The LCD for the fraction 1/3 , 3/4 , 5/32 , and 8/9 Is?
A. 288
B. 24
C. 3,072
D. 64
The correct answer and explanation is:
The correct answer is A. 288.
Explanation:
To find the Least Common Denominator (LCD) of the fractions 13,34,532,89\frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \frac{8}{9}, the LCD is the least common multiple (LCM) of the denominators 3, 4, 32, and 9.
Step 1: List the denominators:
3, 4, 32, 9
Step 2: Find the prime factorization of each denominator:
- 3 = 33
- 4 = 222^2
- 32 = 252^5
- 9 = 323^2
Step 3: To find the LCM, take the highest powers of each prime factor present:
- For prime factor 2: highest power is 252^5 (from 32)
- For prime factor 3: highest power is 323^2 (from 9)
Step 4: Calculate the LCM by multiplying the highest powers:
LCM=25×32=32×9=288\text{LCM} = 2^5 \times 3^2 = 32 \times 9 = 288
Therefore, the LCD of the fractions is 288.
Additional notes:
The LCD is the smallest number into which all the denominators divide evenly. It is useful for adding, subtracting, or comparing fractions, as it allows rewriting each fraction with the same denominator. Here, 288 is the smallest number divisible by 3, 4, 32, and 9, making it the correct least common denominator for the given fractions.