What is the compound interest (in Rs.) on a sum of Rs. 8192 for \(1 \frac{1}{4}\) years at 15% per annum, if interest is compounded 5-monthly ?
1634
Calculating Compound Interest with 5-Monthly Compounding
This problem asks us to find the compound interest on a given principal amount over a specific time period at a certain annual rate, with the interest being compounded every 5 months.
Understanding the Given Information
- Principal amount (P) = Rs. 8192
- Time period (T) = \(1 \frac{1}{4}\) years = \( \frac{5}{4} \) years
- Annual interest rate (R) = 15% per annum
- Compounding frequency: 5-monthly
Adjusting Rate and Time for Compounding Frequency
Since the interest is compounded 5-monthly, we need to find the rate per compounding period and the total number of compounding periods in the given time.
- A year has 12 months.
- Number of 5-month periods in a year = \( \frac{12}{5} \)
- Rate per 5-month period (r) = Annual rate × \( \frac{5}{12} \) = 15% × \( \frac{5}{12} \) = \( \frac{15 \times 5}{12} \) % = \( \frac{75}{12} \) % = \( \frac{25}{4} \) %
- Convert the rate to a decimal: \( r = \frac{25}{400} = \frac{1}{16} \)
Now, let's find the total number of compounding periods in the given time:
- Time in years = \( \frac{5}{4} \) years
- Time in months = \( \frac{5}{4} \) × 12 months = 15 months
- Number of 5-month periods (n) = Total months / Months per period = 15 months / 5 months = 3 periods
Calculating the Amount Using the Compound Interest Formula
The formula for the amount (A) with compound interest is:
\( A = P(1 + r)^n \)
Where:
- P = Principal amount
- r = Rate per compounding period
- n = Number of compounding periods
Substitute the values we found:
\( A = 8192 \left(1 + \frac{1}{16}\right)^3 \)
\( A = 8192 \left(\frac{16 + 1}{16}\right)^3 \)
\( A = 8192 \left(\frac{17}{16}\right)^3 \)
\( A = 8192 \times \frac{17^3}{16^3} \)
\( A = 8192 \times \frac{4913}{4096} \)
Since \( 8192 = 2 \times 4096 \), we can simplify:
\( A = 2 \times 4913 \)
\( A = 9826 \)
The amount after \(1 \frac{1}{4}\) years, with interest compounded 5-monthly, is Rs. 9826.
Calculating the Compound Interest
The compound interest (CI) is the difference between the amount and the principal:
\( CI = A - P \)
\( CI = 9826 - 8192 \)
\( CI = 1634 \)
The compound interest is Rs. 1634.
Let's summarise the key values:
| Item | Value |
|---|---|
| Principal (P) | Rs. 8192 |
| Annual Rate (R) | 15% |
| Time (T) | \(1 \frac{1}{4}\) years |
| Compounding Frequency | 5-monthly |
| Rate per period (r) | \( \frac{25}{4} \)% or \( \frac{1}{16} \) |
| Number of periods (n) | 3 |
| Amount (A) | Rs. 9826 |
| Compound Interest (CI) | Rs. 1634 |
The calculated compound interest is Rs. 1634, which matches one of the given options.
Revision Table: Key Concepts in Compound Interest
| Term | Definition | Formula/Notes |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | Starting amount. |
| Amount (A) | The total sum after interest is added to the principal. | A = P + CI |
| Compound Interest (CI) | Interest calculated on the principal amount and also on the accumulated interest from previous periods. | CI = A - P or \( CI = P[(1 + r)^n - 1] \) |
| Annual Rate (R) | The interest rate charged per year. | Given as a percentage. |
| Compounding Frequency | How often interest is calculated and added to the principal (e.g., annually, semi-annually, quarterly, monthly). | Determines the period length. |
| Rate per period (r) | The interest rate applicable for one compounding period. | \( r = \frac{\text{Annual Rate}}{\text{Number of periods per year}} \) |
| Number of periods (n) | The total count of compounding periods over the entire time duration. | \( n = \text{Time in years} \times \text{Number of periods per year} \) |
Additional Information: Different Compounding Frequencies
The way interest is compounded significantly affects the final amount. Here are common compounding frequencies and how they affect the rate and time conversion:
- Annually: Interest is compounded once a year. Rate per period = Annual Rate. Number of periods = Time in years.
- Semi-Annually: Interest is compounded twice a year (every 6 months). Rate per period = Annual Rate / 2. Number of periods = Time in years × 2.
- Quarterly: Interest is compounded four times a year (every 3 months). Rate per period = Annual Rate / 4. Number of periods = Time in years × 4.
- Monthly: Interest is compounded twelve times a year (every month). Rate per period = Annual Rate / 12. Number of periods = Time in years × 12.
- n-Monthly: Interest is compounded every 'n' months. Number of periods per year = \( \frac{12}{n} \). Rate per period = Annual Rate × \( \frac{n}{12} \). Number of periods = Time in years × \( \frac{12}{n} \). In our problem, n = 5.
Understanding these conversions is crucial for solving compound interest problems with varying compounding frequencies.
Important Questions from Compound Interest
At what rate percent per annum will Rs. 7200 amount to Rs. 7938 in one year, if interest is compounded half yearly?
What is the difference (in Rs.) between the interests on Rs. 50,000 for one year at 8% per annum compounded half yearly and yearly?
A sum of money becomes Rs. 11,880 after 4 years and Rs. 17,820 after 6 years on compound interest, if the interest is compounded annually. What is the half of the sum (in Rs.)?
A sum invested at compound interest amounts to Rs. 7,800 in 3 years and Rs. 11,232 in 5 years. What is the rate per cent?
A sum amounts to Rs. 18,600 after 3 years and to Rs. 27,900 after 6 years, at a certain rate percent p.a., when the interest is compounded annually. The sum is:
