Question

two trains \(A\) and \(B\) start simultaneously in the opposite Two train from two points \(P\) and \(Q\) and arrive at their direction fons 16 and 9 hours respectively after their meeting destinations each other. At what speed does the second train \(B\) travel if the travels at \(120 \mathrm{~km} / \mathrm{h}\) per hour : (a)90km/hr (b) 160km/hr (c) 67.5km/hr (d) none of these \(67.5 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: (d) none of these, as the speed of train B is 120 km/h.

Step-by-step solution

Calculate the distance traveled by train \(A\) after meeting train \(B\)

First, we need to find the distance traveled by train \(A\) after it meets train \(B\). We know that train \(A\) travels at a speed of \(120 \mathrm{~km/h}\) and it takes 16 hours to reach its destination after their meeting. Therefore, the distance traveled can be calculated using the formula: Distance \(=\) Speed \(×\) Time Hence, the distance traveled by train \(A\) after the meeting point is: Distance\(_A = 120 \mathrm{~km/h} × 16 \mathrm{~hours} = 1920 \mathrm{~km}\)

Establish the ratio of distances traveled by trains after meeting

Train \(A\) reaches its destination 16 hours after their meeting, while train \(B\) reaches its destination 9 hours after their meeting. As both trains move in opposite directions, the ratio of their speeds will not change at the meeting point. Therefore, the ratio of distances traveled by the trains after their meeting will be the same as the ratio of the time they take to reach their destinations: Ratio of distances \(= \frac{\text{Distance}_A}{\text{Distance}_B} = \frac{16}{9}\) As we calculated earlier, Distance\(_A = 1920 \mathrm{~km}\) So, we can find the distance traveled by train \(B\) after their meeting point using the ratio: \(\frac{1920}{\text{Distance}_B} = \frac{16}{9}\) Now solve for Distance\(_B\): \(\text{Distance}_B = \frac{1920 × 9}{16}= 1080 \mathrm{~km}\)

Find the speed of train \(B\)

Now, we can calculate the speed of train \(B\). We know the distance traveled by train \(B\) (1080 km) and the time it took to reach its destination after meeting train \(A\) (9 hours). We can use the same speed formula: Speed \(=\) \(\frac{\text{Distance}}{\text{Time}}\) Speed\(_B = \frac{1080 \mathrm{~km}}{9 \mathrm{~hours}} = 120 \mathrm{~km/h}\) Hence, the speed of train \(B\) is \(120 \mathrm{~km/h}\). The correct answer is not in the given choices, so the answer is (d) none of these.

Key concepts

Understanding Ratios in Speed and Distance

Ratios serve as a powerful tool for comparing two quantities, and they are particularly helpful in problems involving speed and distance. When two trains depart from different stations and meet at some point, their speeds and travel times can be related through ratios.

In the given problem, we established that the ratio of distances traveled after their meeting is equal to the ratio of time taken by each train to reach their destination post-meeting. This is because the trains maintain constant speeds, and the distances they travel over time reflect their respective velocities.

To compute the distance, use the formula:

• Distance = Speed × Time

The proportional relationship between the distances is thus:

• Ratio of Distances = Ratio of Times

This understanding allows you to solve for unknown distances or verify consistency in travel parameters between the two trains.

Exploring Relative Speed

Relative speed is a fascinating concept, especially when two objects, like trains, are moving towards or away from each other. When considering the movement of two objects in opposite directions, the relative speed is simply the sum of their individual speeds.

In scenarios where two objects move towards each other, like the trains in our problem, they close the distance between them at a rate equal to the sum of their speeds. If they were moving in the same direction, the relative speed would be the difference between their speeds.

Understanding and determining relative speed helps in calculating how soon two objects will meet, how far apart they are after some time, or analyzing changes when they pass each other. It directly impacts how we approach and solve these kinds of problems.

For any such problem, consider:

• If moving towards each other: Add their speeds
• If moving in the same direction: Subtract their speeds

Problem Solving Strategy Using Speed, Time, and Distance

A strategic approach simplifies a seemingly complex problem involving speed, time, and distance. Begin by clearly defining all known quantities and identify what you need to find. Use well-established formulas such as Distance = Speed × Time and keep an eye on the relationships between quantities.

In this problem, after recognizing that the trains start simultaneously from two points and meet at some common point, we solve it step by step:

• Calculate the distance covered by each train using the distances they travel post-meeting.
• Use ratios to establish relationships and solve for unknowns.
• Verify by re-calculating speeds using derived distances and travel times.

Solve the pieces of the puzzle, and soon you'll connect the dots to find the required answer. This approach not only aids in problem-solving but deepens your understanding of how these elements interact.