An investor needs ​$19,000 in 12 years.

​(b) Find the​ investor's quarterly deposit if the money is deposited at 7.6​% compounded quarterly.


To find the investor's quarterly deposit needed to reach $19,000 in 12 years at an interest rate of 7.6% compounded quarterly, we can use the future value of an annuity formula:

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

Where:
- $ FV $ = future value ($19,000)
- $ P $ = quarterly deposit (what we want to find)
- $ r $ = quarterly interest rate
- $ n $ = total number of deposits

**Step 1: Calculate $ r $ and $ n $**

1. The annual interest rate is 7.6%, so the quarterly interest rate $ r $ is:
   \[
   r = \frac{7.6\%}{4} = \frac{0.076}{4} = 0.019
   \]

2. The total number of deposits over 12 years, with quarterly deposits, is:
   \[
   n = 12 \times 4 = 48
   \]

**Step 2: Substitute values into the formula**

Now we rearrange the formula to solve for $ P $:

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

Substituting the values:

\[
P = \frac{19000}{\frac{(1 + 0.019)^{48} - 1}{0.019}}
\]

**Step 3: Calculate**

1. Calculate $ (1 + 0.019)^{48} $:
   \[
   (1.019)^{48} \approx 2.485
   \]

2. Calculate $ (1.019)^{48} - 1 $:
   \[
   2.485 - 1 = 1.485
   \]

3. Now, calculate the denominator:
   \[
   \frac{1.485}{0.019} \approx 78.684
   \]

4. Finally, calculate $ P $:
   \[
   P = \frac{19000}{78.684} \approx 241.09
   \]

Thus, the investor needs to deposit approximately **$241.09** quarterly.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How does changing the interest rate affect the quarterly deposit?
2. What would the total amount deposited be at the end of 12 years?
3. How would the deposit amount change if the investment period was shorter?
4. What are other investment options with different compounding frequencies?
5. How do inflation rates impact the future value of this investment?

**Tip:** Always consider the impact of fees or taxes on investment returns, as they can significantly alter your final amount.

An investor needs ​$19,000 in 12 years. Find the​ investor's quarterly deposit if the money is deposited at 7.6​% compounded quarterly.

Discover how to calculate the quarterly deposit required to reach $19,000 in 12 years with a 7.6% interest rate compounded quarterly. This step-by-step guide uses the future value of annuity formula, ideal for finance students.

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