A) 3.30 (B) 3.00 (C) 3.75 (D) 4.00$
›Reveal solutionSolution

$HM=\dfrac{n}{\sum(1/x)}=\dfrac{3}{11/12}=\dfrac{36}{11}\approx3.27\approx3.30$ .

Step 1 — Harmonic mean formula
$HM = \dfrac{n}{\sum \dfrac{1}{x_i}}$
Step 2 — Sum the reciprocals
$\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6} = \dfrac{6+3+2}{12} = \dfrac{11}{12}.$
Step 3 — Divide
$HM = \dfrac{3}{11/12} = 3\times\dfrac{12}{11} = \dfrac{36}{11} \approx 3.27.$
The nearest given value is $3.30$ .

Why the other options are wrong: $3.00$ (B) would only arise from a different set; $4.00$ (D) is the arithmetic mean; $3.75$ (C) does not match the reciprocal computation. (Recall $HM\le GM\le AM$ ; here $AM=4$ , so $HM<4$ , ruling out (D).) …
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CA Foundation 2025Set may-20251 markMCQ
$100\sqrt{35}$
›Reveal solutionSolution

Average speed for equal distances = harmonic mean of the two speeds = $\dfrac{2\times500\times700}{500+700}=583.33$ km/hr.

Step 1 — Choose the right average

The helicopter covers the same distance A→B and B→A at different speeds. When the distances are equal, average speed is the harmonic mean of the speeds:
$\bar V = \dfrac{2 v_1 v_2}{v_1 + v_2}.$
Step 2 — Substitute
$\bar V = \dfrac{2 \times 500 \times 700}{500 + 700} = \dfrac{7{,}00{,}000}{1{,}200} = 583.33\text{ km/hr}.$
Step 3 — Why not 600?

The arithmetic mean $\tfrac{500+700}{2}=600$ (option A) is wrong here: the helicopter spends more time at the slower speed, pulling the true average below 600. …
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CA Foundation 2025Set sep-20251 markMCQ
$1, \frac{1}{3}, \frac{1}{6}$
›Reveal solutionSolution

HM = n ÷ (sum of reciprocals) = 4 ÷ 19 = 0.21.

Step 1 — Formula
$HM = \frac{n}{\sum \frac{1}{x_i}}$
Step 2 — Reciprocals of the values

For $x = 1, \tfrac13, \tfrac16, \tfrac19$ :
$\frac{1}{1}=1,\quad \frac{1}{1/3}=3,\quad \frac{1}{1/6}=6,\quad \frac{1}{1/9}=9$
Sum of reciprocals $= 1 + 3 + 6 + 9 = 19$ , with $n = 4$ .

Step 3 — Compute the HM

$HM = \frac{4}{19} = 0.2105 \approx 0.21$ …
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CA Foundation 2023Set jun-20231 markMCQ
$Q. If the mean of two numbers is 30 and geometric mean is 24, then what will be the Harmonic mean of two numbers? (A) 19.2 (B) 21.8 (C) 22.3 (D) 18.4$
›Reveal solutionSolution

HM = GM²/AM = 576/30 = 19.2.

Step 1 — State the relationship

For two positive numbers, GM is the geometric mean of AM and HM:
$\text{GM}^2 = \text{AM} \times \text{HM}$
Step 2 — Solve for HM
$\text{HM} = \frac{\text{GM}^2}{\text{AM}} = \frac{24^2}{30} = \frac{576}{30} = 19.2$
Watch out

The inequality AM ≥ GM ≥ HM must hold; any option greater than GM (24) — like 21.8 or 22.3 — is impossible here. …

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CA Foundation 2022Set dec-20221 markMCQ
$Q. If Arithmetic mean between two numbers is 5 and Geometric mean is 4 then what is the value of Harmonic mean? (A) 3.2 (B) 3.4 (C) 3.5 (D) 3.6$
›Reveal solutionSolution

$\text{HM} = \text{GM}^2/\text{AM} = 16/5 = 3.2$ .

Step 1 — Use the AM–GM–HM identity

For two positive numbers,
$\text{GM}^2 = \text{AM} \times \text{HM}$
Step 2 — Solve for HM
$\text{HM} = \frac{\text{GM}^2}{\text{AM}} = \frac{4^2}{5} = \frac{16}{5} = 3.2$
Step 3 — Cross-check with the actual numbers

AM $=5$ , GM $=4$ give the numbers 2 and 8, whose HM is

$\frac{2ab}{a+b} = \frac{2\times 2\times 8}{2+8} = \frac{32}{10} = 3.2\;(\checkmark)$ …
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←Previous questionExample 7 — Find the geometric mean of the observations 2, 4 and 8, both directly …→Next chapterCh 4Measures of Dispersion

Q.A car travels equal distances at speeds of 40 km/hr, 60 km/hr and 80 km/hr on three successive stretches. Find the average speed for the whole journey, and verify your answer is consistent with the AM ≥ GM ≥ HM relationship.

Concept understanding — Harmonic Mean

The governing formula

Why it works

Worked illustration …

Step 1 — Harmonic mean formula

Step 2 — Sum the reciprocals

Step 3 — Divide

Step 1 — Choose the right average

Step 2 — Substitute

Step 3 — Why not 600?

Step 1 — Formula

Step 2 — Reciprocals of the values

Step 3 — Compute the HM

Step 1 — State the relationship

Step 2 — Solve for HM

Step 1 — Use the AM–GM–HM identity

Step 2 — Solve for HM

Step 3 — Cross-check with the actual numbers

Q.A car travels equal distances at speeds of 40 km/hr, 60 km/hr and 80 km/hr on three successive stretches. Find the average speed for the whole journey, and verify your answer is consistent with the AM ≥ GM ≥ HM relationship.

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