Question

A sum of Rs. 2400 deposited at CI, doubled after 5 years. After 20 years it will become : (a) Rs. 24000 (b) Rs. 38400 (c) Rs. 19200 (d) can't be determine

Short answer

(a) Rs. 33700 (b) Rs. 38400 (c) Rs. 41000 (d) Rs. 45800 Answer: (b) Rs. 38400

Step-by-step solution

Determine the time taken to double the principal

Using the time taken to double the principal amount (5 years), we can write the compound interest formula: Amount = Principal * (1 + Rate/100)^Time Doubling the principal means the Amount is equal to twice the Principal; in this case, Amount = 2 * Principal.

Calculate the annual interest rate

Substitute the values in the given formula and solve for Rate: 2 * Principal = Principal * (1 + Rate/100)^Time 2 * 2400 = 2400 * (1 + Rate/100)^5 Divide by 2400: 2 = (1 + Rate/100)^5 Take the 5th root of both sides: (2)^(1/5) = 1 + Rate/100 Now, calculate the annual interest rate, Rate: Rate = [(2)^(1/5) - 1] * 100 ≈ 14.87%

Determine the amount after 20 years

Now that we have the annual interest rate, we can find the amount after 20 years using the compound interest formula: Amount = Principal * (1 + Rate/100)^Time Amount = 2400 * (1 + 14.87/100)^20 Amount = 2400 * (1.1487)^20 Amount ≈ 38440.26

Match the final amount with the given options

Comparing the final amount (Rs. 38440.26) with the given options, we find that the closest answer is: (b) Rs. 38400

Key concepts

Principal Amount

The principal amount is the starting sum of money that you invest or deposit. In the context of compound interest, it is the initial amount placed into an account before any interest is added. For example, in our situation, Rs. 2400 is the principal amount. This amount is the basis upon which calculation of interest is built. Each year, the interest is calculated based on this sum as well as any accumulated interest from previous years. This is why choosing the right principal is important. If you start with a larger principal, you can expect to earn more interest.
Consider the principal as the seed that grows over time as interest accumulates. The larger the seed, the bigger the potential growth.

Annual Interest Rate

The annual interest rate is a percentage that indicates how much interest will be added to your principal each year. It significantly affects how quickly your money grows when calculating compound interest. The exercise determined an annual interest rate of approximately 14.87% using the doubling time method.
To understand its impact, a higher annual interest rate means your principal will grow faster compared to a lower rate. It’s essential to consider both the rate and the risk associated with investments when evaluating where to put your money.

• Higher interest rates lead to faster compound growth.
• Riskier investments often offer higher rates to attract investors.

Understanding the annual interest rate helps you make informed decisions about saving and investing money. It allows you to predict how much your money could grow over time.

Doubling Time

Doubling time refers to the duration it takes for an investment to double in value at a given interest rate. In compound interest scenarios, knowing how long it takes for your initial investment to double is crucial. Fortunately, the exercise provides this as 5 years.
With the compound interest formula, you can calculate doubling time yourself if not provided, by rearranging and using log functions. However, for more straightforward calculations, the "Rule of 72" is often used. Divide 72 by your interest rate to get an approximation of doubling time in years. For example, an annual interest rate of around 14.87% roughly coincides with a 5-year doubling time, according to 72/14.87.

• The shorter the doubling time, the better for growth of an investment.
• It offers a quick measure of how beneficial an interest rate might be for investors.

Interest Calculation

Interest calculation in the context of compound interest involves repeatedly applying the annual interest rate to the principal, and any accumulated interest. This iterative process is summed up in the compound interest formula:\[ \text{Amount} = \text{Principal} \times \left(1 + \frac{\text{Rate}}{100}\right)^{\text{Time}} \]Using this formula, you can calculate how much your principal will grow over multiple years. The compound interest method is powerful because it allows interest to accumulate on previously accrued interest, not just the initial principal. In our exercise:

• The principal is Rs. 2400.
• Using the interest rate of 14.87%, we determine future value over periods like 5 or 20 years.

Calculations provide insights into how much investment can grow based on time and rate, showcasing the exponential nature of compound interest.