Which of the following statemetns is/are true
I. `(1+1/2) (1+1/3) (1+1/4)………….(1+1/998) gt 497`
II. `14 3/4+5 1/4 -2 1/2 gt 11 1/8 +12 3/8-7 1/4`

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:
\[ (1 + \frac{1}{2})(1 + \frac{1}{3})(1 + \frac{1}{4}) \ldots (1 + \frac{1}{998}) > 497 \]

1. **Simplifying the terms**: Each term can be rewritten as:
\[ 1 + \frac{1}{n} = \frac{n + 1}{n} \]
Therefore, we can express the product as:
\[ (1 + \frac{1}{2})(1 + \frac{1}{3})(1 + \frac{1}{4}) \ldots (1 + \frac{1}{998}) = \frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdots \frac{999}{998} \]

2. **Canceling terms**: Notice that in the product, all intermediate terms will cancel out:
\[ = \frac{999}{2} \]

3. **Calculating the value**: Now, we compute:
\[ \frac{999}{2} = 499.5 \]

4. **Comparing with 497**: Since \(499.5 > 497\), the first statement is true.

### Statement II:
\[ 14 \frac{3}{4} + 5 \frac{1}{4} - 2 \frac{1}{2} > 11 \frac{1}{8} + 12 \frac{3}{8} - 7 \frac{1}{4} \]

1. **Converting mixed numbers to improper fractions**:
- \(14 \frac{3}{4} = \frac{59}{4}\)
- \(5 \frac{1}{4} = \frac{21}{4}\)
- \(2 \frac{1}{2} = \frac{5}{2} = \frac{10}{4}\)
- \(11 \frac{1}{8} = \frac{89}{8}\)
- \(12 \frac{3}{8} = \frac{99}{8}\)
- \(7 \frac{1}{4} = \frac{29}{4}\)

2. **Combining the left side**:
\[ \frac{59}{4} + \frac{21}{4} - \frac{10}{4} = \frac{59 + 21 - 10}{4} = \frac{70}{4} = 17.5 \]

3. **Combining the right side**:
\[ \frac{89}{8} + \frac{99}{8} - \frac{29}{4} = \frac{89 + 99 - 58}{8} = \frac{130}{8} = 16.25 \]

4. **Comparing the two sides**: Now we compare:
\[ 17.5 > 16.25 \]
Therefore, the second statement is also true.

### Conclusion:
Both statements are true.

### Final Answer:
Both statements I and II are true.

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