Calculating Remaining Distance Radhas Village Journey

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Introduction

In this article, we will delve into a mathematical problem involving distance and fractions. The problem revolves around Radha, who is traveling from her village to another, a journey spanning 35 3/4 km. During her journey, she takes a break after covering 24 1/3 km. Our task is to determine the remaining distance Radha needs to travel to reach her destination. This problem is a practical application of fraction subtraction, a fundamental concept in mathematics. Understanding how to solve such problems is crucial for everyday life, whether it's planning a trip, calculating distances, or managing resources. In this detailed explanation, we will break down the steps involved in solving this problem, making it easy to understand for everyone, regardless of their mathematical background. Fraction subtraction can often seem daunting, but with a clear method and step-by-step guidance, it becomes a manageable and even enjoyable task. We will convert mixed fractions to improper fractions, find a common denominator, perform the subtraction, and then convert the result back to a mixed fraction. This process will not only help us solve this specific problem but also provide a solid foundation for tackling similar mathematical challenges in the future. So, let's embark on this mathematical journey with Radha and unravel the solution together, enhancing our understanding of fractions and their applications in real-world scenarios. We will also explore the importance of accurate calculations in travel and how understanding fractions can aid in efficient planning and execution of journeys. This problem highlights the relevance of mathematics in everyday life and underscores the need for a strong grasp of basic mathematical principles.

Problem Statement

Radha is on a journey from her village to another, with the total distance between the villages being 35 3/4 km. After traveling a portion of the distance, specifically 24 1/3 km, she decides to stop and rest. The core question we aim to answer is: How much further does Radha need to travel to reach her destination? This problem is a classic example of a distance calculation problem, where we need to subtract the distance already covered from the total distance to find the remaining distance. The presence of mixed fractions adds a layer of complexity, requiring us to convert these mixed fractions into improper fractions before performing the subtraction. Understanding the problem statement is the first crucial step in solving any mathematical problem. It involves identifying the known quantities (total distance and distance covered), the unknown quantity (remaining distance), and the mathematical operation required to find the solution (subtraction). In this case, the total distance of 35 3/4 km represents the entire journey, while the distance covered of 24 1/3 km represents the portion of the journey that Radha has already completed. The remaining distance is the difference between these two quantities. This problem also emphasizes the importance of attention to detail, particularly when dealing with fractions. A small error in conversion or subtraction can lead to a significant difference in the final answer. Therefore, we will proceed step by step, ensuring accuracy and clarity in each step. This problem is not just about finding a numerical answer; it's about understanding the process of problem-solving and applying mathematical concepts to real-life situations. So, let's break down the problem and approach it with a clear and methodical strategy, making the solution easily understandable and replicable for similar problems.

Solution

To determine the remaining distance Radha needs to cover, we must subtract the distance she has already traveled from the total distance between the villages. This involves subtracting 24 1/3 km from 35 3/4 km. The first step in solving this problem is to convert the mixed fractions into improper fractions. This conversion is necessary because it simplifies the subtraction process. A mixed fraction consists of a whole number and a proper fraction, while an improper fraction has a numerator that is greater than or equal to its denominator. To convert a mixed fraction to an improper fraction, we multiply the whole number by the denominator of the fraction and then add the numerator. The result becomes the new numerator, and the denominator remains the same. For 35 3/4, we multiply 35 by 4, which gives us 140, and then add 3, resulting in 143. So, 35 3/4 becomes 143/4. Similarly, for 24 1/3, we multiply 24 by 3, which gives us 72, and then add 1, resulting in 73. So, 24 1/3 becomes 73/3. Now, we have the problem restated as subtracting 73/3 from 143/4. To subtract fractions, they must have a common denominator. The common denominator is the least common multiple (LCM) of the denominators. In this case, the denominators are 4 and 3. The LCM of 4 and 3 is 12. We need to convert both fractions to have a denominator of 12. To convert 143/4 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3. This gives us (143 * 3) / (4 * 3) = 429/12. To convert 73/3 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 4. This gives us (73 * 4) / (3 * 4) = 292/12. Now, we can subtract the fractions: 429/12 - 292/12. To subtract fractions with a common denominator, we subtract the numerators and keep the denominator the same. So, 429 - 292 = 137. Therefore, the result of the subtraction is 137/12. The final step is to convert the improper fraction 137/12 back to a mixed fraction. To do this, we divide the numerator by the denominator. 137 divided by 12 gives us a quotient of 11 and a remainder of 5. The quotient becomes the whole number part of the mixed fraction, the remainder becomes the new numerator, and the denominator remains the same. So, 137/12 is equal to 11 5/12. Therefore, the remaining distance Radha needs to cover is 11 5/12 km.

