At what rate of interest per annum, compounded annually, will an amount of ₹8,000 yield a compound interest of ₹904.2 in 2 years?
5.5%
Understanding the Compound Interest Problem
This question asks us to find the annual rate of interest when a principal amount of ₹8,000 grows over 2 years, compounded annually, resulting in a compound interest of ₹904.2.
Let's define the key terms involved in this compound interest problem:
- Principal (P): The initial amount of money, which is ₹8,000.
- Compound Interest (CI): The interest earned over the period, which is ₹904.2.
- Time Period (n): The duration for which the money is invested or borrowed, which is 2 years.
- Compounding Frequency: How often the interest is added to the principal. Here, it is compounded annually.
- Rate of Interest (r): The percentage at which the interest is calculated annually. This is what we need to find.
Compound Interest Formula and Calculation
The formula for the amount (A) after \(n\) years when the principal (P) is compounded annually at an interest rate \(r\) (as a decimal) is:
\[A = P \left(1 + r\right)^n\]The compound interest (CI) is the difference between the amount (A) and the principal (P):
\[CI = A - P\]From the CI formula, we can also write \(A = P + CI\).
Given:
- Principal (P) = ₹8,000
- Compound Interest (CI) = ₹904.2
- Time Period (n) = 2 years
First, let's calculate the total Amount (A) after 2 years:
\[A = P + CI\] \[A = ₹8,000 + ₹904.2\] \[A = ₹8,904.2\]Now, we can substitute the values of A, P, and n into the main compound interest formula:
\[₹8,904.2 = ₹8,000 \left(1 + r\right)^2\]To find the rate \(r\), we need to isolate \((1 + r)^2\):
\[\left(1 + r\right)^2 = \frac{8904.2}{8000}\]Let's perform the division:
\[\left(1 + r\right)^2 = 1.113025\]Now, take the square root of both sides to find \((1 + r)\):
\[\sqrt{\left(1 + r\right)^2} = \sqrt{1.113025}\] \[1 + r = 1.055\]Finally, solve for \(r\):
\[r = 1.055 - 1\] \[r = 0.055\]The rate \(r\) is in decimal form. To express it as a percentage, multiply by 100:
\[\text{Rate (in %)} = r \times 100\] \[\text{Rate (in %)} = 0.055 \times 100\] \[\text{Rate (in %)} = 5.5\%\]So, the rate of interest per annum is 5.5%.
Checking the Options
Let's compare our calculated rate with the given options:
| Option | Rate |
|---|---|
| 1 | 6% |
| 2 | 8% |
| 3 | 10% |
| 4 | 5.5% |
Our calculated rate of 5.5% matches Option 4.
Revision Table: Key Concepts Review
| Concept | Definition/Formula | Value in this Problem |
|---|---|---|
| Principal (P) | Initial amount | ₹8,000 |
| Compound Interest (CI) | Interest on interest | ₹904.2 |
| Time (n) | Duration | 2 years |
| Amount (A) | Principal + CI | ₹8,904.2 |
| Compound Interest Formula (Amount) | \(A = P(1+r)^n\) | \(8904.2 = 8000(1+r)^2\) |
| Rate (r) | Annual interest rate (decimal) | 0.055 |
| Rate (%) | Annual interest rate (percentage) | 5.5% |
Additional Information: Compound vs. Simple Interest
It's important to understand the difference between compound interest and simple interest.
- Simple Interest: Interest is calculated only on the initial principal amount. The formula for simple interest is \(SI = \frac{P \times R \times T}{100}\), where P is principal, R is rate (percentage), and T is time. Simple interest is constant every year.
- Compound Interest: Interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means the principal amount grows over time, leading to exponential growth in the amount. In this problem, because it says "compounded annually", we used the compound interest formula.
Compound interest is generally more beneficial for investors than simple interest because the interest earns interest, leading to faster growth of the investment.
Important Questions from Compound Interest
At what rate percent per annum will Rs. 7200 amount to Rs. 7938 in one year, if interest is compounded half yearly?
What is the compound interest (in Rs.) on a sum of Rs. 8192 for \(1 \frac{1}{4}\) years at 15% per annum, if interest is compounded 5-monthly ?
What is the difference (in Rs.) between the interests on Rs. 50,000 for one year at 8% per annum compounded half yearly and yearly?
A sum of money becomes Rs. 11,880 after 4 years and Rs. 17,820 after 6 years on compound interest, if the interest is compounded annually. What is the half of the sum (in Rs.)?
A sum invested at compound interest amounts to Rs. 7,800 in 3 years and Rs. 11,232 in 5 years. What is the rate per cent?
