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# Ratios in Simplest Form

## Simplify ratios using greatest common factors.

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Ratios in Simplest Form

Noah's family is on a road trip across the United States in their van. They started in Buffalo, New York, and they are driving all the way to San Francisco, California. The entire trip is approximately 2,700 miles, and so far they have traveled 2,400 miles. Noah can only think of the 300 miles they have yet to go. How can you express the distance still left to go to the total number of miles as a fraction in its simplest form?

In this concept, you will learn to reduce ratios to their simplest form.

### Reducing Ratios to Simplest Form

A ratio is the relationship, or comparison, of one amount to another.

To simplify a ratio means to reduce it to its smallest, simplest, terms.

In order to reduce a ratio, or fraction, to its simplest form, it is necessary to find the greatest, or highest, common factor for both terms in the ratio or fraction.

The greatest common factor of two numbers is the largest number that will divide into both terms evenly.

Let's look at an example:

Write this ratio in simplest form 2024\begin{align*}\frac{20}{24}\end{align*}.

First, find the factors of the terms 20 and 24.

The factors of 20 are 1, 2, 4, 5, 10, and 20.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Next, find the highest common factor.

The factors that both 20 and 24 have in common are 1, 2 and 4.

The greatest of those common factors is 4.

Then, divide both terms by 4 to write the ratio in its simplest form.

2024=20÷424÷4=56\begin{align*}\frac{20}{24} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6}\end{align*}

The answer is 56\begin{align*}\frac{5}{6}\end{align*}

Simplifying ratios can be helpful in determining equivalence.

Equivalent ratios are those that are equal.

### Examples

#### Example 1

Earlier, you were given a problem about Noah and the family cross-country road trip.

Noah was dismayed as he thought about having to ride yet another 300 miles, but that was all that was left of the 2,700 mile trip. Express this distance as a fraction in its simplest terms.

First, write a fraction.

300 miles to go : 2,700 miles total

3002700\begin{align*}\frac{300}{2700}\end{align*}

Next, recognize the two zeros at the end of each term make 100  a large factor that can be immediately removed.

327\begin{align*}\frac{3}{27}\end{align*}

Then, find the greatest common factor to further simplify and reduce both terms by division.

3÷327÷3=19\begin{align*}\frac{3\div 3}{27\div 3}=\frac{1}{9}\end{align*}

The answer is 19\begin{align*}\frac{1}{9}\end{align*}

Noah's family has only 19\begin{align*}\frac{1}{9}\end{align*}  of the total distance yet to go.

#### Example 2

Simplify this ratio: 14:16

First, write the ratio as a fraction.

1416\begin{align*}\frac{14}{16}\end{align*}

Next, determine the common factors.

The factors of 14 are 1, 2, 7, and 14.

The factors of 16 are 1, 2, 4, 8, and 16.

Then, determine the greatest common factor.

Common factors are 1 and 2.

2 is the greatest common factor.

Next, divide both terms in the fraction by the greatest common factor.

14÷216÷2=78\begin{align*}\frac{14\div 2}{16\div 2}=\frac{7}{8}\end{align*}

The answer is 78\begin{align*}\frac{7}{8}\end{align*} or 7:8

#### Example 3

Simplify this ratio: 3 to 18

First, write the ratio as a fraction.

318\begin{align*}\frac{3}{18}\end{align*}

Next, determine the factors of each term.

The factors of 3 are 1 and 3.

The factors of 18 are 1, 2, 3, 6, 9, and 18.

Then, divide both terms by the highest common factor.

3÷318÷3=16\begin{align*}\frac{3\div 3}{18\div 3}=\frac{1}{6}\end{align*}

The answer is 16\begin{align*}\frac{1}{6}\end{align*} or 1 to 6

#### Example 4

Write an equivalent ratio for 16 to 20 in its lowest terms:

First, write the ratio as a fraction.

1620\begin{align*}\frac{16}{20}\end{align*}

Then, find the factors for each term.

1, 2, 4, 8, 16

1, 2, 4, 5, 10, 20

Next, divide by the greatest common factor.

16÷420÷4=45\begin{align*}\frac{16\div 4}{20\div 4}=\frac{4}{5}\end{align*}

The answer is 45\begin{align*}\frac{4}{5}\end{align*}or 4 to 5.

#### Example 5

Beatrice has her own dog-walking service. Each day, she walks 2 dogs in the morning and 4 in the evening. In simplest terms, express the ratio of dogs that Beatrice walks in the morning.

First, remember that a ratio expresses a part to a whole and realize that the total number of dogs is:

2 in the morning + 4 in the evening = 6 dogs total

Then, write the ratio (2 dogs in the morning, 6 dogs total)  2 to 6 as a fraction.

26\begin{align*}\frac{2}{6}\end{align*}

Next, find the factors.

1, 2

1, 2, 6

Then, divide both terms by the greatest common factor.

2÷26÷2=13\begin{align*}\frac{2\div 2}{6\div 2}=\frac{1}{3}\end{align*}

The answer is 13\begin{align*}\frac{1}{3}\end{align*}

Beatrice walks one third of the total number of dogs in the morning.

### Review

Simplify each ratio. Write the simplified version in the same form of the ratio.

1. 3 to 6
2. 5:20
3. 18 to 22
4. 1820\begin{align*}\frac{18}{20}\end{align*}
5. 2555\begin{align*}\frac{25}{55}\end{align*}
6. 6 to 42
7. 18 to 10
8. 12 to 4
9. 16:8
10. 24 to 16
11. 18 to 36
12. 54 to 9
13. 81:27
14. 72 to 9
15. 1812\begin{align*}\frac{18}{12}\end{align*}

### Vocabulary Language: English

Equivalent Ratios

Equivalent ratios are ratios that can each be simplified to the same ratio.