Alternative Format

Try This

Ratios in Advertising 

There are many different ways to compare quantities. For instance, consider the following three advertisements, which all represent the result of the same survey. In this particular survey, students were asked whether they prefer texting or calling their friends.

1. Advertisement 1: \(\dfrac{3}{4}\) of students surveyed prefer texting
2. Advertisement 2: \(375\) out of \(500\) students surveyed prefer texting
3. Advertisement 3: \(125\) more students prefer texting

For each advertisement, think about how the quantities are being compared.

For a moment, look specifically at Advertisements 1 and 2. These advertisements compare the number of students who prefer texting to the total number of students surveyed. In each, a ratio is used to indicate the population that prefers texting. Think about where you have seen these types of comparisons before.

Lesson Goals

• Explore the meaning of a ratio.
• Write ratios in different forms.
• Solve word problems involving ratios.

Try This!

Consider the following diagram. This diagram is composed of a collection of shapes. Some shapes are squares and some are circles, while some shapes are black and some are white.

Using the diagram, which two quantities are being compared in each of the following ratios? 

• \(3\) to \(12\)
• \(7\) to \(12\)
• \(2\) to  \(7\)
• \(3\) to \(9\)
• \(9\) to \(3\)

If you need help getting started consider the following example: the ratio \(5\) to \(7\) represents the ratio of black shapes to white shapes.

Think about this problem, then move on to the next part of the lesson

The Meaning of Ratio

Uses of Ratios

You have likely seen the idea of a ratio before, because ratios are used in many practical situations.

For example, ratios are needed to make scale drawings or scale models.

They are used to determine the actual distances for maps.

And ratios are important when baking to make sure that goodies taste just right.

Sources: Toy Train - Pichunter/iStock/Getty Images;
City Map - giorgos245/iStock/Getty Images; Baking - Askold Romanov/iStock/Getty Images

Ratio Definition

Before we go any farther, let's review what a ratio is.

You should be fairly comfortable with the idea of a comparison.

But what do we mean when we say "like quantities"? Like quantities could mean that we count the number of different objects and compare the counts. For example, the ratio \(3\) to \(2\) tells us that there are three cowboy hats for every two baseball hats. We are counting the number of hats of each type.

Counting

\(3\) cowboy hats for every
\(2\) baseball hats

Source: Hats - kowalska-art/iStock/Getty Images

The ratio \(24\) to \(4\) tells us the number of hexagons compared to the number of circles. Again, we are counting the number of each type of shape.

Ratio of hexagons to circles is
\(24\) to \(4\)

But it doesn't always make sense to compare relative quantities by counting how many of each object there are.

For example, if we compare the relative amounts of sugar and flour in a recipe, we would not want to count the number of grains of each.

Instead, we measure each quantity, using a more helpful unit of measurement — for example, a cup or a teaspoon. Here, "like quantities" means that we must use the same unit of measurement for each ingredient.

Measuring

\(1\) teaspoon of sugar to
\(2\) teaspoons of flour

In this picture, we chose to use a teaspoon as our unit of measurement, and the ratio is \(1\) teaspoon of sugar to \(2\) teaspoons of flour.

Source: Baking - Askold Romanov/iStock/Getty Images

Ratios in Baking

Let's look more closely at ratios in baking.

To make her famous chocolate chip cookies, Louise starts by mixing \(1\) cup of sugar with \(2\) cups of flour. We say that in this recipe the ratio of sugar to flour is \(1\) cup to \(2\) cups, or \(1\) to \(2\).

Sugar to flour

\(1\) cup to \(2\) cups 

\(1\) to \(2\)

If Louise wanted to make a much smaller batch of cookies she could use a smaller scoop, such as a teaspoon. As long as she puts \(1\) teaspoon of sugar for every \(2\) teaspoons of flour, the cookies will turn out the same.

