Answer :
To solve the question of how much money you need to deposit each quarter to reach [tex]$900,000 in 30 years at a 7.1% annual interest rate compounded quarterly, follow these steps.
### Step-by-Step Solution:
#### a) Determine the quarterly deposit amount
1. Understand the problem:
- Future Value (FV) needed: $[/tex]900,000
- Time (t): 30 years
- Annual interest rate (r): 7.1%
- Compounding periods per year: 4 (quarterly compounding)
2. Calculate the quarterly interest rate:
- Quarterly interest rate (i) = Annual interest rate / number of compounding periods per year
- Therefore, [tex]\( i = \frac{0.071}{4} \approx 0.01775 \)[/tex]
3. Calculate the total number of compounding periods:
- Total periods (n) = number of years * number of compounding periods per year
- Therefore, [tex]\( n = 30 \times 4 = 120 \)[/tex]
4. Use the future value of an ordinary annuity formula to find the quarterly deposit (P):
- The formula for the Future Value (FV) of an ordinary annuity is:
[tex]\[ FV = P \times \frac{(1 + i)^n - 1}{i} \][/tex]
- We need to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{FV \times i}{(1 + i)^n - 1} \][/tex]
5. Substitute the given values into the formula:
- [tex]\( FV = 900000 \)[/tex]
- [tex]\( i = 0.01775 \)[/tex]
- [tex]\( n = 120 \)[/tex]
6. Calculate the quarterly deposit amount:
- First, calculate the denominator:
[tex]\[ (1 + 0.01775)^{120} - 1 \approx 11.140225590716817 \][/tex]
- Next, calculate the numerator:
[tex]\[ 900000 \times 0.01775 = 15975 \][/tex]
- Finally, divide the numerator by the denominator to find the quarterly deposit:
[tex]\[ P = \frac{15975}{11.140225590716817} \approx 2200.68 \][/tex]
- Therefore, the quarterly deposit should be approximately [tex]$2200.68. Answer for (a): \( \$[/tex]2200.68 \)
#### b) Calculate the total money put into the account
1. Calculate the total number of deposits:
- Total deposits = Quarterly deposit * Total compounding periods
- Therefore, total deposits = [tex]\( 2200.68 \times 120 \approx 264081.91 \)[/tex]
Answer for (b): [tex]\( \$264081.91 \)[/tex]
#### c) Calculate the total interest earned
1. Calculate the total interest earned:
- Total interest earned = Future value - Total deposits
- Therefore, total interest = [tex]\( 900000 - 264081.91 \approx 635918.09 \)[/tex]
Answer for (c): [tex]\( \$635918.09 \)[/tex]
### Summary:
a) You should deposit approximately [tex]$2200.68 each quarter. b) The total money you will put into the account is approximately $[/tex]264081.91.
c) The total interest you will earn is approximately $635918.09.
- Time (t): 30 years
- Annual interest rate (r): 7.1%
- Compounding periods per year: 4 (quarterly compounding)
2. Calculate the quarterly interest rate:
- Quarterly interest rate (i) = Annual interest rate / number of compounding periods per year
- Therefore, [tex]\( i = \frac{0.071}{4} \approx 0.01775 \)[/tex]
3. Calculate the total number of compounding periods:
- Total periods (n) = number of years * number of compounding periods per year
- Therefore, [tex]\( n = 30 \times 4 = 120 \)[/tex]
4. Use the future value of an ordinary annuity formula to find the quarterly deposit (P):
- The formula for the Future Value (FV) of an ordinary annuity is:
[tex]\[ FV = P \times \frac{(1 + i)^n - 1}{i} \][/tex]
- We need to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{FV \times i}{(1 + i)^n - 1} \][/tex]
5. Substitute the given values into the formula:
- [tex]\( FV = 900000 \)[/tex]
- [tex]\( i = 0.01775 \)[/tex]
- [tex]\( n = 120 \)[/tex]
6. Calculate the quarterly deposit amount:
- First, calculate the denominator:
[tex]\[ (1 + 0.01775)^{120} - 1 \approx 11.140225590716817 \][/tex]
- Next, calculate the numerator:
[tex]\[ 900000 \times 0.01775 = 15975 \][/tex]
- Finally, divide the numerator by the denominator to find the quarterly deposit:
[tex]\[ P = \frac{15975}{11.140225590716817} \approx 2200.68 \][/tex]
- Therefore, the quarterly deposit should be approximately [tex]$2200.68. Answer for (a): \( \$[/tex]2200.68 \)
#### b) Calculate the total money put into the account
1. Calculate the total number of deposits:
- Total deposits = Quarterly deposit * Total compounding periods
- Therefore, total deposits = [tex]\( 2200.68 \times 120 \approx 264081.91 \)[/tex]
Answer for (b): [tex]\( \$264081.91 \)[/tex]
#### c) Calculate the total interest earned
1. Calculate the total interest earned:
- Total interest earned = Future value - Total deposits
- Therefore, total interest = [tex]\( 900000 - 264081.91 \approx 635918.09 \)[/tex]
Answer for (c): [tex]\( \$635918.09 \)[/tex]
### Summary:
a) You should deposit approximately [tex]$2200.68 each quarter. b) The total money you will put into the account is approximately $[/tex]264081.91.
c) The total interest you will earn is approximately $635918.09.
