Mr. Mueller took a loan of $4200 and arranged to repay the loan with 3 equal
payments of $X at the end of 4, 8, and 12 months respectively. If the lender charges
a rate j12 = 12%, then what is the value of each payment ($X)?
To find the value of each payment ($X), we need to solve the problem using the present value formula.
The present value formula is given by: PV = future payment / (1 + interest rate)^n, where PV is the present value, future payment is the amount to be paid in the future, interest rate is the rate at which interest is charged, and n is the time period.
In this case, we need to find the present value of each payment ($X) at the time of taking the loan. We are given that the loan amount is $4200 and it is to be repaid with 3 equal payments at the end of 4, 8, and 12 months respectively.
Let's solve for the present value of each payment ($X):
For the first payment at 4 months:
PV_1 = X / (1 + 0.12)^(4/12)
For the second payment at 8 months:
PV_2 = X / (1 + 0.12)^(8/12)
For the third payment at 12 months:
PV_3 = X / (1 + 0.12)^(12/12)
Since the loan amount is to be repaid with 3 equal payments, the sum of the present values of all three payments should be equal to the loan amount:
PV_1 + PV_2 + PV_3 = $4200
Now, let's substitute the values and solve the equation:
X / (1 + 0.12)^(4/12) + X / (1 + 0.12)^(8/12) + X / (1 + 0.12)^(12/12) = $4200
Simplifying the equation:
X / (1.04) + X / (1.08) + X / (1.12) = $4200
To solve this equation, we can use a numerical method or trial and error to find the value of X that satisfies the equation.