₹4,800 amounts to ₹6,000 after 4 years if compounded annually. Find out the amount after 12 years at the same rate of interest. |

Options

A. ₹ 1,000
B. ₹9,375
C. ₹9,500
D. ₹9,000

Answer

To solve the problem step by step, we need to find the rate of interest first and then use it to calculate the amount after 12 years.

### Step 1: Identify the given values
- Principal (P) = ₹4,800
- Amount after 4 years (A) = ₹6,000
- Time (T) = 4 years

### Step 2: Use the formula for compound interest
The formula for the amount \( A \) in compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^T \]
Where \( r \) is the rate of interest.

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula, we get:
\[ 6000 = 4800 \left(1 + \frac{r}{100}\right)^4 \]

### Step 4: Divide both sides by 4800
\[ \frac{6000}{4800} = \left(1 + \frac{r}{100}\right)^4 \]
Calculating the left side:
\[ \frac{6000}{4800} = 1.25 \]
So, we have:
\[ 1.25 = \left(1 + \frac{r}{100}\right)^4 \]

### Step 5: Take the fourth root of both sides
To isolate \( 1 + \frac{r}{100} \), we take the fourth root:
\[ 1 + \frac{r}{100} = (1.25)^{\frac{1}{4}} \]

### Step 6: Calculate \( (1.25)^{\frac{1}{4}} \)
Using a calculator, we find:
\[ (1.25)^{\frac{1}{4}} \approx 1.0574 \]
So, we have:
\[ 1 + \frac{r}{100} \approx 1.0574 \]

### Step 7: Solve for \( r \)
Subtract 1 from both sides:
\[ \frac{r}{100} \approx 0.0574 \]
Multiply by 100 to find \( r \):
\[ r \approx 5.74\% \]

### Step 8: Calculate the amount after 12 years
Now we need to find the amount after 12 years using the same rate of interest. We use the formula again:
\[ A = P \left(1 + \frac{r}{100}\right)^T \]
Where \( T = 12 \) years:
\[ A = 4800 \left(1 + \frac{5.74}{100}\right)^{12} \]
This simplifies to:
\[ A = 4800 \left(1.0574\right)^{12} \]

### Step 9: Calculate \( (1.0574)^{12} \)
Using a calculator:
\[ (1.0574)^{12} \approx 1.7493 \]
So, we have:
\[ A \approx 4800 \times 1.7493 \approx 8396.04 \]

### Step 10: Round off the final amount
The final amount after 12 years is approximately:
\[ A \approx ₹8396.04 \]

### Final Answer
The amount after 12 years at the same rate of interest is approximately ₹8396.

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