Page 47 - Mathematics of Business and Finance

Chapter 1 | Review of Basic Arithmetic 27

Fraction in Lowest (or Simplest) Terms

A fraction in which the numerator and denominator have no factors in common (other than 1) is said
to be a fraction in its lowest (or simplest) terms. Any fraction can be fully reduced to its lowest terms
by dividing both the numerator and denominator by the greatest common factor (GCF).

Example 1.3(a) Reducing Fractions to their Lowest Terms
Reduce the following fractions to their lowest terms.
40 63
(i) (ii)
45 84

Solution (i) The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40.
The factors of 45 are: 1, 3, 5, 9, 15, 45.

The common factors are: 1, 5.
Therefore, the GCF is 5.
Therefore, dividing the numerator and denominator by the GCF, 5, results in the fraction in its
40 40 ÷ 5 8
lowest terms: = =
45 45 ÷ 5 9

(ii) The factors of 63 are: 1, 3, 7, 9, 21, 63.
The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

The common factors are: 1, 3, 7, 21.
Therefore, the GCF is 21.
Therefore, dividing the numerator and denominator by the GCF, 21, results in the fraction in
63 63 ÷ 21 3
its lowest terms: = =
84 84 ÷ 21 4

Comparing Fractions

Fractions can easily be compared when they have the same denominator. If they do not have the same
denominator, determine the LCD of the fractions, then convert them into equivalent fractions with
the LCD as their denominators.

When the denominators are the same, the larger fraction is the one with the greater numerator.
7 5
For example, > , >
12 12
When the numerators are the same, the larger fraction is the one with the smaller denominator.
3 3
For example, > , >
4 8

Example 1.3(b) Comparing Fractions
Determine which of the fractions is larger in each set of fractions given below by comparing the
numerators after converting the fractions to equivalent fractions with the same denominator.
9 11 5 3 19 11 15 15
(i) or (ii) or (iii) or (iv) or
25 25 12 8 60 36 22 26