To solve this problem, we need to compare two different interest rates that are compounded differently:

1. 11% compounded semiannually
2. 10.8% compounded daily

The general formula for compound interest is:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $A$ is the final amount
• $P$ is the principal (initial investment)
• $r$ is the annual interest rate (expressed as a decimal)
• $n$ is the number of times the interest is compounded per year
• $t$ is the time the money is invested or borrowed for, in years

Step 1: Calculate for 11% compounded semiannually

For the 11% compounded semiannually, the interest is compounded twice a year, so $n = 2$.

• $r = 0.11$
• $n = 2$
• $t = 1$

The formula becomes:

$A = P \left(1 + \frac{0.11}{2}\right)^{2 \times 1}$

Simplify:

$A = P \left(1 + 0.055\right)^2$ $A = P \times (1.055)^2$ $A = P \times 1.113025$

So, the amount after 1 year is $P \times 1.113025$.

Step 2: Calculate for 10.8% compounded daily

For the 10.8% compounded daily, the interest is compounded 365 times a year, so $n = 365$.

• $r = 0.108$
• $n = 365$
• $t = 1$

The formula becomes:

$A = P \left(1 + \frac{0.108}{365}\right)^{365 \times 1}$

Simplify:

$A = P \left(1 + 0.00029589\right)^{365}$ $A = P \times (1.00029589)^{365}$ $A = P \times 1.113492$

So, the amount after 1 year is $P \times 1.113492$.

Step 3: Compare the two results

• 11% compounded semiannually yields $P \times 1.113025$.
• 10.8% compounded daily yields $P \times 1.113492$.

Thus, 10.8% compounded daily yields a slightly larger amount after 1 year compared to 11% compounded semiannually.

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Do you need more details on any step or clarification?

Related Questions:

1. How do you calculate compound interest for monthly compounding?
2. How do interest rates affect investments over time?
3. What is the formula for continuous compounding?
4. What happens to the final amount as the number of compounding periods increases?
5. How would the results change if the time period was extended to 5 years?

Tip: For comparing compounded rates, remember that the more frequently interest is compounded, the higher the final amount will be, even if the nominal rate is lower.