Chapter 11 - Fractions and Fractional Equations
In this chapter we look at fractions for the fourth and last time. Let’s
review our previous three encounters:
- Common fractions. In section 1.2 we introduced the
fraction notation, a/b,
where a and b were both integers to
describe a fraction or part of a whole object.
For example ¾ meant that we had broken an object into 4 equal
parts and we had 3 of those parts. Note that a/b was a number;
the notation a/b had nothing to do with division.
In section 1.2 we also learned how to reduce a fraction to
how to add and subtract fractions,
divide fractions, and how
to convert an improper fraction
to a mixed fraction using long division.
- Division of numbers. In section 2.4
we defined the division of two numbers in terms of multiplication.
We said that dividing a by b produced a number c such that
c multiplied by b gave back a.
We used the same fraction notation, a/b, to denote the division of
a by b because when a and b were both integers, then the
division a/b resulted in the common fraction a/b.
In any other case, though, the division resulted in a real number.
In section 2.4 we also learned that the division of a by b could be
replaced by the multiplication of a by the
reciprocal of b.
Finally, we learned the rules for division involving minus signs.
- Division of expressions. In section 3.5
we saw that there were three different ways to divide expressions, depending on
whether the numerator, a, and denominator, b, were monomials, multinomials or polynomials.
- If they were monomials, then dividing a by b simply amounted to writing down the algebraic fraction, a/b,
and reducing it to lowest terms, just like a common fraction.
- If they were polynomials, then a could be divided by b using long division,
just like an improper common fraction could be converted to a mixed fraction using long division.
- If a was a multinomial and b was a monomial then we placed each term of
a over b so that the result of the division was a sum of
What remains to be discussed are algebraic fractions, which are fractions whose
numerator and denominator are algebraic expressions.
This chapter discusses algebraic fractions and fractional equations.
It contains the following sections:
- section 11.1 - In this section we talk about
the simplification of algebraic fractions. The main new result is that
since we now know how to factor an expression,
we can factor a numerator or denominator, and this opens up a new way to reduce an
algebraic fraction to lowest terms.
- section 11.2 - In this section we
talk about the multiplication and division of algebraic fractions.
- section 11.3 - In this section we
talk about the addition and subtraction of algebraic fractions.
- section 11.4 - In this section we show how to
solve equations containing algebraic fractions.