To find the LCM of the fractions \( \frac{18}{5}, \frac{6}{5}, \) and \( \frac{18}{125} \), we can use the formula for the LCM of fractions:
\[
\text{LCM}\left(\frac{A}{B}, \frac{C}{D}\right) = \frac{\text{LCM}(A, C)}{\text{GCD}(B, D)}
\]
### Step-by-step Solution:
1. **Identify A, B, C, and D**:
- From the fractions, we have:
- \( A = 18 \)
- \( B = 5 \)
- \( C = 6 \)
- \( D = 125 \)
2. **Calculate LCM of the Numerators (A and C)**:
- We need to find \( \text{LCM}(18, 6) \).
- The multiples of 18 are: 18, 36, 54, 72, ...
- The multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
- The smallest common multiple is 18.
- Thus, \( \text{LCM}(18, 6) = 18 \).
3. **Calculate GCD of the Denominators (B and D)**:
- We need to find \( \text{GCD}(5, 125) \).
- The factors of 5 are: 1, 5.
- The factors of 125 are: 1, 5, 25, 125.
- The greatest common factor is 5.
- Thus, \( \text{GCD}(5, 125) = 5 \).
4. **Combine the Results**:
- Now we can find the LCM of the fractions:
\[
\text{LCM}\left(\frac{18}{5}, \frac{6}{5}, \frac{18}{125}\right) = \frac{\text{LCM}(18, 6)}{\text{GCD}(5, 125)} = \frac{18}{5}
\]
5. **Convert to Decimal**:
- To express \( \frac{18}{5} \) as a decimal:
\[
\frac{18}{5} = 3.6
\]
### Final Answer:
The LCM of \( \frac{18}{5}, \frac{6}{5}, \) and \( \frac{18}{125} \) is \( 3.6 \).
