If the compound interest received on a certain amount in the first year is Rs. 1,440. What will be the compound interest for the second year on the same principal at a 10% rate of interest?
Rs. 1,584
Understanding Compound Interest for the Second Year
The question asks us to find the compound interest earned in the second year, given the compound interest for the first year and the rate of interest.
In compound interest, the interest for each period is calculated on the principal amount plus any interest accumulated from previous periods. This is different from simple interest, where interest is only calculated on the initial principal.
Calculating the Principal Amount
For the first year, compound interest is the same as simple interest because there is no previously accumulated interest. We are given that the compound interest for the first year is Rs. 1,440 at a 10% rate.
Let the principal amount be \(P\). The interest for the first year (\(I_1\)) is given by the formula:
\(I_1 = P \times \text{Rate}\)
We have \(I_1 = 1440\) and Rate = 10% or 0.10.
So, we can write the equation:
\(1440 = P \times 0.10\)
To find the principal \(P\), we rearrange the equation:
\(P = \frac{1440}{0.10}\)
\(P = 14400\)
The principal amount is Rs. 14,400.
Calculating Compound Interest for the Second Year
The compound interest for the second year is calculated on the amount accumulated at the end of the first year. The amount at the end of the first year is the initial principal plus the interest earned in the first year.
- Principal (\(P\)) = Rs. 14,400
- Interest for the first year (\(I_1\)) = Rs. 1,440
Amount at the end of the first year = \(P + I_1\)
Amount at the end of the first year = \(14400 + 1440 = 15840\)
Now, the interest for the second year (\(I_2\)) is calculated on this amount (Rs. 15,840) at the same rate of 10%.
\(I_2 = (\text{Amount at end of 1st year}) \times \text{Rate}\)
\(I_2 = 15840 \times 0.10\)
\(I_2 = 1584\)
The compound interest for the second year is Rs. 1,584.
Summary of Calculations
| Given: | Compound Interest for 1st year | Rs. 1,440 |
| Given: | Rate of Interest | 10% per annum |
| Step 1: | Calculate Principal (\(P\)) | \(P = \frac{I_1}{\text{Rate}} = \frac{1440}{0.10} = 14400\) |
| Step 2: | Calculate Amount at end of 1st year | \(P + I_1 = 14400 + 1440 = 15840\) |
| Step 3: | Calculate Interest for 2nd year (\(I_2\)) | \(I_2 = (\text{Amount}) \times \text{Rate} = 15840 \times 0.10 = 1584\) |
Thus, the compound interest for the second year will be Rs. 1,584.
Revision Table: Key Compound Interest Concepts
| Concept | Description | Formula (A = Amount, P = Principal, r = rate, n = time in years) |
|---|---|---|
| Compound Interest | Interest calculated on the initial principal and also on the accumulated interest from previous periods. | \(A = P(1 + r)^n\) CI = \(A - P\) |
| Simple Interest | Interest calculated only on the initial principal amount. | \(SI = \frac{P \times r \times n}{100}\) (if rate is in %) |
| Interest in 1st Year (Compound) | Same as simple interest on the principal. | \(I_1 = P \times r\) |
| Interest in 2nd Year (Compound) | Calculated on the amount at the end of the 1st year (\(P + I_1\)). | \(I_2 = (P + I_1) \times r\) |
Additional Information on Compound Interest Calculation
Compound interest is a powerful concept often used in finance for investments and loans. The frequency of compounding (annually, semi-annually, quarterly, etc.) affects the total interest earned or paid. In this problem, compounding is annual as we are given yearly rates and interest for specific years.
Understanding the difference between interest earned in a specific period (like the second year) and the total compound interest accumulated over multiple years is important. The question specifically asked for the interest in the second year, not the total interest over two years.
The general formula for the amount at the end of 'n' years with annual compounding at rate 'r' is \(A = P(1+r)^n\). The compound interest for the nth year can be found by taking the difference between the amount at the end of n years and the amount at the end of (n-1) years.
For example, Amount at end of 2 years = \(P(1+r)^2\). Amount at end of 1 year = \(P(1+r)^1\). Interest in the 2nd year = Amount at end of 2 years - Amount at end of 1 year = \(P(1+r)^2 - P(1+r)\) = \(P(1+r)((1+r) - 1)\) = \(P(1+r)r\). Notice that \(P(1+r)\) is the amount at the end of year 1, which is \(P+I_1\), so the formula \(I_2 = (P+I_1)r\) used in the solution is consistent with the general formula.
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