Compound Interest Rate Calculation

The question asks us to find the rate of interest when a sum of money grows to a certain amount in two years and another amount in three years, with compound interest compounded annually.

We are given the following information:

• Amount after 2 years (A₂) = Rs. 1600
• Amount after 3 years (A₃) = Rs. 1700

In compound interest, the interest for any year is calculated on the amount accumulated till the end of the previous year. Therefore, the interest earned during the third year is calculated on the amount at the end of the second year (A₂).

Calculating Interest 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 for 3rd year = A₃ - A₂

Interest for 3rd year = Rs. 1700 - Rs. 1600

Interest for 3rd year = Rs. 100

Determining the Rate of Interest

The interest of Rs. 100 earned in the 3rd year is based on the principal amount at the beginning of the 3rd year, which is Rs. 1600 (the amount after 2 years). We can use the simple interest formula for this specific year to find the rate, as the principal (amount at the end of year 2) remains constant for calculating interest for the next period (year 3).

Let the rate of interest be r% per annum.

Interest = Principal × Rate × Time

Here, Principal = Amount after 2 years = Rs. 1600, Interest = Rs. 100, and Time = 1 year (the 3rd year).

So, we have:

$$100 = 1600 \times \frac{r}{100} \times 1$$

Now, we solve for r:

$$\frac{100}{1600} = \frac{r}{100}$$

Multiply both sides by 100:

$$r = \frac{100}{1600} \times 100$$

$$r = \frac{1}{16} \times 100$$

$$r = \frac{100}{16}$$

$$r = \frac{25}{4}$$

$$r = 6.25$$

Thus, the rate of interest is 6.25% per annum.

Summary of Calculation

• Amount after 2 years = Rs. 1600
• Amount after 3 years = Rs. 1700
• Interest earned in the 3rd year = Rs. 1700 - Rs. 1600 = Rs. 100
• This interest is on the amount at the end of year 2 (Rs. 1600).
• Rate of interest = $\left(\frac{\text{Interest}}{\text{Principal}}\right) \times 100 = \left(\frac{100}{1600}\right) \times 100 = \left(\frac{1}{16}\right) \times 100 = 6.25\%$

The calculated rate of interest is 6.25%.