Step-by-Step Breakdown

  1. Convert Mixed Fractions to Improper Fractions:
    • 35 3/4 = (35 * 4 + 3) / 4 = 143/4
    • 24 1/3 = (24 * 3 + 1) / 3 = 73/3
  2. Find the Least Common Denominator (LCM):
    • The LCM of 4 and 3 is 12.
  3. Convert Fractions to Equivalent Fractions with the Common Denominator:
    • 143/4 = (143 * 3) / (4 * 3) = 429/12
    • 73/3 = (73 * 4) / (3 * 4) = 292/12
  4. Subtract the Fractions:
    • 429/12 - 292/12 = (429 - 292) / 12 = 137/12
  5. Convert the Improper Fraction Back to a Mixed Fraction:
    • 137/12 = 11 5/12

Verification

To ensure the accuracy of our solution, we can verify our answer by adding the distance Radha has already traveled to the remaining distance. If the sum equals the total distance, our solution is correct. We will add 24 1/3 km and 11 5/12 km. First, we convert these mixed fractions into improper fractions:

  • 24 1/3 = 73/3
  • 11 5/12 = 137/12

Next, we find a common denominator, which is 12. We convert 73/3 to an equivalent fraction with a denominator of 12:

  • 73/3 = (73 * 4) / (3 * 4) = 292/12

Now, we add the fractions:

  • 292/12 + 137/12 = (292 + 137) / 12 = 429/12

Finally, we convert the improper fraction 429/12 back to a mixed fraction:

  • 429/12 = 35 9/12

We can simplify the fraction 9/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

  • 9/12 = (9 ÷ 3) / (12 ÷ 3) = 3/4

So, 429/12 is equal to 35 3/4, which is the total distance between the villages. This verifies that our solution of 11 5/12 km for the remaining distance is correct. This verification step is crucial in problem-solving as it helps ensure accuracy and builds confidence in the solution. By adding the distances and obtaining the total distance, we have successfully confirmed the correctness of our answer. This process reinforces the understanding of fraction addition and subtraction and highlights the importance of checking one's work in mathematics. So, we can confidently conclude that Radha needs to cover 11 5/12 km to reach the other village.

Answer

Therefore, the distance Radha needs to cover to reach the other village is 11 5/12 km. This answer represents the solution to the problem and is the result of carefully applying the principles of fraction subtraction. We have successfully navigated the steps involved in converting mixed fractions to improper fractions, finding a common denominator, performing the subtraction, and converting the result back to a mixed fraction. This process not only provides the answer to the specific problem but also enhances our understanding of fraction operations. The answer is a concrete value that provides a clear understanding of the remaining distance in Radha's journey. It is not just a numerical result but a practical representation of the distance that needs to be covered. This problem highlights the relevance of mathematics in everyday life and underscores the importance of a strong foundation in basic mathematical concepts. The ability to solve such problems is crucial for various real-world applications, from planning trips to managing resources. The answer of 11 5/12 km is not just a solution; it's a testament to the power of mathematical reasoning and problem-solving skills. It represents the culmination of a step-by-step process, each step carefully executed to arrive at the correct answer. This journey through the problem has not only provided us with the solution but also reinforced our understanding of fractions and their applications. So, with confidence, we can state that Radha needs to cover 11 5/12 km to reach her destination, a testament to her perseverance and our mathematical prowess.