Questions: How much money will there be in an account at the end of 10 years if 5000 is deposited at 5% interest compounded semiannually? (Assume no withdrawals are made.)

Solution Steps

To solve this problem, we need to use the formula for compound interest. The formula is:

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

where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested for in years.

Given:

• $P = 5000$
• $r = 0.05$
• $n = 2$ (since the interest is compounded semiannually)
• $t = 10$

We will plug these values into the formula to find $A$.

We are given the following values for the compound interest formula:

• Principal amount $P = 5000$
• Annual interest rate $r = 0.05$
• Number of times interest is compounded per year $n = 2$ (semiannually)
• Time in years $t = 10$

The formula for compound interest is given by:

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

Substituting the known values into the formula:

$$A = 5000 \left(1 + \frac{0.05}{2}\right)^{2 \times 10}$$

First, we calculate the term inside the parentheses:

$$1 + \frac{0.05}{2} = 1 + 0.025 = 1.025$$

Next, we calculate the exponent:

$$nt = 2 \times 10 = 20$$

Now we can compute $A$:

$$A = 5000 \times (1.025)^{20}$$

Calculating $(1.025)^{20}$:

$$(1.025)^{20} \approx 1.806111234669$$

Finally, we find $A$:

$$A \approx 5000 \times 1.806111234669 \approx 9030.556173345$$

Rounding to two decimal places, we have:

$$A \approx 9030.56$$

Final Answer