A train running at a constant speed crosses a stationary platform of length 200\text{ mete… — State PSC Exam

A train running at a constant speed crosses a stationary platform of length $200\text{ meters}$ . What is the speed of the train in $\text{km/h}$ ?

Statement I: The train takes $20\text{ seconds}$ to completely cross the platform.
Statement II: The train takes $8\text{ seconds}$ to cross a telegraph pole standing beside the track.

Which of the following options correctly describes the sufficiency of the statements to answer the question?

A
Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.
B
Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.
C
Either Statement I alone or Statement II alone is sufficient to answer the question.
D
Neither Statement I alone nor Statement II alone is sufficient, and both statements together are not sufficient to answer the question.
Both Statement I and Statement II together are necessary and sufficient to answer the question.Answer

Answer

Both Statement I and Statement II together are necessary and sufficient to answer the question.

Evaluating each statement individually shows that neither Statement I nor Statement II alone provides enough information to determine the speed of the train because the length of the train remains unknown. However, combining both statements yields a system of two independent linear equations with two variables (length and speed of the train), allowing us to solve uniquely for the speed of the train (

60\text{ km/h}

). Therefore, both statements together are necessary and sufficient.

Step-by-Step Solution

Analyze Statement I alone.

Let the length of the train be

L\text{ meters}

and its speed be

v\text{ m/s}

. The equation is

L + 200 = 20v

. Since there are two unknowns (

L

and

v

), Statement I alone is not sufficient.

The length of the train is an unknown variable.

Analyze Statement II alone.

Crossing a telegraph pole means distance traveled equals the length of the train. The equation is

L = 8v

. Since there are two unknowns (

L

and

v

), Statement II alone is not sufficient.

The speed cannot be determined without knowing the length of the train.

Evaluate Statement I and Statement II together.

Substitute

L = 8v

into the first equation:

8v + 200 = 20v \implies 12v = 200 \implies v = \frac{50}{3}\text{ m/s}

. Converting to

\text{km/h}

v = \frac{50}{3} \times \frac{18}{5} = 60\text{ km/h}

. Thus, both statements together give a unique answer.

Two independent equations are sufficient to solve for two unknown variables.

Key Concept

Data Sufficiency evaluation for relative motion and distance-speed problems using linear equations.

Estimated Time:1m 30s

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