28. This question requires you to make use of the distributive property to simplify the expression. That property says multiplication can be distributed over addition.
You can think of the contents of parentheses as the contents of a package. The number outside the parentheses tells you how many packages you have.
For example, if you have 3 packages, each of which contains 2 marbles and 1 coin, we might represent this as
3·(2m + c)
The distributive property lets you rearrange this to be
= 3·2m + 3·c
= 6m + 3c
which is to say that your collection of packages contains a total of 6 marbles and 3 coins.
We can use this property in reverse to "collect terms". If we have the expression
-12a + 5a
we notice that the "a" is a multiplier of both terms. We can put it outside parentheses like this
= a·(-12 + 5)
Then we can do the arithmetic on the numbers inside parentheses and simplify this to
= a·(-7) . . . . . remember, the -7 is multiplying "a"
= -7a
When parentheses have a minus sign in front, you can think of it as a multiplier of -1. That is
-(12a +14)
= -1·(12a + 14)
= -1·12a + (-1)·14
= -12a -14 . . . . . . . . the minus sign has been applied to all terms inside the parentheses
Using this, we can rewrite the expression as follows:
(-7b +8c) -(12a +14) +(5a +5b)
= -7b +8c -12a -14 +5a +5b
We recognize terms with the variable a, the variable b, the variable c, and no variable at all. It can be helpful to write terms containing the same variable so they are next to each other. We keep the sign of the term with the rest of it. We usually like to put the variables in alphabetical order, though that is not strictly necessary.
= -12a +5a -7b +5b +8c -14
We can "collect terms" when they have the same variable.
= a(-12 +5) +b(-7 +5) +8c -14
= -7a -2b +8c -14
29. The store clerk will ask you two questions:
• "what variety of cheese do you want?"
• "in what style do you want that prepared?"
The first question can have 3 different answers. For each of those, the second question can have 2 different answers. You are asked to count the outcomes. There are few enough that you can list them. Perhaps by doing so, you can see how the number might be calculated.
cheddar - shredded
cheddar - sliced
Gouda - shredded
Gouda - sliced
Swiss - shredded
Swiss - sliced
Since each of the styles can be applied to each of the varieties, the 6 outcomes can be computed by multiplying the 3 varieties by the 2 styles: 3·2 = 6.
30. The IQR of temperature data for the two cities is identical. That is, half the time, the temperatures in each city fall within a range of 7 degrees. Other things being equal, this would suggest that the weather patterns are equally consistent. (Other things are not equal, as we shall see.)
However, the overall range of temperatures in City 1 is greater (20°) than in City 2 (15°). This tells you that
b) The weather pattern in City 2 is more consistent than the weather pattern in City 1.