A car travels 3/4th of the distance at a speed of 60 km/hr and the remaining 1/4th of the distance at a speed of v km/hr. If the average speed for the full journey is 50 km/hr, then the value of v is

Q: What is the answer to CAPF 2022 Paper I Q32?

C. Correct — C, 100/3. Average speed is total distance over total time, never the mean of the two speeds. Take the journey as D. Time on the first leg is (3D/4)/60 = D/80; on the second it is (D/4)/v. The condition D ÷ (D/80 + D/4v) = 50 cancels D and gives 1/50 = 1/80 + 1/(4v). Subtracting, 1/(4v) = 1/50 − 1/80 = 3/400, so 4v = 400/3 and v = 100/3, about 33.3 km/hr.

(a)40
(b)30
(c)100/3
(d)35

Answer

Why

Correct — C, 100/3. Average speed is total distance over total time, never the mean of the two speeds. Take the journey as D. Time on the first leg is (3D/4)/60 = D/80; on the second it is (D/4)/v. The condition D ÷ (D/80 + D/4v) = 50 cancels D and gives 1/50 = 1/80 + 1/(4v). Subtracting, 1/(4v) = 1/50 − 1/80 = 3/400, so 4v = 400/3 and v = 100/3, about 33.3 km/hr.

Why the others are wrong

(a)40 — At v = 40 the total time for a 4-unit journey is 3/60 + 1/40 = 0.075 hours per 4 km-unit, giving an average of about 53.3 km/hr — too fast.
(b)30 — At v = 30 the average works out to 48 km/hr, below the required 50. The answer must lie between 30 and 40, which is where 100/3 sits.
(d)35 — This is the value you get by averaging arithmetically instead of by time; it produces an average speed of about 51.4 km/hr, not 50.

Concept

Average speed is a harmonic-type average when equal distances are covered at different speeds, because the slower leg occupies more of the total time. Writing time = distance ÷ speed for each leg and adding is the only reliable route.

The tempting shortcut is to notice that three-quarters of the trip runs at 60 and to weight the speeds three-to-one, which suggests something near 20 for the last leg. That fails because the weights belong to distance, not to time — the slow quarter eats a disproportionate share of the clock. Note also that the assumed distance D cancels, so you may set D = 4 units and work with whole numbers instead.

Key facts

Average speed = total distance ÷ total time; it equals the arithmetic mean of the speeds only when the times, not the distances, are equal.
For two equal distances at speeds a and b, the average speed is the harmonic mean 2ab/(a + b).
Here the identity to use is 1/50 = (3/4)/60 + (1/4)/v, the reciprocal form of the same rule.
Setting the unknown total distance to a convenient number such as 4 or 400 removes fractions without changing the answer.

Working with time, not with speed

Let the distance be→D, split as 3D/4 then D/4

Time, first leg→(3D/4) ÷ 60 = D/80

Time, second leg→(D/4) ÷ v = D/4v

Average-speed condition→D ÷ (D/80 + D/4v) = 50, so 1/50 = 1/80 + 1/4v

Solve→1/4v = 1/50 − 1/80 = 3/400, so v = 100/3 ≈ 33.3 km/hr

The distance cancels, which is why no actual length is given in the question.

Study next

Average speed for equal distances versus equal timesHarmonic mean and where it appears in time-speed-distance problemsRelative speed for bodies moving in the same and opposite directionsTrains crossing poles, platforms and each otherBoats and streams — upstream and downstream speeds

Common traps

Averaging the two speeds directly, or weighting them by distance share.
Forgetting that the assumed total distance cancels and hunting for a value that was never given.
Dropping the reciprocal when moving from the time equation to the speed equation.

Asked as a journey split into unequal fractions with one speed unknown and the overall average given, or the reverse.

Related PYQs

UPSC_2003_GS1_Q442003Open & attempt →

A car travels the first one-third of a certain distance with a speed of 10 km/hr, the next one-third distance with a speed of 20 km/hr and the last one-third distance with a speed of 60 km/hr. The average speed for the whole journey is

(a) 18 km/hr
(b) 24 km/hr
(c) 30 km/hr
(d) 36 km/hr

Answer(a) 18 km/hr

The same rule with the journey cut into thirds instead of quarters, and every speed supplied. Reciprocals are added there exactly as they are here, which is why the answer sits far below the arithmetic mean of 30.

Practice

practice — not a real PYQ
A man covers half a journey at 20 km/hr and the other half at 30 km/hr. His average speed for the whole journey is
- (a)24 km/hr
- (b)25 km/hr
- (c)26 km/hr
- (d)27 km/hr
Answer(a) 24 km/hr — equal distances take the harmonic mean, 2 × 20 × 30 ÷ (20 + 30) = 1200/50 = 24.
practice — not a real PYQ
A cyclist rides for 2 hours at 12 km/hr and then for 3 hours at 22 km/hr. The average speed for the whole ride is
- (a)16 km/hr
- (b)17 km/hr
- (c)18 km/hr
- (d)19 km/hr
Answer(c) 18 km/hr — total distance 24 + 66 = 90 km in 5 hours, so 90 ÷ 5 = 18. With equal times the plain weighted mean is correct.