Understanding Compound Interest Calculation Half-Yearly

This problem involves calculating the amount received after borrowing a sum of money with compound interest applied half-yearly. Compound interest means that the interest earned in each period is added to the principal for the next period, leading to interest earning interest.

Breaking Down the Problem

We are given the following information:

• Principal amount (P) = ₹6,500
• Annual Interest Rate (R) = 10% per annum
• Time period (T) = 2 years
• Compounding frequency: Half-yearly

When interest is compounded half-yearly, the following adjustments are made:

• The annual interest rate is divided by 2 to get the rate per compounding period.
• The number of years is multiplied by 2 to get the total number of compounding periods.

Calculating Rate and Number of Periods

• Rate per half-year (r) = Annual Rate / 2 = 10% / 2 = 5%
• In decimal form, r = 5 / 100 = 0.05
• Number of compounding periods (n) = Time in years × 2 = 2 years × 2 = 4 periods

Applying the Compound Interest Formula

The formula for the amount (A) with compound interest is:

\begin{equation*} A = P \left(1 + r\right)^n \end{equation*}

Where:

• A is the amount after n periods
• P is the principal amount
• r is the interest rate per period
• n is the total number of periods

Step-by-Step Calculation

Let's substitute the values we have into the formula:

\begin{equation*} A = 6500 \left(1 + 0.05\right)^4 \end{equation*}

\begin{equation*} A = 6500 \left(1.05\right)^4 \end{equation*}

Now, let's calculate $(1.05)^4$:

• $(1.05)^1 = 1.05$
• $(1.05)^2 = 1.05 \times 1.05 = 1.1025$
• $(1.05)^3 = 1.1025 \times 1.05 = 1.157625$
• $(1.05)^4 = 1.157625 \times 1.05 = 1.21550625$

Now substitute this value back into the formula for A:

\begin{equation*} A = 6500 \times 1.21550625 \end{equation*}

\begin{equation*} A = 7899.790625 \end{equation*}

Finding the Integral Value

The question asks for the integral value only. The calculated amount is ₹7899.790625. The integral part of this number is 7899.

However, looking at the options provided, the closest value to the calculated amount ₹7899.790625 is ₹7900, which would result from rounding the calculated amount to the nearest whole number. This suggests that the intended answer among the options corresponds to the calculated amount rounded to the nearest integer.

Rounding ₹7899.790625 to the nearest whole number gives ₹7900.

Comparing with Options

The calculated amount, when rounded to the nearest integral value among the options, is ₹7900.

• Option 1: ₹8,150
• Option 2: ₹7,900
• Option 3: ₹7,650
• Option 4: ₹8,250

The calculated value of ₹7899.790625 is closest to ₹7900.

Revision Table: Compound Interest Half-Yearly

Additional Information: Compound Interest Concepts

Understanding compound interest is crucial for financial planning. Here are some related concepts:

• Simple Interest: In simple interest, interest is calculated only on the initial principal amount. The formula is \(SI = P \times R \times T\). Compound interest generally yields a higher amount over time compared to simple interest for the same principal, rate, and time, especially over multiple periods.
• Different Compounding Periods: Interest can be compounded annually, half-yearly, quarterly, monthly, daily, or even continuously. The more frequent the compounding, the higher the final amount, assuming the same annual rate.
  • Annually: n=1, r=R
  • Half-yearly: n=2, r=R/2
  • Quarterly: n=4, r=R/4
  • Monthly: n=12, r=R/12
• Effective Annual Rate (EAR): When interest is compounded more than once a year, the actual annual rate earned or paid is higher than the stated annual rate (nominal rate). The EAR accounts for the effect of compounding. The formula is \(\text{EAR} = \left(1 + \frac{R}{\text{n}}\right)^{\text{n}} - 1\), where R is the nominal annual rate and n is the number of compounding periods per year.