If the simple interest on a sum of money for 2 years at 5% per annum is Rs. 50, the compound interest on the same at the same rate and for the same time is:
51.25
Understanding Simple and Compound Interest
This question asks us to first find the principal amount using the given simple interest details and then calculate the compound interest on that same principal for the same rate and time period. Let's break it down step by step.
Step 1: Calculate the Principal Amount using Simple Interest
We are given the simple interest (SI), the rate of interest (R), and the time period (T). The formula for simple interest is:
\(SI = \frac{P \times R \times T}{100}\)
Where:
- SI = Simple Interest = Rs. 50
- R = Rate of Interest = 5% per annum
- T = Time Period = 2 years
- P = Principal Amount (what we need to find)
Let's plug the given values into the formula:
\(50 = \frac{P \times 5 \times 2}{100}\)
Simplify the equation:
\(50 = \frac{10P}{100}\)
\(50 = \frac{P}{10}\)
Now, solve for P:
\(P = 50 \times 10\)
\(P = 500\)
So, the principal amount is Rs. 500.
Step 2: Calculate the Compound Interest
Now we need to find the compound interest (CI) on the principal amount (P = Rs. 500) at the same rate (R = 5%) for the same time period (T = 2 years).
The formula for the amount (A) with compound interest is:
\(A = P \left(1 + \frac{R}{100}\right)^T\)
Plug in the values:
\(A = 500 \left(1 + \frac{5}{100}\right)^2\)
\(A = 500 \left(1 + \frac{1}{20}\right)^2\)
\(A = 500 \left(\frac{20+1}{20}\right)^2\)
\(A = 500 \left(\frac{21}{20}\right)^2\)
\(A = 500 \times \frac{21^2}{20^2}\)
\(A = 500 \times \frac{441}{400}\)
We can simplify this by cancelling common factors:
\(A = \frac{500 \times 441}{400}\)
\(A = \frac{5 \times 441}{4}\)
\(A = \frac{2205}{4}\)
\(A = 551.25\)
The amount after 2 years with compound interest is Rs. 551.25.
To find the compound interest (CI), we subtract the principal amount from the total amount:
\(CI = A - P\)
\(CI = 551.25 - 500\)
\(CI = 51.25\)
The compound interest is Rs. 51.25.
Comparing with Options
Let's compare our calculated compound interest with the given options:
- Option 1: 50.50
- Option 2: 51.25
- Option 3: 51.50
- Option 4: 50.05
Our calculated value of Rs. 51.25 matches Option 2.
Conclusion
Based on the calculations, the compound interest on the sum of money (Rs. 500) for 2 years at 5% per annum is Rs. 51.25. This confirms that Option 2 is the correct answer.
Revision Table
| Concept | Formula | Calculation for this Problem |
|---|---|---|
| Simple Interest (SI) | \(SI = \frac{P \times R \times T}{100}\) | \(50 = \frac{P \times 5 \times 2}{100} \Rightarrow P = 500\) |
| Amount with Compound Interest (A) | \(A = P \left(1 + \frac{R}{100}\right)^T\) | \(A = 500 \left(1 + \frac{5}{100}\right)^2 = 551.25\) |
| Compound Interest (CI) | \(CI = A - P\) | \(CI = 551.25 - 500 = 51.25\) |
Additional Information: Simple vs. Compound Interest
Understanding the difference between simple interest and compound interest is crucial.
- Simple Interest: Interest is calculated only on the initial principal amount. The interest earned each period remains constant. It does not earn interest on itself.
- Compound Interest: Interest is calculated on the initial principal *and* also on the accumulated interest from previous periods. This means the interest earned grows over time, leading to a larger total amount compared to simple interest for the same principal, rate, and time (for periods greater than 1 year).
In this problem, the simple interest for 2 years is Rs. 50, meaning Rs. 25 is earned each year (\(50/2\)). With compound interest, the interest from the first year (Rs. 25) is added to the principal for the second year's calculation. So, in the second year, interest is earned on Rs. 500 + Rs. 25 = Rs. 525.
Let's see the difference year-by-year:
- Year 1: Interest is 5% of Rs. 500 = Rs. 25 (Same for both SI and CI). Amount at end of Year 1 = 500 + 25 = 525.
- Year 2 (SI): Interest is 5% of initial P (Rs. 500) = Rs. 25. Total SI = 25 + 25 = Rs. 50.
- Year 2 (CI): Interest is 5% of amount at end of Year 1 (Rs. 525) = \(0.05 \times 525 = 26.25\). Total CI = Interest Year 1 + Interest Year 2 = \(25 + 26.25 = 51.25\).
This step-by-step breakdown shows why the compound interest (Rs. 51.25) is slightly more than the simple interest (Rs. 50) over 2 years.
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Important Questions from Simple and Compound Intrest
Amit had invested same amount of sums at simple as well as compound interest, compounded annually. The time period of investment for both the sums was 2 years and rate of interest too was the same, 4% per annum. At the end, he found a difference of ₹43 in both the interests received. What were the sums (in ₹) invested?
When the difference between compound interest, compounded annually, and simple interest for three years is ₹186 at 10% interest per annum, the principal is ₹______.
- When the difference between compound interest, compounded annually, and simple interest for three years is ₹217 at 10% interest per annum, the principal is ₹______.
- When the difference between compound interest, compounded annually, and simple interest for three years is ₹228 at 4% interest per annum, the principal is ₹______.
- The difference between the compound interest, compounded annually and the simple interest if ₹17,700 is deposited at 4% rate of interest per annum for 2 years is:
