Which of the following statements (s) is are TRUE ?
I. `(3)/(110) lt (9)/(308) lt (7)/(225)`
II. `99 (1)/(7) + 99 (2)/(7) + 99 (3)/(7) +...99 (6)/(7) = 279`

Options

A. Only I
B. Only II
C. Neither I nor II
D. Both I and II

Answer

To determine which of the statements is true, we will analyze each statement step by step.

### Statement I: `(3)/(110) < (9)/(308) < (7)/(225)`

1. **Compare `(3)/(110)` and `(9)/(308)`**:
- To compare these two fractions, we can cross-multiply:
\[ 3 \times 308 \quad \text{and} \quad 9 \times 110 \]
- Calculate:
\[ 3 \times 308 = 924 \]
\[ 9 \times 110 = 990 \]
- Since \(924 < 990\), we have:
\[ \frac{3}{110} < \frac{9}{308} \]

2. **Compare `(9)/(308)` and `(7)/(225)`**:
- Again, we will cross-multiply:
\[ 9 \times 225 \quad \text{and} \quad 7 \times 308 \]
- Calculate:
\[ 9 \times 225 = 2025 \]
\[ 7 \times 308 = 2156 \]
- Since \(2025 < 2156\), we have:
\[ \frac{9}{308} < \frac{7}{225} \]

3. **Conclusion for Statement I**:
- Since both comparisons hold true, we conclude that:
\[ \frac{3}{110} < \frac{9}{308} < \frac{7}{225} \]
- Therefore, Statement I is **True**.

### Statement II: `99 (1)/(7) + 99 (2)/(7) + 99 (3)/(7) + ... + 99 (6)/(7) = 279`

1. **Calculate the left side**:
- The expression can be simplified as follows:
\[ 99 \left(\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7}\right) \]
- Factor out \(99\):
\[ 99 \cdot \left(\frac{1 + 2 + 3 + 4 + 5 + 6}{7}\right) \]
- Calculate the sum of the integers from 1 to 6:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]
- Substitute back:
\[ 99 \cdot \left(\frac{21}{7}\right) = 99 \cdot 3 = 297 \]

2. **Conclusion for Statement II**:
- The left side equals \(297\), not \(279\).
- Therefore, Statement II is **False**.

### Final Conclusion:
- Statement I is **True**.
- Statement II is **False**.