3. Reducing Fractions to Lowest Terms

Notes:

Properties of quotients or fractions provide a rationale for simplifying fractions, otherwise known as “reducing them to lowest terms.

• The first property is that two fractions, a/b and c/d, are equivalent if ad = bc. For example, 3/4 = 6/8 because 3(8) = 4(6) = 24.
• Next, if we divide –a by b or a by –b we get the same answer: (-a)/ b = a/(-b) = -(a/b). For example, (-3)/4 = 3/(-4) = -(3/4) = -0.75.
• Finally, if we multiply the numerator and denominator of a fraction, a/b, by the same number, c, we get an equivalent fraction: a/b = ac/bc. For example, 3/4 = 3(6)/4(6) = 18/24.

This last property means that for any fraction, a/b, there are an infinite number of equivalent fractions. However, only one of these is said to be in “lowest terms.” a/b is in lowest terms if a and b have no common factor greater than 1. For example, 3/4 is in lowest terms because the only common factor for 3 and 4 is 1.

This begs the question; how do we reduce a fraction to lowest terms? We can apply one of the fraction properties from earlier. We can reduce a fraction to lowest terms by repeatedly dividing by common factors: a/b = (a/d)/(b/d). For example, 18/24 = (18/3)/(24/3) = 6/8 = (6/2)/(8/2) = 3/4. Alternatively, divide the numerator and denominator by 6 to get 18/24 = 18³/24⁴ = 3/4.

Why reduce a fraction to lowest terms?

In general, it is easier to work with small numbers than large numbers. For example, 3/4 is easier to work with than 18/24. The idea of “reducing to lowest terms” comes up again when we consider rational expressions in Video 12.

However, sometimes we don’t want to reduce a fraction to lowest terms. For example, if we want to express 3/4 as a terminating decimal: 3/4 = 3(25)/4(25) = 75/100 = 0.75.

More examples