Calculating Running Rate How Long Does It Take Jose To Run A Mile

In this article, we will delve into a practical mathematical problem involving rate calculation. Specifically, we will analyze the running performance of an individual named Jose. Jose ran $\frac{1}{3}$ of a mile in a time span of 3 minutes and 15 seconds. Our primary objective is to determine Jose's rate in minutes per mile. In simpler terms, we aim to calculate the total time it would take Jose to run a complete mile if he maintained the same pace. This exercise is not just a mathematical calculation; it's a real-world application of understanding speed, time, and distance relationships. This article provides a detailed, step-by-step solution to the problem, making it accessible for anyone interested in improving their problem-solving skills in mathematics. Whether you are a student learning about rates and ratios or simply someone who enjoys mathematical challenges, this guide will offer valuable insights and a clear methodology for tackling similar problems. Understanding Jose's pace involves more than just crunching numbers; it's about grasping the relationship between distance, time, and speed. When we say Jose ran $\frac{1}{3}$ of a mile, we're establishing the distance covered. The 3 minutes and 15 seconds represent the time taken to cover that distance. The key to solving this problem lies in converting these values into a single rate – minutes per mile – that gives us a clear picture of Jose's running speed. To accurately calculate Jose's rate, we'll need to break down the given information into manageable parts. First, we'll convert the time from minutes and seconds into a single unit, either minutes or seconds, to simplify our calculations. Then, we'll use the relationship between distance, time, and rate to find out how long it would take Jose to run a full mile. This process involves basic arithmetic operations, such as division and multiplication, but the core concept is understanding how these operations apply to the context of the problem. We're not just looking for an answer; we're aiming to understand the underlying principles of rate calculation.

Step 1: Converting Time to a Single Unit

The initial step in calculating Jose's running rate is to convert the given time, 3 minutes and 15 seconds, into a single unit. This conversion is crucial for simplifying our calculations and ensuring accuracy in the final result. There are two common approaches we can take: converting the time to minutes or converting it to seconds. For this explanation, we will convert the time into minutes, as it aligns directly with our goal of finding the rate in minutes per mile. To convert 3 minutes and 15 seconds into minutes, we first need to understand the relationship between minutes and seconds. There are 60 seconds in a minute. Therefore, to convert seconds into minutes, we divide the number of seconds by 60. In this case, we have 15 seconds. Dividing 15 by 60 gives us $\frac{15}{60}$, which simplifies to $\frac{1}{4}$ or 0.25 minutes. Now that we have 15 seconds converted to 0.25 minutes, we can add this to the existing 3 minutes. So, 3 minutes plus 0.25 minutes equals 3.25 minutes. This means that Jose ran $\frac{1}{3}$ mile in 3.25 minutes. Converting the time to a single unit is a fundamental step in solving rate problems. It allows us to work with a consistent measurement, making the subsequent calculations more straightforward. Without this conversion, we would be dealing with two different units of time, which could lead to errors in our final answer. In summary, by converting 3 minutes and 15 seconds to 3.25 minutes, we have successfully expressed the time in a single unit, setting the stage for the next steps in calculating Jose's running rate. This conversion highlights the importance of unit consistency in mathematical problem-solving, particularly in scenarios involving rates, speeds, and distances. The ability to fluently convert between different units of measurement is a valuable skill, not only in mathematics but also in everyday situations where time, speed, and distance are considered.

Step 2: Calculating Minutes Per Mile

With the time now expressed in a single unit (3.25 minutes), we can proceed to calculate Jose's rate in minutes per mile. The core concept here is to determine how long it would take Jose to run a full mile if he maintained the same pace as when he ran $\frac{1}{3}$ mile. Since Jose ran $\frac{1}{3}$ mile in 3.25 minutes, we need to find out how many times this $\frac{1}{3}$ mile fits into a full mile. Mathematically, this means we need to determine how many $\frac{1}{3}$ are in 1. This can be calculated by dividing 1 by $\frac{1}{3}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{1}{3}$ is 3. Therefore, 1 divided by $\frac{1}{3}$ is equal to 1 multiplied by 3, which equals 3. This tells us that a full mile is three times the distance Jose initially ran. Now that we know a full mile is three times the distance Jose ran, we can calculate the time it would take him to run a full mile. If it took Jose 3.25 minutes to run $\frac{1}{3}$ mile, it would take him three times that amount to run a full mile. So, we multiply 3.25 minutes by 3. To perform this multiplication, we can break it down: 3 times 3 minutes is 9 minutes, and 3 times 0.25 minutes is 0.75 minutes. Adding these together, 9 minutes plus 0.75 minutes, we get 9.75 minutes. Therefore, Jose's rate is 9.75 minutes per mile. This calculation demonstrates a fundamental principle in rate problems: to find the rate for a whole unit (in this case, a mile), we scale up the rate for a fraction of that unit. By understanding this principle, we can solve a wide range of problems involving rates, speeds, and distances. The final answer, 9.75 minutes per mile, gives us a clear understanding of Jose's running pace. It tells us that, at the same speed, Jose would take approximately 9 minutes and 45 seconds to run a complete mile. This information can be used to compare Jose's performance with others or to set goals for improvement.

