Fraction or ratio?
Sometimes a fraction is called a ratio. These two terms are almost synonymous, but not quite.
The term “fraction” implies that a particular quantity is a part of some other quantity. The
term “ratio” expresses how two quantities are related in terms of their relative size or value.
Think of the Happyville-Bluesdale situation again. If you say, “It’s seven-thirds times as far
to Happyville as it is to Bluesdale,” then you’re using 7/3 as a fraction. If you say, “The ratio
of the distance to Happyville compared with the distance to Bluesdale is seven to three,” then
you’re saying the same thing, but in a different way.
You can write, “The ratio of the distance to Happyville compared with the distance to
Bluesdale is 7/3,” and read “7/3” as “seven to three.” If you want to make clear that you’re
talking about a ratio, you can use a colon instead of a slash to separate the 7 and the 3. You
would then write something like, “The Happyville-to-Bluesdale distance ratio is 7:3,” reading
“7:3” as “seven to three.”
Ratios are always expressed in terms of two integers, one divided by the other. They’re
never expressed as a whole integer plus or minus a proper fraction.
Are you confused?
Proper fractions are always larger than −1 but smaller than 1. A mathematician would use the “strictly
larger than” inequality (>) and the “strictly smaller than” symbol (<) to express this fact. If q is a proper
fraction, then
q>−1 and q< 1You can also write
− 1 <q< 1The proper-fraction interval
Figure 6-2 shows the “realm of proper fractions” on the number line. It contains the points for
all possible fractions between, but not including, −1 and 1. The interval is shown as a shaded
line. It’s gray (not black) for a subtle reason. The set of all proper fractions doesn’t account for
all the geometric points above −1 and below 1. Within that range, there are numbers that can’t
be expressed as fractions. The same thing is true everywhere along the number line. Ratios of
integers can’t account for all the points on a true geometric line. You’ll learn more about that
in Chap. 9.
You should also be acquainted with two other symbols. The “larger than or equal to”
symbol looks like a “strictly larger than” symbol with a line under it (≥). The “smaller than
or equal to” symbol looks like a “strictly smaller than” symbol with a line under it (≤). If you
wanted to include −1 and 1 in the interval above, you would write
q≥−1 and q≤ 1Alternatively, you could write
− 1 ≤q≤ 1“Messy” Quotients 85