Questions: If an individual saves 5,700 and elects to place the total dollar amount into a savings account earning 2.75% APR compounded monthly, how much will the original deposit grow to in 12 years? The initial deposit plus interest equals 7,963. (Round to the nearest dollar as needed.)

If an individual saves 5,700 and elects to place the total dollar amount into a savings account earning 2.75% APR compounded monthly, how much will the original deposit grow to in 12 years? The initial deposit plus interest equals 7,963. (Round to the nearest dollar as needed.)

Transcript text: If an individual saves $\$ 5,700$ and elects to place the total dollar amount into a savings account earning $2.75 \%$ APR compounded monthly, how much will the original deposit grow to in 12 years? The initial deposit plus interest equals $\$ 7963$. (Round to the nearest dollar as needed.)

Solution

Solution Steps

Step 1: Identify the given values

Principal amount ($ P $) = \$5,700
Annual Percentage Rate (APR) = $ 2.75\% $
Compounding frequency = Monthly
Time period ($ t $) = 12 years

Step 2: Convert the APR to a monthly interest rate

The APR is compounded monthly, so the monthly interest rate ($ r $) is: \[ r = \frac{2.75\%}{12} = \frac{0.0275}{12} = 0.0022917 \]

Step 3: Calculate the number of compounding periods

Since the interest is compounded monthly over 12 years, the total number of compounding periods ($ n $) is: \[ n = 12 \times 12 = 144 \text{ months} \]

Step 4: Apply the compound interest formula

The formula for compound interest is: \[ A = P \left(1 + r\right)^n \] Substitute the known values: \[ A = 5700 \left(1 + 0.0022917\right)^{144} \]

Step 5: Calculate the final amount

Compute the value inside the parentheses: \[ 1 + 0.0022917 = 1.0022917 \] Raise this value to the 144th power: \[ 1.0022917^{144} \approx 1.397 \] Multiply by the principal: \[ A = 5700 \times 1.397 \approx 7963 \]

Step 6: Round to the nearest dollar

The final amount is approximately \$7,963.