Express the pair of fractions using the least common denominator of the two. [tex]$\frac{5}{12}, \frac{7}{16}$[/tex]
Answer (1)
Find the least common multiple (LCM) of the denominators 12 and 16, which is 48.
Convert the first fraction, 12 5 , to an equivalent fraction with a denominator of 48: 12 5 = 12 × 4 5 × 4 = 48 20 .
Convert the second fraction, 16 7 , to an equivalent fraction with a denominator of 48: 16 7 = 16 × 3 7 × 3 = 48 21 .
The pair of fractions expressed using the least common denominator is 48 20 , 48 21 .
Explanation
Understanding the Problem We are given two fractions, 12 5 and 16 7 , and we need to express them using their least common denominator (LCD). The least common denominator is the least common multiple (LCM) of the denominators.
Finding the Least Common Multiple First, we need to find the least common multiple (LCM) of the denominators 12 and 16. We can use prime factorization to find the LCM. The prime factorization of 12 is 2 2 × 3 , and the prime factorization of 16 is 2 4 . The LCM is the product of the highest powers of all prime factors present in either number, which is 2 4 × 3 = 16 × 3 = 48 .
Converting the First Fraction Now we need to express each fraction with the LCM, 48, as the new denominator. For the first fraction, 12 5 , we need to find a factor that, when multiplied by 12, gives us 48. Since 48 ÷ 12 = 4 , we multiply both the numerator and the denominator of 12 5 by 4: 12 × 4 5 × 4 = 48 20 .
Converting the Second Fraction For the second fraction, 16 7 , we need to find a factor that, when multiplied by 16, gives us 48. Since 48 ÷ 16 = 3 , we multiply both the numerator and the denominator of 16 7 by 3: 16 × 3 7 × 3 = 48 21 .
Final Answer Therefore, the pair of fractions expressed using the least common denominator is 48 20 and 48 21 .
Examples
When comparing fractions, it's often helpful to express them with a common denominator. For example, if you're trying to determine which of two recipes uses more sugar, and one recipe calls for 12 5 cup of sugar while the other calls for 16 7 cup of sugar, converting both fractions to a common denominator allows for easy comparison. In this case, converting both to fractions with a denominator of 48, we get 48 20 and 48 21 , making it clear that the second recipe uses slightly more sugar.