Step 3: Converting Decimal Minutes to Minutes and Seconds

To provide a more intuitive understanding of Jose's running rate, it's helpful to convert the decimal part of the minutes (9.75 minutes) back into minutes and seconds. This conversion makes the rate easier to visualize and compare with other times. We already know that the whole number part, 9, represents 9 full minutes. The focus now is on converting the decimal part, 0.75 minutes, into seconds. To convert decimal minutes to seconds, we use the fact that there are 60 seconds in a minute. Therefore, we multiply the decimal part of the minutes by 60 to get the equivalent number of seconds. In this case, we multiply 0.75 by 60. To perform this multiplication, we can think of 0.75 as $\frac{3}{4}$. So, we are essentially finding $\frac{3}{4}$ of 60. One way to calculate this is to first find $\frac{1}{4}$ of 60, which is 60 divided by 4, resulting in 15. Then, we multiply this by 3 to get $\frac{3}{4}$ of 60, which is 15 times 3, equaling 45 seconds. Alternatively, we can directly multiply 0.75 by 60. This can be done by multiplying 75 by 60 and then adjusting for the decimal place. 75 times 60 is 4500. Since we multiplied without considering the decimal place (0.75), we need to adjust the result by moving the decimal point two places to the left, giving us 45.00, or simply 45 seconds. Therefore, 0.75 minutes is equivalent to 45 seconds. Now that we have converted the decimal part of the minutes to seconds, we can express Jose's running rate in minutes and seconds. We have 9 full minutes and 45 seconds. So, Jose's rate is 9 minutes and 45 seconds per mile. This conversion from decimal minutes to minutes and seconds provides a more tangible sense of time. It's easier for most people to conceptualize and compare times expressed in minutes and seconds rather than in decimal minutes. This final conversion step underscores the importance of presenting mathematical results in a format that is both accurate and easily understood by the intended audience.

Conclusion: Summarizing Jose's Running Rate

In conclusion, through a step-by-step mathematical analysis, we have successfully calculated Jose's running rate. Starting from the initial information that Jose ran $\frac{1}{3}$ mile in 3 minutes and 15 seconds, we meticulously worked through the problem to determine his pace in minutes per mile. First, we converted the time from minutes and seconds into a single unit, expressing 3 minutes and 15 seconds as 3.25 minutes. This conversion was crucial for simplifying the subsequent calculations. Next, we calculated the time it would take Jose to run a full mile. Since a full mile is three times the distance Jose initially ran, we multiplied 3.25 minutes by 3, resulting in 9.75 minutes. This gave us Jose's rate in decimal minutes per mile. Finally, to provide a clearer understanding of Jose's pace, we converted the decimal minutes back into minutes and seconds. The 0.75 minutes were converted to 45 seconds, giving us a final rate of 9 minutes and 45 seconds per mile. Therefore, Jose's running rate is 9 minutes and 45 seconds per mile. This result provides a comprehensive understanding of Jose's running speed. It tells us that if Jose were to maintain his pace, he would complete a mile in approximately 9 minutes and 45 seconds. This calculation demonstrates the practical application of mathematical principles in real-world scenarios. By understanding how to convert units, calculate rates, and express results in meaningful terms, we can gain valuable insights into various aspects of our daily lives. This exercise not only enhances our mathematical skills but also improves our ability to analyze and interpret information. The process of solving this problem highlights the importance of breaking down complex tasks into smaller, manageable steps. By systematically addressing each step, we can arrive at an accurate and easily understandable solution. This approach is applicable not only in mathematics but also in various problem-solving situations across different disciplines.