## How to Reduce Fractions with the Dividing Method

- Find a common factor between the numerator and the denominator.
- Divide the numerator and denominator by the common factor.
- Repeat steps 1 and 2 until the numerator and denominator are relatively prime.

If you find the greatest common factor of the numerator and denominator in step 1, then you will only have to go through the steps once (as shown in the green and blue examples).

If you don’t find the greatest common factor the first time, it’s OK. You’ll just have to go through the steps a couple times (as shown in the purple example).

### Examples

Reduce:

\[\frac{12}{36}\]

*What is a factor that 12 and 36 have in common?*

12 and 36 can both be divided evenly by 3.

*What is the reduced fraction when you divide both 12 and 36 by 3?*

\[\frac{12\div3}{36\div3}=\frac{4}{12}\]

*Are 4 and 12 relatively prime?*

No, 4 and 12 can both be divided evenly by 2, so I need to repeat steps 1-2 again.

*What is a factor that 4 and 12 have in common?*

4 and 12 can both be divided evenly by 2.

*What is the reduced fraction when you divide both 4 and 12 by 2?*

\[\frac{4\div2}{12\div2}=\frac{2}{6}\]

*Are 2 and 6 relatively prime?*

No, 2 and 6 can both be divided evenly by 2, so I need to repeat steps 1-2 again.

*What is a factor that 2 and 6 have in common?*

2 and 6 can both be divided evenly by 2.

*What is the reduced fraction when you divide both 2 and 6 by 2?*

\[\frac{2\div2}{6\div2}=\frac{1}{3}\]

*Are 1 and 3 relatively prime?*

Yes, 1 and 3 do not have any common factors. So, I am done reducing the fraction.

When you fully reduce \(\frac{12}{36}\), the simplified fraction is \(\frac{1}{3}\).

Reduce:

\[\frac{12}{36}\]

*What is a factor that 12 and 36 have in common?*

12 and 36 can both be divided evenly by 12.

*What is the reduced fraction when you divide both 12 and 36 by 12?*

\[\frac{12\div12}{36\div12}=\frac{1}{3}\]

*Are 1 and 3 relatively prime?*

Yes, 1 and 3 do not have any common factors. So, I am done reducing the fraction.

When you fully reduce \(\frac{12}{36}\), the simplified fraction is \(\frac{1}{3}\).

Reduce:

\[\frac{16}{56}\]

*What is a factor that 16 and 56 have in common?*

16 and 56 can both be divided evenly by 8.

*What is the reduced fraction when you divide both 16 and 56 by 8?*

\[\frac{16\div8}{56\div8}=\frac{2}{7}\]

*Are 2 and 7 relatively prime?*

Yes, 2 and 7 do not have any common factors. So, I am done reducing the fraction.

When you fully reduce \(\frac{16}{56}\), the simplified fraction is \(\frac{2}{7}\).

## How to Reduce Fractions with Prime Factorization

- Find the prime factorization of the numerator and denominator.
- Cancel out the common factors.
- Multiply the remaining factors.

Reduce:

\[\frac{45}{210}\]

*What is the prime factorization of 45 and 210?*

\[\frac{45}{210}=\frac{3 \times 3 \times 5}{2 \times 3 \times 5 \times 7}\]

Click here to learn how to find the prime factorization of a number.

*Which factors do 45 and 210 have in common?*

\[\frac{45}{210}=\frac{{\cancel{\yellow3}} \times 3\times {\cancel{\yellow5}}}{2\times {\cancel{\yellow3}} \times {\cancel{\yellow5}} \times 7}\]

*Which factors were not cancelled out?*

\[\frac{3}{2 \times 7}=\frac{3}{14}\]

When you fully reduce \(\frac{45}{210}\), the simplified fraction is \(\frac{3}{14}\).

Reduce:

\[\frac{720}{5184}\]

*What is the prime factorization of 720 and 5184?*

\[\frac{720}{5184}=\frac{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3}\]

Click here to learn how to find the prime factorization of a number.

*Which factors do 720 and 5184 have in common?*

\[\frac{720}{5184}=\frac{\cancel{\red2} \times \cancel{\red2} \times \cancel{\red2} \times \cancel{\red2} \times \cancel{\red3} \times \cancel{\red3} \times 5}{\cancel{\red2} \times \cancel{\red2} \times \cancel{\red2} \times \cancel{\red2} \times 2 \times 2 \times \cancel{\red3} \times \cancel{\red3} \times 3 \times 3}\]

*Which factors were not cancelled out?*

\[\frac{5}{2 \times 2 \times 3 \times 3}=\frac{5}{36}\]

When you fully reduce \(\frac{720}{5184}\), the simplified fraction is \(\frac{5}{36}\).

## Why It Works

When you are reducing fractions, the goal is to find an equivalent fraction that is as simplified as possible. These two fractions \(\frac{28}{36}\) and \(\frac{7}{9}\) are equivalent because they represent the same amount of shaded space inside the circle.

When you reduce \(\frac{28}{36}\) to \(\frac{7}{9}\), 4 is the common factor that is divided out with the dividing method and it is also the product of the factors that are cancelled out with the prime factorization method.

### Dividing Method

\[\frac{28\div\red4}{36\div\red4}=\frac{7}{9}\]

### Prime Factorization Method

\[\frac{28}{36}=\frac{\cancel{\red{2}} \times \cancel{\red{2}} \times 7}{\cancel{\red{2}} \times \cancel{\red{2}} \times 3 \times 3}=\frac{7}{9}\]

When you divide the denominator of \(\frac{28}{36}\) by 4, you are essentially taking the 36 pieces in the whole circle and merging the slices together in groups of 4 to make 9 merged slices.

The numerator also has to be divided by 4 because when the 36 pieces in the whole are merged into 9 slices, the 28 shaded slices are also merged to make 7 shaded slices.

Any number divided by 1 equals itself. This is true for whole numbers like \(12\div1=12\). And it is also true for fractions. When you reduce fractions with the dividing method, you are basically dividing the fraction by 1.

\[\frac{6}{15}\div\frac{3}{3}=\frac{2}{5}\]

In this example, \(\frac{3}{3}\) is equivalent to 1 because three thirds is the same thing as one whole. When \(\frac{6}{15}\) is divided by \(\frac{3}{3}\), the reduced fraction is still the same amount