Calculating Compound Interest A Sum Of Rs 12000 Doubles In 5 Years

by Scholario Team 67 views

Understanding compound interest is crucial for anyone looking to make sound financial decisions, whether it's for personal savings or investments. Compound interest, often called the eighth wonder of the world, allows your money to grow exponentially over time. In this article, we will delve into a specific scenario involving compound interest and demonstrate how to calculate the future value of an investment. We'll explore a problem where an initial deposit of Rs.12,000 doubles in 5 years and determine its value after 20 years. This example provides a practical understanding of how compound interest works and how it can significantly increase your investment over time. This comprehensive guide will walk you through the step-by-step calculations and provide clear explanations to help you grasp the concepts effectively. Whether you are a student learning about financial mathematics or an individual planning for your future, this article aims to provide you with valuable insights and practical knowledge.

Before diving into the specific problem, it's essential to understand the basics of compound interest. Unlike simple interest, which is calculated only on the principal amount, compound interest is calculated on the principal amount plus the accumulated interest. This means that each year, you earn interest not only on your initial investment but also on the interest earned in previous years. This compounding effect leads to substantial growth over time. The formula for compound interest is:

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

Where:

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

In this article, we'll focus on annual compounding, where interest is added once per year, simplifying the formula and making the calculations more straightforward. Understanding each component of this formula is critical for solving problems related to compound interest and for making informed financial decisions. The power of compound interest lies in the fact that the interest earned each year gets added to the principal, forming a new base for the next year’s interest calculation. This snowball effect is what makes compound interest a powerful tool for wealth accumulation. Now, let's move on to the problem at hand and see how we can apply this formula to find the future value of an investment.

The problem we're addressing states that a sum of Rs.12,000 is deposited at compound interest and doubles after 5 years. Our goal is to determine how much this sum will grow to after 20 years. This involves two main steps: first, we need to find the annual interest rate, and second, we use this interest rate to calculate the amount after 20 years. This problem is a classic example of how compound interest works over different time periods, highlighting the exponential growth that occurs with compounding. The initial doubling period of 5 years gives us a crucial piece of information to calculate the interest rate, which is the key to projecting the future value of the investment. By breaking down the problem into these steps, we can systematically solve it and gain a clear understanding of the underlying financial principles. It’s important to note that the power of compound interest becomes more evident over longer time horizons, making it essential for long-term financial planning and investment strategies. Let’s proceed step-by-step to solve this problem and uncover the potential growth of this investment over 20 years.

Step 1: Finding the Annual Interest Rate

To find the annual interest rate, we use the information that the initial deposit of Rs.12,000 doubles in 5 years. This means that after 5 years, the amount becomes Rs.24,000. We can use the compound interest formula to solve for the interest rate r{ r }. We have:

A=P(1+r)t{ A = P (1 + r)^t }

Where:

  • A=24000{ A = 24000 } (the amount after 5 years)
  • P=12000{ P = 12000 } (the principal amount)
  • t=5{ t = 5 } (the number of years)

Plugging these values into the formula, we get:

24000=12000(1+r)5{ 24000 = 12000 (1 + r)^5 }

Divide both sides by 12000:

2=(1+r)5{ 2 = (1 + r)^5 }

To solve for r{ r }, we take the 5th root of both sides:

25=1+r{ \sqrt[5]{2} = 1 + r }

r=25−1{ r = \sqrt[5]{2} - 1 }

Using a calculator, we find:

r≈1.1487−1{ r ≈ 1.1487 - 1 }

r≈0.1487{ r ≈ 0.1487 }

So, the annual interest rate is approximately 14.87%. This interest rate is the foundation for calculating the future value of the investment after 20 years. The process of isolating the interest rate involves understanding the relationship between the principal, the final amount, and the time period, as defined by the compound interest formula. With this interest rate, we can now move on to the next step, which is calculating the future value of the investment after 20 years. This step will further demonstrate the power of compound interest over a longer duration.

Step 2: Calculating the Amount After 20 Years

Now that we have the annual interest rate (approximately 14.87% or 0.1487), we can calculate the amount after 20 years. We use the same compound interest formula:

A=P(1+r)t{ A = P (1 + r)^t }

Where:

  • P=12000{ P = 12000 } (the principal amount)
  • r=0.1487{ r = 0.1487 } (the annual interest rate)
  • t=20{ t = 20 } (the number of years)

Plugging these values into the formula, we get:

A=12000(1+0.1487)20{ A = 12000 (1 + 0.1487)^{20} }

A=12000(1.1487)20{ A = 12000 (1.1487)^{20} }

Using a calculator, we find:

(1.1487)20≈16.004{ (1.1487)^{20} ≈ 16.004 }

A=12000∗16.004{ A = 12000 * 16.004 }

A≈192048{ A ≈ 192048 }

So, after 20 years, the amount will be approximately Rs. 1,92,048. This calculation demonstrates the significant growth potential of compound interest over an extended period. The initial investment of Rs.12,000 grows substantially, highlighting the benefits of long-term investing and the compounding effect. This result aligns with the exponential nature of compound interest, where the growth accelerates over time as the interest earned in previous years starts to generate its own interest. The precise figure of Rs. 1,92,048 provides a clear picture of the potential returns from a compound interest investment strategy over two decades.

In conclusion, after 20 years, a sum of Rs.12,000 deposited at compound interest, which doubles in 5 years, will become approximately Rs. 1,92,048. Therefore, the correct answer is (B) Rs. 1,92,000. This problem effectively illustrates the power of compound interest and its ability to generate substantial returns over time. The initial doubling in 5 years provides a foundation for significant growth over the subsequent 15 years, demonstrating the long-term benefits of compounding. By breaking down the problem into steps—first calculating the annual interest rate and then using it to find the future value—we gain a clear understanding of the mechanics behind compound interest. This understanding is crucial for making informed financial decisions, whether it's for personal savings, investments, or retirement planning. The example underscores the importance of starting early and staying invested to harness the full potential of compound interest. The final amount, nearly Rs. 1,92,000, is a testament to the exponential growth that compound interest can provide, making it a cornerstone of financial wisdom.

(B) Rs. 1,92,000