Calculating Time in Compound Interest with Half-Yearly Compounding

This problem involves finding the time period over which an initial amount (principal) grows to a final amount under compound interest, with the interest being compounded half-yearly.

Let's identify the given information:

• Principal Amount (P) = Rs. 15000
• Final Amount (A) = Rs. 19965
• Annual Rate of Interest (r) = 20% per annum
• Compounding Frequency = Half-yearly

Since the interest is compounded half-yearly, we need to adjust the annual rate and the time period for the calculation.

• Rate of interest per compounding period (R) = Annual rate / Number of compounding periods per year
• R = 20% / 2 = 10% per half-year
• In decimal form, R = 10 / 100 = 0.10
• Let 'n' be the total number of compounding periods. If the time is 't' years, then n = 2t (since there are 2 half-years in a year).

The formula for compound interest is:

\( A = P \left(1 + R\right)^n \)

Now, let's substitute the given values into the formula:

\( 19965 = 15000 \left(1 + 0.10\right)^n \)

\( 19965 = 15000 \left(1.1\right)^n \)

To find 'n', we first isolate the term \(\left(1.1\right)^n\):

\( \frac{19965}{15000} = \left(1.1\right)^n \)

Let's simplify the fraction:

\( \frac{19965}{15000} = \frac{19965 \div 15}{15000 \div 15} = \frac{1331}{1000} = 1.331 \)

So, we have:

\( 1.331 = \left(1.1\right)^n \)

We need to find the power 'n' to which 1.1 must be raised to get 1.331. Let's calculate the first few powers of 1.1:

• \( \left(1.1\right)^1 = 1.1 \)
• \( \left(1.1\right)^2 = 1.1 \times 1.1 = 1.21 \)
• \( \left(1.1\right)^3 = 1.21 \times 1.1 = 1.331 \)

From the calculation, we see that \(\left(1.1\right)^3 = 1.331\).

Therefore, \( n = 3 \).

The value of 'n' represents the number of compounding periods. Since the interest is compounded half-yearly, 'n = 3' means there are 3 half-year periods.

To find the total time in months, we multiply the number of half-year periods by the duration of each period (6 months):

Total Time = Number of periods \( \times \) Duration per period

Total Time = \( 3 \times 6 \) months

Total Time = 18 months

Let's check this against the options:

• 30 months
• 18 months
• 12 months
• 24 months

Our calculated time of 18 months matches one of the options.

Revision Table: Compound Interest Concepts

Additional Information: Understanding Compounding Frequency

The frequency of compounding significantly impacts the final amount earned. More frequent compounding leads to higher interest earned over the same time period because interest is calculated on an increasingly larger principal amount more often.

• Annual Compounding: Interest is calculated and added once a year. Number of periods per year = 1.
• Half-Yearly Compounding: Interest is calculated and added twice a year (every 6 months). Number of periods per year = 2.
• Quarterly Compounding: Interest is calculated and added four times a year (every 3 months). Number of periods per year = 4.
• Monthly Compounding: Interest is calculated and added twelve times a year (every month). Number of periods per year = 12.

When solving problems involving different compounding frequencies, it is crucial to correctly adjust both the rate per period (R) and the total number of periods (n) before applying the compound interest formula.

In this specific problem, converting the annual rate to a half-yearly rate and ensuring the total time is counted in half-year periods (n) was the key step.