Calculating Compound Interest Years: A Step-by-Step Guide

This problem asks us to find the number of years it takes for a principal amount to grow to a specific amount under compound interest at a given rate. We will use the compound interest formula to solve this.

Understanding Compound Interest

Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. This makes the money grow faster compared to simple interest.

Formula for Compound Interest

The formula used to calculate the future value of an investment with compound interest is:

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

Where:

• $A$ is the future value of the investment/loan, including interest (Amount)
• $P$ is the principal investment amount (Principal)
• $r$ is the annual interest rate (Rate)
• $n$ is the number of years the money is invested or borrowed for (Time Period)

Solving the Compound Interest Problem

We are given the following values:

• Principal ($P$) = Rs. 1875
• Amount ($A$) = Rs. 2187
• Rate ($r$) = 8% per annum
• Number of years ($n$) = ? (This is what we need to find)

We substitute these values into the compound interest formula:

$$2187 = 1875 \left(1 + \frac{8}{100}\right)^n$$

Step-by-Step Calculation to Find the Number of Years

1. Simplify the term inside the parenthesis:

   $$1 + \frac{8}{100} = 1 + 0.08 = 1.08$$

   The equation becomes:

   $$2187 = 1875 (1.08)^n$$

2. Isolate the term with the exponent ($1.08^n$) by dividing both sides by the principal (1875):

   $$\frac{2187}{1875} = (1.08)^n$$

3. Simplify the fraction $\frac{2187}{1875}$. We can divide both the numerator and the denominator by their greatest common divisor. Let's start by dividing by 3:

   $$2187 \div 3 = 729$$

   $$1875 \div 3 = 625$$

   So, the equation is now:

   $$\frac{729}{625} = (1.08)^n$$

4. Recognize the numbers on the left side. We know that $729 = 27^2$ and $625 = 25^2$. So the fraction can be written as:

   $$\left(\frac{27}{25}\right)^2 = (1.08)^n$$

5. Now let's look at the base on the right side, 1.08. As a fraction, $1.08 = \frac{108}{100}$. Simplifying this fraction by dividing the numerator and denominator by 4:

   $$\frac{108 \div 4}{100 \div 4} = \frac{27}{25}$$

6. Substitute this simplified fraction back into the equation:

   $$\left(\frac{27}{25}\right)^2 = \left(\frac{27}{25}\right)^n$$

7. Since the bases on both sides of the equation are equal ($\frac{27}{25}$), the exponents must also be equal for the equation to hold true:

   $$n = 2$$

So, the number of years required is 2.

Verification of the Number of Years

Let's verify the result by calculating the amount after 2 years with a principal of Rs. 1875 and an interest rate of 8% p.a.:

$$A = 1875 \left(1 + \frac{8}{100}\right)^2$$

$$A = 1875 (1.08)^2$$

$$A = 1875 (1.08 \times 1.08)$$

$$A = 1875 (1.1664)$$

$$A = 2187$$

The calculated amount matches the given amount, confirming that the number of years is indeed 2.

Conclusion

A sum of Rs. 1875 will amount to Rs. 2187 at 8 percent p.a. compound interest in 2 years.

Revision Table: Compound Interest Calculations

Additional Information: Compound Interest Concepts

Understanding compound interest is crucial for personal finance and many quantitative exams. Here are a few related concepts:

• Simple Interest: Interest calculated only on the principal amount. The formula is $SI = \frac{P \times R \times T}{100}$. Compound interest yields more than simple interest over time because interest is added to the principal.
• Compounding Frequency: Interest can be compounded annually, semi-annually, quarterly, monthly, or even daily. The more frequent the compounding, the faster the money grows. If compounded $k$ times a year, the formula becomes $A = P \left(1 + \frac{r/k}{100}\right)^{nk}$. In this problem, compounding is annual ($k=1$).
• Effective Annual Rate (EAR): The actual rate of interest earned or paid in a year when compounding occurs more often than annually.
• Rule of 72: A simple way to estimate how long it takes for an investment to double at a fixed annual rate of interest. Divide 72 by the annual interest rate to get the approximate number of years. For 8%, it would be $72/8 = 9$ years to double, which is different from reaching Rs. 2187 from Rs. 1875.

Mastering the compound interest formula and its variables like principal, amount, rate, and number of years is key to solving such problems efficiently.