Basic Math and Pre-Algebra For Dummies (83 page)

 A Cartesian graph is really just two number lines that cross at 0. These number lines are called the
horizontal axis
(also called the
x-axis
) and the
vertical axis
(also called the
y-axis
). The place where these two axes (plural of axis) cross is called the
origin.

Plotting a point (finding and marking its location) on a graph isn't much harder than finding a point on a number line — after all, a graph is just two number lines put together. (Flip to Chapter
1
for more on using the number line.)

 Every point on an
xy
-graph is represented by two numbers in parentheses, separated by a comma, called a set of
coordinates.
To plot any point, start at the origin, where the two axes cross. The first number tells you how far to go to the right (if positive) or left (if negative) along the horizontal axis. The second number tells you how far to go up (if positive) or down (if negative) along the vertical axis.

For example, here are the coordinates of four points called
A, B, C,
and
D:

Figure 
17-5
depicts a graph with these four points plotted. Start at the origin, (0, 0). To plot point
A,
count 2 spaces to the right and 3 spaces up. To plot point
B,
count 4 spaces to the left (the negative direction) and then 1 space up. To plot point
C,
count 0 spaces left or right and then count 5 spaces down (the negative direction). And to plot point
D,
count 6 spaces to the right and then 0 spaces up or down.

When you understand how to plot points on a graph (see the preceding section), you can begin to plot lines and use them to show mathematical relationships.

The examples in this section focus on the number of dollars two people, Xenia and Yanni, are carrying. The horizontal axis represents Xenia's money, and the vertical axis represents Yanni's. For example, suppose you want to draw a line representing this statement:

Xenia has $1 more than Yanni.

Now you have five pairs of points that you can plot on your graph as (Xenia, Yanni): (1,0), (2,1), (3,2), (4,3), and (5,4). Next, draw a straight line through these points, as in Figure 
17-6
.

This line on the graph represents every possible pair of amounts for Xenia and Yanni. For example, notice how the point (6,5) is on the line. This point represents the possibility that Xenia has $6 and Yanni has $5.

Here's a slightly more complicated example:

• Yanni has $3 more than twice the amount that Xenia has.
  

Again, start by making the same type of chart as in the preceding example. But this time, if Xenia has $1, then twice that amount is $2, so Yanni has $3 more than that, or $5. Continue in that way to fill in the chart, as follows:

Now plot these five points on the graph and draw a line through them, as in Figure 
17-7
.

As in the other examples, this graph represents all possible values that Xenia and Yanni could have. For example, if Xenia has $7, Yanni has $17.

Chapter 18

In This Chapter

Solving measurement problems, using conversion chains

Using a picture to solve geometry problems

In this chapter, I focus on two important types of word problems: measurement problems and geometry problems. In a word problem involving measurement, you're often asked to perform a conversion from one type of unit to another. Sometimes you don't have a conversion equation to solve this type of problem directly, so you need to set up a
conversion chain,
which I discuss in detail in the chapter.

Another common type of word problem requires the geometric formulas that I provide in Chapter
16
. Sometimes a geometry word problem gives you a picture to work with. In other cases, you have to draw the picture yourself by reading the problem carefully. Here I give you practice doing both types of problems.

In Chapter
15
, I give you a set of basic conversion equations for converting units of measurement. I also show you how to turn these equations into conversion factors — fractions that you can use to convert units. This information
is useful as far as it goes, but you may not always have an equation for the exact conversion that you want to perform. For example, how do you convert years to seconds?

For more-complex conversion problems, a good tool is the conversion chain. A
conversion chain
links together a sequence of unit conversions.

Here's a problem that shows you how to set up a short conversion chain to make a conversion you won't find a specific equation for:

Vendors at the Fragola County Strawberry Festival sold 7 tons of strawberries in a single weekend. How many 1-ounce servings of strawberries is that?

You don't have an equation to convert tons directly to ounces. But you do have one to convert tons to pounds and another to convert pounds to ounces. You can use these equations to build a bridge from one unit to another. So here are the two equations you want to use:

To convert tons to pounds, note that these fractions equal 1 because the numerator (top number) equals the denominator (bottom number):

To convert pounds to ounces, note that these fractions equal 1:

You could do this conversion in two steps. But when you know the basic idea, you can set up a conversion chain instead to get from tons to ounces:

So here's how to set up a conversion chain to turn 7 tons into pounds and then into ounces. Because you already have tons on top, you want the tons-and-pounds fraction that puts
ton
on the bottom. And because that
fraction puts
pounds
on the top, use the pounds-and-ounces fraction that puts
pound
on the bottom:

The net effect here is to take the expression
7 tons
and multiply it twice by 1, which doesn't change the value of the expression. But now you can cancel out all units of measurement that appear in the numerator of one fraction and the denominator of another:

 If any units don't cancel out properly, you probably made a mistake when you set up the chain. Flip the numerator and denominator of one or more of the fractions until the units cancel out the way you want them to.

Now you can simplify the expression:

 A conversion chain doesn't change the
value
of the expression — just the units of measurement.