Which fraction among `2//3, 4//5` and `7//11` is the largest?

Options

A. `2//3`
B. `4//5`
C. `7//11`
D. All are equal

Answer

To determine which fraction among \( \frac{2}{3}, \frac{4}{5}, \) and \( \frac{7}{11} \) is the largest, we can follow these steps:

### Step 1: Find the LCM of the denominators
The denominators of the fractions are 3, 5, and 11. We need to find the Least Common Multiple (LCM) of these numbers.

- The prime factors of 3 are \( 3^1 \).
- The prime factors of 5 are \( 5^1 \).
- The prime factors of 11 are \( 11^1 \).

Since all these numbers are prime, the LCM is simply the product of these numbers:
\[ \text{LCM} = 3 \times 5 \times 11 = 165. \]

### Step 2: Convert each fraction to have the same denominator
Now, we will convert each fraction to have a denominator of 165.

1. For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 55}{3 \times 55} = \frac{110}{165}. \]

2. For \( \frac{4}{5} \):
\[ \frac{4}{5} = \frac{4 \times 33}{5 \times 33} = \frac{132}{165}. \]

3. For \( \frac{7}{11} \):
\[ \frac{7}{11} = \frac{7 \times 15}{11 \times 15} = \frac{105}{165}. \]

### Step 3: Compare the numerators
Now that we have all fractions with the same denominator, we can compare the numerators:
- \( 110 \) (from \( \frac{2}{3} \))
- \( 132 \) (from \( \frac{4}{5} \))
- \( 105 \) (from \( \frac{7}{11} \))

### Step 4: Determine the largest numerator
Among the numerators \( 110, 132, \) and \( 105 \), the largest is \( 132 \).

### Conclusion
Thus, the largest fraction among \( \frac{2}{3}, \frac{4}{5}, \) and \( \frac{7}{11} \) is:
\[ \frac{4}{5}. \]