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A train X starts from Meerut at 4 p.m. and reaches Ghaziabad at 5 p.m.while another train Y starts from Ghaziabad at 4 p.m. and reaches Meerut at 5:30 p.m. The two trains will cross each other at:
Let's understand what's happening here in simple terms. We have two trains starting at exactly the same time (4 p.m.) from opposite ends of the same route between Meerut and Ghaziabad. They're heading toward each other, so they will definitely meet somewhere in the middle.
Train X: Starts from Meerut at 4 p.m., reaches Ghaziabad at 5 p.m.
Train Y: Starts from Ghaziabad at 4 p.m., reaches Meerut at 5:30 p.m.
We need to find the exact time when these two trains will cross each other.
Process Skill: TRANSLATE - Converting the problem scenario into clear mathematical relationships
Let's think about this like a race where we measure speed by "how much of the total journey each train completes per minute."
Train X completes the entire journey in 1 hour (60 minutes).
So Train X covers \(\frac{1}{60}\) of the total distance every minute.
Train Y completes the entire journey in 1.5 hours (90 minutes).
So Train Y covers \(\frac{1}{90}\) of the total distance every minute.
Think of it this way: if the total distance is like a pizza cut into pieces, Train X eats \(\frac{1}{60}\) of the pizza per minute, while Train Y eats \(\frac{1}{90}\) of the pizza per minute.
Here's the key insight: when two objects move toward each other, they cover the distance between them much faster than if just one was moving.
Since both trains are moving toward each other, together they cover:
\(\frac{1}{60} + \frac{1}{90}\) of the total distance every minute
To add these fractions, let's find a common denominator:
\(\frac{1}{60} = \frac{3}{180}\)
\(\frac{1}{90} = \frac{2}{180}\)
Together they cover: \(\frac{3}{180} + \frac{2}{180} = \frac{5}{180} = \frac{1}{36}\) of the total distance every minute.
This means they will meet when they've collectively covered the entire distance, which takes:
\(1 ÷ \left(\frac{1}{36}\right) = 36\) minutes
Since both trains started at 4:00 p.m., and we found they will meet after 36 minutes:
Meeting time = 4:00 p.m. + 36 minutes = 4:36 p.m.
Let's verify this makes sense: In 36 minutes,
The two trains will cross each other at 4:36 p.m.
This matches answer choice A.
1. Misunderstanding the relative motion concept
Many students approach this as two separate motion problems instead of recognizing it as a relative motion problem where trains are moving toward each other. They might try to find where each train is at various time intervals rather than understanding that the combined speed determines when they meet.
2. Incorrectly setting up the speed relationships
Students often struggle with expressing speeds in terms of fractions of total distance per unit time. They might try to assign actual distance values or get confused about whether to use distance/time or the reciprocal relationship needed for this type of problem.
3. Confusion about simultaneous start times
Some students might miss that both trains start at exactly 4 p.m. and try to account for different start times, or they might get confused about the direction each train is traveling and set up the problem incorrectly.
1. Fraction arithmetic errors when finding common denominators
When adding \(\frac{1}{60} + \frac{1}{90}\), students frequently make mistakes finding the least common multiple (180) or converting to equivalent fractions \(\left(\frac{3}{180} + \frac{2}{180}\right)\). They might incorrectly get \(\frac{2}{150}\) or other wrong results that lead to incorrect meeting times.
2. Incorrect conversion between fractional speeds and time
Students often struggle with the concept that if trains together cover \(\frac{1}{36}\) of distance per minute, then they meet in 36 minutes. They might incorrectly think they need to divide by the combined speed rather than take the reciprocal.
3. Time arithmetic errors
When adding 36 minutes to 4:00 p.m., some students make simple arithmetic mistakes, especially if they try to convert everything to minutes from midnight or use 24-hour time incorrectly.
1. Verification calculation mistakes
Even after getting 4:36 p.m. as an answer, students might second-guess themselves during verification and make arithmetic errors when checking that \(36 × \left(\frac{1}{60}\right) + 36 × \left(\frac{1}{90}\right) = 1\), leading them to select a different answer choice.
2. Time format confusion
Students might calculate the correct number of minutes (36) but make errors in expressing this as a time, potentially getting confused between 4:36 and other nearby times in the answer choices.
Step 1: Choose a convenient distance
Let's assume the distance between Meerut and Ghaziabad is 60 km (chosen because it's divisible by both 60 minutes and 90 minutes, making speed calculations clean).
Step 2: Calculate individual train speeds
Train X: Covers 60 km in 60 minutes (4 p.m. to 5 p.m.)
Speed of Train X = \(60 \text{ km} ÷ 60 \text{ minutes} = 1 \text{ km/minute}\)
Train Y: Covers 60 km in 90 minutes (4 p.m. to 5:30 p.m.)
Speed of Train Y = \(60 \text{ km} ÷ 90 \text{ minutes} = \frac{2}{3} \text{ km/minute}\)
Step 3: Apply relative speed concept
When trains move toward each other, their relative speed = Speed of X + Speed of Y
Relative speed = \(1 + \frac{2}{3} = \frac{5}{3} \text{ km/minute}\)
Step 4: Calculate meeting time
Time to cover total distance at relative speed = \(60 \text{ km} ÷ \left(\frac{5}{3} \text{ km/minute}\right)\)
= \(60 × \frac{3}{5} = 36\) minutes
Step 5: Find exact meeting time
Meeting time = Start time + 36 minutes = 4:00 p.m. + 36 minutes = 4:36 p.m.
Answer: A. 4:36 p.m.