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Let's walk through the solutions step-by-step to each of the questions presented:
### Question 1
Problem: If you want to have \[tex]$60,000 in 8 years, how much do you need to deposit in the bank today if the account pays an interest rate of 9%? Solution: We need to find the present value (PV). The formula to calculate the present value using the future value (FV), interest rate (r), and time (t) is given by: \[ PV = \frac{FV}{(1 + r)^t} \] Given: - Future Value (FV) = \$[/tex]60,000
- Interest Rate (r) = 0.09
- Time (t) = 8 years
Plugging these values into the formula:
[tex]\[ PV = \frac{60000}{(1 + 0.09)^8} \approx 30111.98 \][/tex]
Therefore, you need to deposit approximately \[tex]$30,111.98 in the bank today. ### Question 2 Problem: What will \$[/tex]110,000 grow to be in 9 years if it is invested today at 11%?
Solution: We need to find the future value (FV) using the present value (PV), interest rate (r), and time (t). The formula is:
[tex]\[ FV = PV \times (1 + r)^t \][/tex]
Given:
- Present Value (PV) = \[tex]$110,000 - Interest Rate (r) = 0.11 - Time (t) = 9 years Let's plug in the values: \[ FV = 110000 \times (1 + 0.11)^9 \approx 218900.00 \] Therefore, \$[/tex]110,000 will grow to approximately \[tex]$218,900 in 9 years. ### Question 3 Problem: You would like to have \$[/tex]200,000 in a college fund in 15 years. How much do you need today if you expect to earn 12% while you are investing to pay for your child's college?
Solution: Again, we need to find the present value (PV), using the future value (FV), interest rate (r), and time (t):
[tex]\[ PV = \frac{FV}{(1 + r)^t} \][/tex]
Given:
- Future Value (FV) = \[tex]$200,000 - Interest Rate (r) = 0.12 - Time (t) = 15 years Plugging these values into the formula: \[ PV = \frac{200000}{(1 + 0.12)^15} \approx 36539.25 \] Therefore, you need to deposit approximately \$[/tex]36,539.25 today in order to have \[tex]$200,000 in 15 years. ### Question 4 Problem: You have been offered \$[/tex]3,000 in 4 years for providing \[tex]$2,000 today into a business venture with a friend. If interest rates are 10%, is this a good investment for you? Solution: We need to compare the future value of \$[/tex]2,000 invested at 10% for 4 years with \[tex]$3,000. If the future value is less than \$[/tex]3,000, it is a good investment.
Calculate future value using the formula:
[tex]\[ FV = PV \times (1 + r)^t \][/tex]
Given:
- Present Value (PV) = \[tex]$2,000 - Interest Rate (r) = 0.10 - Time (t) = 4 years Plugging these values into the formula: \[ FV = 2000 \times (1 + 0.10)^4 \approx 2928.20 \] Since \$[/tex]2,928.20 < \[tex]$3,000, the investment is good. Thus, it is a good investment. ### Question 5 Problem: What will \$[/tex]82,000 grow to be in 11 years if it is invested today at 8% and the interest rate is compounded monthly?
Solution: We need to find the future value (FV) using the compound interest formula, which is:
[tex]\[ FV = PV \times \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Given:
- Present Value (PV) = \[tex]$82,000 - Annual Interest Rate (r) = 0.08 - Time (t) = 11 years - Number of compounding periods per year (n) = 12 Plugging these values into the formula: \[ FV = 82000 \times \left( 1 + \frac{0.08}{12} \right)^{(12 \times 11)} \approx 197117.28 \] Therefore, \$[/tex]82,000 will grow to approximately \$197,117.28 in 11 years when compounded monthly at an 8% interest rate.
### Question 1
Problem: If you want to have \[tex]$60,000 in 8 years, how much do you need to deposit in the bank today if the account pays an interest rate of 9%? Solution: We need to find the present value (PV). The formula to calculate the present value using the future value (FV), interest rate (r), and time (t) is given by: \[ PV = \frac{FV}{(1 + r)^t} \] Given: - Future Value (FV) = \$[/tex]60,000
- Interest Rate (r) = 0.09
- Time (t) = 8 years
Plugging these values into the formula:
[tex]\[ PV = \frac{60000}{(1 + 0.09)^8} \approx 30111.98 \][/tex]
Therefore, you need to deposit approximately \[tex]$30,111.98 in the bank today. ### Question 2 Problem: What will \$[/tex]110,000 grow to be in 9 years if it is invested today at 11%?
Solution: We need to find the future value (FV) using the present value (PV), interest rate (r), and time (t). The formula is:
[tex]\[ FV = PV \times (1 + r)^t \][/tex]
Given:
- Present Value (PV) = \[tex]$110,000 - Interest Rate (r) = 0.11 - Time (t) = 9 years Let's plug in the values: \[ FV = 110000 \times (1 + 0.11)^9 \approx 218900.00 \] Therefore, \$[/tex]110,000 will grow to approximately \[tex]$218,900 in 9 years. ### Question 3 Problem: You would like to have \$[/tex]200,000 in a college fund in 15 years. How much do you need today if you expect to earn 12% while you are investing to pay for your child's college?
Solution: Again, we need to find the present value (PV), using the future value (FV), interest rate (r), and time (t):
[tex]\[ PV = \frac{FV}{(1 + r)^t} \][/tex]
Given:
- Future Value (FV) = \[tex]$200,000 - Interest Rate (r) = 0.12 - Time (t) = 15 years Plugging these values into the formula: \[ PV = \frac{200000}{(1 + 0.12)^15} \approx 36539.25 \] Therefore, you need to deposit approximately \$[/tex]36,539.25 today in order to have \[tex]$200,000 in 15 years. ### Question 4 Problem: You have been offered \$[/tex]3,000 in 4 years for providing \[tex]$2,000 today into a business venture with a friend. If interest rates are 10%, is this a good investment for you? Solution: We need to compare the future value of \$[/tex]2,000 invested at 10% for 4 years with \[tex]$3,000. If the future value is less than \$[/tex]3,000, it is a good investment.
Calculate future value using the formula:
[tex]\[ FV = PV \times (1 + r)^t \][/tex]
Given:
- Present Value (PV) = \[tex]$2,000 - Interest Rate (r) = 0.10 - Time (t) = 4 years Plugging these values into the formula: \[ FV = 2000 \times (1 + 0.10)^4 \approx 2928.20 \] Since \$[/tex]2,928.20 < \[tex]$3,000, the investment is good. Thus, it is a good investment. ### Question 5 Problem: What will \$[/tex]82,000 grow to be in 11 years if it is invested today at 8% and the interest rate is compounded monthly?
Solution: We need to find the future value (FV) using the compound interest formula, which is:
[tex]\[ FV = PV \times \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Given:
- Present Value (PV) = \[tex]$82,000 - Annual Interest Rate (r) = 0.08 - Time (t) = 11 years - Number of compounding periods per year (n) = 12 Plugging these values into the formula: \[ FV = 82000 \times \left( 1 + \frac{0.08}{12} \right)^{(12 \times 11)} \approx 197117.28 \] Therefore, \$[/tex]82,000 will grow to approximately \$197,117.28 in 11 years when compounded monthly at an 8% interest rate.
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