Vipul and Manish invested the sum of Rs. 15000 and Rs. 20000 at the rate of 20 percent p.a and 30 percent p.a. respectively on compound interest (compounding annually). If time period is 3 years for both, then what will be the total compound interest earned by Vipul and Manish ?
Rs. 34860
Calculating Total Compound Interest for Multiple Investments
This problem involves calculating the compound interest earned by two different individuals, Vipul and Manish, on their respective investments over a period of 3 years. We need to find the compound interest for each person and then sum them up to get the total compound interest.
The formula for the Amount (A) under compound interest is given by:
\( A = P \left(1 + \frac{R}{100}\right)^T \)
Where:
- \( P \) is the Principal amount
- \( R \) is the Rate of interest per annum
- \( T \) is the Time period in years
The Compound Interest (CI) is then calculated as:
\( CI = A - P \)
Let's calculate the compound interest for Vipul and Manish separately.
Compound Interest Calculation for Vipul
Vipul's investment details:
- Principal (P) = Rs. 15000
- Rate (R) = 20% p.a.
- Time (T) = 3 years
Using the formula for the Amount:
\( A_{Vipul} = 15000 \left(1 + \frac{20}{100}\right)^3 \)
\( A_{Vipul} = 15000 \left(1 + 0.2\right)^3 \)
\( A_{Vipul} = 15000 \left(1.2\right)^3 \)
\( A_{Vipul} = 15000 \times 1.728 \)
\( A_{Vipul} = 25920 \)
Now, calculating the Compound Interest for Vipul:
\( CI_{Vipul} = A_{Vipul} - P_{Vipul} \)
\( CI_{Vipul} = 25920 - 15000 \)
\( CI_{Vipul} = 10920 \)
So, Vipul earned a compound interest of Rs. 10920.
Compound Interest Calculation for Manish
Manish's investment details:
- Principal (P) = Rs. 20000
- Rate (R) = 30% p.a.
- Time (T) = 3 years
Using the formula for the Amount:
\( A_{Manish} = 20000 \left(1 + \frac{30}{100}\right)^3 \)
\( A_{Manish} = 20000 \left(1 + 0.3\right)^3 \)
\( A_{Manish} = 20000 \left(1.3\right)^3 \)
\( A_{Manish} = 20000 \times 2.197 \)
\( A_{Manish} = 43940 \)
Now, calculating the Compound Interest for Manish:
\( CI_{Manish} = A_{Manish} - P_{Manish} \)
\( CI_{Manish} = 43940 - 20000 \)
\( CI_{Manish} = 23940 \)
So, Manish earned a compound interest of Rs. 23940.
Total Compound Interest Earned
The total compound interest earned by Vipul and Manish is the sum of their individual compound interests:
\( \text{Total CI} = CI_{Vipul} + CI_{Manish} \)
\( \text{Total CI} = 10920 + 23940 \)
\( \text{Total CI} = 34860 \)
The total compound interest earned by Vipul and Manish is Rs. 34860.
| Investor | Principal (P) | Rate (R) | Time (T) | Amount (A) | Compound Interest (CI) |
|---|---|---|---|---|---|
| Vipul | Rs. 15000 | 20% | 3 years | Rs. 25920 | Rs. 10920 |
| Manish | Rs. 20000 | 30% | 3 years | Rs. 43940 | Rs. 23940 |
Total Compound Interest = \( CI_{Vipul} + CI_{Manish} = 10920 + 23940 = 34860 \)
Thus, the total compound interest earned is Rs. 34860.
Revision Table: Compound Interest Concepts
| Term | Definition | Formula Snippet |
|---|---|---|
| Principal (P) | The initial amount invested or borrowed. | \( P \) |
| Rate (R) | The percentage at which interest is calculated per period. | \( R\% \) |
| Time (T) | The duration for which the money is invested or borrowed. | \( T \) years |
| Amount (A) | The total sum including principal and interest after a certain time. | \( A = P(1 + R/100)^T \) |
| Compound Interest (CI) | Interest calculated on the principal and the accumulated interest from previous periods. | \( CI = A - P \) |
Additional Information: Compounding Frequency
In this problem, the interest is compounded annually. Compound interest can also be calculated with different frequencies, such as semi-annually, quarterly, monthly, or even daily. When interest is compounded more frequently than annually, the formula is slightly adjusted:
\( A = P \left(1 + \frac{R}{100n}\right)^{nT} \)
Where:
- \( n \) is the number of times interest is compounded per year.
- For semi-annual compounding, \( n=2 \).
- For quarterly compounding, \( n=4 \).
- For monthly compounding, \( n=12 \).
A higher compounding frequency generally leads to a higher amount of compound interest earned over the same time period and interest rate, because interest starts earning interest sooner.
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