Finding Compound Interest Rate

The problem asks us to find the annual rate of interest given the amounts accumulated after two different years under compound interest. We are given the amount after 2 years and the amount after 3 years.

Let P be the principal amount, and r be the annual rate of interest compounded annually.

The amount after n years at compound interest is given by the formula:

\(A = P \left(1 + \frac{r}{100}\right)^n\)

According to the problem:

• Amount after 2 years (\(A_2\)) = 800
• Amount after 3 years (\(A_3\)) = 840

Using the compound interest formula, we can write:

\(A_2 = P \left(1 + \frac{r}{100}\right)^2 = 800 \quad \ldots (1)\)

\(A_3 = P \left(1 + \frac{r}{100}\right)^3 = 840 \quad \ldots (2)\)

To find the rate of interest (r), we can divide equation (2) by equation (1):

\(\frac{A_3}{A_2} = \frac{P \left(1 + \frac{r}{100}\right)^3}{P \left(1 + \frac{r}{100}\right)^2}\)

\(\frac{840}{800} = \left(1 + \frac{r}{100}\right)^{3-2}\)

\(\frac{840}{800} = 1 + \frac{r}{100}\)

Now, we can solve for \(1 + \frac{r}{100}\):

\(\frac{84}{80} = 1 + \frac{r}{100}\)

\(\frac{21}{20} = 1 + \frac{r}{100}\)

Next, isolate \(\frac{r}{100}\):

\(\frac{r}{100} = \frac{21}{20} - 1\)

\(\frac{r}{100} = \frac{21 - 20}{20}\)

\(\frac{r}{100} = \frac{1}{20}\)

Finally, solve for r:

\(r = \frac{1}{20} \times 100\)

\(r = 5\)

So, the rate of interest is 5% per annum.

Alternatively, we can think of the amount at the end of the 2nd year (800) as the principal for the 3rd year. The interest earned during the 3rd year is the difference between the amount after 3 years and the amount after 2 years.

Interest earned in the 3rd year = \(A_3 - A_2 = 840 - 800 = 40\)

This interest of 40 is earned on the amount at the end of year 2, which is 800, over one year.

Using the simple interest formula for one year (which is equivalent to compound interest for a single year):

Interest = Principal × Rate × Time / 100

\(40 = 800 \times r \times \frac{1}{100}\)

\(40 = 8 \times r\)

\(r = \frac{40}{8}\)

\(r = 5\)

The rate of interest is 5%.

Let's check the options provided:

1. 5%
2. 7%
3. 6%
4. 4%

Our calculated rate of 5% matches the first option.

Revision Table - Compound Interest Rate Calculation

Additional Information - Compound Interest Basics

Compound interest is a powerful concept in finance because it allows your money to grow at an accelerating rate. Here are some key points:

• Compounding Period: Interest can be compounded annually, semi-annually, quarterly, monthly, or even daily. The more frequent the compounding, the faster the growth, assuming the same nominal annual rate. In this problem, it is implied to be annual compounding.
• Difference from Simple Interest: Simple interest is calculated only on the initial principal. Compound interest calculates interest on the principal plus all accumulated interest.
• Effect of Rate and Time: Both the rate of interest and the duration of investment significantly impact the final amount under compound interest. A higher rate or a longer period leads to substantially greater growth.
• Real-World Applications: Compound interest is fundamental to understanding savings accounts, loans, mortgages, and investments.

Understanding how the amount grows from one year to the next is crucial for solving problems like this one, where the interest for a specific period is calculated on the amount at the beginning of that period.