Question

A sum of Rs. 400 would become Rs. 441 after 2 years at \(r \%\) compound interest, find the value of ' \(r^{\prime}\) : (a) \(10 \%\) (b) \(5 \%\) (c) \(15 \%\) (d) \(20 \%\)

Short answer

Answer: The rate of compound interest is 5%.

Step-by-step solution

Write down the given values and the formula for compound interest.

We are given: - Principal amount (P) = Rs. 400 - Final amount (A) = Rs. 441 - Number of years (n) = 2 - Rate of interest (r) = ? The compound interest formula is: \(A=P(1+\frac{r}{100})^{n}\)

Plug in the given values into the formula.

Now, we will plug the given values into the compound interest formula and solve for r. \(441=400(1+\frac{r}{100})^{2}\)

Simplify the equation and isolate the variable 'r'.

To simplify the equation, we will first divide both sides by 400. \(\frac{441}{400}=(1+\frac{r}{100})^{2}\) Now, take the square root of both sides. \(\sqrt{\frac{441}{400}}=1+\frac{r}{100}\) Subtract 1 from both sides. \(\sqrt{\frac{441}{400}}-1=\frac{r}{100}\) Now, multiply both sides by 100 to isolate 'r'. \(100(\sqrt{\frac{441}{400}}-1)=r\)

Calculate the value of 'r'.

Now, we will calculate the value of 'r'. \(r=100(\sqrt{\frac{441}{400}}-1)\) \(r=100(\sqrt{1.1025}-1)\) \(r=100(1.05-1)\) \(r=100(0.05)\) \(r=5\)

Choose the correct option.

We found out that the rate of interest (r) is 5%. Therefore, the correct option is: (b) \(5 \%\)

Key concepts

Principal Amount

When calculating compound interest, the principal amount is the initial sum of money deposited or invested, before any interest is added. In our example, the principal amount is Rs. 400.

This is the figure on which the interest calculation begins, serving as the foundation for the compound interest formula. To understand the growth of an investment, recognizing the starting point, which is the principal amount, is crucial. As compound interest accrues, it is calculated on both the initial principal and the accumulated interest from previous periods, leading to a snowball effect on the investment growth.

Rate of Interest

The rate of interest is a critical variable in the context of compound interest, as it determines the growth rate at which the principal amount accrues interest over time. It is usually expressed as a percentage and noted as 'r' in mathematical expressions.

The rate of interest influences how quickly your investment will grow and determines the amount of interest you'll earn each compounding period. In financial scenarios, this rate can vary widely depending on the type of investment or loan, the duration, and market conditions. In the case of our example, the exercise revolves around solving for this unknown rate to understand at what percentage the initial sum grows to reach the final amount.

Compound Interest Formula

The standard compound interest formula is a mathematical expression used to calculate the final amount (A) an initial principal amount (P) will grow to over a certain period (n), at a given rate of interest (r). The formula is expressed as:
\[A = P\left(1 + \frac{r}{100}\right)^n\]
By applying this formula, you can determine how much money you'll have after interest is compounded over the investment period. The formula takes into account that interest is calculated on both the principal and the previously accumulated interest, which makes it a powerful tool in financial planning and understanding the potential growth of savings or investments.

Solving for Interest Rate

Solving for the interest rate in compound interest problems involves finding the unknown rate at which the principal amount will compound to reach a target amount after a set period. In our exercise, we rearrange the compound interest formula to isolate 'r' and determine the rate that turns Rs. 400 into Rs. 441 in two years:

To isolate the rate, we first divide the final amount by the principal to determine how much the initial amount has grown. Then, to reverse the compounding effect, we take the nth root (in this case, the square root, since there are two compounding periods) of the growth factor. Subsequently, we isolate 'r' by subtracting one from the root result and finally multiplying by 100 to convert it to a percentage. Through these steps, we derive that the interest rate in our example is 5%.