Question

I went to Delhi @ speed of \(200 \mathrm{~km} / \mathrm{hr}\) but suddenly I returned to the same place @ speed of \(600 \mathrm{~km} / \mathrm{hr}\). What is my average speed : (a) \(300 \mathrm{~km} / \mathrm{hr}\) (b) \(400 \mathrm{~km} / \mathrm{hr}\) (c) \(366.66 \mathrm{~km} / \mathrm{hr}\) (d) none of these

Short answer

Answer: The average speed of the person is 300 km/hr.

Step-by-step solution

Understand the Formula for Average Speed

Average speed is given by the formula: Average speed = \(\frac{2 \times speed1 \times speed2}{speed1 + speed2}\), where speed1 and speed2 are the individual speeds of the person while going and returning.

Plug in the given values

In this case, speed1 is \(200 \mathrm{ ~km} / \mathrm{ hr}\) and speed2 is \(600 \mathrm{ ~km} / \mathrm{ hr}\). Plug in these values into the formula: Average speed = \(\frac{2 \times 200 \times 600}{200 + 600}\)

Calculate the Average Speed

Now, perform the calculations: Average speed = \(\frac{2 \times 200 \times 600}{800}\) = \(\frac{1200 \times 200}{800}\)

Simplify the expression

Dividing the numerator and the denominator by 200, we get: Average speed = \(\frac{1200}{4}\)

Find the final answer

Finally, calculate the average speed: Average speed = \(300 \mathrm{ ~km} / \mathrm{ hr}\) So, the correct option is (a) \(300 \mathrm{ ~km} / \mathrm{ hr}\).

Key concepts

Quantitative Aptitude

Quantitative aptitude encompasses the ability to reason logically and solve numerical and mathematical problems. This skill is essential in various academic and career fields. Specifically, when tackling problems related to speed, time, and distance, it is crucial to understand the concepts and formulas involved to compute the correct results efficiently.

Enhancing quantitative aptitude requires practice with arithmetic problems, including those that involve calculating average speeds, such as in the exercise provided. Understanding the formula for average speed and being able to apply it to different scenarios is a key component of strong quantitative aptitude. Regularly solving such problems not only prepares students for examinations but also sharpens their problem-solving skills for real-world situations.

Speed, Time, and Distance

The interrelationship between speed, time, and distance is a fundamental concept in physics and essential arithmetic problems. In the context of the given exercise, the average speed is the primary focus. The average speed of a journey is distinct from the average of the speeds, as it takes into account the entire distance covered over the total time taken.

When calculating average speed, remember that it is a measure of the total distance traveled divided by the total time taken. The formula used in the exercise, however, \( \text{Average speed} = \frac{2 \times \text{speed1} \times \text{speed2}}{\text{speed1} + \text{speed2}} \), specifically applies to scenarios where a trip is made to a destination and back at two different speeds. This is a unique solution that leverages the harmonic mean rather than the arithmetic mean to accurately reflect the average speed over the complete journey.

Arithmetic Problems

Arithmetic problems are mathematical questions that involve the basic operations of addition, subtraction, multiplication, and division. In our exercise, the calculation of average speed requires using multiplication and division to solve the problem. A thorough understanding of these operations is essential for accurately and quickly finding solutions to arithmetic problems.

To improve at solving arithmetic problems, students should familiarize themselves with the properties of the operations involved and work on exercises that challenge their understanding in various contexts, such as average speed calculations. This helps in strengthening mental arithmetic skills which are incredibly useful beyond academic assessments, particularly in everyday life scenarios where quick calculation is an advantage.

In summary, mastery of arithmetic operations is not simply about memorizing formulas but also about understanding when and how to apply them in different types of questions, like the average speed exercise showcased here.