LCM of `2/3, 4/9, 5/6` is:

Options

A. `8/27`
B. `20/3`
C. `10/3`
D. `20/27`

Answer

To find the LCM of the fractions \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6} \), we will follow these steps:

### Step 1: Identify the numerators and denominators
The numerators of the fractions are:
- \( 2 \) from \( \frac{2}{3} \)
- \( 4 \) from \( \frac{4}{9} \)
- \( 5 \) from \( \frac{5}{6} \)

The denominators of the fractions are:
- \( 3 \) from \( \frac{2}{3} \)
- \( 9 \) from \( \frac{4}{9} \)
- \( 6 \) from \( \frac{5}{6} \)

### Step 2: Calculate the LCM of the numerators
To find the LCM of the numerators \( 2, 4, \) and \( 5 \):
- The multiples of \( 2 \) are \( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \)
- The multiples of \( 4 \) are \( 4, 8, 12, 16, 20 \)
- The multiples of \( 5 \) are \( 5, 10, 15, 20 \)

The smallest common multiple is \( 20 \). Therefore,
\[ \text{LCM of } 2, 4, 5 = 20 \]

### Step 3: Calculate the HCF of the denominators
To find the HCF of the denominators \( 3, 9, \) and \( 6 \):
- The factors of \( 3 \) are \( 1, 3 \)
- The factors of \( 9 \) are \( 1, 3, 9 \)
- The factors of \( 6 \) are \( 1, 2, 3, 6 \)

The greatest common factor is \( 3 \). Therefore,
\[ \text{HCF of } 3, 9, 6 = 3 \]

### Step 4: Calculate the LCM of the fractions
Using the formula for the LCM of fractions:
\[ \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} \]
Substituting the values we found:
\[ \text{LCM} = \frac{20}{3} \]

### Final Answer
Thus, the LCM of \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6} \) is:
\[ \frac{20}{3} \]

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