Question

There are fuo places \(\mathrm{X}\) and \(Y, 200 \mathrm{~km}\) apart from each other. Initially two persons \(P\) and \(Q\) both are at \(^{\circ} X:\) The speed of \(P\) is \(20 \mathrm{~km} / \mathrm{h}\) and speed of \(Q\) is \(30 \mathrm{~km} / \mathrm{h}\). Later on they starts to move to and fro benveen \(X\) and \(Y\). If they meet third time each other at a point \(C\), somewhere between \(X\) and \(Y\), then the ratio of distances \(C X\) and \(C Y\) is: (a) \(3: 2\) (b) \(1: 3\) (c) \(2: 3\) (d) \(2: 5\)

Short answer

Based on the given problem, the ratio of distances CX and CY when two persons P and Q meet for the third time is 2:3.

Step-by-step solution

Calculate Relative Speed

For P and Q to meet, they must have covered the same distance in the same time, so we calculate their relative speed when they're moving towards each other. Relative speed is found by adding their individual speeds, which are 20 km/h and 30 km/h. Relative Speed (P and Q) = 20 km/h + 30 km/h = 50 km/h

Calculate Time Taken to Meet for the First Time

Now that we have the relative speed, we can find the time taken for them to meet for the first time. They will meet when they cover the total distance of 200 km together. Time Taken to Meet for the First Time = Total Distance / Relative Speed Time Taken = 200 km / 50 km/h = 4 hours

Calculate Time Taken to Meet for the Third Time

They will meet once again after they have covered the total distance once more. Since they've already met once, we just need to add the time taken to meet for the first time twice. Total Time Taken to Meet for the Third Time = 2 * Time Taken to Meet for the First Time Total Time = 2 * 4 hours = 8 hours

Calculate Distances Covered by P and Q

Now that we know the total time taken to meet for the third time, we can calculate the distances covered by P and Q using their individual speeds. Distance Covered by P = Speed of P * Total Time Distance Covered by P = 20 km/h * 8 hours = 160 km Distance Covered by Q = Speed of Q * Total Time Distance Covered by Q = 30 km/h * 8 hours = 240 km

Determine the Ratio of Distances CX and CY

To find the ratio of distances CX and CY, we can divide the distance covered by P (160 km) by the distance covered by Q (240 km) as follows: Ratio of distances = Distance Covered by P / Distance Covered by Q Ratio of distances = 160 km / 240 km Simplifying the ratio, we get: Ratio of distances = 2:3 So, the ratio of distances CX and CY when they meet for the third time is 2:3. Therefore, the correct answer is (c) 2:3.

Key concepts

Understanding Distance in Movement Problems

When tackling problems involving two people or objects moving together or towards each other, it's crucial to understand the concept of distance. Distance is a measure of how far an object has traveled, which in this exercise, is the space between two points, X and Y.

In the scenario given, X and Y are 200 km apart. This distance is vital as it determines how long it will take for two moving objects (in this case, persons P and Q) to meet based on their speeds. When both start moving, they're essentially working on closing this 200 km gap.

One key insight is that even after they first meet, they continue to share total journey distances, contributing towards subsequent meetings. This shared travel distance aids in finding when and where the third meeting will occur, which is calculated through their relative speed. Therefore, grasping how this basic physical distance lays the foundation for further calculations is crucial.

Time Calculation for Meeting Points

Time calculation is central to understanding how P and Q meet at various points. Once the relative speed between two moving objects is known, the time it takes for them to meet can be computed.

In this exercise, the relative speed of P and Q, when moving towards each other, is 50 km/h. This means they close 50 km of the 200 km gap per hour.

• First meeting occurs when the complete 200 km is covered, taking 4 hours.
• Subsequent meetings are symmetric, taking equal intervals of time, so another 4 hours for the second coverage.

So, after 8 hours, during which they've effectively traveled back to the starting crossing point as well, they meet for the third time. This systematic approach demonstrates how distances covered over time are crucial in movement-based problems.

Solving Ratio Problems in Distance

Finally, ratio problems involve comparing parts of a whole, represented by distances in this exercise. Ratio here provides a way to express the relative positions of P and Q when they meet for the third time.

The distances each covers can be calculated by multiplying their speeds by total time, which, for 8 hours, gives:

• P covers 160 km (20 km/h * 8 hours).
• Q covers 240 km (30 km/h * 8 hours).

To find the ratio of their meeting point distances, CX (distance from C to X) and CY (distance from C to Y), their covered distances are simplified to 2:3. This ratio indicates that the point C where they meet has divided the 200 km such that 2 parts are from X and 3 parts from Y. Understanding ratio problems helps in interpreting the relationship of distances between meeting points effectively.