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Imagine you want to borrow some money from a friend or a bank. Usually, when you borrow money, you have to pay back a little extra as a thank you for lending it to you. This extra money is called interest. Similarly, if you lend money to someone or deposit it in a bank, you earn interest as a reward for letting them use your money.
Understanding how interest works is important not only for exams but also for managing money in real life. Whether you are saving money in a fixed deposit or taking a loan, knowing how interest is calculated helps you make smart financial decisions.
Before we learn how to calculate interest, let's understand three important terms:
In India, interest rates are usually given in percentage per annum (per year), and time is counted in years. Sometimes, time may be given in months, which you will need to convert into years before using formulas.
Simple Interest (SI) is the interest calculated only on the original principal amount throughout the entire time period. It does not consider any interest that has been added previously.
For example, if you deposit INR 10,000 in a bank at 5% simple interest per annum for 3 years, the interest is calculated only on INR 10,000 every year.
Here's what each part means:
The total amount (A) you get after the interest period is the sum of the principal and the simple interest:
Step 1: Identify the values:
Step 2: Use the simple interest formula:
\( SI = \frac{P \times R \times T}{100} = \frac{50,000 \times 6 \times 3}{100} \)
Step 3: Calculate:
\( SI = \frac{50,000 \times 18}{100} = \frac{900,000}{100} = 9,000 \) INR
Step 4: Total amount after 3 years:
\( A = P + SI = 50,000 + 9,000 = 59,000 \) INR
Answer: The simple interest earned is INR 9,000, and the total amount after 3 years is INR 59,000.
Compound Interest (CI) is interest calculated on the principal amount plus any interest that has already been added. This means you earn interest on interest, which makes your money grow faster over time.
For example, if you invest INR 10,000 at 5% compound interest per annum, after the first year, you earn interest on INR 10,000. In the second year, interest is calculated on INR 10,000 plus the interest earned in the first year.
The amount (A) is the total money you have after interest is added, and compound interest (CI) is the extra money earned over the principal.
Interest can be compounded at different intervals:
When interest is compounded more than once a year, the formula adjusts to:
Here, the rate is divided by the number of compounding periods, and the time is multiplied by the same number.
Step 1: Identify the values:
Step 2: Use the compound interest amount formula:
\( A = P \left(1 + \frac{R}{100n}\right)^{nT} = 40,000 \left(1 + \frac{8}{100 \times 1}\right)^{1 \times 2} \)
\( A = 40,000 \times (1 + 0.08)^2 = 40,000 \times (1.08)^2 \)
Step 3: Calculate \( (1.08)^2 = 1.1664 \)
\( A = 40,000 \times 1.1664 = 46,656 \) INR
Step 4: Calculate compound interest:
\( CI = A - P = 46,656 - 40,000 = 6,656 \) INR
Answer: The compound interest earned is INR 6,656, and the total amount after 2 years is INR 46,656.
Step 1: Identify the values:
Step 2: Adjust rate and time for half-yearly compounding:
Step 3: Use the formula:
\( A = P \left(1 + \frac{R}{100n}\right)^{nT} = 30,000 \times (1 + 0.05)^6 \)
Step 4: Calculate \( (1.05)^6 \):
\( (1.05)^6 = 1.3401 \) (approx.)
Step 5: Calculate amount:
\( A = 30,000 \times 1.3401 = 40,203 \) INR (approx.)
Step 6: Calculate compound interest:
\( CI = A - P = 40,203 - 30,000 = 10,203 \) INR (approx.)
Answer: The compound interest earned is approximately INR 10,203, and the total amount after 3 years is approximately INR 40,203.
| Feature | Simple Interest (SI) | Compound Interest (CI) |
|---|---|---|
| Interest on | Only principal | Principal + accumulated interest |
| Formula | No | A = P(1 + \frac{R}{100})^T |
| Growth | Linear | Exponential |
| Use | Short-term loans, fixed deposits | Long-term investments, loans |
| Amount | A = P + SI | A = P(1 + \frac{R}{100})^T |
Step 1: Calculate simple interest:
\( SI = \frac{P \times R \times T}{100} = \frac{25,000 \times 7 \times 4}{100} = \frac{700,000}{100} = 7,000 \) INR
Step 2: Calculate amount with simple interest:
\( A_{SI} = P + SI = 25,000 + 7,000 = 32,000 \) INR
Step 3: Calculate compound interest amount:
\( A = P \left(1 + \frac{R}{100}\right)^T = 25,000 \times (1 + 0.07)^4 \)
\( (1.07)^4 = 1.3108 \) (approx.)
\( A = 25,000 \times 1.3108 = 32,770 \) INR (approx.)
Step 4: Calculate compound interest:
\( CI = A - P = 32,770 - 25,000 = 7,770 \) INR (approx.)
Answer: Simple interest earned is INR 7,000 and compound interest earned is approximately INR 7,770. Compound interest yields more money over the same period.
Step 1: Identify known values:
Step 2: Use the simple interest formula and solve for \( R \):
\( SI = \frac{P \times R \times T}{100} \Rightarrow R = \frac{SI \times 100}{P \times T} \)
\( R = \frac{1,800 \times 100}{12,000 \times 3} = \frac{180,000}{36,000} = 5\% \)
Answer: The rate of interest is 5% per annum.
Step 1: Identify known values:
Step 2: Calculate the amount \( A \):
\( A = P + CI = 20,000 + 5,512 = 25,512 \) INR
Step 3: Use compound interest amount formula:
\( A = P \left(1 + \frac{R}{100}\right)^T \Rightarrow \left(1 + \frac{5}{100}\right)^T = \frac{A}{P} \)
\( (1.05)^T = \frac{25,512}{20,000} = 1.2756 \)
Step 4: Find \( T \) such that \( (1.05)^T = 1.2756 \).
We know \( (1.05)^5 = 1.2763 \) (approx.), very close to 1.2756.
Step 5: Therefore, \( T \approx 5 \) years.
Answer: The time period is approximately 5 years.
When to use: For all simple interest calculations.
When to use: When compounding frequency is more than once a year.
When to use: When exact calculations are time-consuming and approximate answers are acceptable.
When to use: When comparing investment options.
When to use: When amount is provided directly.
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