- Identify the variables: Let \(r\) be the speed of the slower car and \(r+20\) be the speed of the faster car.
- Set up the time equality equation using \(\text{Time} = \frac{\text{Distance}}{\text{Rate}}\): \(\frac{165}{r} = \frac{225}{r+20}\).
- Solve the equation by cross-multiplying and simplifying to find \(r = 55\).
- Verify the solution by confirming both cars took 3 hours. The final answer is \(\boxed{55}\) mph.
Explanation
- Understanding the Problem and Defining Variables
The problem involves two cars traveling at constant, different speeds over a certain interval of time. Let \(r\) be the speed of the slower car in miles per hour (mph). Since the faster car travels 20 mph faster than the slower car, its speed is represented as \(r + 20\) mph. We are given that the slower car travels a distance of 165 miles and the faster car travels a distance of 225 miles in the same amount of time. The fundamental relationship between distance, rate, and time is given by the formula \(\text{Time} = \frac{\text{Distance}}{\text{Rate}}\).
- Formulating and Solving the Equation
We can express the time taken for each car to complete its distance using the formula \(\text{Time} = \frac{\text{Distance}}{\text{Rate}}\). For the slower car, the time is \(T_s = \frac{165}{r}\). For the faster car, the time is \(T_f = \frac{225}{r+20}\). Since the cars travel for the same amount of time, we set these two expressions equal to each other:
\[\frac{165}{r} = \frac{225}{r+20}\]
To solve this rational equation for \(r\), we cross-multiply:
\[165(r + 20) = 225r\]
Distributing the 165 on the left side, we obtain:
\[165r + 3300 = 225r\]
Subtracting \(165r\) from both sides gives:
\[3300 = 60r\]
Dividing both sides by 60, we find the speed of the slower car:
\[r = \frac{3300}{60} = 55\]
Thus, the speed of the slower car is 55 mph.
- Verifying the Result
To verify our result, we calculate the time taken by each car. For the slower car, the time is \(\frac{165}{55} = 3\) hours. For the faster car, its speed is \(55 + 20 = 75\) mph, and the time taken is \(\frac{225}{75} = 3\) hours. Since the times are identical, our solution is correct. The speed of the slower car is \(\boxed{55}\) mph.
Examples
Understanding the relationship between distance, rate, and time is essential in many everyday activities, such as planning a road trip, managing logistics for transportation, or even determining how long it will take to complete a commute. By setting up equations based on these constant relationships, we can accurately predict arrival times or compare the performance of different transport methods. For instance, if you are driving to a destination while another friend takes a slightly faster but more direct route, you can use these principles to calculate your relative speeds and predict when each of you will arrive.