Zachary purchased a computer for 1,140 on a payment plan.
Eight months after he purchased the computer, his balance was 320.
Nine months after he purchased the computer, his balance was $185. What is an equation that models the balance y after x months?
Can you please explain I don't know how to solve it!
To find an equation that models the balance y after x months, we can use the information given in the problem.
Let's start by calculating the decrease in balance per month. From month 8 to month 9, the balance decreased by $320 - $185 = $135.
Since the balance decreases by a constant amount every month, we can determine the monthly decrease by dividing $135 by the number of months between the two balances. In this case, it is 9 months - 8 months = 1 month.
So, the monthly decrease in balance is $135 / 1 month = $135.
Now, we can determine the initial balance (the balance at month 0) by subtracting the total decrease from the final balance at month 9.
The final balance at month 9 is given as $185, and the monthly decrease is $135. Therefore, the initial balance is:
Initial balance = $185 + $135 = $320.
To summarize:
Initial balance = $320.
Monthly decrease = $135.
Now, we can construct the equation:
y = Initial balance - (Monthly decrease * x).
Substituting the values we found, the equation becomes:
y = $320 - ($135 * x).
This equation models the balance y after x months, where y is the remaining balance and x is the number of months that have passed since the purchase.
To model the balance y after x months, we can use the formula for linear interpolation.
Linear interpolation is a method used to estimate a value between two known values. In this case, we know the balance in the 8th month ($320) and the 9th month ($185).
The equation for linear interpolation is:
y = y1 + ((y2 - y1) / (x2 - x1)) * (x - x1)
In this equation, y1 and x1 represent the balance and month respectively at the known point (8 months - $320), and y2 and x2 represent the balance and month respectively at the second known point (9 months - $185).
Substituting the given points into the equation:
y = 320 + ((185 - 320) / (9 - 8)) * (x - 8)
Simplifying:
y = 320 - 135 * (x - 8)
This equation models the balance y after x months, taking into account the initial balance, the number of months, and the reduction in balance over time.
You have two unknowns,
1. the monthly payment , let it be p
2. the rate of interest , let i be the monthly rate
after 8 months: balance = 1140(1+i)^8 - p( (1+i)^8 - 1)/i = 320
after 9 months: balance = 1140(1+i)^9 - p( (1+i)^9 - 1)/i = 185
we also know that 320(1+i) - p = 185
p = 320 + 320i - 185
p = 135 +320i
sub that into 1140(1+i)^8 - p( (1+i)^8 - 1)/i = 320
1140(1+i)^8 - (135 +320i)( (1+i)^8 - 1)/i = 320
Not even trying to solve this by some algebraic method and I will send it directly to Wolfram. (had to change the i to x, Wolfram read i as the imaginary number)
https://www.wolframalpha.com/input/?i=1140%281%2Bx%29%5E8+-+%28135+%2B320x%29%28+%281%2Bx%29%5E8+-+1%29%2Fx+%3D+320
got i = .0656167
p = $156.00
Made up a little Excel routine to show my answer is correct, (I am off by 3 cents)
time payment interest outstanding balance
0 $1,140.00
1 $156 $74.80 $1,058.80
2 $156 $69.48 $972.28
3 $156 $63.80 $880.08
4 $156 $57.75 $781.82
5 $156 $51.30 $677.12
6 $156 $44.43 $565.55
7 $156 $37.11 $446.66
8 $156 $29.31 $319.97
9 $156 $21.00 $184.97
10 $156 $12.14 $41.11
Unfortunately, in our format on Jishka it is hard to line up columns
my formula is
y = 1140(1.0656167)^x -156(1.0656167)^x - 1)/.0656167
testing it for x = 6 gave me y = 565.55 , as confirmed in my Excel routine