A sum of `Rs. 2000` amounts to `Rs. 4000` in two years at compound Interest. In how many years will the same amount become `Rs. 8000`?

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem step by step, we need to determine how long it will take for a principal amount of Rs. 2000 to grow to Rs. 8000 at the same compound interest rate that allowed it to grow to Rs. 4000 in 2 years.

### Step 1: Understand the relationship between the amounts
We know that Rs. 2000 amounts to Rs. 4000 in 2 years. This means that the amount has doubled in 2 years.

### Step 2: Determine the growth factor
The growth factor (A) can be calculated as follows:
- Initial amount = Rs. 2000
- Amount after 2 years = Rs. 4000

The growth factor (A) is given by:
\[ A = \frac{\text{Final Amount}}{\text{Initial Amount}} = \frac{4000}{2000} = 2 \]

### Step 3: Set up the equation for the new amount
Now, we need to find out how many years (T2) it will take for Rs. 2000 to become Rs. 8000. The growth factor for this situation (B) is:
\[ B = \frac{8000}{2000} = 4 \]

### Step 4: Use the relationship between the amounts and time
We can use the formula that relates the amounts and time:
\[ A^{\frac{1}{T1}} = B^{\frac{1}{T2}} \]
Where:
- \( A = 2 \) (for the first scenario)
- \( B = 4 \) (for the second scenario)
- \( T1 = 2 \) (the time for the first scenario)

Substituting the values we have:
\[ 2^{\frac{1}{2}} = 4^{\frac{1}{T2}} \]

### Step 5: Simplify the equation
We know that \( 4 \) can be expressed as \( 2^2 \):
\[ 2^{\frac{1}{2}} = (2^2)^{\frac{1}{T2}} \]
This simplifies to:
\[ 2^{\frac{1}{2}} = 2^{\frac{2}{T2}} \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ \frac{1}{2} = \frac{2}{T2} \]

### Step 7: Solve for T2
Cross-multiplying gives:
\[ T2 = 2 \times 2 = 4 \]

### Conclusion
Therefore, it will take **4 years** for the amount to grow from Rs. 2000 to Rs. 8000 at the same compound interest rate.

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