Find the difference between compound interest and simple interest on a sum of Rs. 5000 at 10% per annum for 3 years.
Rs. 155
Calculating the Difference Between Compound Interest and Simple Interest
This problem asks us to find the difference between the compound interest (CI) and the simple interest (SI) on a given sum of money over a specific period at a certain rate.
Understanding Simple Interest (SI)
Simple Interest is calculated only on the principal amount. The interest earned in each period is fixed.
The formula for Simple Interest is:
\( SI = \frac{P \times R \times T}{100} \)
Where:
- \(P\) is the Principal amount
- \(R\) is the Rate of interest per annum
- \(T\) is the Time period in years
Understanding Compound Interest (CI)
Compound Interest is calculated on the principal amount as well as on the accumulated interest from previous periods. The interest grows on the principal plus interest.
The formula for the amount (\(A\)) after \(T\) years at rate \(R\) compounded annually is:
\( A = P(1 + \frac{R}{100})^T \)
The Compound Interest (CI) is then the amount minus the principal:
\( CI = A - P = P(1 + \frac{R}{100})^T - P \)
Given Values
In this question, we are given:
- Principal (P) = Rs. 5000
- Rate (R) = 10% per annum
- Time (T) = 3 years
Step-by-Step Calculation of Simple Interest
Using the formula for SI:
\( SI = \frac{5000 \times 10 \times 3}{100} \)
\( SI = \frac{150000}{100} \)
\( SI = 1500 \)
The simple interest for 3 years is Rs. 1500.
Step-by-Step Calculation of Compound Interest
Using the formula for CI:
\( CI = P(1 + \frac{R}{100})^T - P \)
\( CI = 5000(1 + \frac{10}{100})^3 - 5000 \)
\( CI = 5000(1 + 0.1)^3 - 5000 \)
\( CI = 5000(1.1)^3 - 5000 \)
\( CI = 5000 \times 1.331 - 5000 \)
\( CI = 6655 - 5000 \)
\( CI = 1655 \)
The compound interest for 3 years is Rs. 1655.
Calculating the Difference (CI - SI)
The difference between compound interest and simple interest is:
\( \text{Difference} = CI - SI \)
\( \text{Difference} = 1655 - 1500 \)
\( \text{Difference} = 155 \)
Alternative Method: Using the Formula for CI - SI Difference for 3 Years
There is a specific formula to find the difference between CI and SI for 3 years:
\( \text{Difference} = P (\frac{R}{100})^2 (3 + \frac{R}{100}) \)
Plugging in the given values:
\( \text{Difference} = 5000 (\frac{10}{100})^2 (3 + \frac{10}{100}) \)
\( \text{Difference} = 5000 (0.1)^2 (3 + 0.1) \)
\( \text{Difference} = 5000 \times 0.01 \times 3.1 \)
\( \text{Difference} = 50 \times 3.1 \)
\( \text{Difference} = 155 \)
Both methods yield the same difference.
Conclusion
The difference between compound interest and simple interest on a sum of Rs. 5000 at 10% per annum for 3 years is Rs. 155.
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.
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 difference between the compound interest and simple interest on a certain sum of money in 3 years at 5% p.a. is Rs. 6,100. What is the initial amount?
A sum of money becomes two times of itself in 8 years at simple interest, and it becomes four times of itself in 2 years at compound interest, when interest is compounded annually. Find the ratio of the rate of simple interest to the rate of compound interest offered per year.