\(1\) teaspoon of sugar to \(2\) teaspoons of flour

Similarly, if Louise wants to make a much larger batch of cookies she could mix \(1\) large bowl of sugar for every \(2\) bowls of flour.

\(1\) bowl of sugar to \(2\) bowls of flour

Source: Baking - Askold Romanov/iStock/Getty Images

What matters is that, regardless of the size of the scoop, is that there is always \(1\) scoop of sugar for every \(2\) scoops of flour.

Ratios generally do not mention units, only numbers.

The quantities are measured in the same way and the ratio represents how the quantities compare to one another.

Check Your Understanding 1

Question

Brent is making granola using oats and seeds in the ratio \(6:5\). Which of the following statements gives the correct ratio of oats to seeds? Select all that apply.

1. \(6\) bowls of oats and \(5\) bowls of seeds
2. \(6\) cups of oats and \(5\) cups of seeds
3. \(6\) scoops of oats and \(5\) bowls of seeds
4. \(6\) scoops of oats and \(5\) scoops of seeds

Answer

1. \(6\) bowls of oats and \(5\) bowls of seeds
2. \(6\) cups of oats and \(5\) cups of seeds

4. \(6\) scoops of oats and \(5\) scoops of seeds

Feedback

Remember that a ratio is a comparison of two like quantities. If oats and seeds are being measured in the ratio of \(6:5\), that means we must use the same unit of measurement for each ingredient.

If we choose to use spoons, then we would use \(6\) spoons of oats and \(5\) spoons of seeds.

Thus, the only correct options are:

• \(6\) bowls of oats and \(5\) bowls of seeds
• \(6\) cups of oats and \(5\) cups of seeds
• \(6\) scoops of oats and \(5\) scoops of seeds

Ratios in Diagrams: Finding More Than One

Consider the following diagram. In this diagram there are eight identical pieces. Five are dotted and three are striped. The basic unit of measurement is a piece of the diagram.

We can count the pieces and use ratios to compare the number of pieces of each type. For example, the ratio of dotted pieces to striped pieces is \(5\) to \(3\). We can also write this using ratio notation, which would be \(5:3\). The ratio of dotted pieces to total pieces is \(5\) to \(8\). Again, we can write this in ratio notation which is \(5:8\).

What other ratios can you write that describe a comparison within this diagram?

Perhaps you have already noticed that there are at least six ratios that we can express. The other four ratios you might have noticed are the ratio of striped pieces to dotted pieces is \(3\) to \(5\), the ratio of total pieces to striped pieces is \(8\) to \(3\), the ratio of striped pieces to total pieces is \(3\) to \(8\), and the ratio of total pieces to dotted pieces is \(8\) to \(5\).

Ratios in Diagrams: Order Matters

Order matters when presenting a ratio.

For example, the ratio of dotted pieces to striped pieces in the diagram is \(5\) to \(3\), not \(3\) to \(5\). The word that comes first in the description — in this case, dotted — must correspond to the number that comes first in the ratio. That's \(5\).

Dotted to Striped

\(5:3\)

While the ratio \(3\) to \(5\) compares the same two quantities and may also seem correct, it actually represents a different comparison in context. It declares that there should be \(3\) dotted pieces for every \(5\) striped pieces, which actually describes the inversion of the original diagram.

Dotted to Striped

\(3:5\)

You cannot change the order of the numbers unless you change the order in the description to match.

These two diagrams help to convince us that the ratio \(5\) to \(3\) is different than the ratio \(3\) to \(5\). We can summarize this discussion as an important fact

Check Your Understanding 2

Question — Part 1

What is the ratio of basketballs to total balls in the diagram?

Source: Balls - choness/iStock/Getty Images

Answer — Part 1

\(4:10\)

Feedback — Part 1

There are \(4\) basketballs and \(10\) balls in total so the ratio of basketballs to total balls is \(4:10\).

Question — Part 2

What is the ratio of calculators to staplers in the diagram?

