Question

I have absolutely no idea how to do this problem!!!! It says; In the state of North Carolina, if you make less than $12,750 annually you will pay a 10% state tax. If you make at least $12,750 but less than $60,000 annually, you will pay a flat fee of $765 and a 7% tax on every dollar made over $12,750. If you make at least $60,000 annually, you will pay a flat fee of $4,072.50 and a 7.75% tax on every dollar made over 60,000. Write a function that models these tax brackets.

Answer 1

focus on "if you make less than $12,750 annually you will pay a 10% state tax"

and try to make an equation from that

Answer 2

tell me what you get

Answer 3

10% , x<12,750 ??

Answer 4

Is that even close?

Answer 5

let's say x is the amount of money you made before taxes

Answer 6

if x < 12750, then you take 10% of this to get the amount of tax you have to pay

10% of x = 0.10x

Answer 7

so

x = amount you earn before you pay taxes
y = amount of tax you pay

the equation is

y = 0.10x if x < 12750

Answer 8

notice how I tacked on the " if x < 12750"

that equation is only valid if you make less than $12,750. If you make more than that, then you are not in that tax bracket and another equation applies to you.

Answer 9

making sense so far?

Answer 10

I think I'm following you.

Answer 11

So for if you make at least $60,000, the equation would be 4,072.5+.0775x , and x would greater than or equal to $60,000?

Answer 12

close, but notice "a 7.75% tax on every dollar made over 60,000"

Answer 13

saying ".0775x" means you're taxing all of the income at 7.75% when that isn't the case

you're taxing the dollars that go over $60,000

Answer 14

So would that make it only greater than $60,000?

Answer 15

for example, say you earn $62,000

you'd be taxed a flat $4,072.50 plus an additional amount of 7.75% of $2000 (notice how 62,000 - 60,000 = 2,000)

Answer 16

what do you mean?

Answer 17

Instead of it being greater than or equal to $60,000, it would be only >. I don't think that's right though.

Answer 18

saying "at least $60,000" means "60,000 or more"

so 60,000 is included

Answer 19

Okay that makes sense. So what am I doing wrong? I still don't understand.

Answer 20

let's do this piece by piece

Answer 21

"if you make less than $12,750 annually you will pay a 10% state tax"

the equation here is y = 0.10x if x <12750

Answer 22

"If you make at least $12,750 but less than $60,000 annually, you will pay a flat fee of $765 and a 7% tax on every dollar made over $12,750"


the equation is now

y = 765 + 0.07(x - 12750) if 12750 <= x < 60000

Answer 23

notice how I have x - 12750

this is the amount of dollars you have over 12750 which is taxed at 7%

Answer 24

Okay, I understand now!

Answer 25

and finally,

" If you make at least $60,000 annually, you will pay a flat fee of $4,072.50 and a 7.75% tax on every dollar made over 60,000."

we have this equation

y = 4072.50 + 0.0775(x - 60000) if x >= 60000

Answer 26

WOOHOO!!!! :D I get it now!! Thank you so much!

Answer 27

we have these 3 equations (with conditions attached to each equation)

y = 0.10x if x <12750
y = 765 + 0.07(x - 12750) if 12750 <= x < 60000
y = 4072.50 + 0.0775(x - 60000) if x >= 60000

they then combine to form this piecewise function

\[\large
f(x) =
\begin{cases}
0.10x & \text{if } x < 12,750 \\
765 + 0.07(x - 12750) & \text{if } 12,750 \le x < 60,000 \\
4072.50 + 0.0775(x - 60000) & \text{if } x \ge 60,000 \\
\end{cases}
\]

Answer 28

in reality there are a lot more tax brackets, but this gives a somewhat decent idea of how taxes work