Question

The speeds of Vimal and Kamal are \(30 \mathrm{~km} / \mathrm{h}\) and \(40 \mathrm{~km} / \mathrm{h}\). Initially Kamal is at a place \(L\) and Vimal is at a place \(M\). The distance between \(L\) and \(M\) is \(650 \mathrm{~km}\). Vimal started his journey 3 hours earlier than Kamal to meet each other. If they meet each other at a place \(P\) somewhere between \(L\) and \(M\), then the distance between \(P\) and \(M\) is: (a) \(220 \mathrm{~km}\) (b) \(250 \mathrm{~km}\) (c) \(330 \mathrm{~km}\) (d) \(320 \mathrm{~km}\)

Short answer

Answer: The distance between the meeting point and Vimal's starting point is 330 km.

Step-by-step solution

Calculate the distance Vimal traveled before Kamal started

Since Vimal started 3 hours earlier than Kamal, and his speed is 30 km/h, the distance Vimal traveled before Kamal started can be calculated as: Distance Vimal_traveled = (Vimal's speed) x (time) = 30 km/h * 3 h = 90 km

Calculate the time Vimal and Kamal traveled together before meeting

Let t be the time Vimal and Kamal traveled together before meeting. When they meet, the sum of the distances they have traveled will be equal to the distance between their starting points. So, Vimal's distance + Kamal's distance = Total distance (30 km/h * t) + (40 km/h * t) = 650 km - 90 km

Solve the equation for t

Now, let's solve the equation for t: (30 + 40) km/h * t = 560 km 70 km/h * t = 560 km t = 8 hours.

Calculate the distances Vimal and Kamal traveled during the time they were traveling together.

Now, we can find the distances traveled by Vimal and Kamal during the time t=8h. Distance Vimal_traveled_together = 30 km/h * 8 h = 240 km Distance Kamal_traveled_together = 40 km/h * 8 h = 320 km

Calculate the distance between the meeting point and Vimal's starting point

Now, we can find the distance between the meeting point and Vimal's starting point by adding the distances Vimal traveled before Kamal started and during the time they were together. Distance_PM = 90 km + 240 km = 330 km So, the distance between the meeting point and Vimal's starting point is 330 km. The correct answer is (c) 330 km.

Key concepts

Time, Speed, and Distance

Understanding the relationship between time, speed, and distance is crucial for solving a variety of problems, especially in quantitative aptitude assessments. The basic formula for this relationship is expressed as distance equals speed multiplied by time, or more succinctly
\( \text{Distance} = \text{Speed} \times \text{Time} \).

In practical terms, if you know any two of these variables, you can solve for the third. For example, if a car travels at a speed of \(60 \text{ km/h}\) for \(2 \text{ hours}\), the distance covered is \(120 \text{ km}\). This formula becomes the foundation for more complex distance and speed problems. To improve understanding of this concept, one can practice by charting distances, times, and corresponding speeds, varying two of the parameters to solve for the third. A common mistake is neglecting to ensure that the units of measurement for time and speed are consistent when calculating distance.

Relative Speed

Relative speed is an essential concept when two objects are moving towards each other or in the same direction. It's the speed of one object as observed from the other object, effectively the rate at which the distance between the two objects is changing.

When objects are moving towards each other, like in the case of Vimal and Kamal, their relative speed is the sum of their individual speeds, because they are reducing the distance between them at the rate of both speeds combined. Conversely, if the objects are moving in the same direction, the relative speed is the difference between their speeds. For instance, if a person A is chasing another person B, and A is faster, the relative speed would be \( \text{A's speed} - \text{B's speed} \). Understanding this principle is key to solving problems where the time of interaction or the point of meeting is asked, and it's fundamental in avoiding the common error of using individual speeds when combined speed is needed.

Equations in One Variable

Equations in one variable are algebraic expressions set equal to a value or another expression, with just one unknown variable to solve for. The equation \(70 \text{ km/h} \times t = 560 \text{ km}\) from the given problem is a simple linear equation where \(t\) is the variable.

To solve these, one typically isolates the variable on one side of the equation using basic algebraic operations—adding, subtracting, multiplying, and dividing both sides of the equation by the same number. For instance, if we have \(x + 5 = 20\), subtracting 5 from both sides gives us \(x = 15\). A common challenge when dealing with equations is properly setting them up. It's essential to remember the order of operations and always perform the same operation on each side of the equation to maintain balance. Moreover, when dealing with word problems, it is crucial to translate the scenario correctly into an equation, a skill refined with practice.

Quantitative Aptitude

Quantitative aptitude refers to one's ability to handle numerical and mathematical calculations. It is a measure of an individual's mathematical intelligence and is often tested in academic and professional settings like competitive exams and job interviews.

A good grasp of basic arithmetic operations, algebra, geometry, and number systems is essential for excelling in quantitative aptitude tasks. This area also requires logical thinking and problem-solving skills, helping you break down complex scenarios into solvable equations. For students, enhancing quantitative aptitude can be achieved through regular practice of various types of problems, focusing on accuracy and speed. It's crucial to understand the underlying principles of the problems, such as the time, speed, and distance concept and the approach to relative speed, rather than just memorizing formulas. Developing strategies like identifying patterns, estimating, and eliminating obviously incorrect options can vastly improve performance.