Dhoni invested Rs 22000 at 12% simple interest per annum and Rs 15000 at 10% compound interest per annum, then find the total amount earned by him at the end of 2 years.
Rs 45430
Calculating Total Amount Earned from Investments: Simple and Compound Interest
This problem involves two different types of interest calculations: simple interest and compound interest. We need to calculate the final amount for each investment separately and then sum them up to find the total amount earned at the end of 2 years.
Investment 1: Simple Interest
For the first investment, a principal amount is invested at a simple interest rate for a specific period. The formula for simple interest (SI) is:
\( \text{SI} = \frac{P \times R \times T}{100} \)
Where:
- \(P\) is the Principal amount
- \(R\) is the Rate of interest per annum
- \(T\) is the Time period in years
Given for the first investment:
- Principal (\(P_1\)) = Rs 22000
- Rate (\(R_1\)) = 12% per annum
- Time (\(T_1\)) = 2 years
Let's calculate the simple interest earned:
\( \text{SI}_1 = \frac{22000 \times 12 \times 2}{100} \)
\( \text{SI}_1 = 220 \times 12 \times 2 \)
\( \text{SI}_1 = 220 \times 24 \)
\( \text{SI}_1 = 5280 \)
The amount at the end of 2 years for the first investment (\(A_1\)) is the principal plus the simple interest:
\( A_1 = P_1 + \text{SI}_1 \)
\( A_1 = 22000 + 5280 \)
\( A_1 = 27280 \)
So, the amount from the simple interest investment after 2 years is Rs 27280.
Investment 2: Compound Interest
For the second investment, a principal amount is invested at a compound interest rate. The formula for the amount (\(A\)) with compound interest compounded annually is:
\( A = P \left(1 + \frac{R}{100}\right)^T \)
Where:
- \(P\) is the Principal amount
- \(R\) is the Rate of interest per annum
- \(T\) is the Time period in years
Given for the second investment:
- Principal (\(P_2\)) = Rs 15000
- Rate (\(R_2\)) = 10% per annum
- Time (\(T_2\)) = 2 years
Let's calculate the amount at the end of 2 years (\(A_2\)):
\( A_2 = 15000 \left(1 + \frac{10}{100}\right)^2 \)
\( A_2 = 15000 \left(1 + 0.1\right)^2 \)
\( A_2 = 15000 \left(1.1\right)^2 \)
\( A_2 = 15000 \times 1.21 \)
To calculate \(15000 \times 1.21\):
\( 15000 \times 1.21 = 150 \times 121 \)
\( 150 \times 121 = 150 \times (100 + 20 + 1) \)
\( 150 \times 100 = 15000 \)
\( 150 \times 20 = 3000 \)
\( 150 \times 1 = 150 \)
\( 15000 + 3000 + 150 = 18150 \)
So,
\( A_2 = 18150 \)
The amount from the compound interest investment after 2 years is Rs 18150.
Total Amount Earned
The total amount earned by him at the end of 2 years is the sum of the amounts from both investments.
Total Amount = \(A_1 + A_2\)
Total Amount = \(27280 + 18150\)
Total Amount = \(45430\)
The total amount earned by Dhoni at the end of 2 years is Rs 45430.
Summary of Calculations
| Investment Type | Principal | Rate | Time | Formula Used | Final Amount |
|---|---|---|---|---|---|
| Simple Interest | Rs 22000 | 12% | 2 years | \(A = P + \frac{PRT}{100}\) | Rs 27280 |
| Compound Interest | Rs 15000 | 10% | 2 years | \(A = P \left(1 + \frac{R}{100}\right)^T\) | Rs 18150 |
| Total | Sum of amounts | Rs 45430 |
Therefore, the total amount earned by Dhoni at the end of 2 years is Rs 45430.
Revision Table: Interest Formulas
| Type of Interest | Simple Interest (SI) | Compound Interest (CI) |
|---|---|---|
| Interest Formula | \( \text{SI} = \frac{P \times R \times T}{100} \) | \( \text{CI} = A - P \) or \( \text{CI} = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right] \) |
| Amount Formula | \( A = P + \text{SI} = P + \frac{P \times R \times T}{100} = P\left(1 + \frac{RT}{100}\right) \) | \( A = P \left(1 + \frac{R}{100}\right)^T \) |
| Key Feature | Interest is calculated only on the initial principal amount. | Interest is calculated on the principal amount and also on the accumulated interest of previous periods. |
Understanding these formulas is crucial for solving problems involving simple and compound interest investments.
Additional Information on Simple and Compound Interest
Simple interest and compound interest are two fundamental concepts in finance related to the calculation of interest on loans or investments.
- Simple Interest: In simple interest, the interest is calculated only on the original principal amount. The interest rate is applied solely to the initial investment throughout the entire investment period. This means the interest earned does not contribute to the base for calculating future interest. Simple interest is typically used for short-term loans or specific types of bonds.
- Compound Interest: Compound interest is often referred to as "interest on interest." In compound interest, the interest earned in each period is added to the principal amount for the calculation of interest in the next period. This compounding effect leads to exponential growth over time. The frequency of compounding (annually, semi-annually, quarterly, etc.) affects the total amount earned. In this problem, compounding is annual. Compound interest is widely used in savings accounts, fixed deposits, and most long-term investments and loans.
The choice between simple and compound interest significantly impacts the total return on an investment or the total cost of a loan, especially over longer periods.
Important Questions from Interest
- The difference between compound interest and simple interest on an amount of Rs.25,000 for 2 years is Rs.250. What is the rate of interest per annum? (In %)
- Increase in Debt-Equity ratio in a firm __________ to equity holders.
The difference between the compound interest and simple interest on a certain amount at 5%per annum for 2 years is Rs. 15.The principal amount is equal to :
The difference between the compound interest and the simple interest on a sum of Rs. 8,000 for 2 years at the rate of 5% p.a. is:
Three years ago, the value of a flat was Rs. 65,00,000. Its value depreciated at the rate of 5%, 4% and 3% at the end of the first, the second and the third year, respectively. What is its present value?
