Rate Compounded Annually or Half Yearly?

To determine whether the rate should be compounded annually or half-yearly, we need to understand the concept of compounding and how it affects the growth of an investment.

Understanding Compound Interest:
Compound interest is the interest calculated on both the initial principal and the accumulated interest from previous periods. It allows the investment to grow exponentially over time.

Compounding Annually:
When the interest is compounded annually, the interest is added to the principal at the end of each year. This means that the interest earned during each year is not reinvested until the end of the year.

Compounding Half Yearly:
When the interest is compounded half-yearly, the interest is added to the principal twice a year, at the end of each half-year period. This allows the interest earned during the first half of the year to be reinvested and start earning interest in the second half of the year.

Comparison:
To determine whether compounding annually or half-yearly is more beneficial, we can calculate the amount after one year using both methods.

1. Compounding Annually:
The formula to calculate the amount after one year with annual compounding is:
A = P(1 + r)^n
Where:
A = Amount after one year
P = Principal amount
r = Rate of interest
n = Number of years

In this case, P = Rs4800, r = 10%, and n = 1.
A = 4800(1 + 0.10)^1
A = 4800(1.10)
A = Rs5280

2. Compounding Half Yearly:
The formula to calculate the amount after one year with half-yearly compounding is:
A = P(1 + r/2)^(2n)
Where:
A = Amount after one year
P = Principal amount
r = Rate of interest
n = Number of half-years

In this case, P = Rs4800, r = 10%, and n = 1 1/2.
A = 4800(1 + 0.10/2)^(2 * 1.5)
A = 4800(1.05)^3
A = Rs5292

Conclusion:
Comparing the two amounts calculated, we can see that with compounding half-yearly, the amount after one year is slightly higher (Rs5292) compared to compounding annually (Rs5280). Therefore, it is more beneficial to compound the interest half-yearly in this scenario.

Summary:
- Compounding annually means interest is added to the principal at the end of each year.
- Compounding half-yearly means interest is added to the principal twice a year, at the end of each half-year period.
- To compare the two compounding methods, calculate the amount after one year using the respective formulas.
- In this scenario, compounding half-yearly results in a higher amount (Rs5292) compared to compounding annually (Rs5280).
- Therefore, it is more advantageous to compound the interest half-yearly.