Horse-Drawn Carriage Speed Calculation How To Find Kilometers Per Hour

In this article, we will solve a classic problem involving the speed of a horse-drawn carriage. This problem combines basic arithmetic with the concept of rates, making it a great exercise for understanding how to calculate speed. We'll break down the problem step-by-step, ensuring clarity and comprehension for everyone. Let's dive in!

Understanding the Problem

Horse-drawn carriage speed problems often involve calculating the rate of travel. Our problem states: A horse-drawn carriage travels 47 rac{2}{3} kilometers in 4 rac{1}{3} hours. At this rate, how many kilometers does it travel per hour? This is a rate problem, specifically asking for the speed of the carriage in kilometers per hour. To solve this, we need to divide the total distance traveled by the total time taken. Let's break down each component and then perform the calculation.

First, we need to convert the mixed numbers into improper fractions. This makes the division process much simpler. The total distance is 47 rac{2}{3} kilometers. To convert this to an improper fraction, we multiply the whole number (47) by the denominator (3) and add the numerator (2). This gives us (47 * 3) + 2 = 141 + 2 = 143. So, the improper fraction for the distance is $\frac{143}{3}$ kilometers. Next, we convert the time, 4 rac{1}{3} hours, into an improper fraction. We multiply the whole number (4) by the denominator (3) and add the numerator (1). This gives us (4 * 3) + 1 = 12 + 1 = 13. So, the improper fraction for the time is $\frac{13}{3}$ hours. Now that we have both distance and time as improper fractions, we can proceed with the division to find the speed. Remember, speed is calculated by dividing the distance by the time. In this case, we will divide $\frac{143}{3}$ kilometers by $\frac{13}{3}$ hours. Dividing by a fraction is the same as multiplying by its reciprocal. So, we will multiply $\frac{143}{3}$ by $\frac{3}{13}$.

Step-by-Step Solution

To find the kilometers traveled per hour, we'll use the formula: Speed = Distance / Time. The distance is 47 rac{2}{3} kilometers, and the time is 4 rac{1}{3} hours. As mentioned earlier, converting these mixed numbers to improper fractions is crucial for easy calculation. Converting mixed numbers to improper fractions is a fundamental skill in arithmetic, especially when dealing with division and multiplication. It allows us to work with the numbers more efficiently without the added complexity of whole numbers. So, $\frac{143}{3}$ divided by $\frac{13}{3}$. To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of $\frac{13}{3}$ is $\frac{3}{13}$. Multiplying by the reciprocal transforms the division problem into a multiplication problem, which is generally easier to solve. This step is crucial for simplifying the calculation and arriving at the correct answer. Therefore, the equation becomes: $\frac{143}{3} * \frac{3}{13}$. Now, we can multiply the numerators and the denominators. $(143 * 3) / (3 * 13)$. Before performing the multiplication, we can simplify by canceling out the common factor of 3. Simplifying fractions before multiplying can significantly reduce the size of the numbers we are working with, making the calculation process less prone to errors. This is an important technique to master in fraction manipulation. This leaves us with $\frac{143}{13}$. Now we need to divide 143 by 13. 143 divided by 13 equals 11. Performing the division to arrive at the final answer is the last step in calculating the speed. It's essential to ensure the division is carried out correctly to obtain the accurate result. So, the speed is 11 kilometers per hour. Thus, the horse-drawn carriage travels 11 kilometers per hour.

Detailed Calculation

Let’s perform the calculation step-by-step to ensure clarity on how we arrive at the carriage's speed. We start with the formula: Speed = Distance / Time. The distance is 47 rac{2}{3} kilometers, and the time is 4 rac{1}{3} hours. Breaking down the calculation into smaller steps makes it easier to follow and understand the process. Each step builds upon the previous one, leading to the final solution. First, convert the mixed numbers to improper fractions: 47 rac{2}{3} = \frac{(47 * 3) + 2}{3} = \frac{141 + 2}{3} = \frac{143}{3} kilometers. Converting mixed numbers to improper fractions is a crucial initial step as it simplifies the subsequent division process. This conversion allows us to work with the numbers more efficiently. 4 rac{1}{3} = \frac{(4 * 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} hours. Similarly, converting the time to an improper fraction ensures consistency in our calculations. Now, we divide the distance by the time: Speed = $\frac{143}{3} / \frac{13}{3}$. To divide fractions, we multiply by the reciprocal of the divisor: Speed = $\frac{143}{3} * \frac{3}{13}$. Multiplying by the reciprocal is a fundamental rule in fraction division. It transforms the division problem into a multiplication problem, which is easier to handle. Now, multiply the numerators and the denominators: Speed = $\frac{143 * 3}{3 * 13}$. Before multiplying, we can simplify by canceling the common factor of 3: Speed = $\frac{143}{13}$. Simplifying fractions before multiplying can significantly reduce the complexity of the calculation and prevent errors. Now, divide 143 by 13: Speed = 11 kilometers per hour. Performing the final division gives us the speed of the horse-drawn carriage. Thus, the carriage travels 11 kilometers per hour. This detailed step-by-step calculation ensures a thorough understanding of the process and the accuracy of the result.

Why Convert to Improper Fractions?

The conversion to improper fractions is essential when performing arithmetic operations, especially multiplication and division, with mixed numbers. Mixed numbers consist of a whole number and a proper fraction. While they are easy to understand in terms of quantity, they can complicate calculations. For instance, if we try to directly divide 47 rac{2}{3} by 4 rac{1}{3} without converting to improper fractions, we would need to handle the whole numbers and fractional parts separately, which is cumbersome and increases the risk of errors. Improper fractions, on the other hand, represent the entire quantity as a single fraction where the numerator can be greater than the denominator. This format simplifies the multiplication and division processes significantly. Using improper fractions streamlines the calculations and makes them more manageable. When dividing fractions, we multiply by the reciprocal of the divisor. This operation is straightforward with improper fractions because we simply flip the numerator and the denominator. With mixed numbers, determining the reciprocal is less intuitive. The process of finding the reciprocal is much simpler with improper fractions, reducing the chance of mistakes. Moreover, improper fractions allow us to easily apply the rules of fraction multiplication and division, such as simplifying by canceling common factors before multiplying. This simplification step is crucial for keeping the numbers manageable and reducing the chances of arithmetic errors. Simplifying fractions is a key step in efficient calculation, and improper fractions facilitate this process. In summary, converting mixed numbers to improper fractions before performing multiplication or division is a standard practice in mathematics because it simplifies the calculations, reduces the risk of errors, and aligns with the fundamental rules of fraction arithmetic. This step ensures that the calculations are accurate and efficient, leading to the correct solution.

Identifying the Correct Answer

After performing the calculation, we found that the horse-drawn carriage travels 11 kilometers per hour. Now, we need to identify the correct answer among the given options. The options are: A. 12 kilometers per hour, B. 11 kilometers per hour, C. 11 rac{2}{3} kilometers per hour, and D. 10 rac{2}{3} kilometers per hour. By comparing our calculated speed (11 kilometers per hour) with the options, we can see that option B, 11 kilometers per hour, matches our result. Matching the calculated answer with the provided options is the final step in verifying the solution. Therefore, option B is the correct answer. It's essential to double-check the calculation and ensure that the selected option aligns perfectly with the derived result. Verifying the answer against the options confirms the accuracy of the solution. The other options are incorrect because they do not match the speed calculated from the given distance and time. Option A (12 kilometers per hour) is higher than our calculated speed, while options C (11 rac{2}{3} kilometers per hour) and D (10 rac{2}{3} kilometers per hour) are different values altogether. Identifying and eliminating incorrect options helps reinforce the understanding of the solution. Thus, the correct answer is definitively B. 11 kilometers per hour.

Alternative Approaches to Solving the Problem

While we solved the problem by converting mixed numbers to improper fractions, there are alternative approaches that can be used to arrive at the same answer. Understanding these alternative methods can provide a deeper understanding of the problem and enhance problem-solving skills. One alternative approach involves working with the mixed numbers directly, although this method is generally more complex and prone to errors. Exploring alternative methods can broaden one's mathematical toolkit and improve problem-solving flexibility. Instead of converting to improper fractions, we could perform long division with the mixed numbers. However, this requires careful handling of the whole number and fractional parts and is not recommended for most learners. Direct division with mixed numbers is a less efficient method and increases the likelihood of errors. Another approach is to estimate the answer before performing the exact calculation. This can help in verifying the solution and identifying potential errors. For instance, we know that 47 rac{2}{3} is approximately 48 kilometers, and 4 rac{1}{3} is approximately 4 hours. Estimating, we get 48 kilometers divided by 4 hours, which is 12 kilometers per hour. Estimation can serve as a quick check to ensure the calculated answer is reasonable. This estimate is close to the options provided, but we need to perform the exact calculation to find the correct answer. This estimation also helps us eliminate option D (10 rac{2}{3} kilometers per hour) as it is significantly lower than our estimate. Using estimation as a preliminary step can narrow down the possible answers and improve accuracy. Another method, although less direct, involves converting the fractions to decimals. We could convert 47 rac{2}{3} to 47.67 (approximately) and 4 rac{1}{3} to 4.33 (approximately). Then, we would divide 47.67 by 4.33. This method can be useful, but it requires accurate decimal conversions and can still be more cumbersome than using improper fractions. Converting to decimals is another possible approach, but it may introduce rounding errors and increase complexity. In conclusion, while converting to improper fractions is the most straightforward and accurate method for this type of problem, understanding alternative approaches can enhance mathematical intuition and problem-solving skills. Each method offers a different perspective on the problem and can be valuable in various situations.

Conclusion

In conclusion, the horse-drawn carriage travels 11 kilometers per hour. We arrived at this answer by converting mixed numbers to improper fractions, dividing the total distance by the total time, and simplifying the result. This problem highlights the importance of understanding fractions and rates in mathematics. Mastering fraction operations and rate calculations is crucial for solving a wide range of mathematical problems. By breaking down the problem into manageable steps, we were able to solve it efficiently and accurately. A systematic approach to problem-solving ensures clarity and reduces the chance of errors. Understanding the underlying concepts and applying the correct formulas are key to success in solving mathematical problems. A strong conceptual understanding leads to more effective problem-solving skills. This problem also demonstrates the value of converting mixed numbers to improper fractions when performing multiplication and division. Improper fractions simplify calculations and improve accuracy. Finally, verifying the answer and considering alternative approaches can deepen understanding and enhance problem-solving abilities. Verification and exploration of alternative methods reinforce learning and problem-solving skills. By following these steps, you can confidently tackle similar problems involving rates and fractions. This detailed solution provides a comprehensive understanding of the problem-solving process, from initial analysis to the final answer. This thorough approach ensures that the concepts are clear and the methods are applicable to other mathematical scenarios. Thus, the principles and techniques discussed here can be applied to a variety of similar problems, enhancing mathematical proficiency and confidence.