4.3: Unit Scale and Scale Factor
Introduction
A Model of Everest
Josh is so excited about Mount Everest that he has decided to create a model of the mountain for his geography class. Mr. Watkins approved the project and so Josh is putting together all of the information. During lunch, instead of talking with his friends, he took out his notebook and began figuring out how to measure the mountain. His friend Sasha spotted him and came over.
“Hi Josh, what are you working on?” she asked.
“Well, I am going to build a model of Mount Everest for my geography project. I am reading this really cool book about it. So I’m trying to figure out what scale to use for the model,” he explained grinning from ear to ear.
“What do you mean “scale”?”
“If you are going to build a model, you have to use a scale. You shrink down the real dimensions so that you can actually build a model that everyone will be able to see, think about it, Everest is 29,035 feet high, you can’t build that using the real dimensions,” Josh said.
“Oh I get it, so you use \begin{align*}1^{\prime\prime}\end{align*} for 1 foot.”
“That’s the right idea, but \begin{align*}1^{\prime\prime}\end{align*} is definitely too big. If I used \begin{align*}1^{\prime\prime}\end{align*} equals 1 foot for the scale, I would have a model that is 29,035 feet high. I need to use a smaller unit of measurement,” Josh said.
“What about \begin{align*}\frac{1}{4}\end{align*}?” Sasha suggested.
“That’s possible, but it still will probably be too big,” he stopped talking to scribble some numbers down in his notebook. “I think that \begin{align*}\frac{1}{8}^{\prime\prime} = 2000 \ feet\end{align*} will be perfect.”
Sasha looked puzzled.
Do you understand what Josh was working on? How did he come to that scale? Why do you think Sasha is puzzled? What will the actual dimensions of the model be? To answer these questions, you will need to understand scale, scale factor and ratios. Pay attention to the information presented in this lesson and you will be all ready to answer these questions by the end.
What You Will Learn
In this lesson, you will learn the following skills.
- Express unit scale as the ratio between units of measure of scale dimensions to actual dimensions.
- Use unit scale to find actual dimensions or scale dimensions, given necessary information.
- Express scale factor as the ratio of scale dimensions to actual dimensions, without regard to units.
- Use scale factor to find actual dimensions or scale dimensions, given necessary information.
Teaching Time
I. Express Unit Scale as the Ratio between Units of Measure of Scale Dimensions to Actual Dimensions
In the past two lessons, you have been working with ratios. In this lesson, you are going to use ratios in a new way, but first let’s review what we mean when we talk about a ratio.
A ratio is a comparison between two quantities. We can write a ratio in fraction form, by using a colon or by using the word “to”.
Sometimes in life, we have a real-life object that we want to represent in a smaller form. Think about buildings. We can’t build an actual building to show the dimensions in a smaller way, so we build a model of the building. When we do this, we take the actual dimensions and shrink them down to build a model. When we do this, we create a unit scale for the model. When we create a unit scale, we decide on a measurement to represent an actual measurement.
That’s a great question. First, let’s look at an example of a unit scale.
1 inch = 3 feet
This is an example of a unit scale. I have a unit represented by the one inch. Remember that when we talk about unit, we are talking about a relationship to one. We have one inch represented by three feet.
The one inch is the scale dimension and the three feet is the actual dimension we are measuring.
Now, not all objects that you will create a model of will measure exactly what the unit scale does, so we have to use a unit scale to show the relationship between scale dimensions and actual dimensions. Scale dimensions are the dimensions of the model, and actual dimensions are the real – life dimensions.
Example
Using the unit scale above, what would be the relationship between the scale dimensions and the actual dimensions for an object 24 feet long?
First, let’s think about our unit scale.
1 inch = 3 feet
If we have a building 24 feet long, that is the actual dimensions. We need to represent the dimensions using our unit scale.
We can say that 8 inches = 24 feet. This is our answer.
