Calculate Compound Interest On Deposits Example And Step-by-Step Guide

In this article, we will delve into a common yet crucial financial calculation: compound interest. Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it." - Albert Einstein. We'll break down a specific scenario where a person deposits ₹6,400 in a bank, then deposits the same amount again after six months. The bank offers an 8% interest rate compounded half-yearly. Our goal is to calculate the total interest earned after 12 months from the initial deposit. This step-by-step guide will not only provide the solution but also enhance your understanding of compound interest calculations.

Breaking Down the Problem

To accurately calculate the interest earned, we need to consider the two separate deposits and how they accrue interest over time. The first deposit of ₹6,400 will earn interest for the entire 12-month period, while the second deposit will earn interest only for the remaining six months. Since the interest is compounded half-yearly, we need to adjust the interest rate and time period accordingly. Understanding the principles of compound interest is crucial for making informed financial decisions, whether it's saving, investing, or borrowing. Let's start by defining compound interest.

What is Compound Interest?

Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. In simpler terms, it's interest on interest. This means that your money grows at an accelerating rate because the interest that you've earned also starts earning interest. Compound interest is a powerful tool for wealth creation, as it allows your money to grow exponentially over time. The formula for compound interest is:

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

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (as a decimal)
• n = the number of times that interest is compounded per year
• t = the number of years the money is invested or borrowed for

This formula helps us calculate the future value of an investment, taking into account the effects of compounding. In our scenario, we will use this formula to calculate the interest earned on each deposit separately and then add them together to find the total interest. Understanding the variables in this formula is key to applying it correctly. The principal (P) is the initial amount deposited, the interest rate (r) is the annual rate offered by the bank, the compounding frequency (n) is how often the interest is calculated and added to the principal (in this case, half-yearly), and the time (t) is the duration of the investment.

Calculating Interest on the First Deposit

The first deposit of ₹6,400 is made at the beginning and will accrue interest for the entire 12 months. We need to apply the compound interest formula to calculate the amount after 12 months. Given that the interest is compounded half-yearly, we have two compounding periods within the year. The annual interest rate is 8%, so the half-yearly interest rate is 8%/2 = 4% or 0.04 as a decimal. The time period is 1 year, and the number of compounding periods is 2. We can now plug these values into the compound interest formula:

• Principal (P) = ₹6,400
• Annual interest rate (r) = 8% = 0.08
• Number of times interest is compounded per year (n) = 2
• Number of years (t) = 1

$$A = 6400(1 + \frac{0.08}{2})^{2*1}$$

$$A = 6400(1 + 0.04)^{2}$$

$$A = 6400(1.04)^{2}$$

$$A = 6400 * 1.0816$$

$$A = ₹6,922.24$$

This calculation shows that the first deposit will grow to ₹6,922.24 after 12 months. The interest earned on the first deposit is the difference between the final amount and the principal amount:

Interest earned on the first deposit = ₹6,922.24 - ₹6,400 = ₹522.24

Calculating Interest on the Second Deposit

The second deposit of ₹6,400 is made after 6 months, so it will only accrue interest for the remaining 6 months. This means we need to adjust the time period in our compound interest calculation. The interest rate remains the same at 4% per half-year, but the time period is now 0.5 years or one compounding period. We can use the same compound interest formula, but with the adjusted time period:

• Principal (P) = ₹6,400
• Annual interest rate (r) = 8% = 0.08
• Number of times interest is compounded per year (n) = 2
• Number of years (t) = 0.5

$$A = 6400(1 + \frac{0.08}{2})^{2*0.5}$$

$$A = 6400(1 + 0.04)^{1}$$

$$A = 6400 * 1.04$$

$$A = ₹6,656$$

This calculation shows that the second deposit will grow to ₹6,656 after 6 months. The interest earned on the second deposit is the difference between the final amount and the principal amount:

Interest earned on the second deposit = ₹6,656 - ₹6,400 = ₹256

Total Interest Earned

To find the total interest earned after 12 months, we simply add the interest earned on both deposits:

Total interest = Interest earned on the first deposit + Interest earned on the second deposit Total interest = ₹522.24 + ₹256 Total interest = ₹778.24

Therefore, the total interest earned after 12 months is ₹778.24. This comprehensive calculation demonstrates how compound interest works and how it can be applied to different deposit scenarios.

Importance of Compounding Frequency

The frequency at which interest is compounded can significantly impact the total interest earned. In this case, the interest is compounded half-yearly, meaning it's calculated and added to the principal twice a year. If the interest were compounded more frequently (e.g., quarterly or monthly), the total interest earned would be even higher. This is because the interest earned in each compounding period starts earning interest sooner, leading to faster growth. Understanding the impact of compounding frequency is essential for maximizing returns on investments and savings.

Conclusion

In conclusion, by depositing ₹6,400 initially and another ₹6,400 after six months in a bank offering 8% interest compounded half-yearly, the person will earn a total interest of ₹778.24 after 12 months. This detailed calculation highlights the power of compound interest and its impact on savings and investments. Understanding these concepts is crucial for making informed financial decisions and maximizing your returns over time. Remember, the earlier you start saving and the more frequently your interest compounds, the more your money will grow.

Compound interest is a fundamental concept in finance, and mastering it can empower you to make smarter financial choices. This example illustrates how to break down a problem, apply the compound interest formula, and calculate the total interest earned. By understanding these principles, you can better plan your savings and investments to achieve your financial goals. Whether you're saving for retirement, a down payment on a house, or simply building your wealth, compound interest is your ally in achieving financial success. So, take the time to learn and understand it, and you'll be well on your way to a brighter financial future.

Discussion category :mathematics

Keywords

Keywords: Compound interest, deposits, interest calculation, financial mathematics, bank interest, half-yearly compounding, principal amount, interest rate, time period, future value, wealth creation, financial planning, investment growth, savings, financial decisions.

Repair Input Keyword

Repair Input Keyword: A person deposits ₹6,400 in a bank. After 6 months, they deposit another ₹6,400. If the bank pays 8% interest compounded semi-annually (half-yearly), how much total interest will they earn after 12 months from the first deposit?