Haley won $25,000 in a lottery. She will spend some of her winnings now and
save the rest. The money Haley saves must amount to $45,000 in 25 years.
Haley can invest the money at 6.35% compounded monthly. About how much
could Haley spend now? How long will it take for money to double at 4 1/2% compounded quarterly?
Amount needed to get 45,000 in 25 years at 6.35% comp monthly
i = .0635/12 = .00529166..
n = 25(12) = 300
= 45000(1.005291666..)^-300
= 9.238.21
This is the amount he must invest to get his 45,000 25 years from now,
I suppose he can spend the rest of the $25,000
for money to double at 4.5% comp monthly ....
(1.00375)^n = 2
take log of both sides
log (1.00375)^n = log 2
n log 1.00375 = log 2
n = 185.19 months = appr 15.4 years
To find out how much Haley could spend now, we need to calculate the present value of the $45,000 she wants to save in 25 years.
Let's use the formula for the present value of a future sum of money:
Present Value = Future Value / (1 + (Interest Rate / Number of Compounding Periods))^Number of Compounding Periods * Number of Years
In this case, the future value is $45,000, the interest rate is 6.35% (expressed as a decimal, 0.0635), the number of compounding periods is 12 (monthly compounding), and the number of years is 25.
Present Value = $45,000 / (1 + (0.0635 / 12))^(12 * 25)
Present Value = $45,000 / (1.00529166666667)^(300)
Present Value ≈ $11,373.42
Therefore, Haley could spend approximately $11,373.42 now.
To calculate how long it will take for money to double at 4 1/2% compounded quarterly, we can use the rule of 72.
The rule of 72 states that the number of years it takes for an investment to double is approximately 72 divided by the annual interest rate.
In this case, the annual interest rate is 4.5%, and it is compounded quarterly.
Number of compounding periods per year = 4 (quarterly compounding)
Effective interest rate per compounding period = (1 + (0.045 / 4)) - 1
Years to double = 72 / (Effective interest rate per compounding period)
Years to double = 72 / (((1 + (0.045 / 4)) - 1)
Years to double ≈ 16.0001219
Therefore, it will take approximately 16 years for the money to double at 4 1/2% compounded quarterly.
To find out how much Haley could spend now, we need to calculate the future value of her savings in 25 years. We can use the formula for compound interest:
Future Value = Present Value * (1 + (interest rate / number of compounding periods))^(number of compounding periods * number of years)
Let's plug in the values:
Present Value = $25,000
Interest Rate = 6.35% (or 0.0635)
Number of Compounding Periods = 12 (monthly compounding)
Number of Years = 25
Future Value = $45,000 (the amount Haley wants to save in 25 years)
45,000 = 25,000 * (1 + (0.0635 / 12))^(12 * 25)
Now, we need to solve this equation for the present value, which represents the amount Haley could spend now.
Dividing both sides of the equation by $25,000, we get:
1.8 = (1 + (0.0635 / 12))^(12 * 25)
To isolate the exponential term, we take the natural logarithm (ln) of both sides:
ln(1.8) = ln((1 + (0.0635 / 12))^(12 * 25))
Using a calculator, ln(1.8) ≈ 0.5878
Now, we can solve for the number of years it will take for money to double at 4 1/2% compounded quarterly.
To calculate the doubling time, we can use the formula:
n = (ln(2)) / (interest rate)
Let's plug in the values:
Interest Rate = 4 1/2% (or 0.045)
Number of Compounding Periods = 4 (quarterly compounding)
n = (ln(2)) / (0.045)
Using a calculator, ln(2) ≈ 0.6931
n = 0.6931 / 0.045
n ≈ 15.4 years
Therefore, it would take approximately 15.4 years for the money to double at a 4 1/2% interest rate compounded quarterly.