Source: Supplies - Green_Leaf/iStock/Getty Images

Answer — Part 2

\(4:1\)

Feedback — Part 2

There are \(4\) calculators and \(1\) stapler so the ratio of calculators to staplers is \(4:1\).

Online Version

https://ggbm.at/JYW2WMCV

Ratios in Fractional Form

Try This Problem Revisited

Communicating what two quantities a ratio is comparing is important because often, there are many comparisons that can be made within a single diagram.

To illustrate this, let's revisit the Try This problem.

You may have noticed that there's a lot going on in this diagram. We discussed that it contains a collection of shapes. Some shapes are circles and some are squares, while some shapes are black and some are white. You are asked to identify which two quantities are being compared in each of the following ratios.

• \(3\) to \(12\)
• \(7\) to \(12\)
• \(2\) to \(7\)
• \(3\) to \(9\)
• \(9\) to \(3\)

Solution

Let's start by considering the ratio \(3\) to \(12\). Notice that there are \(3\) circles in the diagram and \(12\) shapes in total. The ratio \(3\) to \(12\) represents the ratio of circles to total shapes.

\(3\) to \(12\)

circles to total shapes

Next, consider the ratio \(7\) to \(12\). Since there are \(7\) white shapes in the diagram and \(12\) shapes in total, the ratio \(7\) to \(12\) represents the ratio of white shapes to total shapes.

\(7\) to \(12\)

white shapes to total shapes

Moving on, since there are \(2\) black circles in the diagram and \(7\) white shapes, the ratio \(2\) to \(7\) represents the ratio of black circles to white shapes.

\(2\) to \(7\)

black circles to white shapes

Since there are \(3\) circles in the diagram and \(9\) squares, the ratio of \(3\) to \(9\) represents the ratio of circles to squares.

\(3\) to \(9\)

circles to squares

Similarly, since there are \(9\) squares and \(3\) circles, the ratio \(9\) to \(3\) represents the ratio of squares to circles.

\(9\) to \(3\)

squares to circles

When you look at these ratios, do you notice that we can actually sort them into two different types?

The first type is ratios that compare part of the diagram to the whole diagram. We call these part to whole ratios.

• \(3\) to \(12\)
• \(7\) to \(12\)

The second type of ratios is those that compare part of the diagram to a different part of the diagram. We call these part to part ratios.

• \(2\) to \(7\)
• \(3\) to \(9\)
• \(9\) to \(3\)

Ratios as Fractions

Let's take a closer look at the part-to-whole ratios.

We determined that the ratio \(3\) to \(12\) represents the ratio of circles to total shapes, but we could also say that \(\dfrac{3}{12}\) of the total shapes are circles. The ratio \(3\) to \(12\) can also be written as the fraction \(\dfrac{3}{12}\). The fraction \(\dfrac{3}{12}\) represents the same information as the ratio. It's just written in a different but equivalent way.

\(3\) to \(12\)  or  \(3:12\)

circles to total shapes

\(\dfrac{3}{12}\)

Similarly, the ratio of white shapes to total shapes is \(7\) to \(12\), or \(7\) over \(12\). The first representation is in ratio form, and the second is in fractional form. These two representations are equivalent, meaning they represent the same information.

\(7\) to \(12\)  or  \(7:12\)

white shapes to total shapes

\(\dfrac{7}{12}\)

We can summarize this as an important fact.

Take a moment to think about why using a fraction to represent a ratio might be helpful. Perhaps there is knowledge that you have about fractions that you could use to do more with ratios?

 

At this point, you might also be wondering if we can use fractions to represent part-to-part ratios. We will explore this connection between ratios and fractions. But for now, we want to remember that, when we write a fraction, we generally view it as part over whole. So for the time being, let's stick to only writing part-to-whole ratios as fractions. The time will come later on when we want to write all ratios in fractional form, so we can discuss these conversations then.

Check Your Understanding 3

Question — Part 1

Which two quantities are being compared in the ratio \(2\) to \(3\)?

