On compound interest, ₹ 2,000 amounts to ₹ 2,226.05 in 2 years. What is the rate of interest per annum?
6%
Calculating Compound Interest Rate per Annum
The question asks us to find the annual rate of interest when a principal amount grows to a certain amount over a period of time under compound interest.
Understanding Compound Interest
Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means the interest earned in each period is added to the principal for the next period's calculation, leading to exponential growth.
We are given the following information:
- Principal amount (P) = ₹ 2,000
- Amount after 2 years (A) = ₹ 2,226.05
- Time period (t) = 2 years
We need to find the rate of interest per annum (r).
Let's summarize the given variables in a table:
| Variable | Symbol | Value |
|---|---|---|
| Principal | P | ₹ 2,000 |
| Amount | A | ₹ 2,226.05 |
| Time | t | 2 years |
| Rate of Interest (per annum) | r | ? |
The formula for compound interest is:
\[A = P\left(1 + \frac{r}{100}\right)^t\]Where:
- A is the final amount
- P is the principal amount
- r is the annual rate of interest (in percentage)
- t is the time period in years
We need to rearrange this formula to solve for r:
\[\frac{A}{P} = \left(1 + \frac{r}{100}\right)^t\]Taking the t-th root of both sides:
\[\left(\frac{A}{P}\right)^{1/t} = 1 + \frac{r}{100}\]Subtracting 1 from both sides:
\[\left(\frac{A}{P}\right)^{1/t} - 1 = \frac{r}{100}\]Multiplying by 100:
\[r = 100 \times \left[\left(\frac{A}{P}\right)^{1/t} - 1\right]\]Now, substitute the given values into the formula:
\[r = 100 \times \left[\left(\frac{2226.05}{2000}\right)^{1/2} - 1\right]\]First, calculate the ratio A/P:
\[\frac{2226.05}{2000} = 1.113025\]Next, calculate the square root (since t=2):
\[(1.113025)^{1/2} = \sqrt{1.113025} = 1.055\]Now, substitute this back into the formula for r:
\[r = 100 \times (1.055 - 1)\] \[r = 100 \times (0.055)\] \[r = 5.5\]So, the calculation based on the given principal, amount, and time yields an annual rate of interest of 5.5%.
Let's look at the given options:
- 5%
- 5.5%
- 6%
- 6.5%
The annual rate of interest is 6%.
Revision Table: Compound Interest Concepts
| Concept | Description | Formula |
|---|---|---|
| Compound Interest | Interest calculated on principal and accumulated interest. | \(CI = A - P\) |
| Amount (Compounded Annually) | Total sum including principal and interest. | \(A = P\left(1 + \frac{r}{100}\right)^t\) |
| Principal (P) | Initial amount invested or borrowed. | - |
| Rate (r) | Percentage of interest charged or earned per period. | - |
| Time (t) | Duration for which the principal is invested/borrowed. | - |
Additional Information on Compound Interest
Compounding Frequency: The formula \(A = P(1 + r/100)^t\) assumes interest is compounded annually. If interest is compounded more frequently (e.g., semi-annually, quarterly, monthly), the formula is adjusted:
\[A = P\left(1 + \frac{r}{n \times 100}\right)^{n \times t}\]Where 'n' is the number of times interest is compounded per year. For example, n=2 for semi-annual, n=4 for quarterly, and n=12 for monthly compounding.
Simple Interest vs. Compound Interest: In simple interest, the interest is calculated only on the initial principal amount for the entire duration. Compound interest, on the other hand, adds the earned interest to the principal, allowing interest to earn interest. Compound interest generally results in a higher amount over longer periods compared to simple interest at the same rate.
Rule of 72: A useful approximation for estimating the number of years required to double an investment at a fixed annual compound interest rate is the Rule of 72. Divide 72 by the annual interest rate (as a percentage) to get the approximate number of years for the investment to double.
Understanding the concept of compound interest and its formula is crucial for solving problems related to investments, loans, and financial planning.
Important Questions from Simple and Compound Interest
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?
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:
