Question

A person \(X\) starts from Lucknow and another persons Y starts from Kanpur to meet each other. Speed of \(X\) is \(25 \mathrm{~km} / \mathrm{h}\), while speed of \(Y\) is \(35 \mathrm{~km} / \mathrm{h}\). If the distance between Lucknow and Kanpur be \(120 \mathrm{~km}\) and both \(X\) and \(Y\) start their journey at the same time, when will they meet? (a) \(1 \mathrm{~h}\) later (b) 2 h later (c) \(\frac{1}{2} \mathrm{~h}\) later (d) 3 h later

Short answer

Answer: (b) 2 hours later

Step-by-step solution

Calculate the Relative Speed

To find the relative speed, we add the individual speeds of X and Y, since they are moving towards each other. So, the relative speed = speed of X + speed of Y = 25 km/h + 35 km/h = 60 km/h.

Calculate the Time Taken

Now that we have the relative speed, we can find the time they take to meet. Using the formula: time = distance / relative speed, we can plug in the values: time = 120 km / 60 km/h = 2 hours. So, the correct answer is (b) 2 hours later.

Key concepts

Time and Distance Problems

Understanding time and distance problems is essential in the realm of quantitative aptitude, a key area for many competitive exams and real-world scenarios. The fundamental formula at the heart of these problems is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \].
To tackle these problems, you need to manipulate the formula, usually to find one of the three aspects: distance, speed, or time if the other two are known. In essence, if you're given the speed and the distance, you can find the time it would take to cover that distance by rearranging the formula to:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \].
It's also essential to understand the units being used. For example, if speed is in kilometers per hour and distance in kilometers, the time will be in hours. Consistency in units across the problem is critical, otherwise, conversions must be made. One common pitfall for students is neglecting to convert minutes into hours or vice versa, leading to incorrect answers.

Practice Example

Imagine a situation where a car travels 90 km in 1.5 hours. To find the speed, you divide distance by time, which is \[ \text{Speed} = \frac{90 \text{ km}}{1.5 \text{ h}} = 60 \text{ km/h} \].This is a straightforward application but can become more complex with additional variables or constraints.

Quantitative Aptitude

Quantitative aptitude encompasses numerical ability and problem-solving skills, often tested in academic and employment settings. It's not just about knowing the formulas, but also about how to apply them to various problems. Basic arithmetic, algebra, geometry, and data interpretation all fall under this domain. For time and distance problems, algebraic skills are particularly important as you'll frequently set up equations based on the problem statements.
To improve your quantitative aptitude, consistent practice is key. Working through various problems will help you recognize patterns and understand the underlying concepts more deeply. Another tip is to simplify complex problems by breaking them into smaller parts and solving each part step by step. For instance, in a scenario where multiple entities are moving at different speeds or in different directions, it's useful to consider each one independently before combining their effects.

Quick Tips for Improvement

• Understand the core concepts and memorize necessary formulas.
• Practice regularly with a variety of problem types.
• Develop good mental math skills to speed up calculations.
• Analyze your mistakes and understand where you went wrong to avoid repeating them.

Becoming proficient in quantitative aptitude involves not only solving problems correctly but also doing so efficiently.

Meeting Point Calculation

The concept of a 'meeting point' arises in time and distance problems when two or more entities move towards each other from different starting points. To calculate the meeting point, we use the concept of relative speed, which refers to the speed of one object as observed from another object. In cases where two bodies are moving towards each other, their relative speeds are added together. Conversely, if they're moving in the same direction, the relative speed is the difference between their speeds.
The solution to these problems relies on the formula:
\[ \text{Time to Meet} = \frac{\text{Distance between them}}{\text{Relative Speed}} \].

Real-World Application

Take for instance, two trains starting from different stations 150 km apart, with train A moving at 60 km/h and train B at 40 km/h. Their relative speed is their sum, 100 km/h, since they're headed towards each other. They'll meet after \[ \text{Time} = \frac{150 \text{ km}}{100 \text{ km/h}} = 1.5 \text{ h} \].This method is crucial for planning, logistics, and even emergency services when it's necessary to determine the time and location where moving entities will converge.