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.
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