Calculating the Rate of Interest from Compound and Simple Interest

The problem asks us to find the annual rate of interest ('r' per cent) given the compound interest (CI) and simple interest (SI) on the same principal sum for the same time period of 2 years.

We are given:

• Simple Interest (SI) for 2 years = Rs. 7800
• Compound Interest (CI) for 2 years = Rs. 8385
• Time (n) = 2 years
• Rate of Interest = r% per annum

Let the principal sum be P.

Understanding Simple Interest for 2 Years

Simple interest is calculated only on the principal amount. For a fixed rate and principal, the simple interest is the same for each year.

Total SI for 2 years = SI for 1st year + SI for 2nd year

Since the SI is the same for each year:

SI for 1 year = \(\frac{\text{Total SI for 2 years}}{2}\)

SI for 1 year = \(\frac{7800}{2} = 3900\)

So, the simple interest earned in the first year is Rs. 3900, and in the second year is also Rs. 3900.

Understanding the Difference Between CI and SI for 2 Years

The compound interest for 2 years is Rs. 8385, and the simple interest for 2 years is Rs. 7800.

The difference between CI and SI for 2 years is:

CI - SI = \(8385 - 7800 = 585\)

This difference arises because, in compound interest, the interest earned in the first year also earns interest in the second year. The simple interest for the first year is Rs. 3900. This Rs. 3900 earns interest in the second year under compound interest. The amount of interest earned on this Rs. 3900 in the second year is exactly the difference between the total CI and total SI for 2 years.

Calculating the Rate 'r'

The difference (Rs. 585) is the simple interest on the first year's simple interest (Rs. 3900) for one year at the rate 'r' per cent per annum.

Using the simple interest formula: \(SI = \frac{P \times r \times n}{100}\)

Here, SI = 585, P = 3900 (which is the interest from the first year), n = 1 year, and the rate is r%.

So, we have:

\(585 = \frac{3900 \times r \times 1}{100}\)

Now, we solve for r:

\(585 = 39 \times r\)

\(r = \frac{585}{39}\)

To calculate \(\frac{585}{39}\):

\(585 \div 39\)

\(585 = 390 + 195\)

\(\frac{585}{39} = \frac{390 + 195}{39} = \frac{390}{39} + \frac{195}{39} = 10 + 5 = 15\)

So, \(r = 15\).

The value of 'r' is 15 per cent.

Final Answer

The rate of interest is 15 per cent per annum.

Revision Table: Key Interest Concepts (2 Years)

Additional Information: Solving Using CI - SI Difference Formula

We can also use the direct formula for the difference between CI and SI for 2 years:

\(CI - SI = P \left(\frac{r}{100}\right)^2\)

We know CI - SI = 585. We need to find P first using the SI information.

\(SI = \frac{P \times r \times 2}{100}\)

\(7800 = \frac{P \times r \times 2}{100}\)

\(780000 = 2Pr\)

\(390000 = Pr\)

\(P = \frac{390000}{r}\)

Now substitute this into the difference formula:

\(585 = \frac{390000}{r} \times \left(\frac{r}{100}\right)^2\)

\(585 = \frac{390000}{r} \times \frac{r^2}{10000}\)

\(585 = \frac{390000 \times r}{10000}\)

\(585 = 39r\)

\(r = \frac{585}{39}\)

\(r = 15\)

This method gives the same result, confirming our answer. The first method, using the interest on the first year's SI, is often quicker when the SI for 2 years is given, as it directly uses the first year's SI amount.