Question

Wray bought a bicycle for $2500 plus GST and PST, to compete in triathlons. He arranged to make a payment to the store at the end of every month for 2 years. The store is charging 11% interest per annum, compounded monthly. (Use GST=5%, PST=8%) How much is each payment?

Answer 1

GST = 2500*0.05 = 125
PST = 2500*0.08 = 200

Now what?

Answer 2

The loan payment formula can be found here:
http://www.1728.org/loanform.htm
So the total is 2825.00 and we need to find the monthly payment for a 2 year loan at 11% annual interest?

Answer 3

Would the formula be the same as PV=R[1-1 + i^-n]/i ?

Answer 4

Okay here's the formula:

Answer 5

This is why a despise formula-based introductions. It's just never clear what we have.

i = .11 -- Annual Interest
j = i/12 -- Monthly Interest
v = 1/(1+j) -- Monthly Discount Factor

Payment? This is what we seek.

Draw a map.

\(2825 = Payment \cdot (v + v^{2} + v^{3} + ... + v^{24})\)

\(2825 = Payment\cdot\dfrac{v-v^{25}}{1-v}\)

\(Payment = \dfrac{2825\cdot (1-v)}{v-v^{25}}\)

If you wish, some of that can be simplified so that it looks more familiar. I tend not to care about that.

Let's see: \(1-v = 1-\dfrac{1}{1+j} = \dfrac{1+j-1}{1+j} = \dfrac{j}{1+j} = jv\)

This gives: \(Payment = \dfrac{2825\cdot (jv)}{v(1-v^{24})} = \dfrac{2825\cdot j}{1-v^{24}}\)

This gives the formula you offered, but you did not write it correctly.

Answer 6

I didn't write the formula correctly? Well I use it in my loan payment calculator and it gives the correct answers.

Answer 7

That was the formula that was given to us... :(

Answer 8

I guess you are talking about the formula you posted?

Answer 9

Yes I am :(

Answer 10

@peekaboopork No, that was not correct. It should be (1+i)^(-n), not 1+i^-n

The parentheses around 1+i are NOT optional.
The parentheses around -n add clarity.

Answer 11

Ohhhh okay, so the PV=2825, n= 12, i= 0.009166? D:

Answer 12

Not quite, n = 24, doesn't it?

Answer 13

My calculator gives an answer of 131.67
I'm not giving away an answer because sometimes it helps to see the answer so you can determine if you have done all the calculations correctly.
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and n=24 for 24 months

Answer 14

oh shoot i meant to put in 24, sorry!

Answer 15

I'll be leaving the computer for a few hours - but I'll be sure to come back here.

Answer 16

Thank you both so much!

Answer 17

My home-made formula also gives 131.667142905182 or $131.67

Answer 18

Rate must be divided by 1,200 so = 11%/1200 = 0.0091666666666...
Time = 24 Months
Amount Borrowed = 2,825.00

Monthly Payment = [0.009166666666...+(0.009166666666.../([(1.009166666666...)^24]-1)]*2,825.00

Monthly Payment = [0.009166666666...+(0.009166666666.../0.244828521440805)]*2,825.00

Monthly Payment = [0.009166666666...+(0.009166666666.../0.244828521440805)]*2,825.00

Monthly Payment = [0.009166666666...+0.0374411715298579]*2,825.00

Monthly Payment = 0.046607838197 * 2,825.00

Monthly Payment = 131.667142905182

rounded = 131.67

Since I was given a medal, I figured I should show the payment all worked out.
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tkhunny - I think it is astounding that our answers match each other exactly - down to 12 decimal places!!

Answer 19

Indeed. It is an exact formula in both cases. If one has a clue about intermediate values, which obviously you do, we can expect that kind of precision.

It is unknown to many students that some of the published formulas are only approximations. Check out some of the published formulas for geometrically increasing payments. In my mind, it's just plain unnecessary to use such approximations. Obviously, not everyone agrees.