Marilyn was supposed to pay $1410 to Bernice on March 1. Some time later Marilyn paid Bernice an equivalent payment of $1530.22, allowing for a time value of money of 4.8% compounded monthly.
See related question below.
To find out the time period between March 1 and the actual payment date, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Ending balance (the actual payment)
P = Principal amount (the intended payment)
r = Annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = time period in years
In this case, let's assume that the time period is in years.
Step 1: Calculate the time value of money
We can use the formula to calculate the time value of money:
FV = PV(1 + r/n)^(nt)
Where:
FV = Future value (actual payment)
PV = Present value (intended payment)
r = Annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = time period in years
Given:
PV = $1410 (intended payment)
FV = $1530.22 (actual payment)
r = 4.8% or 0.048 (annual interest rate)
n = 12 (monthly compounding)
Using the formula, we have:
$1530.22 = $1410(1 + 0.048/12)^(12t)
Step 2: Solve for t
Let's solve for t by isolating the variable in the equation.
Divide both sides by 1410:
$1530.22/$1410 = (1 + 0.048/12)^(12t)
1.0859964546 = (1 + 0.004)/(12t)
Take the natural logarithm ln of both sides:
ln(1.0859964546) = ln((1 + 0.004)/(12t))
Step 3: Calculate the time period (t)
Using a scientific calculator or software, calculate the natural logarithm of 1.0859964546:
ln(1.0859964546) = 0.0808800341
Now the equation becomes:
0.0808800341 = ln((1 + 0.004)/(12t))
Rearrange the equation to solve for t:
ln((1 + 0.004)/(12t)) = 0.0808800341
Take the inverse natural logarithm (e^x) of both sides:
(1 + 0.004)/(12t) = e^0.0808800341
Simplify:
(1 + 0.004)/(12t) = 1.084
Multiply both sides by 12t:
1 + 0.004 = 12t * 1.084
Combine like terms:
1.004 = 13.008t
Divide both sides by 13.008:
t = 1.004 / 13.008
t ≈ 0.077
The time period (t) between March 1 and the actual payment date is approximately 0.077 years, or about 28.1 days.
To calculate the time value of money in this scenario, we need to use the concept of present value and future value.
First, let's find the present value of $1410 on March 1. We'll assume the interest rate of 4.8% compounded monthly is the applicable discount rate. To find the present value, we can use the formula:
Present Value = Future Value / (1 + r)^n
Where:
Future Value = $1410
r = interest rate per period (4.8% per year divided by 12 months)
n = number of periods (the time in months since the payment was made)
Let's calculate the present value:
r = 4.8% / 12 = 0.004 (monthly interest rate)
n = number of months between the payment and the equivalent payment
Next, let's find the number of months that have passed between the two payments. To find this, we need the formula:
Number of Months = Log(Future Value / Present Value) / Log(1 + r)
Where:
Present Value = $1410
Future Value = $1530.22
r = 0.004 (monthly interest rate)
Now, let's calculate the number of months:
Number of Months = Log(1530.22 / 1410) / Log(1 + 0.004)
To evaluate the logarithm in the calculator, use the logarithm function (log), base 10 logarithm (log10), or natural logarithm (ln), depending on the available functions.
Once you have the number of months, you can determine the approximate time.