Solving the Traffic Light Puzzle: A Mathematical Analysis
In this article, we will explore the intricacies of a traffic light cycle and how to calculate the duration of each colored light. We will use a step-by-step mathematical approach to find the solution and provide a clear understanding of the problem.
The Problem Statement
The duration for one complete cycle of a traffic light is 84 seconds. The green light is on for 7 times longer than the yellow light and the red light is on for 13 times longer than the yellow light. We need to determine the duration for each colored light during one cycle.
Mathematical Approach
Let's denote the duration of the yellow light as x seconds. Then, the duration of the green light would be 7x seconds, and the duration of the red light would be 13x seconds. The total time for one cycle is given as 84 seconds.
Formulating the Equation
We can write the equation as follows:
7x x 13x 84
This simplifies to:
21x 84
Now, solve for x:
x 84 / 21 4 seconds
Calculating the Durations
Therefore, the duration for the yellow light is 4 seconds. Using this value, we can find the durations for the green and red lights as follows:
Green light: 7x 7 × 4 28 seconds
Red light: 13x 13 × 4 52 seconds
Verification
To verify, we can add up the durations:
4 (yellow) 28 (green) 52 (red) 84 seconds
Conclusion
In this comprehensive analysis, we have successfully determined the duration of each colored light during one cycle of a traffic light. This method can be applied to similar problems, providing a clear and systematic approach to solving such puzzles.