How Do You Calculate The Unit Rate For 2/5 Mile In 8/10 Hour?

In the realm of mathematics and real-world applications, the concept of unit rate plays a pivotal role in understanding relationships between quantities. A unit rate expresses the amount of one quantity per single unit of another quantity. In simpler terms, it tells us how much of something we get for just one unit of something else. This could be anything from the cost per item in a grocery store to the distance traveled per hour in a car. Understanding how to calculate unit rates is essential for making informed decisions, comparing values, and solving problems efficiently. In this article, we will delve into a specific example of calculating unit rate: determining the miles traveled per hour given a certain distance covered in a specific time. We will break down the steps involved, explain the underlying concepts, and highlight the importance of unit rates in various contexts. By the end of this exploration, you'll have a solid understanding of how to calculate and interpret unit rates, empowering you to tackle similar problems with confidence.

Calculating unit rates often involves division, but it's crucial to understand what is being divided by what. The unit rate we are looking for in this case is miles per hour, meaning we want to know how many miles are covered for every one hour of travel time. Therefore, we need to divide the total distance traveled (in miles) by the total time taken (in hours). This might seem straightforward, but when dealing with fractions, the process requires a bit more attention to detail. We will explore how to divide fractions effectively and accurately to arrive at the correct unit rate. Furthermore, we will discuss the importance of simplifying fractions and expressing the unit rate in its simplest form. This ensures that the result is easily understandable and comparable to other unit rates. The ability to calculate unit rates accurately is a valuable skill in various fields, including physics, engineering, economics, and everyday life. From determining fuel efficiency to comparing prices, unit rates provide a standardized way to analyze and interpret data.

Before we dive into the calculation, let's carefully examine the given information. The problem states that we have traveled ${\frac{2}{5} \text{ mile} }$ in ${\frac{8}{10} \text{ hour} }$ Our goal is to determine the unit rate, which represents the distance traveled per one hour. In other words, we want to find out how many miles are covered in a single hour. This is a classic example of a unit rate problem, where we need to find the rate of change of one quantity (distance) with respect to another quantity (time). The key to solving this problem lies in understanding the relationship between distance, time, and rate. The fundamental formula that connects these three quantities is:

Distance = Rate × Time

In our case, we know the distance ( ${\frac{2}{5} \text{ mile} }$ ) and the time ( ${\frac{8}{10} \text{ hour} }$ ), and we want to find the rate (miles per hour). To do this, we need to rearrange the formula to solve for the rate:

Rate = Distance / Time

This formula provides the foundation for our calculation. We will substitute the given values into this formula and perform the necessary operations to find the unit rate. However, before we proceed with the calculation, it's important to ensure that we understand the units involved. The distance is given in miles, and the time is given in hours, which aligns perfectly with our desired unit rate of miles per hour. If the units were different (e.g., kilometers and minutes), we would need to convert them to a consistent set of units before performing the calculation. This step is crucial for ensuring the accuracy and meaningfulness of the result. Once we have the unit rate in miles per hour, we can use it to make predictions about future travel times or distances. For example, we can use the unit rate to estimate how long it will take to travel a certain distance, or how far we can travel in a given amount of time. This is one of the many practical applications of unit rates in real-world scenarios.

Now that we have set up the problem and identified the formula we need, let's proceed with the calculation. We have the following values:

Distance = ${\frac{2}{5} \text{ mile} }$

Time = ${\frac{8}{10} \text{ hour} }$

We want to find the rate, which is the distance divided by the time:

Rate = ${\frac{2}{5} \div \frac{8}{10} }$

Dividing fractions can sometimes seem daunting, but it's actually a straightforward process once you understand the underlying principle. The key rule to remember is that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by simply flipping the numerator and the denominator. For example, the reciprocal of ${\frac{8}{10} }$ is ${\frac{10}{8} }$ Therefore, to divide ${\frac{2}{5} }$ by ${\frac{8}{10} }$ , we multiply ${\frac{2}{5} }$ by ${\frac{10}{8} }$ :

Rate = ${\frac{2}{5} \times \frac{10}{8} }$

Now we have a multiplication problem involving fractions. To multiply fractions, we simply multiply the numerators together and the denominators together:

