7.4-Compound Interest

7.4-Compound Interest Important Formulae

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7.4 - Compound Interest
  • Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods.
  • Formula for Compound Interest (CI): $A = P \left(1 + \frac{r}{100}\right)^t$, where A = Amount after t years, P = Principal, r = Rate of interest per year, t = Time in years.
  • Compound Interest (CI) = $A - P$
  • CI can be calculated annually, semi-annually, or quarterly depending on the problem.
  • It grows faster than simple interest due to the interest-on-interest effect.

7.4 - Compound Interest

Compound Interest is the interest calculated on the initial principal, which also includes the interest that has been added to the principal from previous periods. Unlike Simple Interest, which is calculated only on the original principal, Compound Interest is calculated on both the principal and the accumulated interest. Compound Interest is commonly used in savings accounts, loans, and investments.

The formula for Compound Interest is:

Compound Interest (CI) = P $(1 + \frac{r}{100})^t - P$

Where:

  • P = Principal amount (the initial amount of money)
  • r = Rate of interest per annum
  • t = Time period in years
  • A = Amount after interest

The formula for the amount after compound interest is:

A = P $(1 + \frac{r}{100})^t$

Where A is the total amount after interest, including both the principal and the compound interest.

Process of Calculation

In compound interest, interest is added at regular intervals, such as annually, semi-annually, quarterly, or monthly. The more frequently interest is compounded, the more interest will accumulate.

Types of Compound Interest
  • Annual Compound Interest: When the interest is compounded once a year.
  • Quarterly Compound Interest: When the interest is compounded four times a year, i.e., every three months.
  • Monthly Compound Interest: When the interest is compounded twelve times a year, i.e., every month.

The formula for compound interest when the interest is compounded n times a year is:

A = P $(1 + \frac{r}{100n})^{nt}$

Where:

  • n = Number of times interest is compounded per year
  • t = Time in years
Examples

1. If ₹1000 is invested at an annual interest rate of 10% for 2 years, the amount after 2 years is:

A = 1000 $(1 + \frac{10}{100})^2 = 1000 $(1 + 0.1)^2 = 1000 $(1.1)^2 = 1000 * 1.21 = ₹1210$

The compound interest is the total amount minus the principal: CI = ₹1210 - ₹1000 = ₹210.

2. If ₹2000 is invested at a quarterly interest rate of 4% for 1 year, the amount after 1 year is:

A = 2000 $(1 + \frac{4}{100 \times 4})^{4 \times 1} = 2000 $(1 + 0.01)^4 = 2000 * 1.04060401 = ₹2081.21$

The compound interest is CI = ₹2081.21 - ₹2000 = ₹81.21.

Important Points to Remember
  • Compound Interest grows faster than Simple Interest because interest is calculated on both the principal and the accumulated interest.
  • Frequency of compounding impacts the total interest earned. The more frequent the compounding, the greater the interest.
  • In Compound Interest, the formula includes the exponentiation of $(1 + \frac{r}{100})$ to the power of time or the product of time and frequency of compounding.
  • When calculating Compound Interest for different time periods and frequencies, the principal and rate may be adjusted accordingly to match the specific case.

7.4-Compound Interest

संयुग्म ब्याज (Compound Interest) वह ब्याज है जो मूलधन के साथ पिछले ब्याज पर भी ब्याज लगाया जाता है। इसका मतलब है कि ब्याज समय के साथ बढ़ता रहता है, क्योंकि ब्याज का गणना पहले के ब्याज और मूलधन दोनों पर होता है।

संयुग्म ब्याज की गणना के लिए निम्नलिखित सूत्र का उपयोग किया जाता है:

संयुग्म ब्याज (CI) के लिए सूत्र:

$A = P \left( 1 + \dfrac{r}{100} \right)^t$

यहाँ:

  • A = कुल राशि (Principal + Interest)
  • P = मूलधन (Principal)
  • r = वार्षिक ब्याज दर (Rate of Interest)
  • t = समय (Time) सालों में

संयुग्म ब्याज (Compound Interest) की गणना करने के लिए, कुल राशि (A) से मूलधन (P) को घटाकर ब्याज निकाला जाता है:

$CI = A - P$

संयुग्म ब्याज का लाभ तब देखा जाता है जब ब्याज समय के साथ बढ़ता है। उदाहरण के लिए, यदि किसी राशि पर 10% सालाना ब्याज लगता है, तो पहले साल के बाद ब्याज उस मूलधन पर लगता है, लेकिन दूसरे साल में ब्याज पहले के ब्याज को भी शामिल करता है।

यदि ब्याज हर साल संयुग्मित (compounded annually) होता है, तो यह सूत्र उपयुक्त होता है:

$A = P \left( 1 + \dfrac{r}{100} \right)^t$

लेकिन यदि ब्याज को महीने के हिसाब से संयुग्मित किया जाता है, तो यह सूत्र उपयोग किया जाता है:

$A = P \left( 1 + \dfrac{r}{100n} \right)^{nt}$

यहाँ n = प्रति वर्ष ब्याज संयुग्मन की संख्या है (उदाहरण के लिए, यदि ब्याज मासिक संयुग्मित होता है, तो n = 12 होगा)।

संयुग्म ब्याज की प्रक्रिया को समझने के लिए एक उदाहरण देखते हैं:

उदाहरण: यदि किसी व्यक्ति ने ₹1000 का निवेश किया है, और ब्याज दर 10% प्रति वर्ष है, तो 2 साल के बाद ब्याज की गणना कैसे की जाएगी?

  • मूलधन (P) = ₹1000
  • ब्याज दर (r) = 10%
  • समय (t) = 2 साल

संयुग्म ब्याज की गणना के लिए:

$A = 1000 \left( 1 + \dfrac{10}{100} \right)^2$

$A = 1000 \left( 1 + 0.1 \right)^2 = 1000 \times 1.21 = ₹1210$

अब, संयुग्म ब्याज (CI) होगा:

$CI = A - P = 1210 - 1000 = ₹210$

इस प्रकार, 2 साल बाद ब्याज ₹210 होगा।

इस उदाहरण से यह स्पष्ट होता है कि संयुग्म ब्याज के द्वारा समय के साथ ब्याज की राशि बढ़ती जाती है, क्योंकि ब्याज पहले के ब्याज को भी शामिल करता है।

संयुग्म ब्याज में ब्याज को समय-समय पर जोड़ने से, यह साधारण ब्याज (Simple Interest) से अधिक होता है।