Questions: How much money should be deposited today in an account that earns 2.5% compounded monthly so that it will accumulate to 11,000 in 3 years? The amount of money that should be deposited is (Round up to the nearest cent.)

How much money should be deposited today in an account that earns 2.5% compounded monthly so that it will accumulate to 11,000 in 3 years?

The amount of money that should be deposited is (Round up to the nearest cent.)

Transcript text: How much money should be deposited today in an account that earns $2.5 \%$ compounded monthly so that it will accumulate to $\$ 11,000$ in 3 years? The amount of money that should be deposited is $\$$ $\square$ (Round up to the nearest cent.)

Solution

Solution Steps

Step 1: Convert the Annual Interest Rate to a Decimal

The annual interest rate $r$ is given as 2.5\%. To convert it to a decimal, we divide by 100: $r = 2.5 / 100 = 0.025$.

Step 2: Identify the Number of Compounding Periods per Year ($n$)

The number of times the interest is compounded per year is given as $n = 12$.

Step 3: Determine the Future Value ($A$)

The future value that the account should accumulate to is given as $A = 11000$.

Step 4: Determine the Number of Years ($t$)

The number of years the money is to be left in the account is given as $t = 3$.

Step 5: Substitute the Values into the Formula to Calculate the Present Value (PV)

Substituting the values into the formula, we get: \[ PV = \frac{A}{(1 + \frac{r}{n})^{n \cdot t}} = \frac{11000}{(1 + \frac{0.025}{12})^{12 \cdot 3}} = 10205.97 \]