High School

E.) Suppose California's population is 37.9 million people, and its population is expected to grow by 3% annually. How long will it take for the population to double? Round your answer to the nearest whole number.

F.) Find the PV (Present Value) of an ordinary annuity that pays $1,000 each of the next 6 years if the interest rate is 18%. Then find the FV (Future Value) of that same annuity. Round your answers to the nearest cent.
- PV of ordinary annuity: $
- FV of ordinary annuity: $

G.) How will the PV and FV of the annuity in part F change if it is an annuity due rather than an ordinary annuity? Round your answers to the nearest cent.
- PV of annuity due: $
- FV of annuity due: $

Answer :

E) It will take approximately 23 years for California's population to double.

F) FV ≈ $7,025.55

G) FV of the annuity due ≈ $8,305.11

E.) To calculate how long it will take for the population to double, we can use the compound interest formula:

Population = [tex]Initial Population * (1 + Growth Rate)^{Time}[/tex]

Where:

Initial Population = 37.9 million

Growth Rate = 3%

= 0.03

Population (after doubling) = 2 × Initial Population

= 2 × 37.9 million

Now, we need to find the time (in years):

2 × 37.9 million

= [tex]37.9 million * (1 + 0.03)^{Time}[/tex]

Divide both sides by 37.9 million:

2 = [tex](1 + 0.03)^{Time}[/tex]

Take the natural logarithm of both sides to solve for Time:

Time = ln(2) / ln(1.03)

Time ≈ 23.1 years

So, it will take approximately 23 years for California's population to double.

F.) To calculate the present value (PV) of the ordinary annuity, we use the formula:

PV = [tex]Payment * [(1 - (1 + Interest Rate)^{(-Number of Periods)) / Interest Rate]}[/tex]

Where:

Payment = $1,000

Interest Rate = 18%

= 0.18

Number of Periods = 6 years

PV =[tex]$1,000 * [(1 - (1 + 0.18)^{(-6)) / 0.18]}[/tex]

PV ≈ $4,044.80 (rounded to the nearest cent)

To calculate the future value (FV) of the ordinary annuity, we use the formula:

FV = [tex]Payment * [(1 + Interest Rate)^{Number of Periods - 1}) / Interest Rate][/tex]

FV = [tex]$1,000 * [(1 + 0.18)^{6 - 1}) / 0.18][/tex]

FV ≈ $7,025.55 (rounded to the nearest cent)

G.) If the annuity is an annuity due instead of an ordinary annuity, the PV and FV formulas will change slightly.

For the PV of the annuity due, we multiply the PV of the ordinary annuity by (1 + Interest Rate):

PV of annuity due = $4,044.80 × (1 + 0.18)

PV of the annuity due ≈ $4,781.86 (rounded to the nearest cent)

For the FV of the annuity due, we multiply the FV of the ordinary annuity by (1 + Interest Rate):

FV of annuity due = $7,025.55 × (1 + 0.18)

FV of the annuity due ≈ $8,305.11 (rounded to the nearest cent)

Learn more about compound interest here:

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