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Compound Interest

Unit: 2

Book Icon Class 10: Mathematics

Compound Interest, Simple Interest vs Compound Interest, Compound Amount, Formula for Compound Amount, Formula for Compound Interest, Example questions (SEE) and Answers

Compound Interest

The interest that is calculated each year or specified time period (quarterly, half-yearly, or yearly) by adding interest to the previous principle is called Compound Interest (CI). Compound interest is the interest calculated on the initial principal amount as well as on the accumulated interest from previous periods. Unlike simple interest, where interest is calculated only on the principal, compound interest allows your money to grow faster over time because it is always calculated by adding interest to the previous principle.

Formula for Compound Interest(C.I.):

\(\begin{aligned} & \text{Compound Interest (CI)} = \text{Compound Amount (CA)} - \text{Principal (P)} \\ \\ & \therefore \text{CI} = P [\left(1 + \frac{R}{100}\right)^{T} - 1] \end{aligned}\)

 

Compound Amount

The compound amount is the total amount of the previous principal plus the interest. It is represented by \(C.A\).

The compound amount (C.A.) is given by:

\(\begin{aligned} & \text{Compound Amount (CA)} = \text{Principal (P)} + \text{Compound Interest (CI)} \\ \\ & \therefore \text{C.A.} = P\left(1 + \frac{R}{100}\right)^{T} \end{aligned}\)

Where:

- \(C.A\) = Compound Amount, \(P\) = Principal amount, \(R\) = Annual interest rate, \(T\) = Time (In years)

 

Simple Interest vs Compound Interest

Let us consider Hupen (payment: Simple Interest) and Ram (payment: Compound Interest) took Rs. 100 for 3 years at the rate of 10% from a commercial bank. Let’s calculate the interest for each of them.

For Hupen (Simple Interest)For Ram (Compound Interest)
1. Simple Interest is calculated with formula \(I = \frac{PTR}{100}\). Principal (P) being same.1. Compound Interest is calculated with the same formula \(I = \frac{PTR}{100}\). But Principal (P) being (Previous principle + Interest).
2. For \(s^{st}\) year: Principal \( P_1 = \text{Rs. 100}\) So, \(I_1 = \frac{(100 \times 10 \times 1)}{100} = \text{Rs. 10}\).2. For \(1^{st}\) year: Principal \( P_1 = \text{Rs. 100}\) So, \(I_1 = \frac{(100 \times 10 \times 1)}{100} = \text{Rs. 10}\).
3. For \(2^{nd}\) year: Principal \( P_2 = \text{Rs. 100}\) So, \(I_2 = \frac{(100 \times 10 \times 2)}{100} = \text{Rs. 20}\).3. For \(2^{nd}\) year: Principal \( P_2 = \text{Rs. 100} + \text{Rs. 10} = \text{Rs. 110}\) So, \(I_2 = \frac{(110 \times 10 \times 2)}{100} = \text{Rs. 22}\).
4. For \(3^{rd}\) year: Principal \( P_3 = \text{Rs. 100}\) So, \(I_3 = \frac{(100 \times 10 \times 3)}{100} = \text{Rs. 30}\).4. For \(3^{rd}\) year: Principal \( P_3 = \text{Rs. 110} + \text{Rs. 22} = \text{Rs. 132}\) So, \(I_3 = \frac{(132\times 10 \times 3)}{100} = \text{Rs. 39.6}\).
Hence, Total SI = \(I_1 + I_2 + I_3 = 10 + 20 + 30 = \text{Rs. 60} \)Hence, Total CI = \(I_1 + I_2 + I_3 = 10 + 22 + 39.6 = \text{Rs. 71.6} \)

Note: As you can see the compound interest is always more than the simple interest.

 

Formulas for different conditions

1. When the interest is compounded annually

\(\begin{aligned} & \text{C.A.} = P\left(1 + \frac{R}{100}\right)^{T} \\ \\ & \text{C.I.} = P [\left(1 + \frac{R}{100}\right)^{T} - 1] \end{aligned}\)

 

2. When the interest is compounded half-annually

\(\begin{aligned} & \text{C.A.} = P\left(1 + \frac{R}{200}\right)^{2T} \\ \\ & \text{C.I.} = P [\left(1 + \frac{R}{200}\right)^{2T} - 1] \end{aligned}\)

 

3. When the interest is compounded quarterly

\(\begin{aligned} & \text{C.A.} = P\left(1 + \frac{R}{400}\right)^{4T} \\ \\ & \text{C.I.} = P [\left(1 + \frac{R}{400}\right)^{4T} - 1] \end{aligned}\)

 

4. When the interest rate is different every year

Let us consider interest rates \(R_1 %\), \(R_2 %\), \(R_3 %\) for three consecutive years.

\(\begin{aligned} & \text{C.A.} = P\left(1 + \frac{R_1}{100}\right) \left(1 + \frac{R_2}{100}\right) \left(1 + \frac{R_3}{100}\right) \\ \\ & \text{C.I.} = P [\left(1 + \frac{R_1}{100}\right) \left(1 + \frac{R_2}{100}\right) \left(1 + \frac{R_3}{100}\right) - 1] \end{aligned}\)

 

5. When the interest is compounded for \(T\) years and \(M\) months

\(\begin{aligned} & \text{C.A.} = P\left(1 + \frac{R}{100}\right)^{T} \left(1 + \frac{MR}{1200}\right) \\ \\ & \text{C.I.} = P [\left(1 + \frac{R}{100}\right)^{T} \left(1 + \frac{MR}{1200}\right) - 1] \end{aligned}\)

 

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