Matilda wishes to retire at age 67 with in her retirement account. When she turns 28 , she decides to begin depositing money into an account with an ApR of
compounded monthly. What is the monthly deposit that Matilda must make in order to reach her goal? Round your answer to the nearest cent. if necessary.
Formulas

Ask by Hart Warner. in the United States

Jan 22,2025

Question

Matilda wishes to retire at age 67 with in her retirement account. When she turns 28 , she decides to begin depositing money into an account with an ApR of
compounded monthly. What is the monthly deposit that Matilda must make in order to reach her goal? Round your answer to the nearest cent. if necessary.
Formulas

Ask by Hart Warner. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

To determine the **monthly deposit** Matilda needs to make to reach her retirement goal of **\$1,700,000** by age **67**, we can use the **Future Value of an Ordinary Annuity** formula. Here's a step-by-step breakdown:

### **Given:**
- **Current Age:** 28
- **Retirement Age:** 67
- **Annual Percentage Rate (ApR):** 9% (compounded monthly)
- **Future Value (FV):** \$1,700,000

### **Calculate the Number of Periods and Monthly Interest Rate:**
1. **Number of Years to Invest:**  
   $$ 67 - 28 = 39 $$ years

2. **Total Number of Monthly Deposits (n):**  
   $$ 39 \text{ years} \times 12 \text{ months/year} = 468 \text{ months} $$

3. **Monthly Interest Rate (r):**  
   $$ \frac{9\%}{12} = 0.75\% $$ or $$ 0.0075 $$ in decimal

### **Future Value of an Ordinary Annuity Formula:**
$$ FV = M \times \left( \frac{(1 + r)^n - 1}{r} \right) $$

Where:
- $$ FV $$ = Future Value (\$1,700,000)
- $$ M $$ = Monthly Deposit
- $$ r $$ = Monthly Interest Rate (0.0075)
- $$ n $$ = Total Number of Deposits (468)

### **Solving for Monthly Deposit (M):**
$$ M = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)} $$

Plugging in the numbers:
$$ M = \frac{1,700,000}{\left( \frac{(1 + 0.0075)^{468} - 1}{0.0075} \right)} $$

Calculating $$ (1 + 0.0075)^{468} $$:
$$ (1.0075)^{468} \approx 32.817 $$

Thus:
$$ M = \frac{1,700,000}{\left( \frac{32.817 - 1}{0.0075} \right)} $$
$$ M = \frac{1,700,000}{4,242.2667} $$
$$ M \approx 400.69 $$

### **Conclusion:**
Matilda needs to make a **monthly deposit of approximately \$400.69** to reach her retirement goal of \$1,700,000 by age 67, assuming a 9% annual interest rate compounded monthly.

**Final Answer:**
$$ \boxed{\$400.69} $$
Matilda needs to deposit approximately \$400.69 each month to reach her retirement goal of \$1,700,000 by age 67.

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