Question 1197902
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Problem 18


x = cost before taxes
<font color=red>t(x) = 1.06x</font> = 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: <font color=red>Choice A</font>


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Problem 19


x = cost before discount
0.10x = discount value, aka the amount you save
<font color=red>d(x) = x - 0.10x</font> = 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: <font color=red>Choice B</font>


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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: <font color=red>Choice B</font>
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