Question 8 of 15 . Step 1 of 1
Emilia and Liam are purchasing a home. They wish to save money for 10 years and purchase a house that has a value of with cash. If they deposit money into
an account paying interest. compounded monthly, how much do they need to deposit each month in order to make the purchase? Round your answer to the
nearest cent, if necessary.
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Question

Question 8 of 15 . Step 1 of 1
Emilia and Liam are purchasing a home. They wish to save money for 10 years and purchase a house that has a value of

with cash. If they deposit money into
an account paying

interest. compounded monthly, how much do they need to deposit each month in order to make the purchase? Round your answer to the
nearest cent, if necessary.
Formulas
Answer
How to enter your answer fopens in new window) Keypad
Keyboard Shortcuts

UpStudy AI · Accepted Answer

To determine how much Emilia and Liam need to deposit each month to save $$ \$190,000 $$ in 10 years with an annual interest rate of $$ 4\% $$ compounded monthly, we can use the **Future Value of an Ordinary Annuity** formula:

$$
FV = PMT \times \left( \frac{(1 + i)^n - 1}{i} \right)
$$

Where:
- $$ FV $$ = Future Value (\$190,000)
- $$ PMT $$ = Monthly Deposit
- $$ i $$ = Monthly Interest Rate
- $$ n $$ = Total Number of Payments

### Step-by-Step Calculation:

1. **Determine the Monthly Interest Rate ($$ i $$):**
   $$
   i = \frac{4\%}{12} = 0.3333\% \text{ or } 0.003333
   $$

2. **Calculate the Total Number of Monthly Payments ($$ n $$):**
   $$
   n = 10 \text{ years} \times 12 \text{ months/year} = 120 \text{ months}
   $$

3. **Rearrange the Future Value Formula to Solve for $$ PMT $$:**
   $$
   PMT = \frac{FV \times i}{(1 + i)^n - 1}
   $$

4. **Plug in the Values:**
   $$
   PMT = \frac{190,000 \times 0.003333}{(1 + 0.003333)^{120} - 1}
   $$

5. **Calculate $$(1 + i)^n$$:**
   $$
   (1 + 0.003333)^{120} \approx 1.4908
   $$

6. **Compute the Denominator:**
   $$
   1.4908 - 1 = 0.4908
   $$

7. **Final Calculation for $$ PMT $$:**
   $$
   PMT = \frac{190,000 \times 0.003333}{0.4908} \approx \frac{633.333}{0.4908} \approx 1,291.71
   $$

### **Answer:**
Emilia and Liam need to deposit **\$1,291.71** each month to save for the $$ \$190,000 $$ home in 10 years.
Emilia and Liam need to deposit **\$1,291.71** each month.

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