A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
1891.5 rupees
Understanding the Problem: Simple and Compound Interest
The question involves two parts related to interest calculations: first, finding the rate of simple interest based on a given increase in principal over a period, and second, using that rate to calculate the compound interest on a different principal amount over a different period.
Step 1: Calculating the Simple Interest Rate
We are given that a sum of money increases by 45% in 9 years at simple interest. This means the simple interest earned is 45% of the original principal amount. Let the original principal be \(P\).
- Simple Interest (SI) = 45% of \(P\) = \(0.45P\)
- Time (T) = 9 years
The formula for Simple Interest is:
\( \text{SI} = \frac{P \times R \times T}{100} \)
Where R is the rate of interest per annum. We can substitute the known values into the formula:
\( 0.45P = \frac{P \times R \times 9}{100} \)
To find the rate (R), we can simplify the equation. Assuming \(P\) is not zero (which it must be for interest to be calculated), we can divide both sides by \(P\):
\( 0.45 = \frac{R \times 9}{100} \)
Now, solve for R:
\( 0.45 \times 100 = R \times 9 \)
\( 45 = 9R \)
\( R = \frac{45}{9} \)
\( R = 5 \)
So, the simple interest rate is 5% per annum.
Step 2: Calculating the Compound Interest
Now we need to calculate the compound interest on Rs. 12,000 after 3 years at the same rate of 5% per annum. The principal for this calculation is Rs. 12,000.
- Principal (P) = Rs. 12,000
- Rate (r) = 5% per annum (same rate found from simple interest)
- Time (n) = 3 years
The formula for the amount (A) after compound interest is:
\( A = P \left(1 + \frac{r}{100}\right)^n \)
Substitute the values:
\( A = 12000 \left(1 + \frac{5}{100}\right)^3 \)
\( A = 12000 \left(1 + 0.05\right)^3 \)
\( A = 12000 \left(1.05\right)^3 \)
Calculate \((1.05)^3\):
\( (1.05)^1 = 1.05 \)
\( (1.05)^2 = 1.05 \times 1.05 = 1.1025 \)
\( (1.05)^3 = 1.1025 \times 1.05 = 1.157625 \)
Now, substitute this back into the formula for A:
\( A = 12000 \times 1.157625 \)
\( A = 13891.5 \)
This is the amount after 3 years. To find the Compound Interest (CI), subtract the original principal from the amount:
\( \text{CI} = A - P \)
\( \text{CI} = 13891.5 - 12000 \)
\( \text{CI} = 1891.5 \)
The compound interest after 3 years is Rs. 1891.5.
Summary of Calculations
| Calculation Step | Details | Result |
|---|---|---|
| Finding Simple Interest Rate | SI = 0.45P, T = 9 years, SI = (P * R * T) / 100 | R = 5% per annum |
| Calculating Compound Amount | P = 12000, r = 5%, n = 3 years, A = P(1 + r/100)n | A = Rs. 13891.5 |
| Calculating Compound Interest | CI = A - P | CI = Rs. 1891.5 |
Final Answer
The compound interest of Rs. 12,000 after 3 years at the same rate (5% per annum) is Rs. 1891.5.
Revision Table: Interest Formulas
| Interest Type | Formula | Variables |
|---|---|---|
| Simple Interest (SI) | \( \text{SI} = \frac{P \times R \times T}{100} \) | P = Principal, R = Rate per annum, T = Time in years |
| Compound Amount (A) | \( A = P \left(1 + \frac{r}{100}\right)^n \) | P = Principal, r = Rate per period, n = Number of periods |
| Compound Interest (CI) | \( \text{CI} = A - P \) | A = Compound Amount, P = Principal |
Additional Information: Simple vs. Compound Interest
It's important to understand the difference between simple interest and compound interest.
- Simple Interest: Interest is calculated only on the initial principal amount for the entire duration. The interest earned does not get added to the principal for calculating future interest.
- 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, we first used the simple interest concept to find the consistent annual rate and then applied the compound interest concept using that rate to find the total interest earned over a different period on a different principal.
Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
Karuna has an option to invest in the following three schemes. If she wants to invest Rs P, then which scheme will be most beneficial for her?
Scheme that comes with lower interest rates and terms;
I. 28% p.a. interest, compounded every quarter, for one year.
II. 28% p.a. interest, compounded every half year, for one year.
III. 28% p.a. interest, for one year, compounded every year.
A sum of money is invested at 40% p.a. compound interest, compounded half-yearly. If the amount at the end of one and half years is Rs. 2,16,000, then find the sum ?
