ACT Math: Reducing Fractions

Reducing fractions is one of those simple operations that we all remember learning from grade school. The only problem is that, when we learn higher-level math, we often forget the basics of arithmetic. Here is a quick refresher course in reducing fractions.
Fractions, as you probably remember, are simply parts of a whole number. Think of a pizza divided into 8 equal slices: each slice is 1/8 of the pizza. Two slices is 2/8, three slices is 3/8, four slices is 4/8, and so on.
You may notice that the fractions 2/8 and 4/8 are not reduced to lowest terms, or simplified. Why do we simplify fractions? For practical purposes, it’s much easier to multiply, divide, and subtract fractions when they’re simplified. The fractions in question, 2/8 and 4/8, are rather easy to simplify. In the first fraction, the numbers 2 and 8 share a common factor of 2, so we can divide both 2 and 8 by 2, getting ¼. In the second fraction, 4 and 8 share a common factor of 4, so if we divide both 4 and 8 by 4, we get ½.

Greatest Common Factor

Most instances of reducing fractions can be solved this way. After a little practice, you can simply identify the “greatest common factor,” that is, the greatest factor that both the numerator and denominator have in common. Still, it helps to be familiar with the method behind identifying the greatest common factor and reducing. Here is an example using the long method.

Example 1

Reduce 6/8 to lowest terms
While most of us will quickly identify the GCF as 2, and thus reduce the fraction to ¾, let’s look at the official steps to arriving at ¾.

Finding the Greatest Common Factor

1.  List the prime factors of both the numerator and denominator
2.  Find the factors common to both numerator and denominator
3.  Divide the numerator and denominator by the common factors (cancelling)

First, we listed the prime factors of each number, then we cancelled the common factors—in this case, one 2 in the numerator and one 2 in the denominator. Finally, we simply multiplied our factors in the numerator and denominator to get our answer.
Notice that this method should only be used when identifying the GCF is rather difficult. If you can identify the GCF quickly, simply divide the numerator and denominator by the GCF. Often the quickest way to identify the GCF is through trial and error. Let’s try that method on these next examples:

Example 2

Reduce to lowest terms: 24/36
First, think of a number that might be the GCF.
Is it 8? 8 goes evenly into 24, but not into 36.
Is it 6? 6 goes evenly into 24 and into 36. Divide both by 6 and you get 4/6.
Notice that 4/6, while a reduction, is not reduced to lowest terms. Simply reduce this fraction to 2/3 (GCF was 2), and now we’re done.
In this case, we overlooked the fact that 12 was the original GCF. In overlooking this, though, we identified a new strategy: sometimes, you can reduce to lowest terms faster by reducing a fraction more than once.

Example 3

Reduce to lowest terms:  56/64
Is it 8? 8 goes into 56 and 64 evenly. Divide both by 8 and we get 7/8. In this case, we’ve correctly identified the GCF, and we’re finished.