What's the least common denominator (LCD) for the group of fractions: [tex]$\frac{8}{5}, \frac{11}{30}$[/tex], and [tex]$\frac{3}{5}$[/tex] ?
Select one of four:
A. 15
B. 45
C. 30
D. 60
Answer (2)
Find the prime factorization of each denominator: 5 = 5 , 30 = 2 × 3 × 5 , 5 = 5 .
Identify the highest power of each prime factor: 2 1 , 3 1 , 5 1 .
Multiply the highest powers of all prime factors to find the LCD: 2 × 3 × 5 = 30 .
The least common denominator is 30 .
Explanation
Understanding the Problem We are asked to find the least common denominator (LCD) of the fractions 5 8 , 30 11 , and 5 3 . The least common denominator is the smallest number that is a multiple of all the denominators.
Identifying Denominators The denominators of the fractions are 5, 30, and 5. To find the LCD, we need to find the least common multiple (LCM) of these numbers.
Prime Factorization First, find the prime factorization of each denominator:
5 = 5 30 = 2 × 3 × 5 5 = 5
Determining Highest Powers The LCM is the product of the highest powers of all prime factors that appear in any of the factorizations. The prime factors are 2, 3, and 5.
The highest power of 2 is 2 1 = 2 .
The highest power of 3 is 3 1 = 3 .
The highest power of 5 is 5 1 = 5 .
Calculating the LCD Therefore, the LCM (and thus the LCD) is 2 × 3 × 5 = 30 .
Final Answer The least common denominator (LCD) for the group of fractions 5 8 , 30 11 , and 5 3 is 30 .
Examples
When adding or subtracting fractions with different denominators, you need to find the least common denominator (LCD). For example, if you want to add 5 1 and 30 1 , the LCD is 30. You would rewrite the fractions as 30 6 and 30 1 , respectively, and then add them to get 30 7 . This concept is also useful in everyday situations, such as when comparing fractions of different units, like comparing 5 1 of an hour to 30 1 of an hour.
The least common denominator (LCD) for the fractions 5 8 , 30 11 , and 5 3 is 30 . This is found by determining the prime factorization of each denominator and multiplying the highest powers of the prime factors. Therefore, the correct choice is C .30 .
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