What's the least common denominator of [tex]$3 / 4, 4 / 5$[/tex], and [tex]$2 / 3$[/tex]?
A) 20
B) 60
C) 12
Answer (2)
Find the prime factorization of each denominator: 4 = 2 2 , 5 = 5 , 3 = 3 .
Identify the highest power of each prime factor: 2 2 , 3 1 , 5 1 .
Multiply these highest powers together: 2 2 ⋅ 3 1 ⋅ 5 1 = 4 ⋅ 3 ⋅ 5 = 60 .
The least common denominator is 60 .
Explanation
Understanding the Problem We need to find the least common denominator (LCD) of the fractions 4 3 , 5 4 , and 3 2 . The least common denominator is the least common multiple (LCM) of the denominators. In this case, the denominators are 4, 5, and 3.
Prime Factorization To find the LCM of 4, 5, and 3, we can list the multiples of each number until we find a common multiple. Alternatively, we can use the prime factorization method. Let's use the prime factorization method. First, we find the prime factorization of each number:
4 = 2 2 5 = 5 1 3 = 3 1
Calculating the LCM The LCM is the product of the highest powers of all prime factors that appear in any of the factorizations. In this case, the prime factors are 2, 3, and 5. The highest power of 2 is 2 2 , the highest power of 3 is 3 1 , and the highest power of 5 is 5 1 . Therefore, the LCM is:
LCM ( 4 , 5 , 3 ) = 2 2 × 3 1 × 5 1 = 4 × 3 × 5
Finding the LCD Now, we calculate the product:
4 × 3 × 5 = 12 × 5 = 60
So, the least common denominator of 4 3 , 5 4 , and 3 2 is 60.
Final Answer Therefore, the least common denominator of the given fractions is 60 .
Examples
When planning a party, you might want to serve pizza, salad, and drinks. If the pizza is cut into 4 slices, the salad is divided into 5 portions, and the drinks come in packs of 3, the least common denominator (LCD) helps you determine the smallest number of guests you can serve so that you use a whole number of pizza slices, salad portions, and drink packs. In this case, the LCD of 4, 5, and 3 is 60, meaning you can serve 60 guests and have a whole number of slices, portions, and packs for everyone.
The least common denominator of the fractions 4 3 , 5 4 , and 3 2 is 60 . This is determined by finding the least common multiple of their denominators using prime factorization. The answer is B) 60.
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