Comparing Quantities Ex 8.3: Understanding Compound Interest

What is Compound Interest and How is it Different?

Imagine you deposit ₹1,000 in a bank. With Simple Interest (SI), if the rate is 10% per year, you earn ₹100 every year. The principal amount (your initial ₹1,000) never changes. After 3 years, you would have earned ₹300 in interest.

Now, let's see how Compound Interest (CI) works. In the first year, you earn ₹100 (10% of ₹1,000), just like with SI. But here's the magic: for the second year, the bank calculates interest not just on your initial ₹1,000, but on the new total of ₹1,100 (Principal + Year 1 Interest). So, in Year 2, you earn 10% of ₹1,100, which is ₹110. Your new total becomes ₹1,210. For Year 3, the interest is calculated on ₹1,210! This is why it's called 'compounding' – the interest gets added to the principal and starts earning its own interest. This method makes money grow exponentially over time.

Calculating Compound Interest: The Formula Method

1. Step 1: Understand the Formula and Variables — The formula to find the final amount (A) is: A = P(1 + R/100)^n. A = Final Amount (Principal + Interest) P = Principal (the initial sum of money) R = Rate of Interest (per year) n = Number of years (the time period)
2. Step 2: Identify Values and Adjust for Compounding Period — Read the question carefully to find P, R, and n. Pay close attention to how often the interest is compounded. If it's 'compounded annually', use R and n as they are. If 'compounded half-yearly', you must divide R by 2 and multiply n by 2 because there are two compounding periods in a year.
3. Step 3: Substitute and Calculate the Amount (A) — Place the values of P, R, and n into the formula. First, calculate the value inside the bracket (1 + R/100). Then, calculate the power (raise the bracket value to the power of 'n'). Finally, multiply this result by the principal (P) to get the total amount (A).
4. Step 4: Calculate the Compound Interest (CI) — The formula gives you the total Amount (A), not just the interest. To find the compound interest, you must subtract the original principal from the final amount. The formula is: CI = A - P.

Solved Example: Understanding Half-Yearly Compounding

• Problem: Calculate the amount and compound interest on Rs 8,000 for 1 year at 10% per annum, compounded half-yearly. Solution: Given: Principal (P) = Rs 8,000; Annual Rate (R) = 10%; Time (n) = 1 year. Adjustment for Half-Yearly Compounding: Since interest is compounded half-yearly, we need to adjust R and n. New Rate (R) = 10% / 2 = 5% per half-year. New Time (n) = 1 year 2 = 2 half-years. Using the formula A = P(1 + R/100)^n: A = 8000 (1 + 5/100)^2 A = 8000 (1 + 1/20)^2 A = 8000 (21/20)^2 A = 8000 (21/20) (21/20) A = 20 21 21 A = Rs 8,820 Calculating Compound Interest (CI): CI = Amount - Principal CI = 8820 - 8000 = Rs 820. Final Answer: The total amount is Rs 8,820 and the compound interest earned is Rs 820.

Exam Tips for Compound Interest

A very common mistake is forgetting the final step! The formula A = P(1 + R/100)^n gives you the Total Amount, not the interest. If the question asks for the Compound Interest (CI), you MUST subtract the principal from the amount: CI = A - P. Always double-check what the question is asking for – 'amount' or 'interest'. Another key point is to be careful with the compounding period. For half-yearly, divide the rate by 2 and multiply time by 2. For quarterly, divide the rate by 4 and multiply time by 4. Getting this adjustment right is crucial for a correct answer.

Practice Questions with Solutions

• Q: Find the amount and the compound interest on Rs 10,000 for 3 years at 10% per annum, compounded annually.
• A: Step 1: Identify the given values. Principal (P) = Rs 10,000, Rate (R) = 10% per annum, Time (n) = 3 years. Step 2: Use the formula for Amount (A) when compounded annually: A = P(1 + R/100)^n. A = 10000(1 + 10/100)^3 A = 10000(1 + 1/10)^3 A = 10000(11/10)^3 A = 10000 (1331/1000) A = 10 1331 A = Rs 13,310. Step 3: Calculate Compound Interest (CI) using the formula CI = A - P. CI = 13310 - 10000 CI = Rs 3,310. Final answer: The amount is Rs 13,310 and the compound interest is Rs 3,310.
• Q: Calculate the amount and compound interest on Rs 8,000 for 1 year at 10% per annum, compounded half-yearly.
• A: Step 1: Identify the given values. Principal (P) = Rs 8,000, Rate (R) = 10% per annum, Time (n) = 1 year. Step 2: Since the interest is compounded half-yearly, the rate is halved and the time is doubled. New Rate (R') = R/2 = 10%/2 = 5% per half-year. New Time (n') = n 2 = 1 2 = 2 half-years. Step 3: Use the formula for Amount (A) when compounded half-yearly: A = P(1 + R'/100)^n'. A = 8000(1 + 5/100)^2 A = 8000(1 + 1/20)^2 A = 8000(21/20)^2 A = 8000 (441/400) A = 20 441 A = Rs 8,820. Step 4: Calculate Compound Interest (CI) using the formula CI = A - P. CI = 8820 - 8000 CI = Rs 820. Final answer: The amount is Rs 8,820 and the compound interest is Rs 820.
• Q: Find the difference between the simple interest and compound interest on Rs 5,000 for 2 years at 8% per annum, compounded annually.
• A: Step 1: Identify the given values. Principal (P) = Rs 5,000, Rate (R) = 8% per annum, Time (T or n) = 2 years. Step 2: Calculate Simple Interest (SI). SI = (P R T) / 100 SI = (5000 8 2) / 100 SI = 50 8 2 SI = Rs 800. Step 3: Calculate Compound Interest (CI). First, find the Amount (A). A = P(1 + R/100)^n A = 5000(1 + 8/100)^2 A = 5000(1 + 2/25)^2 A = 5000(27/25)^2 A = 5000 (729/625) A = (5000/625) 729 A = 8 * 729 A = Rs 5,832. CI = A - P = 5832 - 5000 = Rs 832. Step 4: Find the difference between CI and SI. Difference = CI - SI = 832 - 800 = Rs 32. Final answer: The difference between the simple interest and compound interest is Rs 32.

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