(i) If `(1)/(9!)+(1)/(10!)=(n)/(11!)`, find n.
(ii) If `(1)/(8!)+(1)/(9!)=(x)/(10!)`, find x.

Answer

To solve the given problems step by step, let's break down each part of the question.

### Part (i):
We need to find \( n \) such that:

\[ \frac{1}{9!} + \frac{1}{10!} = \frac{n}{11!} \]

**Step 1:** Rewrite \( \frac{1}{10!} \) in terms of \( 9! \).

\[ \frac{1}{10!} = \frac{1}{10 \times 9!} \]

**Step 2:** Substitute this back into the equation.

\[ \frac{1}{9!} + \frac{1}{10 \times 9!} = \frac{n}{11!} \]

**Step 3:** Factor out \( \frac{1}{9!} \) from the left-hand side.

\[ \frac{1}{9!} \left(1 + \frac{1}{10}\right) = \frac{n}{11!} \]

**Step 4:** Simplify the expression inside the parentheses.

\[ 1 + \frac{1}{10} = \frac{10}{10} + \frac{1}{10} = \frac{11}{10} \]

**Step 5:** Substitute this back into the equation.

\[ \frac{1}{9!} \cdot \frac{11}{10} = \frac{n}{11!} \]

**Step 6:** Rewrite \( \frac{1}{9!} \) in terms of \( 11! \).

\[ \frac{1}{9!} = \frac{11!}{11 \times 10 \times 9!} \]

**Step 7:** Substitute this into the equation.

\[ \frac{11}{10} \cdot \frac{11!}{11 \times 10 \times 9!} = \frac{n}{11!} \]

**Step 8:** Simplify the left-hand side.

\[ \frac{11!}{10 \times 10} = \frac{n}{11!} \]

**Step 9:** Cross-multiply to find \( n \).

\[ n = \frac{11! \cdot 11}{10} = 121 \]

Thus, the value of \( n \) is **121**.

### Part (ii):
We need to find \( x \) such that:

\[ \frac{1}{8!} + \frac{1}{9!} = \frac{x}{10!} \]

**Step 1:** Rewrite \( \frac{1}{9!} \) in terms of \( 8! \).

\[ \frac{1}{9!} = \frac{1}{9 \times 8!} \]

**Step 2:** Substitute this back into the equation.

\[ \frac{1}{8!} + \frac{1}{9 \times 8!} = \frac{x}{10!} \]

**Step 3:** Factor out \( \frac{1}{8!} \) from the left-hand side.

\[ \frac{1}{8!} \left(1 + \frac{1}{9}\right) = \frac{x}{10!} \]

**Step 4:** Simplify the expression inside the parentheses.

\[ 1 + \frac{1}{9} = \frac{9}{9} + \frac{1}{9} = \frac{10}{9} \]

**Step 5:** Substitute this back into the equation.

\[ \frac{1}{8!} \cdot \frac{10}{9} = \frac{x}{10!} \]

**Step 6:** Rewrite \( \frac{1}{8!} \) in terms of \( 10! \).

\[ \frac{1}{8!} = \frac{10!}{10 \times 9 \times 8!} \]

**Step 7:** Substitute this into the equation.

\[ \frac{10}{9} \cdot \frac{10!}{10 \times 9 \times 8!} = \frac{x}{10!} \]

**Step 8:** Simplify the left-hand side.

\[ \frac{10!}{9 \times 9} = \frac{x}{10!} \]

**Step 9:** Cross-multiply to find \( x \).

\[ x = 10 \times 10 = 100 \]

Thus, the value of \( x \) is **100**.

### Final Answers:
(i) \( n = 121 \)
(ii) \( x = 100 \)