Chapter 4 · Ratios and rates
Ratio Language and Writing Ratios in Order
Learn to describe how two quantities compare using ratio language, write ratios with the numbers in the right order, and tell part-to-part from part-to-whole ratios.
Picture a bag that holds red marbles and blue marbles. You can count each color on its own. You can also say how the two colors go together: there are red marbles for every blue marbles.
That second description is a ratio. A quantity is an amount of something that you can count or measure. A ratio is a pair of numbers that compares two quantities. It tells you how much there is of one quantity for a given amount of the other.
In this lesson, you will say ratios in words, write them with symbols, and name two types of ratio.
Ratio language means words that describe a ratio, such as the ratio of ... to ... and for every. Each sentence below is true for the bag:
- The ratio of red marbles to blue marbles is to .
- For every red marbles, there are blue marbles.
- There are red marbles for every blue marbles.
Each sentence names both quantities, and each number goes with the quantity it counts: with red and with blue. The words for every pair the two amounts: a group of red marbles goes with a group of blue marbles.
The sentence "there are more blue marbles than red marbles" also compares the colors, but it is not ratio language. It gives a difference, , which you find by subtracting. It does not pair one amount with the other.
Worked example
Describe a relationship with ratio language
A fish tank holds goldfish and guppies. Use ratio language to describe how the goldfish and the guppies compare. Name the goldfish first.
Name the two quantities and match each number to what it counts. This tells you which amount goes with which kind of fish.
Put the quantities in the order the problem asks for: goldfish first, guppies second. The numbers follow the same order, so 2 comes before 7.
Say it with the phrase "the ratio of ... to ...". Name the quantities in the same order as the numbers, so each number stays with its own quantity: the ratio of goldfish to guppies is 2 to 7.
Say it a second way with the phrase "for every". This phrase pairs the 2 goldfish with the 7 guppies, which matches the tank: for every 2 goldfish, there are 7 guppies.
Answer
The ratio of goldfish to guppies is to . In other words, for every goldfish, there are guppies.
Sentences are clear, but they take time to write. You can write a ratio in three short forms: with the word to, with a colon, or as a fraction. The table below shows each form for the ratio of red marbles to blue marbles.
All three forms keep the same two numbers in the same order, so they all mean the same comparison. That is why you read all three aloud the same way.
Use the fraction form only when the second number is not , because a fraction cannot have as its denominator, the bottom number.
| Form | How you write it | How you read it |
|---|---|---|
With the word "to" | to | 3 to 5 |
With a colon | 3 to 5 | |
As a fraction | 3 to 5 |
Key concept
The order of the words sets the order of the numbers
A ratio compares two quantities with a pair of numbers. The phrase that names the ratio sets the order: the first quantity named gets the first number.
The phrase ratio of red marbles to blue marbles names red first, so the red amount goes first: . The phrase ratio of blue marbles to red marbles names blue first, so that ratio is . Both describe the same bag, because each number still goes with its own quantity.
When the two amounts are different, switching the numbers without switching the words describes a different relationship.
Worked example
Write a ratio when the order is switched
A trail mix recipe says: for every scoops of raisins, use scoops of oats. All the scoops are the same size. Write the ratio of oats to raisins in three ways.
Match each number to its quantity. The recipe measures both amounts in scoops of the same size. It puts 2 with raisins and 5 with oats.
Read the order in the question. "Oats to raisins" names oats first, so the oats amount goes first. This holds even though the recipe sentence mentions raisins first.
Write the ratio with a colon. The colon takes the place of the word "to", so the numbers stay in the same order.
Write the ratio as a fraction. Put the first number on top and the second number on the bottom, so the fraction keeps the same two numbers in the same order. The second number is 2, not 0, so you may use this form.
Answer
The ratio of oats to raisins is to , which you can also write as or .
Many ratios compare quantities that belong to one group. In the bag, the red marbles are one part of the group, and the blue marbles are another part. All the marbles together are the whole. Every marble is either red or blue, so the whole is marbles.
This gives you two types of ratio:
- A part-to-part ratio compares one part of a group to another part of the same group.
- A part-to-whole ratio compares one part of a group to the whole group.
To tell them apart, check what each number counts. Two parts make a part-to-part ratio. A part and the whole make a part-to-whole ratio. A question often points to the whole with a word such as all or total.
| Ratio in words | What the numbers count | Ratio | Type |
|---|---|---|---|
red marbles to blue marbles | red, blue | part-to-part | |
blue marbles to red marbles | blue, red | part-to-part | |
red marbles to all marbles | red, in all | part-to-whole | |
blue marbles to all marbles | blue, in all | part-to-whole |
Worked example
Write a part-to-part and a part-to-whole ratio
In one class, students walk to school and students ride the bus. Every student in the class does exactly one of these. Write the ratio of walkers to bus riders. Then write the ratio of walkers to all students in the class. Name the type of each ratio.
Find the parts. Walkers and bus riders are two parts of one group, the class. Match each number to its part.
Write walkers to bus riders. Both numbers count parts, so this is a part-to-part ratio. The question names walkers first, so 11 goes first.
Find the whole. The problem does not give the number of students in the class, so add the two parts. Adding works because every student is in exactly one of the two parts.
Write walkers to all students. The second quantity is now the whole class, so use 25. Do not use 14, which counts only the bus riders. A part compared with the whole is a part-to-whole ratio.
Answer
The ratio of walkers to bus riders is , a part-to-part ratio. The ratio of walkers to all students is , a part-to-whole ratio.
Worked example
Find a missing part first
A soccer team played games. It won games and lost the rest. No game ended in a tie. Write the ratio of wins to losses and the ratio of losses to games played.
Find the missing part. The games played are the whole, and wins and losses are its only two parts. Wins plus losses equal the whole, so subtracting the wins from the whole leaves the losses. Adding the parts back gives 9 again, which checks the result.
Write wins to losses. Both are parts, so this is a part-to-part ratio. The question names wins first.
Write losses to games played. Losses are a part and the games played are the whole, so this is a part-to-whole ratio. The question names losses first.
Answer
The ratio of wins to losses is , so for every wins there are losses. The ratio of losses to games played is , so out of the games were losses.
You can now describe a comparison in words and in symbols. A ratio compares two quantities with a pair of numbers, and the first quantity named gets the first number. You can say a ratio with phrases such as for every, and you can write it with the word to, with a colon, or as a fraction when the second number is not . Check what each number counts: two parts give a part-to-part ratio, and a part with the whole gives a part-to-whole ratio.
Next, you will identify the model that represents a ratio.