Math Test
This should show a math equation: \( a^2 + b^2 = c^2 \)
🚆 Train & Platform Problem
Question: A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?
- 120 m
- 240 m
- 300 m
- None of these
Solution
📘 Step-by-Step Solution:
Speed of train = \[ 54 \, \text{km/hr} = 54 \times \frac{1000}{3600} = 15 \, \text{m/s} \]
Time taken to pass man = 20 sec ⇒ Length of train = \[ 15 \times 20 = 300 \, \text{m} \]
Time taken to pass platform = 36 sec ⇒ Total length = \[ 15 \times 36 = 540 \, \text{m} \]
So, platform length = \[ 540 – 300 = \boxed{240 \, \text{m}} \]
✅ Final Answer: \(\boxed{240 \, \text{metres}}\)
🚄🚄 Two Trains Crossing Each Other
Question: Two trains 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time (in seconds) which they take to cross each other is:
- 9
- 9.6
- 10
- 10.8
Show Solution
📘 Step-by-Step Solution:
Length of first train = 140 m
Length of second train = 160 m
Total distance to be covered = \[ 140 + 160 = 300 \, \text{m} \]
Speed of first train =
\[
60 \, \text{km/hr} = 60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}
\]
Speed of second train =
\[
40 \, \text{km/hr} = 40 \times \frac{1000}{3600} = 11.11 \, \text{m/s}
\]
Relative speed (opposite direction) = \[ 16.67 + 11.11 = 27.78 \, \text{m/s} \]
Time to cross each other = \[ \frac{300}{27.78} \approx \boxed{10.8 \, \text{seconds}} \]
✅ Final Answer: \(\boxed{10.8 \, \text{seconds}}\)
🚆 Train Speed Problem
Question: A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?
- 120 metres
- 180 metres
- 324 metres
- 150 metres
Show Solution
📘 Step-by-Step Solution:
We use the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
First, convert speed from km/hr to m/s:
\[
60 \, \text{km/hr} = 60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}
\]
Now calculate distance using the time = 9 seconds:
\[
\text{Distance} = 16.67 \times 9 = 150 \, \text{metres}
\]
✅ Final Answer: \(\boxed{150 \, \text{metres}}\)
🚆 Train & Bridge Problem
Question: The length of the bridge, which a train 130 metres long and travelling at 45 km/hr can cross in 30 seconds, is:
- 200 m
- 225 m
- 245 m
- 250 m
Show Solution
📘 Step-by-Step Solution:
Total distance covered in crossing the bridge = Length of train + Length of bridge
Speed = 45 km/hr = \[ 45 \times \frac{1000}{3600} = 12.5 \, \text{m/s} \]
Time = 30 seconds
Total Distance = Speed × Time \[ = 12.5 \times 30 = 375 \, \text{m} \]
Train length = 130 m
\[ \text{Bridge length} = 375 – 130 = \boxed{245 \, \text{m}} \]
✅ Final Answer: \(\boxed{245 \, \text{metres}}\)
🚄 Two Trains from Howrah and Patna
Question: Two trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. What is the ratio of their speeds?
- 2 : 3
- 4 : 3
- 6 : 7
- 9 : 16
Show Solution
📘 Step-by-Step Solution:
Let the two trains meet at point M.
🧠 Key Concept:
If two objects start at the same time and move towards each other, and after meeting, take t₁ and t₂ hours respectively to complete their journeys, then:
\[ \text{Ratio of their speeds} = \sqrt{t_2} : \sqrt{t_1} \]
Here,
Time taken by train from Howrah after meeting = 9 hours
Time taken by train from Patna after meeting = 16 hours
So the ratio of speeds is:
\[
\sqrt{16} : \sqrt{9} = 4 : 3
\]
✅ Final Answer: \(\boxed{4 : 3}\)