What is the LCM of 18/5 and 20/9?

Options

A. 60
B. 12
C. 30
D. 180

Answer

To find the LCM of the fractions \( \frac{18}{5} \) and \( \frac{20}{9} \), we can use the formula for the LCM of fractions:

\[ \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)} \]

Where:
- \( a = 18 \)
- \( b = 5 \)
- \( c = 20 \)
- \( d = 9 \)

### Step 1: Find the LCM of the numerators (18 and 20)

To find the LCM of 18 and 20, we can list the multiples of each number or use the prime factorization method.

- Prime factorization of 18: \( 2 \times 3^2 \)
- Prime factorization of 20: \( 2^2 \times 5 \)

The LCM is found by taking the highest power of each prime number:

- For 2: the highest power is \( 2^2 \)
- For 3: the highest power is \( 3^2 \)
- For 5: the highest power is \( 5^1 \)

Thus, the LCM of 18 and 20 is:

\[ \text{LCM}(18, 20) = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \]

### Step 2: Find the HCF of the denominators (5 and 9)

To find the HCF of 5 and 9, we can list the factors of each number:

- Factors of 5: 1, 5
- Factors of 9: 1, 3, 9

The only common factor is 1, so:

\[ \text{HCF}(5, 9) = 1 \]

### Step 3: Apply the LCM and HCF to the formula

Now we can substitute the values we found into the formula:

\[ \text{LCM}\left(\frac{18}{5}, \frac{20}{9}\right) = \frac{\text{LCM}(18, 20)}{\text{HCF}(5, 9)} = \frac{180}{1} = 180 \]

### Final Answer

The LCM of \( \frac{18}{5} \) and \( \frac{20}{9} \) is \( 180 \).