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?
Rs. 10248.7
Understanding Simple and Compound Interest Calculation
This problem involves two parts: first, using the given simple interest data to find the original principal amount and the rate of interest; second, using these calculated values to determine the amount if the same principal were lent at compound interest for the same duration and rate.
Step 1: Calculate Principal and Rate from Simple Interest Data
We are given the amount after 4 years at simple interest and the total simple interest earned in those 4 years.
- Amount after 4 years = Rs. 9800
- Simple Interest for 4 years = Rs. 2800
- Time (T) = 4 years
The Amount (A) in simple interest is the sum of the Principal (P) and the Simple Interest (SI):
\(A = P + SI\)
We can find the Principal (P) by subtracting the total simple interest from the total amount:
\(P = A - SI\)
\(P = 9800 - 2800\)
\(P = 7000\)
So, the original principal amount is Rs. 7000.
Now, we use the simple interest formula to find the rate of interest (R):
\(SI = \frac{P \times R \times T}{100}\)
We know SI = 2800, P = 7000, and T = 4. Plugging these values in:
\(2800 = \frac{7000 \times R \times 4}{100}\)
Multiply both sides by 100:
\(2800 \times 100 = 7000 \times R \times 4\)
\(280000 = 28000 \times R\)
Now, solve for R:
\(R = \frac{280000}{28000}\)
\(R = 10\)
So, the rate of interest is 10% per annum.
| Parameter | Value |
|---|---|
| Principal (P) | Rs. 7000 |
| Rate (R) | 10% per annum |
| Time (T) | 4 years |
Step 2: Calculate Amount at Compound Interest
Now we need to calculate the amount at the end of 4 years if the same principal (Rs. 7000) were lent at the same rate (10% per annum), compounded annually for the same time (4 years).
The formula for the amount (ACI) under compound interest compounded annually is:
\(A_{CI} = P \left(1 + \frac{R}{100}\right)^T\)
Plugging in the values P = 7000, R = 10, and T = 4:
\(A_{CI} = 7000 \left(1 + \frac{10}{100}\right)^4\)
\(A_{CI} = 7000 \left(1 + 0.1\right)^4\)
\(A_{CI} = 7000 \left(1.1\right)^4\)
Now, we calculate \((1.1)^4\):
\(1.1^1 = 1.1\)
\(1.1^2 = 1.1 \times 1.1 = 1.21\)
\(1.1^3 = 1.21 \times 1.1 = 1.331\)
\(1.1^4 = 1.331 \times 1.1 = 1.4641\)
Substitute this value back into the formula for ACI:
\(A_{CI} = 7000 \times 1.4641\)
\(A_{CI} = 10248.7\)
So, the amount at the end of four years, if lent at compound interest compounded annually, would be Rs. 10248.7.
Conclusion
By first determining the principal and rate from the simple interest information, we were able to calculate the amount that would accumulate under compound interest for the same period and rate. The calculated amount at the end of four years with compound interest is Rs. 10248.7.
Revision Table: Key Formulas
| Concept | Formula |
|---|---|
| Simple Interest (SI) | \(SI = \frac{P \times R \times T}{100}\) |
| Amount (Simple Interest) | \(A = P + SI\) |
| Amount (Compound Interest, Annually) | \(A_{CI} = P \left(1 + \frac{R}{100}\right)^T\) |
Additional Information on Interest Calculations
Understanding the difference between simple and compound interest is fundamental in finance.
- Simple Interest: The interest is calculated only on the initial principal amount for the entire duration. The interest earned does not get added back to the principal for subsequent periods. It's a fixed amount added to the principal each period.
- Compound Interest: The interest for each period is calculated on the principal amount plus any accumulated interest from previous periods. This means the interest earns interest, leading to exponential growth over time. Compound interest can be compounded annually, semi-annually, quarterly, monthly, or even daily. The more frequently it's compounded, the higher the final amount will be, assuming the same nominal rate.
- Comparing Simple vs. Compound Interest: For the same principal, rate, and time (greater than one period), compound interest will always result in a higher amount than simple interest because of the effect of earning interest on interest.
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?
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?
