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Commercial Mathematics - Compound Interest (without using formula)

Grade 9ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of Compound Interest (CI): Unlike Simple Interest where the principal remains constant, Compound Interest is the interest calculated on the initial principal plus the accumulated interest of previous periods. Think of it as 'interest on interest,' visualized as a snowball gathering more snow (interest) as it rolls down a hill.

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The Iterative Process: To calculate CI without a formula, you must calculate the Simple Interest for one year at a time. The calculated interest is added to the principal of that year to get the amount, which then becomes the principal for the following year. This forms a staircase-like growth where each step is taller than the last.

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The Year-by-Year Principal Shift: The most crucial rule is that the Amount (AA) at the end of Year 1 (A1A_1) becomes the Principal (PP) for Year 2 (P2P_2). Similarly, A2A_2 becomes P3P_3, and so on. This chain continues until the end of the given time period.

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Compounding Periods: While usually annual, interest can be compounded half-yearly or quarterly. If compounded half-yearly, the year is divided into two 6-month intervals. Visualize a timeline split into equal segments; the interest is added to the principal at the end of every segment rather than just at the end of the year.

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Simple Interest vs. Compound Interest: For the very first conversion period (e.g., the first year), SISI is equal to CICI. However, from the second period onwards, CICI is always greater than SISI because the principal increases. On a graph, SISI represents a straight diagonal line, while CICI represents an upward-curving exponential line.

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Final Calculation of CI: After finding the final amount at the end of the total time duration, the total Compound Interest earned is determined by subtracting the original Initial Principal from the final accumulated Amount. This represents the total 'extra' money earned over the duration.

📐Formulae

Simple Interest for one period: I=P×R×T100I = \frac{P \times R \times T}{100}

Amount after a period: A=P+IA = P + I

Principal for the next period: Pnew=ApreviousP_{new} = A_{previous}

Total Compound Interest: CI=Final Amount−Original PrincipalCI = \text{Final Amount} - \text{Original Principal}

Interest for any specific year nn: In=An−An−1I_n = A_n - A_{n-1}

💡Examples

Problem 1:

Calculate the compound interest on ₹6,000 for 2 years at 5% per annum, without using the formula.

Solution:

Step 1: Calculate interest for the 1st year. Principal (P1P_1) = ₹6,000 Rate (RR) = 5% Time (TT) = 1 year Interest for the 1st year (I1I_1) = 6000×5×1100=₹300\frac{6000 \times 5 \times 1}{100} = ₹300 Amount at the end of the 1st year (A1A_1) = 6000+300=₹6,3006000 + 300 = ₹6,300

Step 2: Calculate interest for the 2nd year. New Principal (P2P_2) = A1A_1 = ₹6,300 Rate (RR) = 5% Time (TT) = 1 year Interest for the 2nd year (I2I_2) = 6300×5×1100=₹315\frac{6300 \times 5 \times 1}{100} = ₹315 Amount at the end of the 2nd year (A2A_2) = 6300+315=₹6,6156300 + 315 = ₹6,615

Step 3: Calculate the total Compound Interest. Total CI=Final Amount−Initial PrincipalCI = \text{Final Amount} - \text{Initial Principal} CI=6615−6000=₹615CI = 6615 - 6000 = ₹615

Explanation:

We calculate the interest for each year separately using the Simple Interest formula. The amount at the end of the first year is treated as the starting principal for the second year. Finally, we subtract the original sum from the final amount to find the total interest.

Problem 2:

Calculate the compound interest on ₹8,000 for 1 year at 10% per annum, interest being compounded half-yearly.

Solution:

Since interest is compounded half-yearly, there are 2 conversion periods of 6 months each. The rate for 6 months = 102=5%\frac{10}{2} = 5\%.

Step 1: Calculate interest for the first 6 months. P1=₹8,000,R=5%,T=1P_1 = ₹8,000, R = 5\%, T = 1 (half-year unit) I1=8000×5×1100=₹400I_1 = \frac{8000 \times 5 \times 1}{100} = ₹400 A1=8000+400=₹8,400A_1 = 8000 + 400 = ₹8,400

Step 2: Calculate interest for the next 6 months. P2=₹8,400,R=5%,T=1P_2 = ₹8,400, R = 5\%, T = 1 (half-year unit) I2=8400×5×1100=₹420I_2 = \frac{8400 \times 5 \times 1}{100} = ₹420 A2=8400+420=₹8,820A_2 = 8400 + 420 = ₹8,820

Step 3: Total CI calculation. CI=A2−P1=8820−8000=₹820CI = A_2 - P_1 = 8820 - 8000 = ₹820

Explanation:

When compounding half-yearly, we divide the annual rate by 2 and double the number of periods. We then apply the step-by-step calculation for each 6-month block, updating the principal halfway through the year.