What is the compound interest on Rs. 16400 for 1 year at 9% per annum when compounded half-yearly?
Rs. 1509.21
Calculating Compound Interest Compounded Half-Yearly
This problem requires us to calculate the compound interest on a principal amount when the interest is compounded half-yearly. Understanding how the compounding frequency affects the interest rate and time period is crucial.
Understanding the Key Terms
- Principal (P): The initial amount of money deposited or borrowed. Here, P = Rs. 16400.
- Annual Interest Rate (R): The rate at which interest is charged or earned per year. Here, R = 9% per annum.
- Time (T): The duration for which the money is invested or borrowed. Here, T = 1 year.
- Compounding Frequency: How many times the interest is calculated and added to the principal within a year. Here, it is compounded half-yearly, meaning twice a year (n=2).
Adjusting Rate and Time for Half-Yearly Compounding
When interest is compounded half-yearly, we need to adjust the annual rate and the total time period:
- Rate per compounding period (r): The annual rate is divided by the number of compounding periods per year. $$ r = \frac{\text{Annual Rate}}{\text{Number of compounding periods per year}} = \frac{9\%}{2} = 4.5\% \text{ per half-year} $$ In decimal form, $r = 4.5 / 100 = 0.045$.
- Total number of compounding periods (N): The total time in years is multiplied by the number of compounding periods per year. $$ N = \text{Time in years} \times \text{Number of compounding periods per year} = 1 \text{ year} \times 2 = 2 \text{ half-years} $$
Compound Interest Formula for Half-Yearly Compounding
The formula for the amount (A) after N periods when the rate per period is r is:
$$ \text{A} = \text{P} (1 + r)^N $$The compound interest (CI) is the difference between the amount and the principal:
$$ \text{CI} = \text{A} - \text{P} $$Step-by-Step Calculation of Compound Interest
Let's plug in the values into the formula:
- Calculate the amount (A): $$ \text{A} = 16400 (1 + 0.045)^2 $$ $$ \text{A} = 16400 (1.045)^2 $$ First, calculate $(1.045)^2$: $$ (1.045)^2 = 1.045 \times 1.045 = 1.092025 $$ Now, calculate the amount: $$ \text{A} = 16400 \times 1.092025 $$ $$ \text{A} = 17909.21 $$ The total amount after 1 year compounded half-yearly is Rs. 17909.21.
- Calculate the Compound Interest (CI): $$ \text{CI} = \text{A} - \text{P} $$ $$ \text{CI} = 17909.21 - 16400 $$ $$ \text{CI} = 1509.21 $$ The compound interest is Rs. 1509.21.
Alternatively, you can calculate the interest for each half-year period:
- Interest for the 1st half-year: $$ \text{I}_1 = \text{Principal} \times \text{Rate per period} = 16400 \times 0.045 = 738 $$ Amount after 1st half-year = $16400 + 738 = 17138$.
- Interest for the 2nd half-year: $$ \text{I}_2 = \text{Amount after 1st half-year} \times \text{Rate per period} = 17138 \times 0.045 = 771.21 $$ Amount after 2nd half-year = $17138 + 771.21 = 17909.21$.
- Total Compound Interest (CI): $$ \text{CI} = \text{I}_1 + \text{I}_2 = 738 + 771.21 = 1509.21 $$
Both methods confirm that the compound interest on Rs. 16400 for 1 year at 9% per annum compounded half-yearly is Rs. 1509.21.
Conclusion
Based on the calculations, the compound interest is Rs. 1509.21.
| Particulars | Value |
|---|---|
| Principal (P) | Rs. 16400 |
| Annual Rate (R) | 9% |
| Time (T) | 1 year |
| Compounding Frequency | Half-yearly (n=2) |
| Rate per period (r) | 4.5% or 0.045 |
| Number of periods (N) | 2 |
| Amount (A) | Rs. 17909.21 |
| Compound Interest (CI) | Rs. 1509.21 |
Revision Table: Compound Interest Concepts
| Concept | Description | Formula Example (Annual Compounding) |
|---|---|---|
| Simple Interest (SI) | Interest calculated only on the principal amount. | $\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$ |
| Compound Interest (CI) | Interest calculated on the principal amount and accumulated interest from previous periods. | $\text{A} = \text{P} (1 + \frac{\text{R}}{100})^\text{T}$, $\text{CI} = \text{A} - \text{P}$ |
| Amount (A) | The total sum including principal and interest. | $\text{A} = \text{P} + \text{CI}$ |
| Compounding Frequency | How often interest is added to the principal (e.g., annually, half-yearly, quarterly, monthly). | Adjusts rate and time period in CI formula. |
Additional Information: Compounding Frequency Impact
The frequency of compounding significantly impacts the total compound interest earned over a period. The more frequently interest is compounded, the higher the total interest will be, assuming the same annual rate.
- Annual Compounding: Interest calculated once a year. N = T, r = R/100.
- Half-Yearly Compounding: Interest calculated twice a year. N = 2T, r = R/200.
- Quarterly Compounding: Interest calculated four times a year. N = 4T, r = R/400.
- Monthly Compounding: Interest calculated twelve times a year. N = 12T, r = R/1200.
In this problem, compounding half-yearly means the interest earned in the first six months is added to the principal, and then the interest for the next six months is calculated on this new, larger principal. This process leads to higher interest compared to annual compounding over the same period.
Important Questions from Compound Interest
- A sum of money triples itself at a certain rate of compound interest in 11 years. In how many years will it amount to 27 times of itself?
A sum of ₹10,000 amounts to ₹12,321 in 2 years at a certain rate of interest per annum, compounded annually. The rate of interest per annum is:
- If the interest earned during the 2nd year on a certain sum is ₹2,352, and the rate of interest is 5% per annum compounded annually, then the sum is:
- A sum of money triples itself at a certain rate of compound interest in 17 years. In how many years will it amount to 81 times of itself?
If the interest earned during the $2^{nd}$ year on a certain sum is ₹3,342, and the rate of interest is 20% per annum compounded annually, then the sum is:
