8.1: Ratios
Introduction
The Milk Comparison
On the way home from soccer practice, Casey’s mom sends her into the grocery store to get a half gallon of milk. Casey is hungry after practice, so she isn’t paying attention to what kind her Mom has asked her to get. In Casey’s house they drink both whole milk and skim milk.
Casey runs to the dairy section of the grocery store and stops short. She isn’t sure what to get. There are five different kinds of milk. There is whole milk, reduced fat milk, lowfat milk, skim milk and organic milk. There are also different brands to choose from: Hood, Eagle Brand, and Garelic for non-organic milk, and Organic Valley and Nature’s Valley for organic brands.
Casey notices that there are three non-organic brands to the two organic brands. Then she notices that the supermarket has its own brand of non-organic milk as well. That makes four non-organic brands to two organic brands.
Casey is sure that this means something. She has been reading about organic food in school and is interested in organic milk and food. Casey wishes that there were more organic brands than non organic brands. She decides to make a note of this to show her teacher at school.
If Casey wants to document this information as a ratio, how could she do it?
Does simplifying the ratio change anything? What conclusions can Casey figure out by working with these ratios?
This lesson is all about ratios, by the end of the lesson, you will know how to help Casey write her milk ratios. Pay close attention and what you learn will be very helpful!!
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills:
- Identify and write different forms of equivalent ratios.
- Write ratios in simplest form.
- Write and compare ratios in decimal form.
- Solve real-world problems involving ratios.
Teaching Time
I. Identify and Write Different Forms of Equivalent Ratios
This lesson focuses on ratios. Ratios are everywhere in everyday life. In fact, we work with ratios so much that we probably don’t even realize that we are working with them. In this lesson, you will learn how to write ratios, simplify ratios and compare ratios, but there is a question that we must answer first.
What is a ratio?
A ratio is a comparison of two quantities. The quantities can be nearly anything; people, cars, dollars... even two groups of things!
Let’s look at a picture.
We can write ratios that compare the boys in this picture, but how?
How do we write a ratio?
A ratio is written in three different ways. It can be written as a fraction, with the word “to” or with a colon.
Let’s take a look at this in action by writing ratios that compare the boys in the picture.
Example
What is the ratio of boys with striped shirts to boys with solid shirts?
There are two boys with striped shirts and two boys with solid shirts.
Let’s write the ratio in three ways.
\begin{align*}&\frac{2}{2}\\ &2:2\\ &2 \ \text{to}\ 2\end{align*}
Each of these ratios is correct. Notice that we are comparing an individual quality here.
What about comparing a category to the group?
Example
What is the ratio of boys that are holding binders to all of the boys?
There are two boys holding binders and four boys in the group.
Let’s write the ratio three different ways. Notice that the first thing being compared comes first when writing the ratio. Or the first thing becomes the numerator in the fraction form of the ratio.
\begin{align*}&\frac{2}{4}\\ &2:4\\ &2 \ \text{to}\ 4\end{align*}
Each of these ratios is equivalent, meaning that they are all equal. Each ratio, though written in a different form, is an equivalent ratio.
We can write many different ratios by comparing these figures. Let’s list some and use the word “to” to write our ratio form.
Stars to circles = 3 to 2
Red stars to total stars = 2 to 3
Red stars to blue stars = 2 to 1
Blue stars to red stars = 1 to 2
Blue stars to total stars = 1 to 3
We could continue making this list.
We can also write the same ratios using a colon or a fraction.
Practice on your own. Use the picture to write each ratio three different ways.
- What is the ratio of orange marbles to green marbles?
- What is the ratio of yellow marbles to total marbles?
- What is the ratio of orange marbles to total marbles?
Take a few minutes to check your work with a peer.
II. Write Ratios in Simplest Form
Sometimes, a ratio does not represent a clear comparison. If you look at one of the ratios in the practice problems you just finished you will see what I mean. The ratio of orange marbles to total marbles was 2 to 22.
We can simplify a ratio just as we would a fraction. Let’s look at the ratio 2 to 22 in the fraction form of the ratio.
\begin{align*}\frac{2}{22}\end{align*}
We simplify a ratio in fraction form in the same way that we would simplify a fraction.
We use the greatest common factor of both the numerator and the denominator. By dividing the numerator and the denominator by the GCF we can simplify the fraction.
The GCF of both 2 and 22 is 2.
\begin{align*}\frac{2 \div 2}{22 \div 2} = \frac{1}{11}\end{align*}
The simplest form of the ratio is 1 to 11. We can write this in three ways 1 to 11, 1:11 and \begin{align*}\frac{1}{11}\end{align*}.
When we simplify a ratio in fraction form, we also write an equivalent form of the original ratio.
\begin{align*}\frac{1}{11} = \frac{2}{22}\end{align*}
Simplify these ratios on your own. If the ratio is not written in fraction form, you will need to do that first.
- \begin{align*}\frac{2}{10}\end{align*}
- 6 to 8
- 5:20
Check your answers with a partner.
