Answer :
To determine how much money Investor A would need to invest now to receive $50,000 after 6 years with an annual interest rate of 5%, we need to calculate the present value of this future amount using the present value formula for compound interest. Here's a step-by-step solution:
### Step-by-Step Solution:
1. Identify the given values:
- Future Value (\(FV\)): $50,000
- Annual Interest Rate (\(r\)): 5% or 0.05
- Number of Years (\(n\)): 6
2. Using the Present Value formula:
The formula to calculate the present value (\(PV\)) is:
[tex]\[ PV = \frac{FV}{(1 + r)^n} \][/tex]
3. Substitute the known values into the formula:
[tex]\[ PV = \frac{50,000}{(1 + 0.05)^6} \][/tex]
4. Calculate the compound factor:
[tex]\[ (1 + r)^n = (1 + 0.05)^6 \][/tex]
[tex]\[ (1.05)^6 \][/tex]
5. Compute \((1.05)^6\):
[tex]\[ (1.05)^6 \approx 1.3401 \][/tex]
6. Divide the future value by the compound factor:
[tex]\[ PV = \frac{50,000}{1.3401} \approx 37,310.77 \][/tex]
7. Round to the nearest cent:
[tex]\[ PV \approx 37,310.77 \][/tex]
So, Investor A would need to invest approximately [tex]$37,310.77 today to receive $[/tex]50,000 after 6 years, given an annual interest rate of 5% compounded annually.
### Step-by-Step Solution:
1. Identify the given values:
- Future Value (\(FV\)): $50,000
- Annual Interest Rate (\(r\)): 5% or 0.05
- Number of Years (\(n\)): 6
2. Using the Present Value formula:
The formula to calculate the present value (\(PV\)) is:
[tex]\[ PV = \frac{FV}{(1 + r)^n} \][/tex]
3. Substitute the known values into the formula:
[tex]\[ PV = \frac{50,000}{(1 + 0.05)^6} \][/tex]
4. Calculate the compound factor:
[tex]\[ (1 + r)^n = (1 + 0.05)^6 \][/tex]
[tex]\[ (1.05)^6 \][/tex]
5. Compute \((1.05)^6\):
[tex]\[ (1.05)^6 \approx 1.3401 \][/tex]
6. Divide the future value by the compound factor:
[tex]\[ PV = \frac{50,000}{1.3401} \approx 37,310.77 \][/tex]
7. Round to the nearest cent:
[tex]\[ PV \approx 37,310.77 \][/tex]
So, Investor A would need to invest approximately [tex]$37,310.77 today to receive $[/tex]50,000 after 6 years, given an annual interest rate of 5% compounded annually.
