William owes $9,000.00 to a friend who is charging him interest of 1.60% p.m. he is required to settle the amount with 2 equal payments, one today and the other in 2 months. Calculate the size of the payments using 2 months as the focal date. Round your answer to 2 decimal places.
To calculate the size of the payments, we can use the formula for the present value of an annuity:
P = (A / (1 + r)^n) + (A / (1 + r)^(2n))
Where:
P is the present value of the annuity (the amount William owes),
A is the amount of each payment,
r is the interest rate per period (1.60% per month),
n is the number of periods (2 months).
We can rearrange the formula to solve for A:
P = (A / (1 + r)^n) + (A / (1 + r)^(2n))
P = A * (1 + r)^(-n) + A * (1 + r)^(-2n)
Rearranging further, we have:
P = A * [(1 + r)^(-n) + (1 + r)^(-2n)]
Since William owes $9,000, we have:
9,000 = A * [(1 + 0.016)^(-2) + (1 + 0.016)^(-4)]
Simplifying the right side of the equation:
9,000 = A * [0.984*0.984 + 0.984*0.984*0.984*0.984]
Calculating the values inside the brackets:
9,000 = A * [0.968256 + 0.937963817]
Adding the values inside the brackets:
9,000 = A * 1.906219817
Further simplifying:
A = 9,000 / 1.906219817
Calculating A:
A ≈ 4,719.61
Therefore, the size of each payment, rounded to 2 decimal places, is approximately $4,719.61.
To calculate the size of the payments, we need to consider the principal amount owed, the interest rate, and the time period.
First, let's calculate the interest accrued over the 2-month period. We can use the formula:
Interest = Principal * Interest Rate * Time
Plugging in the values:
Principal = $9,000.00
Interest Rate = 1.60% per month
Time = 2 months
Interest = $9,000.00 * 0.016 * 2 = $288.00
Now, let's add the interest to the principal amount:
Total Amount = Principal + Interest
Total Amount = $9,000.00 + $288.00 = $9,288.00
Since we have 2 equal payments, we will divide the Total Amount by 2 to find the size of each payment:
Payment Size = Total Amount / Number of Payments
Payment Size = $9,288.00 / 2 = $4,644.00
Therefore, the size of each payment, rounded to 2 decimal places, is $4,644.00.
Note: Make sure to clarify with your friend if the interest gets compounded monthly or if it's a simple interest calculation.