Are you confused?
Any ratio of two integers, where the denominator is nonzero, is known as a rational number. The set of
rational numbers, symbolized R, contains all such integer ratios that can exist.
It’s always possible to get any ratio of this kind into lowest terms. If you have some rational number r,
you can always convert it to the form m/p, where m is an integer and p is a positive integer, with the result-
ing fraction is in lowest terms. The term rational in this context comes from the word “ratio.”
If you stumble across the ratio (−6)/(−7), for example, you can write it as 6/7 and it represents the same
number. If you see 6/(−7), you can rewrite it as −6/7.
Here’s a challenge!
Leta, b, c, d, e, and f all be nonzero integers. Suppose you start with a/b and divide it by c/d, and then
divide that result by e/f. Write an expression for the final quotient.
Solution
Table 6-1 shows how this can be done, in the form of an S/R derivation.
Adding and Subtracting Fractions
When you want to add or subtract two integers, the process is straightforward. Fractions are
more involved. You’ve probably had plenty of practice adding and subtracting fractions in
arithmetic courses. Let’s look at these problems from a point of view a little closer to algebra,
using variables instead of specific numerical examples.
Getting a common denominator
When you have two fractions that you want to add or subtract, you should be sure that neither
fraction has a negative denominator. If one of them does, convert it into the equivalent form
that has a positive denominator. Then you can modify the fractions so they have the same
denominator, called the common denominator.
Suppose you have two fractions a /b and c/d in which a and c are integers, and b and d are
positive integers. You want to add them, so you write
a /b+c/d
Table 6-1. Derivation of a formula for repeated division of fractions. As you read
down the left-hand column, each statement is equal to all the statements above it.
Statements Reasons
[(a/b)/(c/d)]/(e/f ) Begin here
(ad/bc) / (e/f ) Apply the formula for division of a/b by c/d
(g/h)/(e/f ) Temporarily let ad = g and bc = h, and substitute the new names
in the previous expression
gf /he Apply the formula for division of g/h by e/f
adf /bce Substitute ad for g and bc for h in the previous expression
Adding and Subtracting Fractions 91
