Which fractions have 24 as the LCD (lowest common denominator)?
$\frac{7}{9}$
$\frac{1}{8}$
$\frac{3}{7}$
$\frac{4}{5}$
$\frac{5}{6}$
Answer (2)
The fractions that have 24 as the lowest common denominator are 8 1 and 6 5 . This conclusion is drawn from checking the divisibility of each denominator by 24. Only 8 and 6 are divisible by 24, confirming them as parts of the LCD set.
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Check if 24 is a multiple of each denominator: 9, 8, 7, 5, and 6.
Identify that 24 is a multiple of 8 and 6.
Confirm that the least common multiple (LCM) of 8 and 6 is 24.
Conclude that the fractions with 24 as the LCD are 8 1 and 6 5 .
Explanation
Understanding the Problem We are given the fractions 9 7 , 8 1 , 7 3 , 5 4 , and 6 5 , and we want to determine which of these fractions could have 24 as their lowest common denominator (LCD). The LCD of a set of fractions is the smallest positive integer that is divisible by all the denominators of the fractions in the set.
Checking Denominators To determine which of the given fractions can have 24 as the LCD, we need to check if 24 is a multiple of each fraction's denominator. In other words, we need to see if 24 is divisible by each of the denominators: 9, 8, 7, 5, and 6.
Divisibility Check Let's check each denominator:
9: Is 24 divisible by 9? 24 ÷ 9 = 2.666... No, 24 is not a multiple of 9.
8: Is 24 divisible by 8? 24 ÷ 8 = 3 . Yes, 24 is a multiple of 8.
7: Is 24 divisible by 7? 24 ÷ 7 = 3.428... No, 24 is not a multiple of 7.
5: Is 24 divisible by 5? 24 ÷ 5 = 4.8 No, 24 is not a multiple of 5.
6: Is 24 divisible by 6? 24 ÷ 6 = 4 . Yes, 24 is a multiple of 6.
Confirming the LCD From the divisibility checks, we found that 24 is a multiple of 8 and 6. This means that the fractions 8 1 and 6 5 could be part of a set of fractions that has 24 as the LCD. To confirm that 24 is indeed the LCD for these two fractions, we can find the least common multiple (LCM) of their denominators, 8 and 6.
Multiples of 8: 8, 16, 24, 32, ... Multiples of 6: 6, 12, 18, 24, 30, ...
The smallest common multiple of 8 and 6 is 24. Therefore, the LCD of 8 1 and 6 5 is 24.
Final Answer Therefore, the fractions that have 24 as the LCD are 8 1 and 6 5 .
Examples
Understanding the lowest common denominator (LCD) is crucial when planning events with recurring schedules. For instance, imagine you're coordinating two different promotional activities: one occurs every 8 days and another every 6 days. To determine when both activities will happen on the same day, you need to find the LCD of 8 and 6, which is 24. This means both promotions will align every 24 days, allowing for strategic planning and maximizing impact.
