A sum of Rs. 5760 amount to Rs 7200 in 4 years and to Rs x in 12 years at a certain rate per cent per annum, when the interest is compounded yearly in both the cases. What is the value of x ?

Options

A. 11500
B. 11250
C. 12500
D. 12250

Answer

To solve the problem step by step, we will use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{100}\right)^n \]

Where:
- \( A \) is the amount after time \( n \),
- \( P \) is the principal amount (initial sum),
- \( r \) is the rate of interest per annum,
- \( n \) is the number of years.

### Step 1: Find the rate of interest

Given:
- Principal \( P = 5760 \)
- Amount after 4 years \( A = 7200 \)
- Time \( n = 4 \)

Using the compound interest formula:

\[ 7200 = 5760 \left(1 + \frac{r}{100}\right)^4 \]

### Step 2: Simplify the equation

Divide both sides by 5760:

\[ \frac{7200}{5760} = \left(1 + \frac{r}{100}\right)^4 \]

Calculating the left side:

\[ \frac{7200}{5760} = 1.25 \]

So, we have:

\[ 1.25 = \left(1 + \frac{r}{100}\right)^4 \]

### Step 3: Take the fourth root

To solve for \( 1 + \frac{r}{100} \), we take the fourth root of both sides:

\[ 1 + \frac{r}{100} = (1.25)^{\frac{1}{4}} \]

Calculating \( (1.25)^{\frac{1}{4}} \):

Using a calculator or estimation, we find:

\[ (1.25)^{\frac{1}{4}} \approx 1.0574 \]

Thus:

\[ 1 + \frac{r}{100} \approx 1.0574 \]

### Step 4: Solve for \( r \)

Now, subtract 1 from both sides:

\[ \frac{r}{100} \approx 0.0574 \]

Multiplying by 100 to find \( r \):

\[ r \approx 5.74 \]

### Step 5: Calculate the amount after 12 years

Now we need to find the amount \( x \) after 12 years using the same formula:

\[ A = 5760 \left(1 + \frac{r}{100}\right)^{12} \]

Substituting \( r \approx 5.74 \):

\[ A = 5760 \left(1.0574\right)^{12} \]

Calculating \( \left(1.0574\right)^{12} \):

Using a calculator, we find:

\[ \left(1.0574\right)^{12} \approx 1.8009 \]

Now substituting back into the amount formula:

\[ A \approx 5760 \times 1.8009 \approx 10365.184 \]

### Step 6: Round to find \( x \)

Thus, the value of \( x \) is approximately:

\[ x \approx 10365.18 \]

### Final Answer

The value of \( x \) is approximately **Rs. 10365.18**.

---