Determine the least common denominator of the above by inspection 1/9 5/18 three /6 1/90

To determine the least common denominator (LCD) of the fractions \( \frac{1}{9}, \frac{5}{18}, \frac{3}{6}, \frac{1}{90} \), we need to find the least common multiple (LCM) of their denominators: 9, 18, 6, and 90.

### Step 1: Prime Factorization
Let's find the prime factorizations of each denominator:
- \( 9 = 3^2 \)
- \( 18 = 2 \times 3^2 \)
- \( 6 = 2 \times 3 \)
- \( 90 = 2 \times 3^2 \times 5 \)

### Step 2: Identify the highest powers of each prime factor
- For the factor 2, the highest power is \( 2^1 \) (from 18 and 90).
- For the factor 3, the highest power is \( 3^2 \) (from 9, 18, and 90).
- For the factor 5, the highest power is \( 5^1 \) (from 90).

### Step 3: Calculate the LCM
Now, multiply the highest powers of all prime factors:
\[
\text{LCM} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90
\]

Thus, the least common denominator (LCD) of the fractions is **90**.

Would you like any further details or have any questions?

Here are five follow-up questions:
1. How do you convert each fraction to have the LCD of 90?
2. What is the general method for finding the least common denominator?
3. Can you simplify fractions before finding their LCD?
4. Why is it important to use the least common denominator when adding or subtracting fractions?
5. How does the prime factorization method help in finding the LCM?

**Tip:** The least common denominator is always the smallest number that all denominators divide into evenly!

Learn how to determine the least common denominator (LCD) for the fractions 1/9, 5/18, 3/6, and 1/90 using prime factorization. This detailed step-by-step solution demonstrates how to find the LCD by calculating the least common multiple of the denominators. Ideal for students in Grades 5-7.

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