III. Write and Compare Ratios in Decimal Form
We just finished writing ratios in fraction form and simplifying them. What about decimal form? Fractions and decimals are related, in fact a fraction can be written as a decimal and a decimal can be written as a fraction.
Is it possible to write a ratio as a decimal too?
Yes! Because a ratio can be written as a fraction, it can also be written as a decimal. To do this, you will need to remember how to convert fractions to decimals.
Now we can apply this information to our work with ratios.
Example
Convert 2:4 into a decimal.
First, write it as a ratio in fraction form.
\begin{align*}2:4 = \frac{2}{4}\end{align*}
Next, simplify the fraction if possible.
\begin{align*}\frac{2}{4} = \frac{1}{2}\end{align*}
Finally, convert the fraction to a decimal.
\begin{align*} \overset{\quad .5}{2\overline{)1.0 \;}}\end{align*}
Our answer is .5.
Practice converting ratios to decimal form.
- 4 to 5
- \begin{align*}\frac{5}{20}\end{align*}
- 6 to 10
Take a few minutes to check your work with a partner.
Real Life Example Completed
The Milk Comparison
Remember Casey and her milk comparison? Well, now you know all that you need to know about ratios to help her with her comparison. Here is the problem once again.
On the way home from soccer practice, Casey’s mom sends her into the grocery store to get a half gallon of milk. Casey is hungry after practice, so she isn’t paying attention to what kind her Mom has asked her to get. In Casey’s house they drink both whole milk and skim milk.
Casey runs to the dairy section of the grocery store and stops short. She isn’t sure what to get. There are five different kinds of milk. There is whole milk, reduced fat milk, lowfat milk, skim milk and organic milk. There are also different brands to choose from: Hood, Eagle Brand, and Garelic for non-organic milk, and Organic Valley and Nature’s Valley for organic brands.
Casey notices that there are three non-organic brands to the two organic brands. Then she notices that the supermarket has its own brand of non-organic milk as well. That makes four non-organic brands to two organic brands.
Casey is sure that this means something. She has been reading about organic food in school and is interested in organic milk and food. Casey wishes that there were more organic brands than non organic brands. She decides to make a note of this to show her teacher at school.
If Casey wants to document this information as a ratio, how could she do it?
Does simplifying the ratio change anything? What conclusions can Casey figure out by working with these ratios?
First, let’s underline the important information. (This has been done for you above.)
A ratio is a comparison. We can write a ratio to compare two quantities in three different ways. In this problem, Casey wants to compare organic and non-organic brands of milk.
She notices that there are four non-organic brands and two organic brands.
Casey can write this comparison three different ways.
\begin{align*}4\ \text{to}\ 2 \qquad \frac{4}{2} \qquad 4:2\end{align*}
If Casey simplifies these ratios, what conclusions can she draw?
4 to 2 simplifies to 2 to 1
\begin{align*}\frac{4}{2} &= \frac{2}{1}\\ 4 : 2 &= 2 : 1\end{align*}
Casey concludes that there are twice as many non-organic brands as there are organic. When she shows her teacher, Ms. Gilson challenges Casey to do some research about organic brands of milk to bring to the grocery store manager. Casey rises to the challenge!!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Ratio
- a comparison between two quantities; can be written three different ways.
- Equivalent
- equal
- Simplify
- to make smaller
- Greatest Common Factor
- the largest number that will divide into two or more numbers evenly.
Technology Integration
Khan Academy, Introduction to Ratios
James Sousa, Introduction to Ratios
James Sousa, Example of Writing a Ratio as a Simplified Fraction
James Sousa, Another Example of Writing a Ratio as a Simplified Fraction
Other Videos:
http://www.mathplayground.com/howto_ratios.html – This is a great video that explains the concept of ratios.
http://www.mathplayground.com/howto_equalratios.html – This is a nice basic video on understanding equal ratios.
Time to Practice
Directions: Use the picture to answer the following questions. Write each ratio three ways.
1. What is the ratio of hens to chicks?
2. What is the ratio of green chicks to yellow chicks?
3. What is the ratio of white chicks to total chicks?
4. What is the ratio of green chicks to total chicks?
5. What is the ratio of yellow chicks to total chicks?
6. What is the ratio of green chicks to white chicks?
Directions: Simplify each ratio. Write your answer in fraction form.
7. 2 to 4
8. 3:6
9. 5 to 15
10. 2 to 30
11. 10 to 15
12. \begin{align*}\frac{4}{6}\end{align*}
13. 3:9
14. 6:8
15. \begin{align*}\frac{2}{8}\end{align*}
16. \begin{align*}\frac{4}{16}\end{align*}
17. 10 to 12
18. 7:21
19. 12:24
20. 25 to 75
Directions: Convert the following ratios into decimals.
21. 3 to 4
22. 2 to 4
23. \begin{align*}\frac{1}{5}\end{align*}
24. 25 to 100
25. 16 to 32
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