Question:
A person travels 444 km by train, 260 km by bike and 112 km by auto. It took a total of 15 hours. If the speed of the train is 3 times the speed of an auto and \(1\frac12\) times the speed of bike, then the speed of the bike, in kmph, is
A person travels 444 km by train, 260 km by bike and 112 km by auto. It took a total of 15 hours. If the speed of the train is 3 times the speed of an auto and \(1\frac12\) times the speed of bike, then the speed of the bike, in kmph, is
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When multiple speed relations are given, express all speeds in terms of one variable first.
Updated On: Jul 15, 2026
- \(52\)
- \(26\)
- \(78\)
- \(39\)
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The Correct Option is A
Solution and Explanation
Concept:
Use the relation:
\[
\text{Time}=\frac{\text{Distance}}{\text{Speed}}
\]
Step 1: Assume bike speed.
Let bike speed be: \[ x \text{ kmph} \] Given: Train speed is \(1\frac12\) times bike speed. \[ =\frac32x \] Train speed is also \(3\) times auto speed. So auto speed: \[ =\frac{\frac32x}{3} \] \[ =\frac x2 \]
Step 2: Write time for each journey.
Train: \[ \frac{444}{\frac32x}=\frac{296}{x} \] Bike: \[ \frac{260}{x} \] Auto: \[ \frac{112}{x/2}=\frac{224}{x} \]
Step 3: Use total time.
\[ \frac{296}{x}+\frac{260}{x}+\frac{224}{x}=15 \] \[ \frac{780}{x}=15 \]
Step 4: Solve for \(x\).
\[ x=\frac{780}{15} \] \[ x=52 \] Thus, the speed of the bike is: \[ \boxed{52 \text{ kmph}} \]
Step 1: Assume bike speed.
Let bike speed be: \[ x \text{ kmph} \] Given: Train speed is \(1\frac12\) times bike speed. \[ =\frac32x \] Train speed is also \(3\) times auto speed. So auto speed: \[ =\frac{\frac32x}{3} \] \[ =\frac x2 \]
Step 2: Write time for each journey.
Train: \[ \frac{444}{\frac32x}=\frac{296}{x} \] Bike: \[ \frac{260}{x} \] Auto: \[ \frac{112}{x/2}=\frac{224}{x} \]
Step 3: Use total time.
\[ \frac{296}{x}+\frac{260}{x}+\frac{224}{x}=15 \] \[ \frac{780}{x}=15 \]
Step 4: Solve for \(x\).
\[ x=\frac{780}{15} \] \[ x=52 \] Thus, the speed of the bike is: \[ \boxed{52 \text{ kmph}} \]
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