How much money did X invest ?
(i) An increase in the rate in interest from `4(7)/(8)%` to `5(1)/(8)%` per annum increases his yearly income by Rs 25.
(ii) The sum invested gets doubled, when invested at `8%` p.a. for `12(1)/(2)` years.

Answer

To determine how much money X invested, we will analyze the two statements provided and see if either one or both are sufficient to answer the question.

### Step 1: Analyze Statement (i)
Statement (i) states that an increase in the interest rate from \(4\frac{7}{8}\%\) to \(5\frac{1}{8}\%\) increases X's yearly income by Rs 25.

1. Convert the mixed fractions to improper fractions:
- \(4\frac{7}{8} = \frac{39}{8}\%\)
- \(5\frac{1}{8} = \frac{41}{8}\%\)

2. Find the difference in interest rates:
\[ \text{Difference} = \frac{41}{8} - \frac{39}{8} = \frac{2}{8} = \frac{1}{4}\% \]

3. The increase in yearly income of Rs 25 corresponds to this difference in interest rates. Therefore, we can set up the equation:
\[ \frac{1}{4}\% \text{ of } X = 25 \]

4. To find X, we rearrange the equation:
\[ X = \frac{25 \times 100}{\frac{1}{4}} = 25 \times 400 = 10000 \]

Thus, from statement (i), we can determine that X invested Rs 10,000.

### Step 2: Analyze Statement (ii)
Statement (ii) states that the sum invested gets doubled when invested at \(8\%\) per annum for \(12\frac{1}{2}\) years.

1. Convert \(12\frac{1}{2}\) years to an improper fraction:
- \(12\frac{1}{2} = \frac{25}{2}\) years

2. The formula for simple interest is:
\[ \text{Interest} = \frac{P \times R \times T}{100} \]
where \(P\) is the principal amount, \(R\) is the rate of interest, and \(T\) is the time in years.

3. Since the sum doubles, the interest earned is equal to the principal \(P\):
\[ P = \frac{P \times 8 \times \frac{25}{2}}{100} \]

4. Simplifying this gives:
\[ P = \frac{200P}{100} \implies P = 2P \]
This leads to a contradiction, indicating that we cannot determine the value of \(P\) from this statement alone.

### Conclusion
From the analysis:
- Statement (i) is sufficient to determine how much money X invested.
- Statement (ii) is not sufficient.

Thus, the answer is that **only statement (i) is sufficient** to answer the question.

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