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Simplifying Fractions

Think, for a minute, about the fractions 1/2 and 3/6. At first glance, they look different but, if you get out your calculator and calculate "1 divided by 2" and "3 divided by 6" you'll see that they both equal 0.5. This tells us that both of the fractions myst mean the same number. What we need is a method to reduce a fraction down to its "simplest" form.

In Math . . .

A fraction is simplified or reduced if the numerator and denominator have no factors in common (except 1).

In English . . .

Look back at our 1/2 and 3/6 example. Because 3 and 6 have a factor in common, both of them are divisible by 3, 3/6 can't be the simplest form of the fraction. On the other hand, because 1 and 2 don't have any common factors 1/2 is simplified.

There's a very simple method you can follow for simplifying a fraction.

1. Find the greatest common factor of the numerator and the denominator.
2. Divide both the numerator and the denominator by the number you found in step one.

Example 1

Simplify 38 / 14.

The greatest common factor of 38 and 14 is 2 so our fraction would reduce to:

$$\frac{38}{14} = \frac{38/2}{14/2} = \frac{19}{7}$$

Example 3

Simplify 98 / 42.

There's another way to approach this kind of problem that's a little less direct than finding the greatest common factor but it may be easier for people who are a little uncomfortable with finding the greatest common factor. It's based on looking at the two numbers and finding numbers that divide both of them. For example, when I look at 98 and 42 I see that they're both even so I know I can divide both of them by 2.

$$\frac{98}{42} = \frac{98/2}{42/2} = \frac{49}{21}$$

Now when I look at 49 and 21, I see that they're both divisible by 7 so that gives us:

$$\frac{98}{42} = \frac{98/2}{42/2} = \frac{49}{21}$$ $$\frac{98}{42} = \frac{98/2}{42/2} = \frac{49/7}{21/7} = \frac{7}{3}$$

At this point, I see that 7 and 3 are both prime numbers. That tells me that there's nothing else I can divide into them so 7/3 must be the simplified version I was looking for.

Example 5

Simplify 16 / 4.

The greatest common factor of 16 and 4 is 4 so our fraction would reduce to:

$$\frac{16}{4} = \frac{16/4}{4/4} = \frac{4}{1} = 4$$

Notice what happened when the denominator reduced all the way down to 1. That reduced our fraction all the way to an integer.

Example 2

Simplify 72 / 78.

The greatest common factor of 72 and 78 is 6 so our fraction would reduce to:

$$\frac{72}{78} = \frac{72/6}{78/6} = \frac{12}{13}$$

Example 4

Simplify 84 / 108.

We'll try doing this one the same way as Example 3. When I look at 84 and 108, they're both even so I could divide both of them by 2 but when I look a little closer, I see that they're actually both divisible by 4:

$$\frac{84}{108} = \frac{84/4}{108/4} = \frac{21}{27}$$

Now when I look at 49 and 21, I see that they're both divisible by 7 so that gives us:

$$\frac{84}{108} = \frac{84/4}{108/4} = \frac{21/3}{27/3} = \frac{7}{9}$$

There aren't any numbers that go into both 7 and 9 so 7/9 must be the simplified form of 84/108.

Example 6

Simplify 27 / 34.

If you try to find the greatest common factor of 27 and 34, you'll see that it's 1. That tells us that there are no numbers (greater than 1) that divide both 27 and 34 so the fraction must already be in its simplest form.

Videos

Dynamic Practice - Simplifying Fractions