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 first time at a point \(M\) somewhere between \(X\) and \(Y\), then what is the distance travelled by \(P ?\) (a) \(160 \mathrm{~km}\) (b) \(150 \mathrm{~km}\) (c) \(200 \mathrm{~km}\) (d) \(210 \mathrm{~km}\)

Short answer

Answer: (a) 160 km

Step-by-step solution

Calculate the time they both meet

Both person P and Q need to travel the same amount of time to meet at point M. We will denote this time as t. Since they have different speeds, we know that distance = speed * time. Therefore, the distance traveled by P is: 1. distance_P = 20t And the distance traveled by Q is: 2. distance_Q = 30t Since they meet at point M, the sum of their distances must be equal to the distance between X and Y (i.e., 200 km). So: 3. distance_P + distance_Q = 200

Create an equation and solve for t

Substitute the equations (1) and (2) into equation (3): 20t + 30t = 200 Combining the terms on the left side, we get: 50t = 200 Now, divide both sides by 50 to solve for t: t = 4 hours

Calculate the distance traveled by P

Now that we have the time at which they meet, we can find the distance traveled by P using the equation (1): distance_P = 20t Plug in t=4 hours: distance_P = 20 * 4 Calculate the distance: distance_P = 80 km Since P travels 80 km during the first meeting, and they are moving back and forth between X and Y, we have to add 80 km for each round trip back to X. So, P's total distance traveled will be: 80 km (first 80 km toward point M) + 80 km (return trip) = 160 km Hence, the distance traveled by P when they first meet at a point M is 160 km. The correct answer is (a) 160 km.

Key concepts

Quantitative Aptitude

Quantitative aptitude encompasses the ability to reason quantitatively and solve mathematical problems. It is often tested in educational settings and forms a core part of many competitive examinations. Distinct from raw numerical processing, it includes understanding logical numerical relationships, and applying mathematical concepts to find solutions.

The problem at hand is a classic example where quantitative aptitude comes into play. It requires basic understanding of concepts such as speed, distance, and time, as well as the ability to form and solve a linear equation. Students must marshal their aptitude to represent the situation mathematically and manipulate this representation to find the desired quantity, which in this case is the distance travelled by person P.

Relative Speed

Relative speed is a concept used when two objects are moving in the same or opposite directions. In essence, it finds how fast one object is moving relative to the other. When two objects are moving in the same direction, the relative speed is the difference of their speeds; conversely, when moving towards each other, as in this problem, the relative speed is the sum of their individual speeds.

Understanding relative speed is crucial for solving many transportation or motion problems. While the textbook problem doesn't explicitly mention 'relative speed', it demonstrates a situation where P and Q move towards each other, implying their relative speed is the sum of their individual speeds. This is indirectly used in forming the linear equation that leads to the solution.

Linear Equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. Linear equations are fundamental in algebra and appear frequently in various quantitative problems, including those that deal with distance, speed, and time.

The solution to the stated problem involves creating a linear equation based on the given data. By setting up the equation \(20t + 30t = 200\), we are leveraging the concept of linear equations to relate the distances traveled by P and Q to the total distance between points X and Y. Solving this equation provides the value of t, the time at which P and Q meet, which is the cornerstone for completing the rest of the problem.

Problem Solving

Problem solving is a multi-step process that involves understanding the problem, devising a plan, carrying out the plan, and retrospectively evaluating the solution for accuracy. This skill is of paramount importance in mathematics but is also applicable across disciplines.

In approaching the problem, one must first comprehend the relationship between speed, distance, and time. After understanding, the next step is to represent these relationships in a systematic way through mathematical equations. By following logical steps to solve the linear equations derived, we reach a solution that reveals the distance travelled by person P. Problem solving not only involves the steps taken to reach the answer but also the ability to check if the solution makes sense within the context of the given problem.