Suppose A is the amount borrowed, r is the interest rate (in decimal form), and m is the total number of monthly payments.
Let w = (r)/(12)
Then the formula to determine the monthly payment amount for a loan is given by
(Aw)/1-(1)/(1+w)^m
Suppose Alice buys a car and obtains a 5 year loan for $25,000 at an interest rate of 6%.
(a) What is the numerical value of w?
(b) What is the monthly payment? (Show values substituted in the formula, and calculate the numerical amount.) Note: Since the loan is an amount in dollars and cents, it’s important to maintain a high degree of precision in intermediate calculations, and round to the nearest cent at the end.
(c) If Alice makes all 60 payments, how much will have been paid altogether?
The posted expression for the monthly payment has missing parentheses. Perhaps that's why there are difficulties with getting the answer.
When transposing a complicated fraction to a single line, as you have done above, it is important to provide parentheses to delimit the numerator (which was done above) and the denominator (parentheses missing). In fact, the expression above has an embedded denominator which also requires parentheses.
If the expression is not transcribed correctly to a single line, it usually means that the operations will not be executed according to the proper priorities, and will result in incorrect answers.
Let's start with putting the expression straight, using the given notations
Let
A=Amount borrowed.
P=Monthly payment
m=number of payments
r="annual" interest (as advertised)
w=monthly interest, compounded monthly.
then
w=r/12
(The denominators are enclosed in square brackets)
P(A,w,m)=(Aw)/[1-1/[(1+w)^m]]
Note that this formula will be indefinite if the rate of interest is zero, in which case the payment is simply
P=A/m
Also, this formula assumes that the interest is compounded monthly, which increases the advertised 6% interest to 6.16778%.
Given numerical data for Alice:
A=25000
r=0.06
m=5*12=60
(a) The numerical value of w is as given by the formula, w=r/12=0.06/12=0.005
(b) the monthly payment P is given by
P(A,w,m)
=P(25000,0.005,60)
=(25000*0.005)/(1-1/(1.005^60))
=(125)/(1-1/(1.348850152549304))
=(125)/(1-0.74137219624435)
=(125)/(0.25862780375565)
=483.320038235716
=$483.32
(c) The total amount Alice would have paid after 60 months is 60*$483.32=$28999.20
(a) To find the numerical value of w, we need to divide the interest rate r (in decimal form) by 12.
Given that r = 6%, we convert it to decimal form by dividing by 100, so r = 0.06.
Now, we find w = r/12.
w = 0.06/12 = 0.005.
Therefore, the numerical value of w is 0.005.
(b) To find the monthly payment, we use the formula:
P = (Aw) / (1 - (1 + w)^(-m)).
Given that A = $25,000, w = 0.005, and m = 5 years (60 months), we can substitute these values into the formula:
P = (25,000 * 0.005) / (1 - (1 + 0.005)^(-60)).
Now, we calculate (1 + 0.005)^(-60):
(1 + 0.005)^(-60) ≈ 0.888122.
Substituting this back into the formula:
P = (25,000 * 0.005) / (1 - 0.888122) = (125) / 0.111878 ≈ $1,117.63.
Therefore, the monthly payment amount is approximately $1,117.63.
(c) To find out how much Alice will have paid altogether after making all 60 payments, we multiply the monthly payment by the total number of payments:
Total amount paid = Monthly payment * Number of payments.
Total amount paid = $1,117.63 * 60 = $67,057.80.
Therefore, if Alice makes all 60 payments, she will have paid a total of approximately $67,057.80.