Comparing Coloring Times Who Worked Longer And By How Much
Introduction
In this mathematical exploration, we delve into a scenario involving two individuals, Sanchit and Kanchan, who are engaged in a coloring activity. Our objective is to determine who took a longer time to complete the task and by what fraction their coloring times differ. This exercise provides a practical application of fraction comparison and subtraction, skills that are fundamental in mathematics and everyday life. Let's unravel the details of their coloring endeavors and arrive at a clear and concise answer.
Understanding the Problem: Sanchit and Kanchan's Coloring Times
To effectively tackle this problem, we need to first understand the information provided. Sanchit completed the coloring of a picture in 2 3/9 hours, while Kanchan finished the same picture in 7/6 hours. The core question here is: who dedicated more time to coloring, and what is the fractional difference in their efforts? This involves comparing mixed fractions and improper fractions, and then finding the difference between them. We'll break down the steps involved in converting these times into a common format, comparing them accurately, and finally calculating the fractional difference. This comparison will reveal not just who worked longer, but the extent of the difference in their time commitments, highlighting the importance of precise calculation in problem-solving.
Converting Mixed Fractions to Improper Fractions: A Crucial Step
To accurately compare Sanchit and Kanchan's coloring times, the initial step involves converting Sanchit's time, which is given as a mixed fraction (2 3/9 hours), into an improper fraction. This conversion is essential because it allows us to perform mathematical operations like subtraction with greater ease and precision. A mixed fraction combines 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. This result becomes the new numerator, and the denominator remains the same. Let's apply this process to Sanchit's time: 2 3/9 hours. We multiply the whole number 2 by the denominator 9, which gives us 18. Then, we add the numerator 3 to this result, yielding 21. Therefore, the improper fraction equivalent of 2 3/9 is 21/9. This conversion is a cornerstone of fraction arithmetic and is crucial for solving problems that involve comparing or combining fractions. With Sanchit's time now expressed as an improper fraction, we can proceed to compare it with Kanchan's time, which is already in fraction form.
Finding a Common Denominator: Essential for Fraction Comparison
Once we've converted Sanchit's time to an improper fraction (21/9 hours) and we have Kanchan's time (7/6 hours), the next crucial step is to find a common denominator. This is a fundamental requirement for comparing and performing arithmetic operations on fractions. A common denominator is a number that is a multiple of the denominators of all the fractions being considered. It provides a standardized basis for comparing the fractions, as it allows us to directly compare the numerators. To find the least common denominator (LCD), which is the smallest common multiple, we can list the multiples of each denominator or use prime factorization. In this case, our denominators are 9 and 6. The multiples of 9 are 9, 18, 27, and so on, while the multiples of 6 are 6, 12, 18, 24, and so on. We can see that the smallest number that appears in both lists is 18. Therefore, the least common denominator for 9 and 6 is 18. Finding the LCD is a cornerstone of fraction arithmetic, as it enables accurate comparison and calculation. With the common denominator identified, we can now proceed to rewrite each fraction with this denominator, preparing them for comparison and subtraction.
Rewriting Fractions with the Common Denominator: Preparing for Comparison
Having determined that the least common denominator (LCD) for the fractions 21/9 and 7/6 is 18, our next step involves rewriting each fraction with this new denominator. This process ensures that we can accurately compare the fractions, as they will both be expressed in terms of the same whole. To rewrite a fraction with a different denominator, we need to multiply both the numerator and the denominator by the same number. This maintains the value of the fraction while changing its representation. For the fraction 21/9, we need to determine what number to multiply 9 by to get 18. The answer is 2. So, we multiply both the numerator and the denominator of 21/9 by 2, resulting in (21 * 2) / (9 * 2), which equals 42/18. Similarly, for the fraction 7/6, we need to find the number that, when multiplied by 6, gives us 18. This number is 3. Multiplying both the numerator and the denominator of 7/6 by 3 gives us (7 * 3) / (6 * 3), which equals 21/18. Now that both fractions, 21/9 and 7/6, have been rewritten with the common denominator of 18, they are expressed as 42/18 and 21/18, respectively. This crucial step sets the stage for a direct and meaningful comparison of the fractions, allowing us to determine which represents a larger quantity of time spent coloring.
