Which one of the following is the correct statement?

Options

A. `2/3lt3/5lt14/15
B. `3/5lt2/3lt14/15`
C. `14/15lt3/5lt2/3`
D. none of these

Answer

To determine which statement is correct among the given options, we will compare the fractions: \( \frac{2}{3} \), \( \frac{3}{5} \), and \( \frac{14}{15} \).

### Step-by-step Solution:

1. **Identify the fractions to compare**:
We have three fractions: \( \frac{2}{3} \), \( \frac{3}{5} \), and \( \frac{14}{15} \).

2. **Find a common denominator**:
The denominators are 3, 5, and 15. The least common multiple (LCM) of these numbers is 15.

3. **Convert each fraction to have the common denominator**:
- For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]
- For \( \frac{3}{5} \):
\[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \]
- For \( \frac{14}{15} \):
\[ \frac{14}{15} \text{ remains } \frac{14}{15} \]

4. **List the converted fractions**:
Now we have:
- \( \frac{2}{3} = \frac{10}{15} \)
- \( \frac{3}{5} = \frac{9}{15} \)
- \( \frac{14}{15} = \frac{14}{15} \)

5. **Compare the fractions**:
- \( \frac{9}{15} < \frac{10}{15} < \frac{14}{15} \)
- This means \( \frac{3}{5} < \frac{2}{3} < \frac{14}{15} \).

6. **Evaluate the options**:
- **Option 1**: \( \frac{2}{3} < \frac{3}{5} < \frac{14}{15} \) (False)
- **Option 2**: \( \frac{3}{5} < \frac{2}{3} < \frac{14}{15} \) (True)
- **Option 3**: \( \frac{14}{15} < \frac{3}{5} < \frac{2}{3} \) (False)
- **Option 4**: None of these (False)

7. **Conclusion**:
The correct statement is **Option 2**: \( \frac{3}{5} < \frac{2}{3} < \frac{14}{15} \).