Solving Rate Problems Hildas Running Distance Explained Step By Step
In this article, we will delve into a classic rate problem involving Hilda's running distance. Rate problems are a fundamental concept in mathematics, often appearing in various standardized tests and real-life scenarios. Understanding how to solve these problems is crucial for developing strong analytical and problem-solving skills. We'll break down the problem step-by-step, providing a clear and concise explanation to help you grasp the underlying principles. Our goal is to not only solve the specific problem at hand but also equip you with the tools and knowledge to tackle similar rate problems with confidence.
Hilda's running rate is the core concept of this problem. The problem states that Hilda runs 2/3 of a mile in a half-hour (30 minutes). The question asks: If she continues to run at a constant rate, how many miles will she run in 3 hours? This is a typical rate problem where we need to determine the distance covered over a given time, knowing the rate of travel. To solve this, we need to first find Hilda's speed, which is the distance she runs per unit of time. Then, we can use this speed to calculate the distance she will cover in 3 hours.
To solve this problem effectively, a fundamental understanding of rates is essential. A rate is a ratio that compares two quantities with different units. In this case, the rate is Hilda's running speed, which compares the distance she runs (miles) to the time she takes (hours). Understanding the concept of rates is crucial not only for solving mathematical problems but also for real-world applications such as calculating travel time, understanding fuel efficiency, and determining the speed of various processes. To calculate a rate, we divide the quantity representing the change or outcome by the time it takes for that change to occur. For example, if a car travels 100 miles in 2 hours, its rate (speed) is 100 miles / 2 hours = 50 miles per hour. In this problem, Hilda's rate is given as 2/3 of a mile in 1/2 hour. To find her speed per hour, we need to divide the distance by the time.
Let's embark on a step-by-step solution to unravel Hilda's running journey. To solve this problem, we will follow these steps:
- Find Hilda's speed: Divide the distance she runs in a half-hour by the time (1/2 hour) to find her speed per hour.
- Calculate the total distance: Multiply her speed by the total time (3 hours) to find the total distance she will run.
Step 1: Find Hilda's speed
Hilda runs 2/3 of a mile in 1/2 hour. To find her speed per hour, we divide the distance by the time:
Speed = Distance / Time
Speed = (2/3 mile) / (1/2 hour)
To divide fractions, we multiply by the reciprocal of the divisor:
Speed = (2/3) * (2/1) miles per hour
Speed = 4/3 miles per hour
So, Hilda's speed is 4/3 miles per hour.
Step 2: Calculate the total distance
Now that we know Hilda's speed, we can calculate the distance she will run in 3 hours. We multiply her speed by the time:
Distance = Speed * Time
Distance = (4/3 miles per hour) * (3 hours)
Distance = (4/3) * 3 miles
Distance = 4 miles
Therefore, Hilda will run 4 miles in 3 hours.
Let's provide a detailed explanation of the solution to ensure complete understanding. The key to solving this problem lies in understanding the relationship between distance, rate, and time. The formula that connects these three quantities is:
Distance = Rate * Time
In this problem, we are given the distance Hilda runs in a certain amount of time (2/3 of a mile in 1/2 hour) and we are asked to find the distance she will run in a different amount of time (3 hours), assuming she runs at a constant rate. To solve this, we first need to find Hilda's rate (speed). We can find the rate by dividing the distance by the time:
Rate = Distance / Time
In Hilda's case, the distance is 2/3 of a mile and the time is 1/2 hour. So, her rate is:
Rate = (2/3 mile) / (1/2 hour)
To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of 1/2 is 2/1, so we have:
Rate = (2/3) * (2/1) miles per hour
Rate = 4/3 miles per hour
This means Hilda runs 4/3 miles in one hour. Now that we know her rate, we can find the distance she will run in 3 hours by multiplying her rate by the time:
Distance = Rate * Time
Distance = (4/3 miles per hour) * (3 hours)
Distance = (4/3) * 3 miles
Distance = 4 miles
Therefore, Hilda will run 4 miles in 3 hours. This detailed explanation breaks down each step of the solution, providing a clear understanding of the mathematical concepts involved and how they are applied to solve the problem.
Let's explore why the other options are incorrect, reinforcing our understanding of the correct solution. Understanding why incorrect answers are wrong is just as important as knowing why the correct answer is right. It helps to solidify your understanding of the problem and the concepts involved. In this case, the incorrect options likely arise from common mistakes in rate problems, such as misinterpreting the given information or performing incorrect calculations.
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Option B: 2 miles
This answer might be obtained if we incorrectly assume that the distance Hilda runs doubles when the time doubles. Since she runs 2/3 of a mile in 1/2 hour, one might think she would run 4/3 miles in 1 hour, and then simply multiply 4/3 by 3 to get 4 miles. However, this approach skips the crucial step of correctly calculating the rate and applying it over the total time. It's a common mistake to overlook the fractional nature of the initial distance and time and not calculate the speed properly before extending it to the 3-hour period. This option fails to account for the constant rate of running, which is essential for solving the problem correctly.
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Option C: 2/9 miles
This answer is likely the result of a misunderstanding of how to handle the fractions in the problem. It's possible that this answer comes from incorrectly dividing the initial distance (2/3 miles) by the total time (3 hours) instead of using the rate to find the distance. This would be a misapplication of the distance, rate, and time formula. The correct approach involves finding the rate first (miles per hour) and then multiplying that rate by the total time. This option demonstrates a fundamental misunderstanding of the steps required to solve a rate problem and the order in which the operations should be performed.
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Option D: 9 miles
This answer might be obtained through a series of incorrect calculations or a misunderstanding of the relationship between distance, rate, and time. One possible error could be inverting the rate calculation or incorrectly multiplying the fractions. For instance, one might mistakenly multiply the distance (2/3 miles) by the time (3 hours) and then divide by the initial time (1/2 hour) without properly considering the units and the concept of rate. This option suggests a significant error in the application of mathematical operations and a lack of understanding of the problem's structure. It's a reminder of the importance of carefully reviewing each step in the solution process to avoid such errors.
Let's consider alternative approaches to solving the problem, expanding our problem-solving toolkit. While the step-by-step method we used is effective, exploring different approaches can provide a deeper understanding of the problem and improve our problem-solving skills. Alternative methods can also offer a way to check our answer and ensure accuracy. In this section, we will discuss two alternative methods: using proportions and using a table.
Method 1: Using Proportions
Proportions are a powerful tool for solving rate problems. A proportion is an equation that states that two ratios are equal. In this case, we can set up a proportion to relate the distance Hilda runs to the time she takes. We know that Hilda runs 2/3 of a mile in 1/2 hour. Let's denote the distance she runs in 3 hours as 'x' miles. We can set up the following proportion:
(2/3 mile) / (1/2 hour) = (x miles) / (3 hours)
To solve for x, we can cross-multiply:
(2/3) * 3 = (1/2) * x
2 = (1/2) * x
Now, multiply both sides by 2 to isolate x:
2 * 2 = x
x = 4
So, Hilda will run 4 miles in 3 hours. This method provides a direct way to relate the given information to the unknown quantity, making it a useful technique for rate problems.
Method 2: Using a Table
Another helpful approach is to use a table to organize the information and find the solution. This method is particularly useful for visualizing the relationship between distance, rate, and time. We can create a table with columns for time and distance:
| Time (hours) | Distance (miles) |
|---|---|
| 1/2 | 2/3 |
| 1 | ? |
| 3 | ? |
We know that Hilda runs 2/3 of a mile in 1/2 hour. To find the distance she runs in 1 hour, we can double the distance she runs in 1/2 hour:
Distance in 1 hour = 2 * (2/3) = 4/3 miles
Now we can fill in the second row of the table:
| Time (hours) | Distance (miles) |
|---|---|
| 1/2 | 2/3 |
| 1 | 4/3 |
| 3 | ? |
To find the distance she runs in 3 hours, we multiply the distance she runs in 1 hour by 3:
Distance in 3 hours = 3 * (4/3) = 4 miles
So, Hilda will run 4 miles in 3 hours. This method helps to break down the problem into smaller steps and visualize the relationship between the quantities.
