The compound interest accrued on Rs. 18000 in two years is Rs. 2995.2. What will be the simple interest accrued at the same rate of interest for the same sum for three years?
Rs. 4320
Finding Simple Interest from Compound Interest Data
The problem asks us to first determine the rate of interest from the given compound interest data and then use that rate to calculate the simple interest for a different time period but the same principal amount.
Step 1: Calculate the Rate of Interest from Compound Interest
We are given the principal amount (P), the time period (n), and the compound interest (CI) accrued. We can use the formula for compound interest to find the rate of interest (R).
The formula for Compound Interest is:
\(\text{CI} = \text{P} \left[ \left(1 + \frac{\text{R}}{100}\right)^\text{n} - 1 \right]\)
Alternatively, we can first calculate the amount (A) after n years, where \(\text{A} = \text{P} + \text{CI}\). Then use the formula:
\(\text{A} = \text{P} \left(1 + \frac{\text{R}}{100}\right)^\text{n}\)
Given:
- Principal (\(\text{P}\)) = Rs. 18000
- Time (\(\text{n}\)) = 2 years
- Compound Interest (\(\text{CI}\)) = Rs. 2995.2
Calculate the Amount (\(\text{A}\)):
\(\text{A} = \text{P} + \text{CI} = 18000 + 2995.2 = 20995.2\)
Now substitute the values into the amount formula:
\(20995.2 = 18000 \left(1 + \frac{\text{R}}{100}\right)^2\)
Divide both sides by 18000:
\(\frac{20995.2}{18000} = \left(1 + \frac{\text{R}}{100}\right)^2\)
\(1.1664 = \left(1 + \frac{\text{R}}{100}\right)^2\)
Take the square root of both sides:
\(\sqrt{1.1664} = 1 + \frac{\text{R}}{100}\)
\(1.08 = 1 + \frac{\text{R}}{100}\)
Subtract 1 from both sides:
\(1.08 - 1 = \frac{\text{R}}{100}\)
\(0.08 = \frac{\text{R}}{100}\)
Multiply by 100 to find R:
\(\text{R} = 0.08 \times 100 = 8\)
So, the rate of interest is 8% per annum.
Step 2: Calculate the Simple Interest
Now that we have the rate of interest (R), we can calculate the simple interest for the same sum and rate for a period of three years.
The formula for Simple Interest (SI) is:
\(\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\)
Given:
- Principal (\(\text{P}\)) = Rs. 18000
- Rate (\(\text{R}\)) = 8% per annum (calculated above)
- Time (\(\text{T}\)) = 3 years
Substitute the values into the SI formula:
\(\text{SI} = \frac{18000 \times 8 \times 3}{100}\)
\(\text{SI} = \frac{180 \times 8 \times 3}{1}\)
\(\text{SI} = 180 \times 24\)
\(\text{SI} = 4320\)
The simple interest accrued will be Rs. 4320.
Summary of Calculations
| Parameter | Value |
|---|---|
| Principal (P) | Rs. 18000 |
| CI for 2 years | Rs. 2995.2 |
| Amount after 2 years (A) | Rs. 20995.2 |
| Calculated Rate (R) | 8% per annum |
| Time for SI (T) | 3 years |
| Calculated Simple Interest (SI) | Rs. 4320 |
Therefore, the simple interest accrued at the same rate of interest for the same sum for three years is Rs. 4320.
Revision Table: Compound Interest and Simple Interest Concepts
| Concept | Formula | Description |
|---|---|---|
| Simple Interest (SI) | \(\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\) | Interest calculated only on the principal amount. |
| Compound Interest (CI) | \(\text{A} = \text{P} \left(1 + \frac{\text{R}}{100}\right)^\text{n}\) \(\text{CI} = \text{A} - \text{P}\) |
Interest calculated on the principal amount and the accumulated interest from previous periods. |
| Amount (A) | \(\text{A} = \text{P} + \text{Interest}\) | The total sum after adding the interest to the principal. |
| Rate of Interest (R) | % per annum | The percentage at which interest is charged on the principal. |
| Time (T or n) | Years | The duration for which the money is borrowed or invested. |
Additional Information on Interest Calculations
Understanding the difference between simple interest and compound interest is crucial in financial calculations. Simple interest is easier to calculate but provides lower returns over time compared to compound interest, especially for longer durations. Compound interest leads to exponential growth because the interest earned in each period is added to the principal for calculating the interest in the next period. The frequency of compounding (annually, half-yearly, quarterly, etc.) also affects the total compound interest accrued; higher frequency leads to higher interest.
In problems like this, where you are given compound interest and need to find simple interest (or vice versa) for the same principal and rate, the first step is always to find the rate of interest using the given information. Once the rate is known, you can apply the formula for the required type of interest and time period.
Important Questions from Simple and Compound Both
X took a loan of Rs.1400 with simple interest for as many years as the rate of interest. If he paid Rs.1210 as interest at the end of that period, what was the rate of interest?
The S.I. on a certain sum of money for 4 years at 4 percent per annum exceeds the C.I. on the same sum for 3 years at 5 percent per annum by Rs. 57. Find the approximate sum.
The simple and compound interest that can be earned in two years at the same rate on a certain sum is Rs. 1,000 and Rs. 1,040 respectively. What is the rate (percent per annum) of interest?
Compound interest on a certain sum of money for 2 years at a rate of 'r' per cent per annum (compounding annually) is Rs. 8385. Simple interest on the same sum at the same rate for 2 years is Rs 7800. What is the value of r?
Sam lent a sum of money at simple interest which amounted to Rs. 9800 in four years and the interest that he earned was Rs. 2800. Had he lent it at compound interest compounded annually for the same time and at the same rate, what would be the amount at the end of four years?
