An automobile travels from city A to city B and returns to city A by the same route. The
speed of the vehicle during the onward and return journeys were constant at 60 km/h and
90 km/h, respectively. What is the average speed in km/h for the entire journey?
(A) 72 (B) 73 (C) 74 (D) 75
(GA) Set-4
Options:
(A) 72
(B) 73
(C) 74
(D) 75
GATE 2018 General Aptitude (GA) Set-4
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.06)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Speed, Time and Distance (Harmonic Mean of Speeds)
**2. Detailed Step-by-Step Derivation**:
- Let the distance between city A and city B be $d$ km.
- Time taken for the onward journey from A to B at speed $v_1 = 60 ext{ km/h}$:
$t_1 = rac{d}{60} ext{ hours}$
- Time taken for the return journey from B to A at speed $v_2 = 90 ext{ km/h}$:
$t_2 = rac{d}{90} ext{ hours}$
- Total distance traveled for the entire round trip:
$D_{ ext{total}} = d + d = 2d ext{ km}$
- Total time taken:
$T_{ ext{total}} = t_1 + t_2 = rac{d}{60} + rac{d}{90} = d \left( rac{3 + 2}{180}
ight) = rac{5d}{180} = rac{d}{36} ext{ hours}$
- Average speed for the entire journey:
$v_{ ext{avg}} = rac{D_{ ext{total}}}{T_{ ext{total}}} = rac{2d}{ rac{d}{36}} = 2 imes 36 = 72 ext{ km/h}$
Alternatively, using the harmonic mean for equal distances:
$v_{ ext{avg}} = rac{2 v_1 v_2}{v_1 + v_2} = rac{2 imes 60 imes 90}{60 + 90} = rac{10800}{150} = 72 ext{ km/h}$
- **Option-by-Option Evaluation**:
- **Option (A) [CORRECT]**: 72 km/h is the exact harmonic mean of the round trip speeds.
- **Option (B) [INCORRECT]**: Arithmetic distractor.
- **Option (C) [INCORRECT]**: Arithmetic distractor.
- **Option (D) [INCORRECT]**: 75 km/h is the simple arithmetic mean $(60 + 90)/2$, which is invalid because travel times for the two legs are unequal.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **A**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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