Question

Ingrid wants to buy a 17,000 car in 5 years. How much money must she deposit at the end of each quarter in an account paying 5.6% Compounded quarterly so that she will have enough money to pay for her car? Round to the nearest as needed

Answer 1

To determine how much Ingrid needs to deposit at the end of each quarter to accumulate \(17,000\) in 5 years with an account that pays \(5.6\%\) interest compounded quarterly, we can use the formula for the future value of an ordinary annuity:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \(FV\) = future value of the annuity (the amount they want to accumulate, in this case \(17,000\))
- \(P\) = payment per period (the amount Ingrid needs to deposit each quarter)
- \(r\) = interest rate per period
- \(n\) = total number of payments

**Step 1: Define the values**
- The future value \(FV = 17,000\)
- The annual interest rate is \(5.6\%\), so the quarterly interest rate \(r = \frac{5.6\%}{4} = \frac{5.6}{100 \times 4} = 0.014\)
- The total number of quarters in 5 years is \(n = 5 \times 4 = 20\)

**Step 2: Plug values into the future value formula and solve for \(P\)**

\[
17,000 = P \times \frac{(1 + 0.014)^{20} - 1}{0.014}
\]

Calculating \((1 + 0.014)^{20}\):

\[
(1 + 0.014)^{20} \approx 1.348850
\]

Now substituting it back into the equation:

\[
17,000 = P \times \frac{1.348850 - 1}{0.014}
\]

Calculating the fraction:

\[
\frac{1.348850 - 1}{0.014} = \frac{0.348850}{0.014} \approx 24.9907
\]

Now, substituting this value back:

\[
17,000 = P \times 24.9907
\]

**Step 3: Solve for \(P\)**

\[
P = \frac{17,000}{24.9907} \approx 680.40
\]

Thus, Ingrid must deposit approximately **$680.40** at the end of each quarter to accumulate enough money to buy the car in 5 years.