Geeta deposited a certain sum of money in her bank account which amounted to Rs. 27,783 in 3 years at 5% per annum. the interest being compounded annually. What amount did she deposit?
Rs. 24,000
Understanding the Compound Interest Problem
The question asks us to find the initial amount of money (the principal) Geeta deposited in her bank account. We are given the final amount after 3 years, the annual interest rate, and that the interest is compounded annually.
This is a classic problem involving compound interest. Compound interest means that the interest earned in each period is added to the principal, and the interest for the next period is calculated on this new, larger principal.
Key Information from the Question
- Final Amount (A): Rs. 27,783
- Time period (n): 3 years
- Annual Interest Rate (r): 5% per annum
- Compounding Frequency: Annually
- Principal Amount (P): Unknown (What we need to find)
Formula for Compound Interest
The formula to calculate the final amount (A) when interest is compounded annually is:
\(A = P(1 + \frac{r}{100})^n\)
Where:
- \(A\) is the final Amount
- \(P\) is the Principal amount (the initial deposit)
- \(r\) is the annual interest rate in percentage
- \(n\) is the number of years
In our case, the rate is 5%, so \(\frac{r}{100} = \frac{5}{100} = 0.05\). We can also write the formula as:
\(A = P(1 + r_{decimal})^n\)
Where \(r_{decimal}\) is the rate as a decimal (0.05).
Solving for the Principal Amount (P)
We have the formula \(A = P(1 + r_{decimal})^n\) and we want to find \(P\). We can rearrange the formula to solve for \(P\):
\(P = \frac{A}{(1 + r_{decimal})^n}\)
Step-by-Step Calculation
Let's substitute the given values into the formula:
- \(A = 27783\)
- \(r_{decimal} = 0.05\)
- \(n = 3\)
\(P = \frac{27783}{(1 + 0.05)^3}\)
\(P = \frac{27783}{(1.05)^3}\)
Now, let's calculate \((1.05)^3\):
\((1.05)^3 = 1.05 \times 1.05 \times 1.05\)
\(1.05 \times 1.05 = 1.1025\)
\(1.1025 \times 1.05 = 1.157625\)
So, \((1.05)^3 = 1.157625\).
Now, substitute this value back into the equation for \(P\):
\(P = \frac{27783}{1.157625}\)
Performing the division:
\(P = 24000\)
The principal amount Geeta deposited was Rs. 24,000.
Verification
Let's verify the result by calculating the final amount if the principal was Rs. 24,000:
\(A = 24000(1 + 0.05)^3\)
\(A = 24000(1.05)^3\)
\(A = 24000 \times 1.157625\)
\(A = 27783\)
This matches the given final amount, confirming our calculation is correct.
Summary of Calculation
| Description | Value |
|---|---|
| Final Amount (A) | Rs. 27,783 |
| Time (n) | 3 years |
| Rate (r) | 5% or 0.05 |
| Formula | \(P = \frac{A}{(1+r)^n}\) |
| Calculation | \(P = \frac{27783}{(1.05)^3} = \frac{27783}{1.157625}\) |
| Principal (P) | Rs. 24,000 |
Therefore, the amount Geeta deposited was Rs. 24,000.
Revision Table: Compound Interest Concepts
| Term | Definition | Formula (Annual Compounding) |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | - |
| Amount (A) | The total sum after adding interest to the principal. | \(A = P(1 + r)^n\) |
| Interest Rate (r) | The percentage at which interest is calculated. | - |
| Time (n) | The period for which the money is invested or borrowed. | - |
| Compound Interest (CI) | The interest calculated on the principal and the accumulated interest of previous periods. | \(CI = A - P\) or \(CI = P[(1+r)^n - 1]\) |
Additional Information: Compound Interest Variations
Compound interest calculations can vary based on the compounding frequency. While this problem uses annual compounding, interest can be compounded semi-annually, quarterly, monthly, or even daily.
- Semi-annually: Interest is calculated twice a year. The rate per period is \(r/2\) and the number of periods is \(2n\). Formula: \(A = P(1 + \frac{r/2}{100})^{2n}\).
- Quarterly: Interest is calculated four times a year. The rate per period is \(r/4\) and the number of periods is \(4n\). Formula: \(A = P(1 + \frac{r/4}{100})^{4n}\).
- Monthly: Interest is calculated twelve times a year. The rate per period is \(r/12\) and the number of periods is \(12n\). Formula: \(A = P(1 + \frac{r/12}{100})^{12n}\).
Understanding the compounding frequency is crucial for accurate compound interest calculations. In this specific problem, the clear mention of "compounded annually" simplifies the calculation using the standard formula.
Important Questions from Compound Interest
- A sum of money triples itself at a certain rate of compound interest in 11 years. In how many years will it amount to 27 times of itself?
A sum of ₹10,000 amounts to ₹12,321 in 2 years at a certain rate of interest per annum, compounded annually. The rate of interest per annum is:
- If the interest earned during the 2nd year on a certain sum is ₹2,352, and the rate of interest is 5% per annum compounded annually, then the sum is:
- A sum of money triples itself at a certain rate of compound interest in 17 years. In how many years will it amount to 81 times of itself?
If the interest earned during the $2^{nd}$ year on a certain sum is ₹3,342, and the rate of interest is 20% per annum compounded annually, then the sum is:
