Question

please help =(

The sales tax in Illinois is 8.75%. The cashier rings up your purchase and tells you that your total bill is $103. What was the amount of your bill before taxes were added?

Answer 1

Let's do this together! Ready?

Answer 2

yea

Answer 3

i did .0875*103

Answer 4

got 9.0125

Answer 5

then subtracted from 103 and got 93.9875

Answer 6

Hmm... that's very close to what I got. Lemme just double check and I'll see what's up.

Answer 7

http://www.docstoc.com/docs/80455051/Percentages

Answer 8

yea i saw that but i don't understand where they're gettin the 1+r from

Answer 9

makes no sense why you would add a 1.....

Answer 10

One thing I do need to point out is that we can't do the 103*taxrate because the 103 already includes the tax rate.

Here's an example why:

Let's say you took a test and you got an 80. The teacher is nice and tells everyone they get a 10% curve. 10% of 80 is 8, right? So if you were to get an 88 and do 88*10%, you'd get 8.8.

Now if we did 88 - 8.8, you'd get 79.2, which isn't what we started with. :(

Answer 11

So what we need to do is work backwards:


final amount = pre tax amount * (1 + tax rate)

final amount is the $103, pretax amount will be "x".

The tricky part is the (1+tax rate).

Look at what would happen in our test example if we DIDN'T include the 1:

(88)(.10) = 8.8

You'd actually get less than a ten on your test if the teacher forgot to do the +1!

The +1 is there to say "do 88(1+.10)" so that you add the bonus points TO the initial score". In other words, if you only did 88(.10), your answer would be 8.8 which is ONLY the bonus points that you got on your test.

Please lemme know if this makes sense the way I worded it. It's very important to know before we continue.

Answer 12

i'm still a bit lost as to why the 1 is necessary

Answer 13

Thanks for letting me know. ^_^

Alright, so let's come up with a similar example.

Let's say you want to buy a $1.00 hotdog, but the tax rate is a whopping 10%. Can you set me up a formula that calculates only the price of the tax?

Answer 14

yea it would be 1/.1

Answer 15

which is 10

Answer 16

Close. You don't want to divide by the tax, though. $10 is too high for the tax.

Hint: Try the opposite of dividing.

Answer 17

oh oops so 1*.1

Answer 18

which is .1

Answer 19

Perfect! That comes out to like ten cents of tax (a realistic amount, so we know we're correct).

So what you basically did just now was:

total tax paid = purchase price * tax rate

Sound fair so far?

Answer 20

yea

Answer 21

Awesome. But here's the problem:


We can't use just that formula to figure out how much we owe the hotdog guy, right? There's no way he's going to accept ten cents for our hotdog, right?

Answer 22

yea but isn't it implicit that we pay $1.10?

Answer 23

It is, but in a formula we have to include everything. So what would we add to the $.10 to get the correct answer? (Sorry, parent walked in and is nagging me to do stuff, so my mind went off topic lol)

Answer 24

its ok... oh ok so we add the 1... but what if the amount was $2? would it be plus 2 then?

Answer 25

Interestingly, it wouldn't. I'll show ya why.

Answer 26

So we agree that this formula is perfect for calculating just the tax:

total tax paid = purchase price * tax rate

Answer 27

yea

Answer 28

And to figure out how much the entire total is, we simply add the price of the hotdog again:

total price = hotdog + (hotdog * tax rate)

Please lemme know if that formula makes total sense so far. Note that the stuff in parentheses was what we used earlier to do just the tax rate; all I did was put in the "hotdog + " so that we could add the price of the hotdog itself to the tax paid.

We're almost done! But we have to make sure this step made sense first.

Answer 29

yea it makes sense

Answer 30

BTW: I noticed that when I wrote this:

total tax paid = purchase price * tax rate

that was worded a little badly. By purchase price, I meant to say "item price without tax".

Answer 31

oh ok gotcha

Answer 32

Excellent!

Now lets make a few small changes to the variables to make it more 'math friendly'.

Let's say

x = price of an item (hotdog) without tax
r = tax Rate (notice the r for rate)
y = price with tax included

Could you translate our formula into a formula using those letters?

Our existing formula:

total price = hotdog + (hotdog * tax rate)

Answer 33

I promise we're almost done. :P

Answer 34

yea so x+(x*r)

Answer 35

Beautiful!

And once we drop the parentheses (we can do that because we multiplied the stuff inside and 'did the work', we are left with:

y = x+xr

But that looks kinda ugly, doesn't it? We need to factor. Do you see any common factors on the right side?

Answer 36

(this is probably the trickiest part of the whole thing, btw)

Answer 37

hey but is it a (1*r) or (1+r)?

Answer 38

Well, let's see it done right here:

y = x+xr

Sorry if it seems like I'm being too talkative, but it's important to see these steps so that it makes sense. :P

Answer 39

oh not at all you're fine

Answer 40

So, x is obviously our common factor.

y = x+xr

We pull out one x, then start out a parenthesis. We put a 1 in the parenthesis to account for that initial x:

y = x(1

So that took care of the first x that was by itself.

Now we deal with "how do we get x to turn into xr?"

Well, we simply multiply an r to our x:

y = x(1 + r)

And if you distribute it again, you can confirm it's the same formula:

y = x + rx

Answer 41

And if you want to go back to "English" instead of "Math", we can change

y = x(1 + r)

into

total_price_with_tax = price_without_tax (1 + tax_rate)

Did this make sense so far? If so, we can go ahead and solve the initial HW problem.

Answer 42

yea lets do it

Answer 43

Okie dokie. Lemme copy the problem down here.

Answer 44

The sales tax in Illinois is 8.75%. The cashier rings up your purchase and tells you that your total bill is $103. What was the amount of your bill before taxes were added?

total_price_with_tax = price_without_tax (1 + tax_rate)

Go ahead and set up the values in their appropriate spots, please, and we'll solve from there.

Answer 45

And I know you already know this, but just in case you're as sleepy as me, make sure you put the 8.75% in as .0875 and not 8.75 :P

Answer 46

103=price_without_tax (1+.0875)

Answer 47

Excellent! So 103 = price_without_tax(1.0875)

Answer 48

so 103/1.0875

Answer 49

And to make it math friendly again (sorry about switching in and out so much lol):

103 = 1.0875x

So what's the final step?

Answer 50

Yup, you've got it!

Answer 51

lol i think you circumvented why the 1 is necessary i still dont get that part to be honest ... well if this is the formula then i guess i'll just stick to it

Answer 52

And that comes out to like $94.71.

And something I always recommend: check your answer if it's possible. If we do

$94.72 times 1.0875 (notice I included the +1 with the .0875), we should get $103, or at least pretty dang close to it (within a penny or two is OK).

Answer 53

o ok thank u

Answer 54

No problemo! Thanks for participating! It means a lot to me when someone actually replies and whatnot instead of just demanding an answer from me. Heh.

And by the way, when I checked our answer, it came out to like $102.98, which means we've got the right answer. BOO YA!

Have fun; good night (morning?) :p