- The student identified a valid common denominator (\(36\)), but not the least one.
- By listing the multiples of \(9\), \(3\), and \(6\), we determine that the smallest shared multiple is \(18\).
- Using the smallest common denominator (\(18\)) makes subsequent arithmetic significantly easier.
- The correct least common denominator is \(\boxed{18}\).
Explanation
- Analyzing the Problem
The student is tasked with adding three numbers involving fractions with denominators \(9\), \(3\), and \(6\). The first step in this process is finding a common denominator, which the student identified as \(36\). The student correctly noted that \(36\) is divisible by \(9\), \(3\), and \(6\), making it a valid common denominator. However, the task specifically calls for the Least Common Denominator (LCD). This means we must identify the smallest positive integer that is a multiple of all the given denominators.
- Identifying the Least Common Denominator
To find the least common denominator, we can list the multiples of each denominator until we find the smallest value they share:
- Multiples of \(9\): \(9, 18, 27, 36, \dots\)
- Multiples of \(3\): \(3, 6, 9, 12, 15, 18, 21, 24, \dots\)
- Multiples of \(6\): \(6, 12, 18, 24, 30, 36, \dots\)
Comparing these lists, we see that the numbers \(18\) and \(36\) are both common denominators. However, \(18\) is the smallest positive integer present in all three lists. Thus, the least common denominator is \(18\).
- Correcting the Student's Mistake
The student's error lies in assuming that any common denominator is the least common denominator. While \(36\) can be used as a common denominator to add the fractions, it is not the least common denominator because \(18\) is smaller and also divisible by all the denominators (\(18 \div 9 = 2\), \(18 \div 3 = 6\), and \(18 \div 6 = 3\)). Using the LCD of \(18\) simplifies calculations by working with smaller numbers. The correct least common denominator is \(\boxed{18}\).
Examples
Knowing how to find the least common denominator is essential for everyday tasks involving measurements, such as when you are following a recipe. If a recipe calls for 1/9 of a cup of sugar, 1/3 of a cup of flour, and 5/6 of a cup of oats, finding the least common denominator allows you to convert these into equivalent fractions with the same denominator—in this case, 18—making it much easier to determine the total volume of ingredients needed by simply adding the numerators. Using the smallest common denominator simplifies the arithmetic and reduces the likelihood of errors.