Henry Devine bought a new dishwasher for 320. He paid 20 down and made 10 monthly payments of 34. What actual yearly interest rate did Henry pay?
using this formula r=2mI/p(n+1)
i got this r=2(12)(20)/300(10+1)=
but i don't know what to do next . can you please show me how to do it and correct me it am wrong.
Wow, haven't seen or used that formula in almost 50 years.
I don't think you are substituting the correct values.
First of all since Henry paid $20 down the actual principal is 300
in r=2mI/p(n+1) m is the possible number of payments per year, in this case m=12
I is the actual interest paid, in this case it is 10(34) - 300 = 40
and n is the actual number of payments
p is the principal
so rate = 2(12)(40)/(300(11))
= .2909 or 29.09%
I would say that the rate is 12.5%.
btw, this task is just nonsense
He borrowed 300 dollars to buy the 320 dollar washer.(since he paid 20 down)
Using the mortgage payment formula:
payment = principal * r/[ 1-(1+r)^-n]
where r is the monthly rate and n is ten months
34 = 300 *r / [1-(1+r)^-10]
[1-(1+r)^-10] = 8.82 r
1 = 8.82 r + (1+r)^-10
I have to solve that by iteration
table of r versus right side, must be 1
.10 .882 + .386 = 1.26
.09 .794 + .422 = 1.22
.08 .706 + .463 = 1.16
.07 .617 + .508 = 1.12
.05 .441 + .614 = 1.05
.04 .353 + .676 = 1.02
.03 .265 + .744 = 1.01
.02 .176 + .820 = .996
so around r = .025 per month or .3 per year
30%
wicked high
check
pmt = 300 (.025 /(1 - (1.025)^-10) ) ??
= 34.27 close enough
I guess it doesn't hurt to have one more opinion.
I will try to work from basic principles, where the principal will accumulate over the 10 months, and the payments will accumulate at the same rate of interest, separately. At the end of 10 months, the future value of the principal and the payments should be equal.
Let
P=principal, $300
r=rate of interest per month, about 0.025
R=1+r, about 1.025
n=number of periods, 10
A=monthly payment, $34
Future value of the principal
= PR^n
Future value (FV) of the first payment (payable after at the end of the first month)
= AR^(n-1)
FV of second payment
= AR^(n-2)
...
Future value of last payment
=A
Thus future value of monthly payments
=A(1+R+R^2+....+R^(n-1))
=A(R^n-1)/(R-1)
Equating future values,
PR^n = A(R^n-1)/(R-1)
Substituting numerical values:
300R^10=34(R^10 -1)/(R-1)
This can be solved by iteration if we arrange the equation as:
R=(34(R^10-1)/(300(R-1)))^(1/10)
With an initial value of R=1.025, we get successively
R=1.02417
1.02377
1.02359
1.02350
...
and finally
R=1.0234293 at the 8th iteration.
or 2.343% per month, which translates to 1.0234293^12
=1.3204, or
32.4% p.a. simple interest
Check using Damon's formula
monthly payment
= principal * r/[ 1-(1+r)^-n]
=300*.02343/(1-1/1.02343^10)
=34.000
OK
To calculate the actual yearly interest rate (r) that Henry paid, you can use the formula: r = (2mI)/(p(n+1)).
Let's break down the formula and substitute the given values:
m = number of payments per year = 12 (since he made 10 monthly payments)
I = interest payment per period = ?
p = principal amount borrowed = $320
n = total number of payments = 10
r = actual yearly interest rate = ?
Initially, you correctly calculated the early part of the formula:
r = (2 * 12 * I) / (320 * (10 + 1))
Now, to determine the interest payment per period (I), we need to calculate the total amount paid over the financing period and then deduct the principal amount (p).
Total amount paid = down payment + (number of payments * payment amount)
Total amount paid = $20 + (10 * $34) = $20 + $340 = $360
To find the interest payment per period (I), we subtract the principal amount from the total paid and divide it by the number of payments:
I = (total amount paid - principal amount) / number of payments
I = ($360 - $320) / 10
I = $40 / 10
I = $4
Now, substituting the values back into the formula:
r = (2 * 12 * $4) / (320 * (10 + 1))
Simplifying further:
r = 96 / (320 * 11)
r = 96 / 3520
r = 0.0273 (rounded to four decimal places)
So, the actual yearly interest rate that Henry paid is approximately 0.0273 or 2.73%.
Note: The formula used in this calculation assumes simple interest, so if there are any additional fees or compounding interest, this calculation may not give the exact value.