Source: Balls - choness/iStock/Getty Images

1. Tennis balls to soccer balls
2. Baseballs to all balls
3. Basketballs to tennis balls
4. Baseballs to basketballs

Answer — Part 1

4. Baseballs to basketballs

Question — Part 2

Which two quantities are being compared in the ratio \(3\) to \(17\).

Souce: Balls - choness/iStock/Getty Images

1. Tennis balls to soccer balls
2. Baseballs to all balls
3. Basketballs to all balls
4. Basketballs to baseballs

Answer — Part 2

3. Basketballs to all balls

Online Version

https://ggbm.at/xp87AsCK

Solving Ratio Problems

Let's now take what we know about ratios and use it to solve some problems.

Example 1

Solution

Since the question refers to the total number of games our first step is to write a ratio that compares the number of wins, which we represent using check marks, to the total number of games played.

Step 1

If there are \(5\) wins for every \(2\) losses and we can represent these losses using an 'X.'

Then there are \(5\) wins for every \(7\) games played.

Wins to Total Games

\(5:7\)

That is, the ratio of wins to total games is \(5\) to \(7\).

Step 2

In our second step, we apply the ratio \(5: 7\) to the total games played.

Let's start by considering the total number of games, which is \(28\). If the ratio \(5\) to \(7\) must hold, then for every \(7\) games \(5\) must be wins. We can show this using our diagram.

In the first group of \(7\) games \(5\) are wins. This means that \(2\) games are also losses.

In another group of \(7\) games, we have \(5\) wins and \(2\) losses.

From here we can form \(2\) more groups of \(7\) and represent \(5\) wins and \(2\) losses in each group.

Essentially, we have divided \(28\) into groups of \(7\) games and, within each group of \(7\), identified \(5\) wins and \(2\) losses.

To determine the number of games the team won, we just need to count the number of check marks in the diagram. Therefore, after playing \(28\) games, the team has won \(20\) games this year.

Check Your Understanding 4

Question

A school's basketball and soccer teams are both playing now.

The ratio of basketball spectators to soccer spectators is \(4:1\). If there are \(20\) spectators in total, how many are watching the soccer game?

1. \(4\)
2. \(5\)
3. \(16\)
4. \(15\)

Answer

1. \(4\)

Feedback

Pictures can represent the ratio \(4:1\).

So for every \(5\) spectators, \(4\) are basketball spectators and \(1\) is a soccer spectator.

Apply this thinking to a group of \(20\) spectators to get an answer of \(4\).

Using a picture, we can see there are \(4\) soccer spectators.

Source: Balls - choness/iStock/Getty Images

Online Version

https://ggbm.at/yxenPGgK

Example 2

Solution

When you put a scale on a drawing, you're trying to communicate to another person how to get accurate, real-world information from the diagram. So let's consider the diagram of the fire station.

\(50:2\)

If this diagram has a scale of \(50\) to \(2\), then we would read this as "every \(50\) units on the diagram represents \(2\) of the same units in the real world."

If we chose to measure the diagram in centimetres, then this would mean that \(50\) centimetres on the diagram represents \(2\) centimetres in the real world.

This doesn't match the information that we were given. Also, if you think about what this is actually telling us, it doesn't make any sense because a real station should be much larger than the diagram, not smaller.

The ratio \(50:2\) does not communicate that the measurements on the diagram and the measurements in the real world were taken using different units. 

In other words, we made a ratio comparing centimetres to meters, which are not like quantities. We must take the different units of measurement into consideration before we find a ratio.

Source: Blueprint - stock_shoppe/iStock/Getty Images

Take It With You

Consider the scale drawing of the fire station from the previous example. We discussed why \(50:2\) is not an appropriate ratio to use as a scale when \(50\) cm on the diagram represents \(2\) m in the real world.

Give \(2\) ratios that are appropriate to use as a scale for this diagram.

Source: Blueprint - stock_shoppe/iStock/Getty Images