Question

Harsha takes 3 hours more than Ashok, who drives his car \(5 \mathrm{~km} / \mathrm{h}\) faster than Harsha drives, to cover \(180 \mathrm{~km}\) distance. What is the speed of Harsha? (a) \(12 \mathrm{~km} / \mathrm{h}\) (b) \(15 \mathrm{~km} / \mathrm{h}\) (c) \(30 \mathrm{~km} / \mathrm{h}\) (d) \(40 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: Harsha's speed is 20 km/h.

Step-by-step solution

Understand the problem

Ashok's speed is \(5 \mathrm{~km} / \mathrm{h}\) faster than Harsha's speed, which we'll call \(x\). We know Ashok takes 3 hours less than Harsha to cover the distance, and they both cover \(180 \mathrm{~km}\). We need to create an equation based on the information and solve for Harsha's speed (\(x\)). Step 2: Forming the equations

Form the equations

We'll use the relation: distance = speed × time. For Harsha: \(180 = x \times t\) For Ashok: \(180 = (x + 5) \times (t - 3)\) Step 3: Solve the first equation for t

Solve first equation

From the equation: \(180 = x \times t\) We can solve for time, t: \(t = \frac{180}{x}\) Step 4: Substitute t into the second equation

Substitute time

From the equation: \(180 = (x + 5) \times (t - 3)\) Substitute t with the expression found in step 3: \(180 = (x + 5) \times (\frac{180}{x} - 3)\) Step 5: Solve the equation

Solve the equation

We'll solve for x in the equation: \(180 = (x + 5) \times (\frac{180}{x} -3)\) Multiply by x to get rid of the fraction: \(180x = (x + 5)(180 - 3x)\) Distribute: \(180x = 180x - 3x^2 + 900 - 15x\) Move all terms to one side: \(3x^2 - 15x - 900 = 0\) Divide by 3: \(x^2 - 5x - 300 = 0\) Factor: \((x - 20)(x + 15) = 0\) Step 6: Find Harsha's speed

Harsha's speed

From the factored equation, we get two possible solutions for x: \(x - 20 = 0 \Rightarrow x = 20\) \(x + 15 = 0 \Rightarrow x = -15\) Since the speed can't be negative, Harsha's speed is \(20 \mathrm{~km} / \mathrm{h}\). However, this speed is not provided in the given options, so there might be a mistake in the question or options presented. It's important to double-check the problem and discuss the discrepancy with the teacher or instructor.

Key concepts

Speed Calculation

Calculating speed is an essential skill in many real-world scenarios. In problems involving time and distance, the speed determines how long it takes to reach a certain destination. The basic formula to keep in mind here is:

• Speed = Distance / Time

This formula allows you to determine one of the three variables if you know the other two. For Harsha and Ashok's situation, we have a specific distance that they both cover, which is 180 km. Harsha's speed is denoted as \( x \), while Ashok's speed is \( x + 5 \), since Ashok drives 5 km/h faster. By knowing this, you can use the speed formula to set up equations that capture both their travel situations. This helps lay the groundwork for solving the problem accurately.

Quadratic Equations

Quadratic equations are a crucial mathematical concept that can solve a variety of problems, including the one involving Harsha and Ashok's speeds. These equations take the general form:

• \( ax^2 + bx + c = 0 \)

In this problem, we derived the quadratic equation \( x^2 - 5x - 300 = 0 \) through logical steps based on distances and time differences between Harsha and Ashok.To solve this equation, you look for two numbers that multiply to give you the constant term (in this case, -300) and add to give you the coefficient of \( x \) (which is -5). Factoring quadratic equations often yields a simple solution, as it did in this case, where the equation factors into \( (x - 20)(x + 15) = 0 \). This step is pivotal as it simplifies your problem-solving process by giving you potential solutions for \( x \).

Problem Solving Steps

Breaking down a problem into manageable steps is essential for effective problem-solving. In our exercise of determining Harsha's speed, we begin by understanding what is being asked—for this problem, it was necessary to compare and calculate speeds based on given differences in time and speed. Next, forming equations is crucial. By leveraging known relationships like speed, distance, and time, we systematically set up equations. This step includes expressing one variable in terms of another, as done by calculating the time \( t = \frac{180}{x} \). Finally, substituting and solving equations demands attention, leading us through the problem until we find our answer. While solving, checking each step ensures the accuracy of our solution. For Harsha and Ashok's case, arriving at a valid speed option, and recognizing any discrepancies with the given choices, wraps up the process. This step-by-step method enhances clarity and efficiency in finding solutions.