Question

A train travelling with constant speed crosses a 80 m long platform in 10 seconds and a 105 m long platform in 12 seconds. Suppose that L denotes the length (in metres) of the train and S denotes the speed (in km/hr) of the train. Then :
A. L = 30, S = 40
B. L = 45, S = 45
C. 10% of 20% of $\frac{2}{3}\text{rd}$ of L = $\frac{3}{5}$
D. 50% of S = 22.5
Choose the correct answer from the options given below :

• A. A and B Only
• B. B and C Only
• C. B, C and D Only
• D. A and D Only

Hint:

Notice that the train takes $2 \text{ seconds}$ longer ($12 \text{ s} - 10 \text{ s}$) to cover the difference in platform lengths.
Difference in length $= 105 \text{ m} - 80 \text{ m} = 25 \text{ m}$.
Thus, the speed of the train is directly:
\[ v = \frac{25 \text{ m}}{2 \text{ s}} = 12.5 \text{ m/s} = 45 \text{ km/hr} \] Using this observation saves setting up complex simultaneous equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem deals with a train crossing platforms of different lengths.
When a train crosses a platform, the total distance covered by the train is equal to the sum of the length of the train and the length of the platform.
Using the given data of two different platforms crossed in different time intervals, we can set up a system of equations to solve for the train's length ($L$) and its speed ($S$).

Step 2: Key Formula or Approach:
Let $L$ be the length of the train in meters, and $v$ be its constant speed in meters per second ($\text{m/s}$).
The general relation is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \] For a platform of length $P$:
\[ L + P = v \times t \]

Step 3: Detailed Explanation:

• Set up the Equations:
- For the first platform ($P_1 = 80 \text{ m}$, $t_1 = 10 \text{ s}$):
\[ L + 80 = 10v \quad \implies \quad L = 10v - 80 \quad \text{---(Eq 1)} \]
- For the second platform ($P_2 = 105 \text{ m}$, $t_2 = 12 \text{ s}$):
\[ L + 105 = 12v \quad \implies \quad L = 12v - 105 \quad \text{---(Eq 2)} \]

• Solve for Speed ($v$ in m/s):
Equating Eq 1 and Eq 2:
\[ 10v - 80 = 12v - 105 \] \[ 2v = 25 \quad \implies \quad v = 12.5 \text{ m/s} \]

• Calculate Train Length ($L$ in meters):
Substitute $v = 12.5$ into Eq 1:
\[ L = 10(12.5) - 80 = 125 - 80 = 45 \text{ meters} \]

• Calculate Speed ($S$ in km/hr):
Convert $v = 12.5 \text{ m/s}$ to $\text{km/hr}$ by multiplying by $\frac{18}{5}$:
\[ S = 12.5 \times \frac{18}{5} = 2.5 \times 18 = 45 \text{ km/hr} \] Therefore:
\[ L = 45 \text{ and } S = 45 \] This proves Statement B is True, and Statement A is False.

• Evaluate Statement C:
We calculate:
\[ 10\% \text{ of } 20\% \text{ of } \frac{2}{3} \text{ of } L \] \[ = \frac{10}{100} \times \frac{20}{100} \times \frac{2}{3} \times 45 \] \[ = 0.1 \times 0.2 \times 30 = 0.02 \times 30 = 0.6 \] Since $0.6 = \frac{3}{5}$, Statement C is True.

• Evaluate Statement D:
We calculate:
\[ 50\% \text{ of } S = 0.5 \times 45 = 22.5 \] Since this matches the statement, Statement D is True.

Step 4: Final Answer:
Statements B, C, and D are true, while Statement A is false.
Therefore, the correct choice is option (C).