Simple interest on a certain sum at 8% per annum for 5 years is ₹2400. What will be the compound interest on the same sum at the same rate for 2 years?
₹998.40
Understanding Simple Interest and Compound Interest
This problem involves two key concepts in finance: simple interest and compound interest. Simple interest is calculated only on the initial principal amount, while compound interest is calculated on the principal amount and also on the accumulated interest from previous periods. This means compound interest grows faster than simple interest over time.
Calculating the Principal Sum from Simple Interest
We are given the simple interest earned on a certain sum over a specific period at a fixed rate. We can use the formula for simple interest to find the original sum (principal). The formula for simple interest is:
\(\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\)
Where:
- SI = Simple Interest
- P = Principal amount
- R = Rate of interest per annum
- T = Time period in years
We are given:
- SI = ₹2400
- R = 8% per annum
- T = 5 years
Let's plug these values into the formula to find P:
\(2400 = \frac{\text{P} \times 8 \times 5}{100}\)
Simplify the equation:
\(2400 = \frac{40\text{P}}{100}\)
\(2400 = \frac{2\text{P}}{5}\)
Now, solve for P:
\(\text{P} = \frac{2400 \times 5}{2}\)
\(\text{P} = 1200 \times 5\)
\(\text{P} = 6000\)
So, the principal sum is ₹6000.
Calculating Compound Interest on the Same Sum
Now we need to calculate the compound interest on this principal amount (₹6000) at the same rate (8% per annum) for a period of 2 years. The formula for the amount (A) under compound interest is:
\(\text{A} = \text{P}\left(1 + \frac{\text{R}}{100}\right)^{\text{T}}\)
Where:
- A = Amount after T years
- P = Principal amount
- R = Rate of interest per annum
- T = Time period in years
We know:
- P = ₹6000
- R = 8% per annum
- T = 2 years
Substitute these values into the formula:
\(\text{A} = 6000\left(1 + \frac{8}{100}\right)^{2}\)
\(\text{A} = 6000\left(1 + 0.08\right)^{2}\)
\(\text{A} = 6000\left(1.08\right)^{2}\)
Calculate \((1.08)^2\):
\(1.08 \times 1.08 = 1.1664\)
Now, calculate the Amount (A):
\(\text{A} = 6000 \times 1.1664\)
\(\text{A} = 6998.40\)
The amount after 2 years with compound interest is ₹6998.40.
To find the compound interest (CI), subtract the principal from the amount:
\(\text{CI} = \text{A} - \text{P}\)
\(\text{CI} = 6998.40 - 6000\)
\(\text{CI} = 998.40\)
The compound interest on ₹6000 at 8% per annum for 2 years is ₹998.40.
Summary of Calculation Steps
- Use the simple interest information (SI, R, T) to find the Principal (P).
- Use the calculated Principal (P) and the new time period (T') with the same rate (R) to find the Amount (A) under compound interest.
- Subtract the Principal (P) from the Amount (A) to find the Compound Interest (CI).
Final Answer
The compound interest on the same sum (₹6000) at the same rate (8% per annum) for 2 years is ₹998.40.
| Detail | Simple Interest Calculation | Compound Interest Calculation |
|---|---|---|
| Principal (P) | Calculated as ₹6000 | ₹6000 |
| Rate (R) | 8% | 8% |
| Time (T) | 5 years | 2 years |
| Interest Earned | SI = ₹2400 (Given) | CI = ₹998.40 (Calculated) |
| Formula Used | \(\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\) | \(\text{A} = \text{P}\left(1 + \frac{\text{R}}{100}\right)^{\text{T}}\), CI = A - P |
Revision Table: Simple vs. Compound Interest Formulas
| Concept | Formula | Notes |
|---|---|---|
| Simple Interest (SI) | \(\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\) | Interest is calculated only on the principal. |
| Amount (A) in Simple Interest | \(\text{A} = \text{P} + \text{SI} = \text{P}\left(1 + \frac{\text{R} \times \text{T}}{100}\right)\) | Total amount after T years. |
| Amount (A) in Compound Interest | \(\text{A} = \text{P}\left(1 + \frac{\text{R}}{100}\right)^{\text{T}}\) | Interest is compounded periodically (usually annually). |
| Compound Interest (CI) | \(\text{CI} = \text{A} - \text{P} = \text{P}\left[\left(1 + \frac{\text{R}}{100}\right)^{\text{T}} - 1\right]\) | Total interest earned after T years. |
Additional Information on Interest Calculations
Interest calculations are fundamental in personal finance, banking, and economics. Understanding the difference between simple and compound interest is crucial. Compound interest is often referred to as 'interest on interest' and is the basis for growth in savings, investments, and loans over longer periods.
- Effective Rate: When interest is compounded more than once a year (e.g., quarterly, monthly), the effective annual rate is higher than the nominal annual rate.
- Time Value of Money: Interest calculations are linked to the concept of the time value of money, which states that a sum of money is worth more now than the same sum will be at a future date due to its earning potential.
- Applications: Simple interest is often used for short-term loans. Compound interest is used for savings accounts, fixed deposits (FDs), recurring deposits (RDs), and long-term loans like mortgages.
Important Questions from Simple and Compound Interest
Find the compound interest on ₹64000 at 10% per annum for 9 months when interest is compounded quarterly.
A man invested on simple interest, 1/4 of his capital at 7% p.a., another 1/4 of the capital at 8% p.a. and the remaining capital at 10% p.a. He earned Rs. 700 as interest in one year. Find his total capital invested.
A sum of money doubles itself in 10 years on compound interest. In how many years will it become four times?
A sum of ₹1500 is invested at 5% p.a. simple interest. If the interest is added to the principal after every 10 years, the amount will become ₹3000 after:
Find the compound interest on ₹42000 for 1½ years at 10% p.a. compounded annually.
