What is the least common denominator for the fractions below

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What is the least common denominator for the fractions below? (Round the answer to the nearest integer)

1/2, 2/3, 4/5

A.
30

B.
25

C.
7

D.
19

The Correct answer and Explanation is:

To find the least common denominator (LCD) for the fractions 1/2, 2/3, and 4/5, you need to determine the least common multiple (LCM) of the denominators 2, 3, and 5. Here’s a step-by-step explanation:

Step-by-Step Solution

  1. Identify the Denominators: The denominators of the fractions are 2, 3, and 5.
  2. Prime Factorization:
    • 2 is a prime number, so its prime factorization is simply 2.
    • 3 is also a prime number, so its prime factorization is 3.
    • 5 is another prime number, so its prime factorization is 5.
  3. Find the Least Common Multiple (LCM): To find the LCM, you need to take each prime number that appears in the factorization of the denominators at the highest power it appears. Here are the prime factorizations again:
    • 2 = 2^1
    • 3 = 3^1
    • 5 = 5^1
    The LCM will be the product of these primes raised to their highest powers:LCM=21×31×51\text{LCM} = 2^1 \times 3^1 \times 5^1LCM=21×31×51 LCM=2×3×5\text{LCM} = 2 \times 3 \times 5LCM=2×3×5 LCM=30\text{LCM} = 30LCM=30
  4. Verification: To ensure that 30 is indeed the least common multiple:
    • 30 ÷ 2 = 15 (an integer)
    • 30 ÷ 3 = 10 (an integer)
    • 30 ÷ 5 = 6 (an integer)
    Since 30 is divisible by all the original denominators, it is indeed the least common multiple.

Conclusion

The least common denominator for the fractions 1/2, 2/3, and 4/5 is 30. Therefore, the correct answer is:

A. 30

This LCD allows us to compare or combine these fractions easily by converting them to equivalent fractions with a common denominator.

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