Question

A business permits its custimers to pay with either a credit card or to receive a percentage discount r for paying cash. For credit card purchases, the business receives 95% of the purchase price one-half month later. At an effective annual rate of 12%, the two payment methods are equivalent. Determine r.

Answer 1

If i could just get help starting, I would appreciate it. Even just the formula.

Answer 2

lets try with numbers

Answer 3

hm?

Answer 4

also, is percentage discount the same retriceeffective rate or discount?

Answer 5

actually the more i read this the more i don't understand it. but i was going to say lets imagine that you buy something for $100
then if payed by credit card the business receives $95 two weeks later, but then i am lost after that, because i don't know who is being charged 12% interest

Answer 6

I KNOW!!! f-ck theories of interest.... would it be the extra 5%? HA! do you know the formula for percentage discount by any chance?

Answer 7

this line
At an effective annual rate of 12%, the two payment methods are equivalent
makes no sense to me.
does it make sense to you?

Answer 8

none whatsoever.

Answer 9

does it mean the business lost 12% annually on the $95 it took two weeks to get ?

Answer 10

that i can compute

Answer 11

r = (1+i/n)^n - 1 i just looked up the formula for the eff. annual int rate

Answer 12

yeah i get that, but i don't know how it applies to this problem

Answer 13

neither do i. i have multiple choice answers of 4.55, 4.85, 5.15, 5.45, and 5.75. but i don't know how to even approach this

Answer 14

5.45 is correct.

Answer 15

how so?

Answer 16

This is how it works:
Using Satellite73's approach on a purchase of $100.

Answer 17

If the customer pays by credit card, the merchant will get $95 (1/24) of a year after.

Effective interest rate of 12% translates to 0.4733195% per (1/24) of a year, using your formula.

If the customer pays cash, merchant get X dollars right away, which becomes $95 (1/24) of a year after.

Answer 18

ok i think i see it. the money you lose is what you would have got if you invest the 100 at .12 effective for two weeks right?

Answer 19

So X*1.004733195=95, or
X=$94.55246
Which means that the merchant would have given a discount of 5.45%

Answer 20

it is \(\frac{1}{26}\) of the year later, but i think the calculation is correct

Answer 21

It said half a month later, so I used 1/24. It would be very close anyway.

Answer 22

sorry, where did the (1/24) come in? is it 24 payments in the year?

Answer 23

oh silly me yes, you are right
for some reason i used 2 weeks
i am wrong

Answer 24

The question says merchant gets money 1/2 a month later. So for calcuations, we have to assume 24 periods.

Answer 25

ok, i am beginning to see

Answer 26

lot of assumptions here

Answer 27

yea

Answer 28

That's the world of business! :)

Answer 29

we are also making the assumption that "the two payment methods are equivalent" means after one year they are equivalent

Answer 30

thank you

Answer 31

@mathmate i have a different method, let me know if it is correct

Answer 32

I guess the merchant has to look at the long term. In any case, 1/2 month could easily mean anything form 13 to 17 days, when we count the weekends.

@IStutts you're welcome.
Let's look at Satellite73's alternative method.

Answer 33

no nvm that didn't work sorry

Answer 34

i was thinking you lost $5 and also the 12% of the $100 for two weeks, but that comes to a loss of $5.50

Answer 35

Actually, we only lost $95 for two weeks, maybe that would help.

Answer 36

Actually, approximately, we lost $95 for 1/24th of a year at 0.004733%, which comes to be 0.45, added to $5 we lost in the first gives $5.45, yay!

Answer 37

The interest rate for (1/24)th of a year is obtained using your formula:
r = (1+i/n)^n - 1
with n=24, i=0.12
Check: 1.04733195^24=1.12 (exactly what we want).

Answer 38

Sorry, it's the other way round:
i=(1+r)^(1/n)-1
where i=rate per period, n=24
and your formula should probably read:
r = (1+i)^n - 1

Answer 39

@IStutts
i=(1+0.12)^(1/24)-1=0.004733195
Check: 1.004733195^24=1.12000

Answer 40

ooooh! okay. thank you. sorry, i'm one of those unfortunates that needs a step-by-step.