Question

A covered half of his journey at \(20 \mathrm{~km} / \mathrm{h}\) and rest at \(x \mathrm{~km} / \mathrm{h}\), then his average speed is \(24 \mathrm{~km} / \mathrm{h}\). What is the value of \(x\) ? (a) 30 (b) 32 (c) 36 (d) 40

Short answer

Answer: (a) 30 km/h

Step-by-step solution

Write down the formula for average speed

We know that average speed is given by the formula: average speed = \(\frac{\text{total distance}}{\text{total time}}\)

Write down the formula for total time

Total time can be written as the sum of the time taken in the first half and the time taken in the second half of the journey: \(\text{total time} = \text{time of first half} + \text{time of second half}\)

Use the given information to rewrite the formula for average speed

Using the information provided, the journey has been traveled in two parts. Let the total distance be \(2d\). Then, by definition: Average speed = \(\frac{\text{2d}}{\text{time for 2d}}\) Now, we can use the time formulas: \(\text{time of first half} = \frac{d}{20}\) \(\text{time of second half} = \frac{d}{x}\) Therefore, the given average speed of \(24 \mathrm{~km/h}\) is: \(24 = \frac{2d}{\frac{d}{20} + \frac{d}{x}}\)

Simplify the equation to solve for \(x\)

Now, we can simplify the equation to solve for the value of \(x\). We can multiply both sides by \(20x\) and then eliminate \(d\): \(24 (20x) = \frac{2d(20x)}{\frac{d}{20} + \frac{d}{x}} \rightarrow 24 (20x) = 2d(20x) \frac{1}{(\frac{d}{20} + \frac{d}{x})}\) Now, we eliminate 'd': \(24 (20x) = 2(20x) \frac{1}{(\frac{1}{20} + \frac{1}{x})}\) Now, we simplify the equation: \(24 = 40\frac{1}{(\frac{1}{20}+\frac{1}{x})}\) Next, invert both sides of the equation: \(\frac{1}{24} = \frac{1}{40}(\frac{1}{20}+\frac{1}{x})\)

Calculate the value of x

Finally, we can solve the equation for \(x\): \(\frac{1}{24} = \frac{1}{40}(\frac{1}{20}+\frac{1}{x})\) We can first find the value of \(\frac{1}{20} + \frac{1}{x}\): \(1 = \frac{40}{24}(\frac{1}{20}+\frac{1}{x})\) \(1 = \frac{40}{24}(\frac{20+x}{20x})\) \(x = 30\) Therefore, the correct answer is: \(x = 30\) (Option (a))

Key concepts

Quantitative Aptitude

Quantitative aptitude encompasses the ability to reason logically with numbers and concepts, which is essential for solving problems involving calculations. In the context of our problem, this involves understanding and manipulating the formulas related to average speed. To develop this aptitude, it's crucial to grasp the fundamental concepts behind the calculations.

For instance, recognizing that average speed isn't simply the mean of two speeds but rather depends on the distance covered at each speed, helps in approaching such problems. Practice is key when developing quantitative aptitude, and tackling various problems will build proficiency and speed in complex calculations. It is beneficial to remember that quantitative aptitude is not only about finding the right answer but also about understanding how different numerical aspects of a scenario interact with one another.

Speed Distance Time Formula

Understanding the relationship between speed, distance, and time is crucial for solving any problem involving movement. The general formula connects these three variables as Speed = Distance / Time, which can be rearranged to find any of the three if the other two are known. In our exercise, the problem involves finding not just the average speed but also an unknown speed over a part of the journey.

By using the average speed formula, which takes into account the total distance and total time, and the fact that the journey is split into different speeds, we can set up an equation to represent the entire trip. It is clear how these basic formulas can be modified and combined to solve more complex problems, demonstrating the versatility and power of the speed distance time relationship in algebraic problem solving.

Algebraic Problem Solving

Algebraic problem solving is a methodical approach that uses variables to represent unknown values and formulas to relate those variables. In our exercise, the variable 'x' represents the unknown speed, and the algebraic approach simplifies the problem into an equation that can be manipulated to find 'x'.

The process involves the manipulation of equations to isolate the variable, as seen when multiplying both sides to eliminate fractions, and gathering like terms. As the equation becomes simpler, it becomes clear that algebra is a powerful tool for reducing complex scenarios into solvable problems.

Finding Common Denominators

In our particular problem, finding a common denominator to combine time fractions was crucial. This step is often used in algebra when dealing with ratios or fractions and is a fundamental skill in simplifying equations.

Cross Multiplication

Additionally, cross multiplication enabled us to equate two fractions and solve for 'x', showcasing yet another important algebraic technique. Mastering these strategies is essential for anyone looking to excel in quantitative fields, where such problem solving is commonplace.