If `P = (96)/(95 xx 97), Q = (97)/(96 xx 98) and R = (1)/(97)`, then which of the following is TRUE ?

Options

A. `P lt Q lt R`
B. `R lt Q lt P`
C. `Q lt P lt R`
D. `R lt P lt Q`

Answer

To solve the problem, we need to analyze the values of \( P \), \( Q \), and \( R \) given by:

- \( P = \frac{96}{95 \times 97} \)
- \( Q = \frac{97}{96 \times 98} \)
- \( R = \frac{1}{97} \)

### Step 1: Calculate \( P \)

We start with \( P \):

\[ P = \frac{96}{95 \times 97} \]

This can be simplified as follows:

\[ P = \frac{96}{95 \times 97} = \frac{96}{95 \times 97} = \frac{96}{95 \times 97} \]

### Step 2: Calculate \( Q \)

Next, we calculate \( Q \):

\[ Q = \frac{97}{96 \times 98} \]

This can also be simplified:

\[ Q = \frac{97}{96 \times 98} = \frac{97}{96 \times 98} \]

### Step 3: Calculate \( R \)

Now, we look at \( R \):

\[ R = \frac{1}{97} \]

### Step 4: Compare \( P \), \( Q \), and \( R \)

To compare these values, we can express them in a common form.

1. **Finding a common denominator**:
- The denominators are \( 95 \times 97 \), \( 96 \times 98 \), and \( 97 \).
- The least common multiple of these denominators will help us compare the fractions.

2. **Cross-multiplication**:
- To compare \( P \) and \( Q \):
\[ P < Q \text{ if } 96 \times 96 \times 98 < 97 \times 95 \times 97 \]
- To compare \( Q \) and \( R \):
\[ Q < R \text{ if } 97 \times 97 < 96 \times 98 \times 97 \]
- To compare \( P \) and \( R \):
\[ P < R \text{ if } 96 < 95 \times 97 \]

### Conclusion

After performing the comparisons, we find that:

- \( R < Q < P \)

Thus, the correct option is:

**Option 1: \( R < Q < P \)**