Find the least common denominator for these fractions. Enter your answer in the space provided.
$\frac{3}{9}$ and $\frac{2}{5}$
Answer (1)
Identify the denominators: 9 and 5.
Find the prime factorization of each denominator: 9 = 3 2 and 5 = 5 .
Calculate the least common multiple (LCM) by taking the highest power of each prime factor: 3 2 × 5 = 45 .
The least common denominator (LCD) is 45 .
Explanation
Understanding the Problem We are given two fractions, 9 3 and 5 2 , and we need to find their least common denominator (LCD). The least common denominator is the smallest number that is a multiple of both denominators.
Identifying Denominators The denominators of the given fractions are 9 and 5. To find the LCD, we need to find the least common multiple (LCM) of 9 and 5.
Prime Factorization First, let's find the prime factorization of each denominator:
9 = 3 × 3 = 3 2 5 = 5
Calculating the LCM Now, to find the LCM, we take the highest power of each prime factor that appears in either factorization:
LCM(9, 5) = 3 2 × 5 = 9 × 5 = 45
Final Answer Therefore, the least common denominator of the fractions 9 3 and 5 2 is 45.
Examples
Imagine you are baking a cake and need to measure ingredients using different sized scoops. One scoop measures 9 1 of a cup, and another measures 5 1 of a cup. To easily measure a whole number of both scoops into the mixing bowl, you need to find a common unit. The least common denominator (LCD) helps you determine the smallest amount of batter you can make so that you use a whole number of both the 9 1 cup scoop and the 5 1 cup scoop. In this case, the LCD is 45, meaning you would need to make at least 45 45 (or 1) cup of batter to use a whole number of both scoops.
