Select the best answer for the question.
13. Express the fractions [tex]$3 / 4,7 / 16$[/tex], and [tex]$5 / 8$[/tex] with the LCD.
A. [tex]$3 / 4,2 / 4,3 / 4$[/tex]
B. [tex]$24 / 32,14 / 32,24 / 32$[/tex]
C. [tex]$9 / 16,49 / 16,36 / 16$[/tex]
D. [tex]$12 / 16,7 / 16,10 / 16$[/tex]
Answer (1)
Find the least common denominator (LCD) of the fractions, which is 16.
Convert each fraction to an equivalent fraction with the LCD: 4 3 = 16 12 , 16 7 = 16 7 , and 8 5 = 16 10 .
Express the fractions with the LCD: 16 12 , 16 7 , 16 10 .
The correct answer is: D
Explanation
Understanding the Problem We need to express the fractions 4 3 , 16 7 , and 8 5 with their least common denominator (LCD). This means we need to find a common denominator for all three fractions and rewrite each fraction with that denominator.
Finding the Least Common Denominator (LCD) First, let's find the least common multiple (LCM) of the denominators 4 , 16 , and 8 . The prime factorizations of these numbers are:
4 = 2 2 16 = 2 4 8 = 2 3
The LCM is the highest power of all prime factors present in the factorizations. In this case, the only prime factor is 2, and the highest power is 2 4 = 16 . So, the LCD is 16.
Converting Fractions to Equivalent Fractions with LCD Now, we'll convert each fraction to an equivalent fraction with a denominator of 16:
4 3 = 4 × 4 3 × 4 = 16 12 16 7 already has a denominator of 16, so it remains 16 7 .
8 5 = 8 × 2 5 × 2 = 16 10
Identifying the Correct Answer So, the fractions expressed with the LCD are 16 12 , 16 7 , and 16 10 . Comparing this to the given options, we see that option D matches our result.
Final Answer Therefore, the correct answer is D. 16 12 , 16 7 , 16 10 .
Examples
When comparing the nutritional content of different food items, it's often helpful to express the quantities of nutrients as fractions with a common denominator. For example, if you want to compare the amount of fat in three different snacks, where one snack has 3/4 of its calories from fat, another has 7/16, and the third has 5/8, expressing these fractions with a common denominator (16 in this case) allows for an easy comparison: 12/16, 7/16, and 10/16. This makes it clear which snack has the highest proportion of calories from fat.
