7.6-Applications of Compound Interest Formula

7.6-Applications of Compound Interest Formula Important Formulae

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7.6 - Applications of Compound Interest Formula
  • Compound Interest is used in calculating the growth of investments, savings, and loans.
  • The formula for Compound Interest is $A = P \left( 1 + \frac{r}{100} \right)^t$, where:
    • $A$ = Amount after $t$ years
    • $P$ = Principal amount
    • $r$ = Annual interest rate
    • $t$ = Time in years
  • Compound Interest is applied in financial sectors like banks, insurance, and business investments.
  • It is different from Simple Interest, as the interest is calculated on the principal plus accumulated interest.

7.6 - Applications of Compound Interest Formula

In this section, we explore the various applications of the compound interest formula. Compound interest is a method of calculating interest where the interest earned over time is added to the principal amount, and the interest for the next period is calculated on this new total. The compound interest formula is particularly useful in real-life situations like investments, loans, and savings accounts. Let's go through some common applications.

The compound interest formula is given by:

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

Where:

  • A = Amount (Principal + Interest)
  • P = Principal (Initial Investment)
  • r = Rate of Interest per period
  • n = Number of periods (years, months, etc.)

Let's now discuss some of the applications:

1. Investment Growth

When you invest money in a savings account or a fixed deposit, the bank calculates interest using the compound interest formula. The interest earned adds to the principal amount, and the new amount earns interest in the next period. The longer the investment duration, the higher the growth due to the effect of compounding.

2. Loans and Mortgages

When individuals or businesses take loans from banks or financial institutions, the loan is often repaid with interest. Compound interest is applied to loans, where the interest on the loan amount is added to the total debt. This results in the total loan amount increasing over time, depending on the interest rate and the time duration of the loan.

3. Population Growth

In demographic studies, compound interest models are used to estimate the growth of a population over time. The population at a given time can be calculated using the compound interest formula, where the population grows at a constant rate, similar to how an investment grows with compound interest.

4. Depreciation of Assets

Just as compound interest is used for growth, it can also be used to model depreciation, or the reduction in value of assets over time. In this case, the value of an asset decreases over time according to a fixed percentage. The formula for depreciation is similar to compound interest, but the rate is negative, indicating a decrease in value.

5. Continuous Compound Interest

In some cases, interest is compounded continuously. The formula for continuously compounded interest is different from the regular compound interest formula. It is given by:

$A = P e^{rt}$

Where:

  • A = Amount after time t
  • P = Principal
  • r = Rate of interest
  • t = Time
  • e = Euler's constant (approximately 2.718)

This formula shows that as the number of compounding periods increases, the amount approaches a limit. Continuous compounding is used in various financial and scientific applications.

6. Compound Interest on Half-Yearly or Quarterly Basis

In many cases, the interest is compounded not annually but on a half-yearly, quarterly, or monthly basis. The formula changes slightly to account for the different number of compounding periods. If the interest is compounded m times a year, the formula becomes:

$A = P \left(1 + \frac{r}{100m}\right)^{mn}$

Where m represents the number of times the interest is compounded in one year. For example, if the interest is compounded quarterly, then m = 4.

7. Interest Earned from Mutual Funds

In investments such as mutual funds, interest is often compounded periodically. The value of an investment grows over time due to compounding, and investors can calculate the future value of their investment using the compound interest formula.

The understanding and application of the compound interest formula are important in various fields such as finance, economics, and business, where it helps to make better decisions related to investments, loans, and growth calculations.

7.6 - अनुपात ब्याज सूत्र के अनुप्रयोग

इस उपविषय में हम अनुपात ब्याज सूत्र के विभिन्न अनुप्रयोगों के बारे में जानेंगे। अनुपात ब्याज का प्रयोग वित्तीय लेन-देन, निवेश, ऋण, और अन्य आर्थिक क्षेत्रों में होता है। हम इसके कुछ महत्वपूर्ण अनुप्रयोगों पर चर्चा करेंगे।

1. निवेश पर ब्याज (Interest on Investment): यदि कोई व्यक्ति किसी बैंक में धन निवेश करता है, तो उस पर ब्याज मिलता है। अनुपात ब्याज सूत्र का उपयोग यह जानने के लिए किया जाता है कि निवेश पर समय के साथ कितना ब्याज अर्जित होगा।

किसी राशि पर ब्याज की गणना के लिए, अनुपात ब्याज सूत्र होता है:

ब्याज = $P \left(1 + \frac{r}{100}\right)^t - P$

यहाँ:

  • P = प्रारंभिक राशि (Principal)
  • r = वार्षिक ब्याज दर (Rate of Interest)
  • t = समय (Time)

2. ऋण पर ब्याज (Interest on Loans): जब किसी व्यक्ति को ऋण लिया जाता है, तो उस पर ब्याज चुकाना पड़ता है। अनुपात ब्याज सूत्र का उपयोग यह जानने के लिए किया जाता है कि कुल ऋण राशि पर कितना ब्याज लगेगा और चुकाने की कुल राशि कितनी होगी।

ऋण पर ब्याज की गणना भी उसी सूत्र से की जाती है:

ब्याज = $P \left(1 + \frac{r}{100}\right)^t - P$

3. निवेश की भविष्य मूल्य (Future Value of Investment): यदि हम किसी निवेश को कुछ वर्षों के लिए करते हैं, तो अनुपात ब्याज सूत्र का उपयोग निवेश के भविष्य मूल्य को निकालने के लिए किया जाता है। यह निवेश के बढ़ने से संबंधित है और यह बताता है कि एक निश्चित समय के बाद निवेश कितनी राशि में परिवर्तित होगा।