Let's summarize the key takeaways for solving rate problems, ensuring we can apply these principles to future challenges. Rate problems can seem daunting at first, but with a clear understanding of the underlying concepts and a systematic approach, they become much more manageable. Here are some key takeaways to remember when tackling rate problems:
- Understand the Relationship: The fundamental relationship in rate problems is Distance = Rate * Time. Make sure you understand how these three quantities are related and how they affect each other. If you know any two of these quantities, you can find the third.
- Identify the Rate: The rate is the key to solving rate problems. It is the amount of change per unit of time. In this problem, the rate is Hilda's speed, which is the distance she runs per hour. Identifying the rate is the first step in solving the problem.
- Use Consistent Units: Ensure that the units of time and distance are consistent throughout the problem. For example, if the rate is given in miles per hour, make sure the time is also in hours and the distance is in miles. Converting units if necessary is crucial for avoiding errors.
- Set Up the Problem Correctly: Whether you choose to use the formula, proportions, or a table, setting up the problem correctly is essential. Make sure you are placing the correct values in the correct places and that your equations or proportions are set up logically.
- Check Your Answer: After solving the problem, take a moment to check your answer. Does it make sense in the context of the problem? If the answer seems too large or too small, review your calculations and make sure you haven't made any errors.
- Practice Regularly: Like any mathematical skill, solving rate problems becomes easier with practice. Work through a variety of problems to build your confidence and develop your problem-solving skills.
Let's explore the real-world applications of rate problems, demonstrating their practical significance. Rate problems are not just abstract mathematical exercises; they have numerous applications in everyday life. Understanding how to solve rate problems can help you make informed decisions in various situations. Here are some examples of real-world applications:
- Travel Planning: Rate problems are essential for planning trips. You can use them to calculate how long it will take to travel a certain distance at a given speed. For example, if you know your average speed on a road trip, you can estimate the travel time to your destination. Similarly, you can use rate problems to calculate fuel consumption and plan your fuel stops.
- Financial Planning: Rate problems are used in financial planning to calculate interest rates, investment returns, and loan payments. For example, you can use rate problems to determine how much interest you will earn on a savings account over a certain period or to calculate the monthly payments on a loan.
- Cooking and Baking: Rate problems can be applied in cooking and baking to adjust recipes for different serving sizes. For example, if a recipe calls for a certain amount of ingredients for a specific number of servings, you can use rate problems to scale the recipe up or down.
- Sports and Fitness: Rate problems are relevant in sports and fitness for calculating speed, pace, and distance. For example, runners can use rate problems to track their pace and estimate their finishing time in a race. Similarly, cyclists can use rate problems to calculate their average speed and distance traveled.
- Business and Economics: Rate problems are used in business and economics to analyze growth rates, market trends, and economic indicators. For example, economists use rate problems to calculate the rate of inflation or the rate of economic growth.
- Healthcare: Rate problems have applications in healthcare for calculating medication dosages, infusion rates, and patient monitoring. For example, nurses use rate problems to calculate the correct dosage of a medication based on a patient's weight and other factors.
In conclusion, we have successfully navigated Hilda's running scenario, demonstrating the power and versatility of rate problems. This article has provided a comprehensive guide to solving rate problems, starting with a clear understanding of the problem statement, a step-by-step solution, detailed explanations, and alternative approaches. We have also explored the real-world applications of rate problems, highlighting their relevance in various aspects of life. By mastering the concepts and techniques discussed in this article, you can confidently tackle rate problems and apply them to practical situations.
Remember, the key to success in solving rate problems lies in understanding the relationship between distance, rate, and time, using consistent units, setting up the problem correctly, and practicing regularly. With these skills, you will be well-equipped to handle any rate problem that comes your way.
