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Number - Compound Interest and Amount

Grade 8IB

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

🔑Concepts

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Compound Interest (CICI) is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.

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The Principal (PP) is the original amount of money invested or borrowed.

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The Amount (AA) is the total sum of money at the end of the time period, including interest.

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The Rate (RR) is the percentage interest charged or earned per annum.

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The Time (nn) is the number of years or conversion periods for which the money is invested.

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In Compound Interest, the amount at the end of the first year becomes the principal for the second year, unlike Simple Interest where the principal remains constant.

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Interest can be compounded annually, half-yearly (semi-annually), or quarterly. If compounded half-yearly, the rate is halved (R/2R/2) and the time is doubled (2n2n).

📐Formulae

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

CI=A−PCI = A - P

A=P(1+R2×100)2n (Compounded half-yearly)A = P \left(1 + \frac{R}{2 \times 100}\right)^{2n} \text{ (Compounded half-yearly)}

A=P(1+R4×100)4n (Compounded quarterly)A = P \left(1 + \frac{R}{4 \times 100}\right)^{4n} \text{ (Compounded quarterly)}

💡Examples

Problem 1:

Calculate the amount and compound interest on Rs 8000 for 22 years at 5%5\% per annum compounded annually.

Solution:

Given: P=8000P = 8000, R=5%R = 5\%, n=2n = 2. Using the formula A=P(1+R100)nA = P \left(1 + \frac{R}{100}\right)^n, we get: A=8000(1+5100)2A = 8000 \left(1 + \frac{5}{100}\right)^2 A=8000(105100)2A = 8000 \left(\frac{105}{100}\right)^2 A=8000×1.05×1.05=8820A = 8000 \times 1.05 \times 1.05 = 8820. To find Compound Interest (CICI): CI=A−PCI = A - P 8820−8000820\begin{array}{r} 8820 \\ - 8000 \\ \hline 820 \end{array}

Explanation:

Substitute the values into the compound interest amount formula. First, calculate the bracket (1+0.05)(1 + 0.05), then square it since n=2n=2, and multiply by the principal. Finally, subtract the principal from the total amount to find the interest.

Problem 2:

Find the amount of Rs 10000 for 11 year at 10%10\% per annum compounded half-yearly.

Solution:

Given: P=10000P = 10000, R=10%R = 10\% per annum, n=1n = 1 year. Since it is compounded half-yearly, we use: Rate per half-year = 10/2=5%10/2 = 5\% and Number of periods = 1×2=21 \times 2 = 2. A=10000(1+5100)2A = 10000 \left(1 + \frac{5}{100}\right)^2 A=10000(1.05)2A = 10000 \left(1.05\right)^2 A=10000×1.1025=11025A = 10000 \times 1.1025 = 11025.

Explanation:

When compounding half-yearly, the annual interest rate is divided by 22 and the number of years is multiplied by 22 because there are two 6-month periods in a year.

Problem 3:

What is the difference between Simple Interest and Compound Interest on Rs 5000 for 22 years at 10%10\% per annum?

Solution:

  1. Simple Interest (SISI): SI=P×R×T100=5000×10×2100=1000SI = \frac{P \times R \times T}{100} = \frac{5000 \times 10 \times 2}{100} = 1000. 2. Compound Interest (CICI): A=5000(1+10100)2=5000×(1.1)2=5000×1.21=6050A = 5000 \left(1 + \frac{10}{100}\right)^2 = 5000 \times (1.1)^2 = 5000 \times 1.21 = 6050. CI=6050−5000=1050CI = 6050 - 5000 = 1050. 3. Difference: 1050−100050\begin{array}{r} 1050 \\ - 1000 \\ \hline 50 \end{array}

Explanation:

Calculate the Simple Interest using the linear formula. Then calculate the Compound Interest by finding the Amount first. Finally, find the difference between the two results. The CICI is always higher than SISI for the same period (greater than 1 year) because of interest earned on interest.