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, let's calculate the value of each payment, or $X, using a bit of humor. Remember, laughter is the best currency!
So, Mr. Mueller took a loan of $4200. Now, that's an amusing number to start with! Must have been quite the prank.
Next, he arranged to repay the loan with 3 equal payments at the end of 4, 8, and 12 months respectively. Ah, the repayment timeline is like a comedy routine!
Now, the lender charges a rate j12 of 12%, which gives this situation a fun twist. It's like paying 12% interest for a front-row seat at a comedy show!
To find the value of each payment, we need to solve this humorous equation. Now, prepare for some mathematical humor!
4200 = X/(1+0.12/12)^4 + X/(1+0.12/12)^8 + X/(1+0.12/12)^12
By performing the calculations with a smile on our face, we find that the value of each payment, $X, is approximately $1719.02. Ah, the punchline!
So, the answer is C. $1719.02. I hope this brought a smile to your face as we solved the equation together!
To find the value of each payment ($X), we need to calculate the amount that Mr. Mueller needs to repay at the end of the loan.
Using the formula for the future value of an ordinary annuity:
FV = A * [(1 + r)^n - 1] / r
Where:
FV = Future value of the loan
A = Amount of each payment
r = Interest rate per period
n = Number of periods
In this case, Mr. Mueller will make 3 equal payments of $X.
First, we need to calculate the interest rate per period (r) by dividing the annual interest rate by the number of periods in a year:
r = j12 / 12 = 12% / 12 = 1%
Now, we can calculate the future value of the loan (FV) using the given information:
FV = $4200 = X * [(1 + 0.01)^4 - 1] / 0.01
4200 = X * (1.01^4 - 1) / 0.01
4200 = X * (1.04060401 - 1) / 0.01
4200 = X * 0.04060401 / 0.01
4200 = X * 4.060401
X = 4200 / 4.060401
X ≈ $1032.46
Therefore, the value of each payment ($X) is approximately $1032.46.
None of the given answer options match this result, so none of the provided options is correct.
To find the value of each payment, we can use the formula for equal payments on a loan. The formula is given by:
X = (P * j) / (1 - (1+j)^(-n))
Where:
X is the value of each payment
P is the principal amount (loan amount) - $4200
j is the annual interest rate adjusted for compounding - j12 = 12% / 12 = 1% per month = 0.01
n is the total number of payment periods - 12 months
Plugging the values into the formula, we have:
X = (4200 * 0.01) / (1 - (1 + 0.01)^(-12))
X = 42 / (1 - (1.01)^(-12))
X ≈ 42 / (1 - 0.887497)
X ≈ 42 / 0.112503
X ≈ 373.37
Therefore, each payment is approximately $373.37.
None of the given answer choices matches exactly with $373.37. However, by comparing the closest answer choices, the closest value to $373.37 is option E. $1400.00.