Solving Compound Interest with Half-Yearly Compounding

The problem asks us to find the amount after 18 months when a principal amount of Rs. 2,000 is invested at an annual interest rate of 20%, compounded half-yearly. Understanding the terms like principal, interest rate, compounding frequency, and time period is crucial for solving compound interest problems.

Understanding the Given Information

• Principal (P): Rs. 2,000
• Annual Interest Rate: 20% per annum
• Compounding Frequency: Half-yearly
• Time Period: 18 months

Adjusting Rate and Time for Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate and time period need to be adjusted to reflect the compounding periods.

• The annual rate is 20%. Since there are two half-years in a year, the rate per half-year is half of the annual rate.
  • Rate per half-year (r) $= \frac{20\%}{2} = 10\%$
• The time period is 18 months. We need to express this in terms of half-yearly periods.
  • 18 months is equal to 1.5 years.
  • Since each year has 2 half-yearly periods, 1.5 years will have $1.5 \times 2 = 3$ half-yearly periods.
  • Number of periods (n) = 3

Compound Interest Formula for Amount

The formula to calculate the amount (A) when interest is compounded is:

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

Where:

• P = Principal amount
• r = Rate of interest per compounding period
• n = Number of compounding periods

Calculating the Amount

Now, we substitute the values we have calculated into the formula:

• P = 2000
• r = 10 (10%)
• n = 3

Let's calculate the amount:

$A = 2000 \left(1 + \frac{10}{100}\right)^3$

$A = 2000 \left(1 + 0.1\right)^3$

$A = 2000 \left(1.1\right)^3$

First, calculate $(1.1)^3$:

$(1.1)^3 = 1.1 \times 1.1 \times 1.1$

$(1.1)^2 = 1.21$

$(1.1)^3 = 1.21 \times 1.1 = 1.331$

Now, substitute this back into the amount formula:

$A = 2000 \times 1.331$

$A = 2662$

So, the amount after 18 months, compounded half-yearly, will be Rs. 2,662.

Final Answer

The calculated amount is Rs. 2,662.

Revision Table: Compound Interest Basics

Additional Information: Different Compounding Frequencies

The method of calculating compound interest varies depending on the compounding frequency. Here are some common types:

• Annually: Interest is compounded once a year. The rate per period is the annual rate, and the number of periods is the number of years.
• Half-Yearly: Interest is compounded twice a year. The rate per period is half the annual rate, and the number of periods is twice the number of years. (As in this problem)
• Quarterly: Interest is compounded four times a year. The rate per period is one-fourth of the annual rate, and the number of periods is four times the number of years.
• Monthly: Interest is compounded twelve times a year. The rate per period is one-twelfth of the annual rate, and the number of periods is twelve times the number of years.

The general formula $A = P \left(1 + \frac{r}{100}\right)^n$ applies to all these cases, where 'r' and 'n' are adjusted based on the compounding frequency per year.