Which of the following fractions does not lie between `(5)/(6) and (8)/(15)` ?

Options

A. `(2)/(3)`
B. `(3)/(4)`
C. `(4)/(5)`
D. `(6)/(7)`

Answer

To determine which of the following fractions does not lie between \( \frac{5}{6} \) and \( \frac{8}{15} \), we will first convert these fractions into decimal form and then compare them with the decimal forms of the given fractions.

### Step-by-Step Solution:

1. **Convert \( \frac{5}{6} \) to Decimal:**
\[ \frac{5}{6} = 0.8333\ldots \approx 0.83 \]

2. **Convert \( \frac{8}{15} \) to Decimal:**
\[ \frac{8}{15} = 0.5333\ldots \approx 0.53 \]

3. **Identify the Range:**
The range we are considering is between \( 0.53 \) and \( 0.83 \).

4. **Convert the Given Fractions to Decimal:**
- For \( \frac{2}{3} \):
\[ \frac{2}{3} = 0.6666\ldots \approx 0.67 \]
- For \( \frac{3}{4} \):
\[ \frac{3}{4} = 0.75 \]
- For \( \frac{4}{5} \):
\[ \frac{4}{5} = 0.8 \]
- For \( \frac{6}{7} \):
\[ \frac{6}{7} = 0.8571\ldots \approx 0.86 \]

5. **Compare Each Decimal with the Range:**
- \( 0.67 \) (from \( \frac{2}{3} \)) is between \( 0.53 \) and \( 0.83 \).
- \( 0.75 \) (from \( \frac{3}{4} \)) is between \( 0.53 \) and \( 0.83 \).
- \( 0.8 \) (from \( \frac{4}{5} \)) is between \( 0.53 \) and \( 0.83 \).
- \( 0.86 \) (from \( \frac{6}{7} \)) is **not** between \( 0.53 \) and \( 0.83 \).

6. **Conclusion:**
The fraction that does not lie between \( \frac{5}{6} \) and \( \frac{8}{15} \) is \( \frac{6}{7} \).

### Final Answer:
The fraction that does not lie between \( \frac{5}{6} \) and \( \frac{8}{15} \) is \( \frac{6}{7} \).