Suppose that $2000 is placed in an account that pays 9% interest compounded each year. Assume that no withdrawals are made from the account.
Follow the instructions below. Do not do any rounding.
(a) Find the amount in the account at the end of 1 year.
(b) Find the amount in the account at the end of 2 years.
The correct answer and explanation is :
To solve this problem, we will use the formula for compound interest, which is:
$$
A = P \left( 1 + \frac{r}{n} \right)^{nt}
$$
Where:
- $A$ is the amount of money accumulated after interest.
- $P$ is the principal amount (the initial deposit).
- $r$ is the annual interest rate (decimal form).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the number of years the money is invested or borrowed for.
Given:
- Principal ($P$) = \$2000
- Annual interest rate ($r$) = 9% = 0.09 (as a decimal)
- Interest is compounded yearly, so $n = 1$
- Time ($t$) = 1 year for part (a) and 2 years for part (b)
(a) Find the amount in the account at the end of 1 year.
Using the formula:
$$
A = 2000 \left( 1 + \frac{0.09}{1} \right)^{1 \times 1}
$$
Simplifying:
$$
A = 2000 \left( 1 + 0.09 \right)^1
$$
$$
A = 2000 \times 1.09
$$
$$
A = 2180
$$
So, at the end of 1 year, the amount in the account is \$2180.
(b) Find the amount in the account at the end of 2 years.
Now, for 2 years ($t = 2$):
$$
A = 2000 \left( 1 + \frac{0.09}{1} \right)^{1 \times 2}
$$
Simplifying:
$$
A = 2000 \left( 1 + 0.09 \right)^2
$$
$$
A = 2000 \times 1.09^2
$$
$$
A = 2000 \times 1.1881
$$
$$
A = 2376.2
$$
So, at the end of 2 years, the amount in the account is \$2376.20.
Explanation:
The compound interest formula calculates how much the investment grows over time with the application of interest. In part (a), for the first year, the principal amount of \$2000 is increased by 9% of itself, giving a total of \$2180. In part (b), after two years, the interest has been applied to the initial principal as well as the interest accrued in the first year, leading to a total of \$2376.20. The key idea behind compound interest is that interest is calculated not only on the original principal but also on the interest that has already been added to the account, leading to exponential growth over time.