Question 1197902: You go to a local mechanic to get your tires changed. The tires cost $300. There is a 6% sales tax, but you get a 10% discount. There is also a $10 non-taxable disposal fee for your old tires, which the mechanic tells you is not subject to discount.
18. Write a function, t(x) for the total purchase amount after taxes, ignoring any discounts and fees.
A. t(x)=1.06x
B. t(x)=6 + x
C. t(x)=x + .06
D. t(x)=6x
E. t(x)=6x
F. t(x)=x + 6x
19. Write a function, d(x) to account for the total after discounts on purchase amount x, ignoring any taxes and fees.
A. d(x)=x - 10
B. d(x)=x - .10x
C. d(x)=x/.10x
D. d(x)=x + 10x
E. d(x)=x - .10
F. d(x)=x - 10x
20. Does it matter whether the mechanic adds the tax first or takes the discount first?
A. Yes
B. No
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39630)   (Show Source):
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Problem 18
x = cost before taxes
t(x) = 1.06x = cost after taxes
This ignores any discounts or fees
1.06 represents an increase of 6% because 100% + 6% = 1 + 0.06 = 1.06
Answer: Choice A
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Problem 19
x = cost before discount
0.10x = discount value, aka the amount you save
d(x) = x - 0.10x = cost after discount
That can be simplified to d = 0.90x to reflect that idea if you save 10%, then you pay the remaining 90%
This ignores taxes and other fees which question 20 will address later.
Answer: Choice B
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Problem 20
The previous problems calculated the tax and discount functions (t(x) and d(x) respectively).
Use function composition to apply the two ideas in a specified order.
t(d(x)) = apply discount first, then tax later
d(t(x)) = tax first, discount later
The innermost function is what goes first. Then you work your way to the outermost function.
Let's compute t(d(x))
t(x) = 1.06x
t(d(x)) = 1.06*d(x)
t(d(x)) = 1.06*(0.90x)
t(d(x)) = 0.954x
Now do d(t(x))
d(x) = 0.90x
d(t(x)) = 0.90*t(x)
d(t(x)) = 0.90*1.06x
d(t(x)) = 0.954x
Either way we get the same result.
This happens simply because multiplication is commutative. The order of multiplication doesn't matter: a*b = b*a
Therefore, the order of "tax then discount" vs "discount then tax" doesn't matter. We arrive at the same final cost either way.
Take your pick which you prefer is the better route.
For either scenario, we haven't added on the $10 disposal fee since it's non-taxable and it's not subject to discount.
Meaning that 0.954x+10 represents the final cost after that pesky extra $10 is added on.
Once again, the x represents the cost before any taxes, fees, or discounts are applied. In this case, x = 300.
Note: The coefficient 0.954 subtracts from 1 to get 1-0.954 = 0.046 to represent a total net discount of 4.6%
In effect, the 10% discount and 6% tax play a game of tug-of-war to settle on a 4.6% discount to the customer.
This is fairly close to 10% - 6% = 4% which is a naive calculation or estimate. These calculations ignore the $10 disposal fee.
Answer: Choice B
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