Solving the Car Speed Problem using Algebra and Simplified Methods

In this article, we will explore a classic problem involving car speeds and how to solve it using algebra and simplified methods. We will cover the step-by-step process to find the speeds of the two cars, along with the underlying algebraic principles and explanations.

Algebraic Solution for the Car Speed Problem

Consider the following scenario: Two cars are 640 km apart and are traveling towards each other. They meet after 4 hours, and one car travels 10 km/h faster than the other. How fast did each car travel?

Algebraic Setup and Solution

We denote the speed of the slower car as x km/h. Therefore, the speed of the faster car is x 10 km/h. Since both cars travel towards each other and meet after 4 hours, we can express the distances traveled by each car as:

Distance traveled by the slower car in 4 hours: 4x Distance traveled by the faster car in 4 hours: 4(x 10)

The total distance they cover together is 640 km:

4x 4(x 10) 640

Now, simplify and solve the equation:

4x 4x 40 640

8x 40 640

Subtract 40 from both sides:

8x 600

Divide by 8:

x 75 km/h

The slower car travels at 75 km/h, and the faster car travels at 85 km/h.

Summary of Speeds

Slower Car Speed: 75 km/h

Faster Car Speed: 85 km/h

Simplified Approach for the Speed Problem

Another way to approach this problem involves understanding the combined speed of the cars and then splitting it based on the given difference in speeds. Let's go through the steps:

Step-by-Step Simplified Solution

The combined speed of the cars is 400 km/5hrs / 80 km/hr Let n be the speed of the slower car; the other car's speed is n 10 n (n 10) 80 2n 10 80 2n 70 n 35 km/hr Faster car speed: 35 10 45 km/hr

Summary of Speeds (Simplified Method)

Slower Car Speed: 35 km/hr

Faster Car Speed: 45 km/hr

Another Method of Solving the Speed Problem

We can also use the formula: D ST, where D is distance, S is speed, and T is time. Here is the step-by-step process:

Rational Calculation of Speeds

Distance (D) 400 km Time (T) 5 hrs Average speed (S) D / T 400 / 5 80 kph The difference in speeds is given as 10 km/hr Divide the average speed by 2 to account for the differential: 80 / 2 40 km/hr Decrease the speed for the slower car: 40 - 5 35 km/hr Increase the speed for the faster car: 40 5 45 km/hr

Summary of Speeds (Rational Calculation)

Slower Car Speed: 35 km/hr

Faster Car Speed: 45 km/hr

Conclusion

Both algebraic and simplified methods provide a clear and efficient way to solve the car speed problem. Understanding the underlying principles and applying them correctly ensures accurate and consistent solutions. This problem not only helps in practicing algebraic skills but also in developing a deeper understanding of speed, distance, and time relationships.

Additional Resources

Algebraic Equations for Real-Life Problems Distance and Time Calculations Speed and Velocity Analysis

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