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 they continue to move to and fro between \(X\) and \(Y\) then what is the distance travelled by \(Q\), when they meet each other for the third time? (a) \(375 \mathrm{~km}\) (b) \(225 \mathrm{~km}\) (c) \(350 \mathrm{~km}\) (d) \(445 \mathrm{~km}\)

Short answer

Answer: 375 km

Step-by-step solution

Calculate the Relative Speed

When P and Q move in opposite directions, we need to find their relative speed. This is found by adding up their speeds: V_rel = V_P + V_Q = 20 + 60 = 80 km/h

Find the Time to First Meeting

Since we have their relative speed (V_rel), we can find the time they take to meet for the first time. The distance between X and Y is given as 100 km, so the time to first meeting (t1) can be calculated as: t1 = Distance / V_rel = 100 / 80 = 1.25 hours

Calculate the Distance Travelled by Q

At the first meeting, we can find the distance travelled by Q (D_Q1) by multiplying its speed by the time taken: D_Q1 = V_Q * t1 = 60 * 1.25 = 75 km

Find the Time for Next Meetings

After their first meeting, P and Q will keep meeting again after a fixed interval (T). This interval comes from their full round trip to X and Y. Since their relative speed is 80 km/h and they have to travel twice the distance between X and Y: T = (2 * 100) / 80 = 2.5 hours

Calculate the Distance Travelled by Q for the Next Meetings

We can now find the distance travelled by Q for the second and third meetings (D_Q2 and D_Q3) by multiplying its speed with the intervals: D_Q2 = V_Q * T = 60 * 2.5 = 150 km; D_Q3 = D_Q2 (Since D_Q2 is constant for each meeting)

Calculate the Total Distance Travelled by Q

To find the total distance travelled by Q when they meet for the third time, we can add up the distances travelled in each meeting: D_Total = D_Q1 + D_Q2 + D_Q3 = 75 + 150 + 150 = 375 km Therefore, the distance travelled by person Q when they meet for the third time is 375 km, which corresponds to option (a).

Key concepts

Relative Speed

The concept of relative speed plays a crucial role when two moving objects are either heading towards each other or moving in the same direction. Imagine two trains or cars moving on parallel tracks; their relative speed determines how quickly they would approach one another or separate.

In the case of objects moving towards each other, like person P and person Q in our exercise, their relative speed is the sum of their individual speeds. That's because both are moving in opposite directions, effectively reducing the time it takes for them to meet. Mathematically, we express this as: \( V_{rel} = V_P + V_Q \).

However, if they were moving in the same direction, the relative speed would be the difference between their speeds, as one would be trying to catch up to the other. Understanding this concept is essential for solving various problems involving motion, including those in competitive exams.

Distance Time Calculation

Distance time calculation is at the heart of many problems in physics and mathematics, especially when dealing with motion. The basic formula connecting distance, speed, and time is: \( \text{Distance} = \text{Speed} \times \text{Time} \).

This equation implies that if you know any two of these variables, you can calculate the third. In our textbook example, we used this relationship to determine how long it took for P and Q to meet by dividing the distance between them by their relative speed.

Key Steps in Distance Time Calculation

• Determine the speeds of the moving objects.
• Identify if they are moving towards each other or in the same direction.
• Calculate relative speed accordingly.
• Use the formula to find out the remaining unknown (distance, speed, or time).

This straightforward approach simplifies complex problems and is a handy tool in many quantitative exams.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and logical problem solving. It's a profound component of many competitive exams, aptitude tests, and even job interviews. This discipline includes various mathematical concepts such as algebra, geometry, arithmetic, and number systems.

Problems such as the one involving persons P and Q enable students to sharpen their quantitative aptitude by applying several mathematical tools. These problems not only test one's ability to perform calculations but also to reason logically and make decisions based on given numerical data.

Improving Quantitative Aptitude

• Practice regularly to improve speed and accuracy.
• Understand the underlying concepts, don't just memorize formulas.
• Take timed quizzes to simulate test conditions.
• Review mistakes and ensure you learn from them.

By consistently working on problems such as distance time calculations and relative speed, one can greatly enhance their quantitative aptitude skills.