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Ratios in Simplest Form

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Have you ever had a goal that you wanted to meet?

Joanna has a goal of reading 16 books in the first semester of the new school year. By Halloween, she had read four out of 16 books.

Joanna wants to use a ratio to express her progress. What is the simplest way to write this ratio?

This Concept is about simplifying ratios. By the end of it, you will be able to complete this task.

Guidance

Previously we worked on how to simplify ratios that are in fraction form to see if they are equivalent or not. We will explore this further in this section.

What does it mean to simplify?

To simplify means to make smaller. When we simplify, we make a fraction smaller. It is still equivalent to the larger form of the fraction, but it is simpler.

How do we simplify a ratio in fraction form?

To write a ratio in simplest form, find the greatest common factor of both terms in the ratio. The greatest common factor of two numbers is the greatest number that divides both numbers evenly. Then, divide both terms of the ratio by the greatest common factor.

This is basically the same procedure you use to rewrite a fraction in simplest form.

Write this ratio in simplest form \frac{20}{24} .

First, find the greatest common factor 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.

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

The greatest of those common factors is 4, so divide both terms by 4 to write the ratio in simplest form.

\frac{20}{24} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6}

So, the simplest form of the ratio \frac{20}{24} is \frac{5}{6} .

How is this useful when working with ratios?

It is useful because we can simplify ratios in fraction form to see if they are equivalent. We can also use a simplified ratio to find an equivalent ratio. To do this, you can multiply both terms of the original ratio by the same number to find an equivalent ratio.

Write three equivalent ratios for this ratio 1:9.

The ratio 1:9 or \frac{1}{9} is already in simplest form. Notice that we wrote the ratio into fraction form so that it is easier to work with.

Now, we will write three equivalent ratios by multiplying both terms by the same number.

It does not matter which numbers we choose to multiply the terms. Let's multiply by 2, first.

\frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18} \ \text{or} \ 2:18

Let's multiply by 5, next.

\frac{1}{9} = \frac{1 \times 5}{9 \times 5} = \frac{5}{45} \ \text{or} \ 5:45

Let's multiply by 100, next.

\frac{1}{9} = \frac{1 \times 100}{9 \times 100} = \frac{100}{900} \ \text{or} \ 100:900

So, three ratios that are equivalent to 1:9 are 2:18, 5:45, and 100:900.

Now it's time for you to try a few on your own.

Example A

Simplify this ratio: 14:16

Solution: 7:8

Example B

Simplify this ratio: 3 to 18

Solution: 1 to 6

Example C

Write an equivalent ratio: 4 to 5

Solution: 8 to 10

Now back to Joanna and the books.

Joanna has a goal of reading 16 books in the first semester of the new school year. By Halloween, she had read four out of 16 books.

Joanna wants to use a ratio to express her progress. What is the simplest way to write this ratio?

First, let's write the ratio of books that Joanna has completed.

\frac{4}{16}

This is the fraction form of the ratio, but we could have also written it with a colon or with the word "to".

Now we simplify by dividing the numerator and the denominator by the greatest common factor, 4.

\frac{1}{4}

This is our answer.

Vocabulary

Ratio
a comparison between two quantities. Ratios can be written as a fraction, with a colon or by using the word "to".
Equivalent Ratios
when two ratios are equal.
Simplify
to write in a simpler form by using the greatest common factor to divide the numerator and the denominator of a fraction by the same number.
Greatest Common Factor
the largest common factor between two numbers.

Guided Practice

Here is one for you to try on your own.

Elena and Jake have a box with only two colors of marbles in it. There are 28 blue marbles and 16 gray marbles in the box. Elena says that the ratio of gray marbles to blue marbles is \frac{16}{28} . Jake says that the ratio of gray marbles to blue marbles is \frac{4}{7} . Who is correct, or are they both correct? Then find the ratio of gray marbles to total marbles in the box. Write your answer in simplest form.

Answer

Consider the first question. It is a question about comparing and determining equivalence.

Write the ratio of gray marbles to blue marbles as a fraction. Be careful that the ratio you write compares gray marbles to blue marbles, not blue marbles to gray marbles.

Since there are 16 gray marbles and 28 blue marbles, the ratio is:

\frac{gray}{blue} = \frac{16}{28}

So, Elena is correct that the ratio of gray marbles to blue marbles is \frac{16}{28} .

If that ratio is equivalent to \frac{4}{7} , then Jake is correct, too. One way to determine if those two ratios are equivalent is to cross multiply.

\frac{16}{28} & \overset{?}{=} \frac{4}{7}\\28 \times 4 & \overset{?}{=} 16 \times 7\\112 & \overset{?}{=} 112\\112 & = 112

Since the cross products are equal, the two ratios are equivalent.

So, the answer to the first question is that Elena and Jake are both correct.

Now let’s think about the second part of the question. To figure this out, we need to figure out the ratio of gray marbles to total marbles in the box.

\frac{gray}{total} = \frac{gray}{gray + blue} = \frac{16}{16 + 28} = \frac{16}{44}

Write that ratio in simplest form.

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

The factors of 44 are: 1, 2, 4, 11, 22, and 44.

The greatest common factor of 16 and 24 is 4. So, divide both terms by 4.

\frac{16}{44} = \frac{16 \div 4}{44 \div 4} = \frac{4}{11}

The ratio of gray marbles to total marbles, written in simplest form, is \frac{4}{11} .

Video Review

- This James Sousa video is about writing ratios in simplest form. It includes fractions and decimals.

Practice

Directions: 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. \frac{18}{20}

5. \frac{25}{55}

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. 56 to 9

15. \frac{18}{12}

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