krit.club logo

Compound Interest - Formula for Compound Interest

Grade 8ICSE

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

🔑Concepts

•

Compound Interest (CI) is the interest calculated on the initial principal and also on the accumulated interest of previous periods. Think of it as 'interest on interest,' which can be visualized as a snowball effect where the amount grows faster over time compared to simple interest.

•

The Principal (PP) is the original sum of money borrowed or invested. In compound interest, the principal for the second year is the amount (principal + interest) at the end of the first year. This creates a rising step-ladder effect in your account balance.

•

The Conversion Period is the fixed interval of time after which the interest is calculated and added to the principal. On a timeline, if interest is compounded annually, the marks are at 1-year intervals; if half-yearly, the marks are every 6 months, doubling the number of times interest is applied.

•

The Rate of Interest (RR) is usually expressed as a percentage per annum (p.a.). When the conversion period is not a year, the rate must be adjusted accordingly (e.g., divided by 2 for half-yearly or 4 for quarterly).

•

The Amount (AA) is the total money (Principal + Compound Interest) at the end of the time period (nn). Visualizing this on a growth graph, the line for compound interest is a curve that bends upwards (exponential growth), while simple interest is a straight line.

•

Compound Interest can also be applied to non-financial situations like Population Growth or Depreciation. In growth, the value increases (++ sign in formula), whereas in depreciation—like the decreasing value of a car over time—the value decreases (−- sign in formula).

•

Difference between CI and SI: For the first conversion period (usually the first year), Compound Interest is equal to Simple Interest. From the second period onwards, CI is always greater than SI because the interest itself starts earning more interest.

📐Formulae

A=P(1+R100)nA = P\left(1 + \frac{R}{100}\right)^{n}

CI=A−P=P[(1+R100)n−1]CI = A - P = P\left[\left(1 + \frac{R}{100}\right)^{n} - 1\right]

When interest is compounded half-yearly: A=P(1+R2×100)2nA = P\left(1 + \frac{R}{2 \times 100}\right)^{2n}

When interest is compounded quarterly: A=P(1+R4×100)4nA = P\left(1 + \frac{R}{4 \times 100}\right)^{4n}

When rates are different for different years (R1,R2,R3R_1, R_2, R_3): A=P(1+R1100)(1+R2100)(1+R3100)A = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)

Formula for Depreciation: V=P(1−R100)nV = P\left(1 - \frac{R}{100}\right)^{n}

💡Examples

Problem 1:

Find the compound interest on ₹12,000₹ 12,000 for 2 years at 10%10\% per annum compounded annually.

Solution:

Given: P=12000P = 12000, R=10%R = 10\%, n=2n = 2 years. Step 1: Use the amount formula A=P(1+R100)nA = P(1 + \frac{R}{100})^n. A=12000(1+10100)2A = 12000(1 + \frac{10}{100})^2 A=12000(1+110)2=12000(1110)2A = 12000(1 + \frac{1}{10})^2 = 12000(\frac{11}{10})^2 A=12000×121100A = 12000 \times \frac{121}{100} A=120×121=₹14,520A = 120 \times 121 = ₹ 14,520. Step 2: Calculate CI using CI=A−PCI = A - P. CI=14520−12000=₹2,520CI = 14520 - 12000 = ₹ 2,520.

Explanation:

To solve this, we first calculate the total amount accumulated after 2 years using the standard formula and then subtract the original principal to find only the interest component.

Problem 2:

Calculate the amount for ₹8,000₹ 8,000 for 1 year at 20%20\% per annum compounded half-yearly.

Solution:

Given: P=8000P = 8000, R=20%R = 20\% p.a., n=1n = 1 year. Since interest is compounded half-yearly: New Rate (rr) = 202=10%\frac{20}{2} = 10\% per half-year. New Time (tt) = 1×2=21 \times 2 = 2 half-years. Step 1: Calculate Amount. A=P(1+r100)tA = P(1 + \frac{r}{100})^t A=8000(1+10100)2A = 8000(1 + \frac{10}{100})^2 A=8000(1.1)2=8000×1.21A = 8000(1.1)^2 = 8000 \times 1.21 A=₹9,680A = ₹ 9,680.

Explanation:

When compounding happens half-yearly, we must divide the annual rate by 2 and multiply the number of years by 2 to get the total number of conversion periods.