Understanding Compound Interest and Doubling Time

This problem involves calculating the future amount based on compound interest principles, specifically focusing on the concept of doubling time.

We are given:

• Principal Amount ($P$): Rs. 950
• Time for the amount to double: 4 years
• Target Time: 12 years

Calculating the Amount After 12 Years

Compound interest means that the interest earned also earns interest over time. When an amount doubles in a certain period, it signifies a specific growth factor over that duration.

Let the principal amount be $P$. According to the question, the amount doubles itself after 4 years. This means the amount after 4 years is $2P$.

Using the compound interest formula, $A = P(1+r)^t$, where:

• $A$ is the future value of the investment/loan, including interest
• $P$ is the principal investment amount (the initial deposit or loan amount)
• $r$ is the annual interest rate (as a decimal)
• $t$ is the number of years the money is invested or borrowed for

From the information given, after 4 years ($t=4$), the amount $A_4 = 2P$. So, we can write:

$$2P = P(1+r)^4$$

Dividing both sides by $P$, we get:

$$2 = (1+r)^4$$

This equation tells us that the growth factor over 4 years is 2.

Now, we need to find the amount after 12 years ($A_{12}$). Using the same formula:

$$A_{12} = P(1+r)^{12}$$

We can rewrite the term $(1+r)^{12}$ using the information we have:

$$ (1+r)^{12} = (1+r)^{4 \times 3} = ((1+r)^4)^3 $$

Since we know that $(1+r)^4 = 2$, we can substitute this value:

$$ ((1+r)^4)^3 = (2)^3 = 8 $$

Now substitute this back into the formula for $A_{12}$:

$$ A_{12} = P \times 8 $$

We are given the principal amount $P = 950$. Plugging this value in:

$$ A_{12} = 950 \times 8 $$

$$ A_{12} = 7600 $$

Therefore, the amount after 12 years will be Rs. 7600.

Final Answer Summary

The initial deposit is Rs. 950. The investment doubles every 4 years. Over 12 years, there are $12 \div 4 = 3$ doubling periods.

• After 4 years: Rs. 950 * 2 = Rs. 1900
• After 8 years: Rs. 1900 * 2 = Rs. 3800
• After 12 years: Rs. 3800 * 2 = Rs. 7600

Alternatively, the amount after 12 years is $P \times 2^{(12/4)} = 950 \times 2^3 = 950 \times 8 = 7600$.