Find the least common denominator for these fractions. Enter your answer in the space provided.
[tex]$\frac{1}{5}$[/tex] and [tex]$\frac{3}{6}$[/tex]
Answer (2)
Identify the denominators: The denominators of the fractions are 5 and 6.
Find the prime factorization of each denominator: 5 = 5 and 6 = 2 × 3 .
Calculate the least common multiple (LCM) by multiplying the highest powers of all prime factors: L CM ( 5 , 6 ) = 2 × 3 × 5 = 30 .
The least common denominator (LCD) is the LCM of the denominators, so the LCD is 30 .
Explanation
Understanding the Problem We are asked to find the least common denominator (LCD) of the fractions 5 1 and 6 3 . The least common denominator is the least common multiple (LCM) of the denominators. In this case, the denominators are 5 and 6.
Choosing a Method To find the least common multiple (LCM) of 5 and 6, we can list the multiples of each number until we find a common multiple. Alternatively, we can use the prime factorization method.
Prime Factorization Let's use the prime factorization method. First, we find the prime factorization of each number:
5 = 5
6 = 2 × 3
Calculating the LCM The LCM is the product of the highest powers of all prime factors that appear in either factorization. In this case, the prime factors are 2, 3, and 5. Therefore, the LCM of 5 and 6 is:
L CM ( 5 , 6 ) = 2 × 3 × 5 = 30
Final Answer Thus, the least common denominator of the fractions 5 1 and 6 3 is 30 .
Examples
Imagine you are planning a party and want to serve pizza and soda. If you buy pizza in slices of 5 and soda in packs of 6, the least common denominator (LCD) helps you determine the smallest number of slices and packs you need to buy so that you have an equal amount of each for your guests. In this case, the LCD of 5 and 6 is 30, meaning you would need to buy enough pizza to have 30 slices and enough soda to have 30 cans. This ensures that you have a balanced supply of both items without any leftovers.
The least common denominator of the fractions 5 1 and 6 3 is 30. This is found by calculating the least common multiple (LCM) of the denominators 5 and 6. The LCM is determined to be 30, which serves as the least common denominator.
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