Comparing the Fractions: Determining Who Worked Longer
With both Sanchit's and Kanchan's coloring times expressed as fractions with a common denominator, 42/18 hours and 21/18 hours respectively, we can now directly compare the fractions to determine who spent more time coloring the picture. When fractions share a common denominator, comparing them becomes straightforward – we simply compare the numerators. The fraction with the larger numerator represents a greater quantity. In this case, we are comparing 42/18 and 21/18. Clearly, 42 is greater than 21. This indicates that the fraction 42/18 is larger than 21/18. Therefore, Sanchit, whose coloring time is represented by 42/18 hours, spent more time coloring the picture than Kanchan, whose time is represented by 21/18 hours. This comparison highlights the importance of establishing a common denominator when working with fractions, as it allows for a clear and accurate assessment of their relative sizes. Having identified that Sanchit worked longer, the next step is to calculate exactly how much longer he worked, which involves subtracting the two fractions.
Calculating the Difference: Finding the Fractional Amount of Extra Time
Having established that Sanchit worked longer on the coloring task, the next step is to quantify the difference in time spent compared to Kanchan. This involves subtracting Kanchan's coloring time from Sanchit's coloring time. Both times are now expressed as fractions with a common denominator: Sanchit's time is 42/18 hours, and Kanchan's time is 21/18 hours. To subtract fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. Therefore, the difference in time is calculated as (42 - 21) / 18. Performing the subtraction, we get 21/18 hours. This fraction represents the additional time Sanchit spent coloring compared to Kanchan. However, to present the difference in its simplest form, we should reduce the fraction to its lowest terms. Both the numerator and the denominator, 21 and 18, are divisible by 3. Dividing both by 3, we get 7/6. This means Sanchit worked 7/6 hours longer than Kanchan. The fraction 7/6 is an improper fraction, which can also be expressed as a mixed number to provide a more intuitive understanding of the time difference. Converting 7/6 to a mixed number gives us 1 1/6 hours. Thus, Sanchit worked 1 1/6 hours longer than Kanchan, providing a precise answer to the question of the fractional difference in their coloring times.
Expressing the Difference as a Mixed Number: Providing Clarity
After calculating that Sanchit worked 7/6 hours longer than Kanchan, expressing this difference as a mixed number provides a clearer understanding of the time duration. An improper fraction, like 7/6, has a numerator greater than its denominator, which can be less intuitive to grasp than a mixed number, which combines a whole number and a proper fraction. To convert the improper fraction 7/6 into a mixed number, we divide the numerator (7) by the denominator (6). The quotient represents the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part, with the original denominator remaining the same. When we divide 7 by 6, the quotient is 1, and the remainder is 1. This means that 7/6 can be expressed as the mixed number 1 1/6. In the context of our problem, this means Sanchit worked 1 whole hour and 1/6 of an hour longer than Kanchan. Expressing the difference in this way helps to contextualize the additional time Sanchit spent coloring, making it easier to comprehend and relate to real-world time measurements. This conversion underscores the importance of being able to move between different forms of fractions to effectively communicate mathematical results.
Conclusion: Sanchit's Extra Effort Quantified
In conclusion, through careful comparison and calculation, we've determined that Sanchit worked longer on coloring the picture than Kanchan. By converting their coloring times into fractions with a common denominator, we were able to accurately compare their efforts. The analysis revealed that Sanchit spent 42/18 hours coloring, while Kanchan spent 21/18 hours. This direct comparison highlighted that Sanchit indeed dedicated more time to the task. Furthermore, by subtracting Kanchan's time from Sanchit's, we quantified the difference, initially finding it to be 21/18 hours. Simplifying this fraction to its lowest terms, we arrived at 7/6 hours. To provide a more intuitive understanding, we converted this improper fraction into a mixed number, which showed that Sanchit worked 1 1/6 hours longer than Kanchan. This comprehensive exploration not only answers the question of who worked longer but also precisely quantifies the additional time spent, demonstrating the practical application of fraction arithmetic in real-world scenarios. The process underscores the importance of understanding fractions, their comparison, and their manipulation in solving everyday problems.