निवेश का भविष्य मूल्य निम्नलिखित सूत्र से निकाला जाता है:

भविष्य मूल्य = $P \left(1 + \frac{r}{100}\right)^t$

4. बैंक के द्वारा दिए जाने वाले ब्याज की गणना: अनुपात ब्याज का प्रयोग यह जानने के लिए किया जाता है कि एक बैंक किसी खाते पर ब्याज किस दर से दे रहा है। इससे यह भी पता चलता है कि एक निश्चित समय के बाद खाताधारक को कितने पैसे मिलेंगे।

इसके लिए हम अनुपात ब्याज का सूत्र उपयोग करते हैं, जैसे कि:

ब्याज = $P \left(1 + \frac{r}{100}\right)^t - P$

5. किश्तों में ऋण का भुगतान: अनुपात ब्याज का एक और महत्वपूर्ण अनुप्रयोग किश्तों में ऋण चुकाने के मामले में होता है। यह विशेष रूप से उन ऋणों के लिए उपयोगी है जिन्हें मासिक या वार्षिक किश्तों में चुकाया जाता है। इसमें हर किश्त में शामिल ब्याज और मूलधन दोनों की गणना की जाती है।

किश्तों में चुकाने का सूत्र कुछ इस प्रकार होता है:

किश्त = $\frac{P \times r \times (1 + r)^t}{(1 + r)^t - 1}$

यहाँ:

  • P = ऋण राशि (Principal)
  • r = ब्याज दर (Rate of Interest per period)
  • t = समय (Number of periods)

6. अन्य अनुप्रयोग: अनुपात ब्याज का उपयोग अन्य कई वित्तीय क्षेत्रों में भी किया जाता है, जैसे पेंशन योजनाओं, बीमा पॉलिसियों, और अन्य निवेश योजनाओं में। इन सभी में, अनुपात ब्याज यह निर्धारित करने में मदद करता है कि समय के साथ पूंजी में कितनी वृद्धि होगी।

इन अनुप्रयोगों के माध्यम से, अनुपात ब्याज का महत्व और उपयोगिता स्पष्ट रूप से सामने आता है, और यह वित्तीय गणनाओं के लिए एक महत्वपूर्ण उपकरण बन जाता है।

The population of a place increased to 54,000 in 2003 at a rate of 5% per annum.

(i) Find the population in 2001.
(ii) What would be its population in 2005?

Solution:

(i) Find the population in 2001:

The population of a place increased to 54,000 in 2003 at a rate of 5% per annum. To find the population in 2001, we can use the formula for population growth:

Population in a given year = Initial population × (1 + rate of increase)^(number of years)

Let the population in 2001 be $P$.

We know the population in 2003 is 54,000, and the rate of increase is 5% per annum, i.e., $r = 5\% = 0.05$.

The number of years between 2001 and 2003 is 2 years. So, using the formula:

$54000 = P × (1 + 0.05)^2$

$54000 = P × (1.05)^2$

$54000 = P × 1.1025$

$P = \frac{54000}{1.1025}$

$P ≈ 49,000$

So, the population in 2001 was approximately 49,000.

(ii) What would be its population in 2005?

Now, to find the population in 2005, we will use the same formula. The number of years between 2001 and 2005 is 4 years. So:

Population in 2005 = $P × (1 + 0.05)^4$

Substitute $P = 49,000$:

Population in 2005 = $49000 × (1.05)^4$

Population in 2005 = $49000 × 1.21550625$

Population in 2005 ≈ $59,548$

So, the population in 2005 would be approximately 59,548.

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In a Laboratory, the count of bacteria in a certain experiment was increasing at the rate of 2.5% per hour. Find the bacteria at the end of 2 hours if the count was initially 5,06,000.

Solution:

Question:

In a laboratory, the count of bacteria in a certain experiment was increasing at the rate of 2.5% per hour. Find the bacteria at the end of 2 hours if the count was initially 5,06,000.

Solution:

The formula to calculate the compound interest is given by:

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

Where:

  • $A$ = Amount (final count of bacteria)
  • $P$ = Initial count of bacteria = 5,06,000
  • $r$ = Rate of growth per hour = 2.5%
  • $t$ = Time in hours = 2

Substituting the given values into the formula:

$A = 5,06,000 \left(1 + \frac{2.5}{100}\right)^2$

$A = 5,06,000 \left(1 + 0.025\right)^2$

$A = 5,06,000 \times (1.025)^2$

$A = 5,06,000 \times 1.050625$

$A = 5,31,759.375$

The bacteria count at the end of 2 hours is approximately 5,31,759.

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A scooter was bought at Rs. 42,000. Its value depreciated at the rate of 8% per annum. Find its value after one year.

Solution:

Value of the Scooter After One Year

The initial value of the scooter is Rs. 42,000.

The rate of depreciation is 8% per annum.

We can use the formula for depreciation to find the value of the scooter after one year:

$\text{Depreciated Value} = \text{Initial Value} \times \left(1 - \frac{\text{Rate of Depreciation}}{100}\right)$

Substitute the given values:

$\text{Depreciated Value} = 42000 \times \left(1 - \frac{8}{100}\right)$

$\text{Depreciated Value} = 42000 \times \left(1 - 0.08\right)$

$\text{Depreciated Value} = 42000 \times 0.92$

$\text{Depreciated Value} = 38,640$

Therefore, the value of the scooter after one year is Rs. 38,640.

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