Rate = ${\frac{2 \times 10}{5 \times 8} = \frac{20}{40} }$

We have now obtained the unit rate as ${\frac{20}{40} }$ miles per hour. However, this fraction can be simplified to its lowest terms. Simplifying fractions makes them easier to understand and compare. To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and the denominator and divide both by the GCD. In this case, the GCD of 20 and 40 is 20. Dividing both the numerator and the denominator by 20, we get:

Rate = ${\frac{20 \div 20}{40 \div 20} = \frac{1}{2} }$

Therefore, the simplified unit rate is ${\frac{1}{2} }$ miles per hour. This means that for every one hour of travel, ${\frac{1}{2} }$ a mile is covered.

After performing the division of fractions, we arrived at the unit rate of ${\frac{20}{40} }$ miles per hour. However, as mentioned earlier, it's crucial to simplify fractions to their lowest terms for clarity and ease of interpretation. Simplifying a fraction involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In our case, the numerator is 20, and the denominator is 40. The GCD of 20 and 40 is 20. This means that 20 is the largest number that divides both 20 and 40 evenly. To simplify the fraction, we divide both the numerator and the denominator by 20:

${\frac{20 \div 20}{40 \div 20} = \frac{1}{2} }$

This simplification process transforms the fraction ${\frac{20}{40} }$ into its equivalent form, ${\frac{1}{2} }$ . The value of the fraction remains the same, but the simplified form is much easier to understand and work with. A simplified fraction provides a clearer picture of the relationship between the numerator and the denominator. In this case, ${\frac{1}{2} }$ indicates that the numerator is half of the denominator. This makes it immediately apparent that the rate is one-half mile per hour. Simplifying fractions is not just a mathematical exercise; it's a practical skill that enhances our ability to interpret and communicate numerical information effectively. In various real-world scenarios, simplifying fractions can help us make quick comparisons, estimate values, and avoid errors. For example, when comparing prices at a store, simplified fractions can make it easier to determine which option offers the best value. Similarly, in scientific calculations, simplified fractions can reduce the complexity of the equations and make the results more transparent. Therefore, mastering the art of simplifying fractions is an essential component of mathematical proficiency.

After simplifying the fraction, we arrive at the final answer for the unit rate: ${\frac{1}{2} }$ miles per hour. This means that for every one hour of travel, a distance of ${\frac{1}{2} }$ mile is covered. This unit rate provides a clear and concise way to express the relationship between distance and time in this particular scenario. It tells us exactly how much distance is covered for each unit of time, which in this case is one hour. The unit rate of ${\frac{1}{2} }$ miles per hour can be interpreted in several ways. One way is to say that the object or person is traveling at a speed of half a mile per hour. Another way is to say that for every two hours of travel, one mile is covered. This unit rate can be used to make predictions about future travel times or distances. For example, if we wanted to know how far we could travel in three hours at this rate, we could simply multiply the rate by the time:

Distance = Rate × Time

Distance = ${\frac{1}{2} \text{ mile/hour} \times 3 \text{ hours} = \frac{3}{2} \text{ miles} }$ This calculation tells us that we could travel ${\frac{3}{2} }$ miles, or 1.5 miles, in three hours. Similarly, if we wanted to know how long it would take to travel a certain distance, we could divide the distance by the rate:

Time = Distance / Rate

For example, if we wanted to travel 2 miles, the time required would be:

Time = ${2 \text{ miles} \div \frac{1}{2} \text{ mile/hour} = 4 \text{ hours} }$ This calculation tells us that it would take 4 hours to travel 2 miles at this rate. These examples illustrate the practical utility of unit rates in making predictions and solving real-world problems. By understanding the concept of unit rate and how to calculate it, we can gain valuable insights into the relationships between different quantities and make informed decisions based on those relationships.

In conclusion, the unit rate for this problem is ${\frac{1}{2} }$ miles per hour. This means that for every hour of travel, ${\frac{1}{2} }$ a mile is covered. Understanding unit rates is crucial for solving various mathematical and real-world problems.