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)?
A. $1515.20 B. $1290.30
C. $1719.02 D. $1764.77
E. $1400.00
Well, Mr. Mueller sure loves his multi-installment repayment plans! Let's get clowning and calculate the value of each payment, shall we?
First, we need to figure out what the interest rate is per period. To do that, we need to take the annual interest rate (j12 = 12%) and divide it by the number of periods in a year (12 months). So, the monthly interest rate (j) will be 12%/12 = 1%.
Now, let's apply this interest rate to calculate the value of each payment. We have 3 equal payments, so we can assume each payment is X.
For the first payment at 4 months, we need to discount it back to present value. We divide by (1 + j)^4, where j is the monthly interest rate. So, the present value of the first payment is X/(1 + j)^4.
For the second payment at 8 months, we divide by (1 + j)^8, and for the third payment at 12 months, we divide by (1 + j)^12.
Now, we know that the present value of the loan is $4200. So, we can set up the equation:
X/(1 + j)^4 + X/(1 + j)^8 + X/(1 + j)^12 = $4200.
Solving this equation might make your head spin, but fear not, my friend! The answer is C. $1719.02.
So, each payment should be $1719.02 to repay the loan over the specified time periods. Happy financing!
To find the value of each payment ($X), we can use the formula for the present value of an annuity:
PV = X * (1 - (1 + j)^(-n)) / j
Where:
PV is the present value (loan amount) = $4200
X is the value of each payment (unknown)
j is the interest rate per period = j12/12 = 12%/12 = 1% per month
n is the number of periods = 12/4 = 3
Let's plug in the values and solve for X:
$4200 = X * (1 - (1 + 0.01)^(-3)) / 0.01
$4200 * 0.01 = X * (1 - 1.01^(-3))
$42 = X * (1 - 1.010301)
$42 = X * 0.029684
X = $42 / 0.029684
X ≈ $1,415.77
Therefore, the value of each payment ($X) is approximately $1,415.77.
None of the answer choices exactly match this amount, so there may be a rounding error or a mistake in the provided options. However, the closest option is E. $1400.00.
To find the value of each payment ($X), we can use the formula for the present value of an annuity. Here's how we can get to the solution:
1. Use the formula for the present value of an annuity:
PV = X * (1 - (1 + r)^(-n)) / r
where PV is the present value, X is the payment amount, r is the interest rate per period, and n is the number of payment periods.
2. Plug in the given information into the formula:
PV = $4200 (the loan amount)
r = j12 (the interest rate per period) = 12% = 0.12
n = 12 months / 4 months per payment = 3 payment periods
3. Substitute the values into the equation:
$4200 = X * (1 - (1 + 0.12)^(-3)) / 0.12
4. Simplify the equation:
$4200 = X * (1 - 1.12^(-3)) / 0.12
5. Calculate 1.12^(-3):
1.12^(-3) ≈ 0.89286
6. Substitute this value back into the equation:
$4200 = X * (1 - 0.89286) / 0.12
7. Simplify further:
$4200 = X * 0.10714 / 0.12
$4200 * 0.12 = X * 0.10714
$504 = 0.10714X
8. Solve for X:
X = $504 / 0.10714
X ≈ $4700
Therefore, the value of each payment ($X) is approximately $4700. None of the given answer choices match this value, so it seems there may be an error in the question or the answer choices.