Question 14 of 15, Step 1 of 1 Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $$ \$ 23,000 $$ over the next 3 years. What amount must be deposited quarterly into an account that earns $$ 4.4 \% $$ interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary. Formulas Answer How to enter your answer (opens in new window) Keyboard Shortcuts Correct Keypad

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To solve this problem, we need to determine the amount that must be deposited quarterly into an account that earns $$ 4.4\% $$ interest, compounded quarterly, in order to save $$ \$ 23,000 $$ over 3 years.

### Known Conditions:
- Future Value ($$ FV $$) = $$ 23000 $$
- Annual Interest Rate ($$ r $$) = $$ 4.4\% = 0.044 $$
- Number of years ($$ t $$) = $$ 3 $$
- Compounding frequency ($$ n $$) = $$ 4 $$ (quarterly)

### Step-by-Step Solution:

1. **Calculate the total number of deposits**:
   $$
   N = n \times t = 4 \times 3 = 12
   $$

2. **Calculate the interest rate per period**:
   $$
   r_p = \frac{r}{n} = \frac{0.044}{4} = 0.011
   $$

3. **Use the Future Value of an Annuity formula**:
   The formula for the future value of an annuity is given by:
   $$
   FV = P \times \frac{(1 + r_p)^N - 1}{r_p}
   $$
   where $$ P $$ is the amount deposited each period.

4. **Rearranging the formula to solve for $$ P $$**:
   $$
   P = \frac{FV \times r_p}{(1 + r_p)^N - 1}
   $$

5. **Substituting the known values**:
   $$
   P = \frac{23000 \times 0.011}{(1 + 0.011)^{12} - 1}
   $$

Now, let's calculate $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{23000\times 0.011}{\left(\left(1+0.011\right)^{12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{23000\times 0.011}{\left(1+0.011\right)^{12}-1}$$
- step2: Add the numbers:
  $$\frac{23000\times 0.011}{1.011^{12}-1}$$
- step3: Convert the expressions:
  $$\frac{23000\times 0.011}{\left(\frac{1011}{1000}\right)^{12}-1}$$
- step4: Multiply the numbers:
  $$\frac{253}{\left(\frac{1011}{1000}\right)^{12}-1}$$
- step5: Subtract the numbers:
  $$\frac{253}{\frac{1011^{12}-1000^{12}}{1000^{12}}}$$
- step6: Multiply by the reciprocal:
  $$253\times \frac{1000^{12}}{1011^{12}-1000^{12}}$$
- step7: Multiply:
  $$\frac{253\times 1000^{12}}{1011^{12}-1000^{12}}$$
The calculated value for $$ P $$ is approximately $$ 1803.46 $$.

### Final Answer:
To reach your goal of saving $$ \$ 23,000 $$ over the next 3 years, you must deposit approximately **\$ 1,803.46** quarterly into the account that earns $$ 4.4\% $$ interest, compounded quarterly.
You need to deposit approximately \$1,803.46 each quarter into the account to save \$23,000 over 3 years.

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