Question

A person P is at \(X\) and another persan \(Q\) is at \(Y .\) The distance between \(X\) and \(Y\) is I00 kn. The speed of \(P\) is \(20 \mathrm{~km} / \mathrm{h}\). While the speed of \(Q\) is \(60 \mathrm{~km} / \mathrm{h} ?\) If \(P\) and \(Q\) continue to move between \(X\) and \(Y\), then the ratio of distances covered by \(P\) and \(Q\) when they meet for the \(5^{\text {th }}\) time? (a) \(1: 4\) (b) \(1: 3\) (c) \(2: 3\) (d) \(3: 4\)

Short answer

Answer: (b) 1:3

Step-by-step solution

Determine the relative speed

P has a speed of 20 km/h and Q has a speed of 60 km/h. To find their relative speed, we can simply subtract the lower speed from the higher speed: \(60 - 20 = 40\) km/h.

Calculate time taken for the first meeting

Since the distance between X and Y is 100 km, the time taken for the first meeting can be calculated by dividing the distance between the points by their relative speed, that is \(\frac{100}{40}\) hours. We can simplify this to get \(\frac{5}{2}\) hours for the time of their first meeting.

Calculate total number of meetings

They meet for the 5th time, so we need to multiply the time taken for the first meeting by the total number of meetings before this one. Since they meet twice in every round trip between X and Y, the total number of meetings before the 5th one is \({5 - 1 = 4}\), so the total time taken is \({\frac{5}{2} * 4 = 10}\) hours.

Calculate distances covered by P and Q

Now that we have the total time taken of 10 hours, we can calculate the distances covered by P and Q at the time of their 5th meeting using their speeds: - Distance covered by P: \(20 \mathrm{~km} / \mathrm{h} * 10 \mathrm{~h} = 200 \mathrm{~km}\) - Distance covered by Q: \(60 \mathrm{~km} / \mathrm{h} * 10 \mathrm{~h} = 600 \mathrm{~km}\)

Calculate the ratio of distances covered

Now we have the distances covered by P and Q, which are 200 km and 600 km respectively. To find the ratio, we can divide both distances by their greatest common divisor: - GCD (200, 600) = 200 - Ratio: \(\frac{200}{200} : \frac{600}{200} = 1 : 3\) Thus, the correct answer is (b) \(1: 3\).

Key concepts

Relative Speed in Motion Problems

Understanding the concept of relative speed is crucial in solving motion problems involving two or more moving objects. Relative speed pertains to how fast one object is moving in comparison to another. It's like having a race between two friends: if one is running really fast and the other is just walking, their speeds in relation to each other is the speed difference between them.

When two objects are moving towards each other, like in the case of persons P and Q from our exercise, their relative speed is the sum of their individual speeds. However, if they were moving in the same direction, you'd subtract the slower speed from the faster one to get the relative speed. In our exercise, person P is moving towards person Q with a speed of 20 km/h, and Q is approaching P at 60 km/h. So, we subtract P's speed from Q's to get a relative speed of 40 km/h.

It's a bit like being on a moving walkway at an airport: if you're standing still and your friend is walking on the adjacent walkway, they seem to be moving faster relative to you. But if both of you are walking side by side on separate moving walkways that are going in opposite directions, you'll be separating faster compared to if you were just walking on solid ground.

Time and Distance

The relationship between time, speed, and distance is a cornerstone of motion problems and forms part of the foundation for physics and travel calculations. To put it simply, 'distance' is how far an object travels, 'speed' tells us how fast it goes, and 'time' is how long it has been moving. Their connection is neatly summed up in the basic formula: distance = speed × time.

This fundamental equation plays a significant role in solving our example problem. Once we figured out the relative speed of persons P and Q, we used the total distance between points X and Y to calculate the time it takes for P and Q to meet for the first time. Understanding how to manipulate this formula allows you to solve for any of the three variables, given the other two, which is a valuable skill for various quantitative problems.

This straightforward relationship also helps in planning – whether you're forecasting how long a road trip will take, scheduling flights, or even timing a morning jog. Knowing how speed and distance interact over time can be incredibly practical in everyday life as well as in academic endeavors.

Ratio and Proportion in Quantitative Aptitude

Ratio and proportion are powerful mathematical concepts used to compare quantities and their relative sizes. Simply put, a ratio shows the relative size of two quantities, while a proportion states that two ratios are equal. It's a bit like comparing slices of a cake: if you get one slice and your friend gets three, the ratio of your slices is 1:3.

Our problem asked for the ratio of distances covered by P and Q after the fifth meeting. Once we calculated the distances using the speed and time ((speed of P × total time) and (speed of Q × total time)), we compared the two using a ratio. Finding the greatest common divisor between these distances helped us to simplify this ratio.

Grasping these concepts is particularly useful for students in various competitive exams and real-life scenarios, such as analyzing financial statements, converting recipes, or even splitting the bill at a restaurant. Being able to express one quantity in terms of another and then simplify it to its most basic form enables clear and concise communication, especially in fields involving data and statistics.