Find the difference between compound interest (compounding annually) and simple interest on a sum of Rs. 56,000 for two years at 15% per annum.
Rs. 1260
Understanding Compound Interest vs Simple Interest Difference
The question asks us to find the difference between the compound interest (CI) and the simple interest (SI) accrued on a principal sum over a specific period at a given annual rate. This is a common type of problem involving basic interest calculations.
We are given the following information:
- Principal amount (P) = Rs. 56,000
- Rate of interest (R) = 15% per annum
- Time period (T) = 2 years
- Compounding frequency = Annually
We need to calculate both the simple interest and the compound interest for this period and then find their difference.
Calculating Simple Interest (SI)
Simple interest is calculated only on the principal amount for the entire duration. The formula for simple interest is:
\( SI = \frac{P \times R \times T}{100} \)
Let's plug in the given values:
\( SI = \frac{56000 \times 15 \times 2}{100} \)
\( SI = \frac{56000 \times 30}{100} \)
\( SI = 560 \times 30 \)
\( SI = 16800 \)
So, the simple interest for two years is Rs. 16,800.
Calculating Compound Interest (CI)
Compound interest is calculated on the principal amount as well as on the accumulated interest from previous periods. For annual compounding, the formula for the amount (A) after T years is:
\( A = P\left(1 + \frac{R}{100}\right)^T \)
Once we find the amount, the compound interest is calculated as \(CI = A - P\).
Let's calculate the amount first:
\( A = 56000\left(1 + \frac{15}{100}\right)^2 \)
\( A = 56000\left(1 + \frac{3}{20}\right)^2 \)
\( A = 56000\left(\frac{20+3}{20}\right)^2 \)
\( A = 56000\left(\frac{23}{20}\right)^2 \)
\( A = 56000 \times \frac{23}{20} \times \frac{23}{20} \)
\( A = 56000 \times \frac{529}{400} \)
\( A = \frac{56000}{400} \times 529 \)
\( A = 140 \times 529 \)
\( A = 74060 \)
The amount after two years is Rs. 74,060.
Now, calculate the compound interest:
\( CI = A - P \)
\( CI = 74060 - 56000 \)
\( CI = 18060 \)
So, the compound interest for two years is Rs. 18,060.
Finding the Difference (CI - SI)
The difference between compound interest and simple interest is:
\( \text{Difference} = CI - SI \)
\( \text{Difference} = 18060 - 16800 \)
\( \text{Difference} = 1260 \)
The difference between the compound interest and the simple interest is Rs. 1,260.
Alternative Method for CI - SI Difference for 2 Years
For a period of 2 years, the difference between compound interest and simple interest can be directly calculated using the formula:
\( CI - SI = P\left(\frac{R}{100}\right)^2 \)
Let's use this formula to verify our result:
\( CI - SI = 56000\left(\frac{15}{100}\right)^2 \)
\( CI - SI = 56000\left(\frac{3}{20}\right)^2 \)
\( CI - SI = 56000 \times \frac{9}{400} \)
\( CI - SI = \frac{56000}{400} \times 9 \)
\( CI - SI = 140 \times 9 \)
\( CI - SI = 1260 \)
Both methods give the same result: the difference is Rs. 1,260.
Summary of Results
| Interest Type | Calculated Value |
|---|---|
| Simple Interest (SI) | Rs. 16,800 |
| Compound Interest (CI) | Rs. 18,060 |
| Difference (CI - SI) | Rs. 1,260 |
The difference between the compound interest and simple interest is Rs. 1,260.
Revision Table: Key Interest Concepts
| Concept | Description | Formula (Annual) |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | - |
| Rate (R) | The percentage at which interest is charged or earned per period (usually per year). | - |
| Time (T) | The duration for which the money is invested or borrowed. | - |
| Simple Interest (SI) | Interest calculated only on the principal amount. | \( SI = \frac{P \times R \times T}{100} \) |
| Compound Interest (CI) | Interest calculated on the principal and accumulated interest from previous periods. | \( A = P\left(1 + \frac{R}{100}\right)^T \) \( CI = A - P \) |
| Amount (A) | The total sum after adding interest to the principal (Principal + Interest). | \( A = P + SI \) (for SI) \( A = P + CI \) (for CI) |
Additional Information on Interest Calculations
Understanding simple interest and compound interest is fundamental in finance. Simple interest is easier to calculate but compound interest is more common in real-world scenarios like savings accounts, loans, and investments because it accounts for the compounding effect, where interest earns interest. This leads to significantly higher returns (or costs) over longer periods compared to simple interest.
The difference between compound interest and simple interest is always non-negative for T > 1 and R > 0, and it grows larger as the principal, rate, and time period increase. For a 1-year period, simple interest and compound interest (compounding annually) are the same.
When the compounding frequency is not annual (e.g., half-yearly, quarterly, monthly), the compound interest formula changes slightly. If interest is compounded n times per year, the formula for the amount becomes:
\( A = P\left(1 + \frac{R/n}{100}\right)^{nT} \)
In our problem, the compounding was annual, meaning n=1, which simplifies the formula back to the one we used.
Important Questions from Simple and Compound Both
What is the compound interest (in Rs.) at the rate of 10%, compounded annually, for 3 years on the principal which in 8 years at the rate of 12% per annum gives Rs. 4,800 as simple interest?
A certain sum amounts to Rs. 15,500 in 2 years at 12% p.a. simple interest. If the same sum is compounded half-yearly at 10% per annum for \(1 \frac{1}{2}\) years, what will be the amount received?
The difference in compound interest on a certain sum at 10% p.a. for one year, when the interest is compounded half-yearly and yearly, is Rs. 88.80. What is the simple interest on the same sum for \(1\frac{2}{3}\) years at the same rate?
A sum of Rs. 7500 amounts to Rs. 9075 at 10% p.a, interest being compounded yearly in a certain time. The simple interest (in Rs.) on the same sum for the same time and the same rate is:
A certain sum amounts to Rs.291600 in 2 years and to Rs.314928 in 3 years on compound interest compounded annually. How much will be the simple interest (in Rs.) on Rs.40000 at the same rate for 2 years?