Write down the definitions for unit scale, scale dimension and actual dimension in your notebook. Include the example unit scale: 1 inch = 3 feet
II. Use Unit Scale to Find Actual Dimensions or Scale Dimensions,
Given Necessary Information
In the last section, we began using unit scale to find scale dimensions if we had been given the actual dimensions. We can also work the other way around. If we know the scale dimensions and the unit scale, then we can find the actual dimensions.
It’s a little like figuring out a puzzle. We have to have the pieces of the puzzle to put the puzzle together. What are the pieces that we need?
Necessary Information
- To find the scale dimensions, we need the unit scale and the actual dimensions
- To find the actual dimensions, we need the unit scale and the scale dimensions.
Write this necessary information down in your notebooks.
Now let’s apply this information with a few examples.
Example
What is the scale length of the object if the unit scale is 2 inches : 4 feet and the actual dimensions of the object is 20 feet?
First, let’s make sure that we have been given all of the necessary information. First, the unit scale has been given.
2 inches : 4 feet
So for every four feet of the building, we are going to have 2 inches of our model.
The actual length of the building has also been given. It is 20 feet. We can figure this out using a proportion.
\begin{align*}\frac{scale \ dimension}{actual \ dimension} = \frac{2 \ in}{4 \ ft} = \frac{x}{20 \ ft}\end{align*}
Notice that we used a different form of the ratio to solve this proportion. Now we can solve it. Four times five equals twenty, so we can do this to the top measurement. Two times five is equal to 10.
The scale dimension for length would be 10 inches.
Example
Using a unit scale of 1 inch : 8 feet, what is the actual dimension of an object with a scale dimension for length of 5 inches?
First, let’s make a note of the unit scale.
1 inch = 8 feet
We have been given the scale dimension and not the actual dimension. We are going to need to solve for the actual dimension. Let’s write a proportion to do this.
\begin{align*}\frac{scale \ dimension}{actual \ dimension} = \frac{1 \ in}{8 \ ft} = \frac{5 \ in}{x}\end{align*}
We can see that one times five is equal to five. We can do this to the actual dimension as well. Eight times five is equal to forty.
The actual length of the building is 40 feet.
III. Express Scale Factor as the Ratio of Scale Dimensions to Actual Dimensions, without Regard to Units
Once you know what information you need and what information you are looking for, you can work on figuring out the actual dimensions of different items. If we know the scale factor, or the relationship between the measurement and the actual size, we can figure out the missing measurements by using proportions. Notice that we don’t have to know the units to do this.
For example, let’s say that the scale is 1 : 2.
We can use this information to determine the scale factor. The scale factor is the relationship between the scale dimension and the measurement comparison between the scale measurement of the model and the actual length.
In this case, it is \begin{align*}\frac{1}{2}\end{align*}.
Example
What is the scale factor if 3 inches is equal to 12 feet?
We can write a ratio to show the scale factor.
\begin{align*}\frac{3}{12} = \frac{1}{4}\end{align*}
The scale factor is 1 : 4. It is expressed in simplest form.
Now let’s look at applying this information further.
If the scale dimension is 4, then we can figure out the actual dimension. Here is a proportion to show these two ratios.
\begin{align*}1 : 2 = 4 : x\end{align*}
Let’s use fraction form of the ratios to make this clearer.
\begin{align*}\frac{1}{2} = \frac{4}{x}\end{align*}
See the units aren’t necessary for figuring out the missing part of the proportion. We can simply use what we have learned to find the actual dimension.
1 times 4 = 4
2 times 4 = 8
\begin{align*}\frac{1}{2} = \frac{4}{8}\end{align*}
This is the answer.
Let’s look at another one.
Example
Find the missing actual dimension if the scale factor is 2 : 3 and the scale measurement is 6.
First, we can set up a proportion.
\begin{align*}2 : 3 = 6 : x\end{align*}
Now we can use fraction form to make it easier to solve this proportion.
\begin{align*}\frac{2}{3} = \frac{6}{x}\end{align*}
\begin{align*}2 \times 3 = 6\end{align*}
\begin{align*}3 \times 3 = 9\end{align*}
\begin{align*}\frac{2}{3} = \frac{6}{9}\end{align*}
This is the answer.
Notice that while units such as inches, feet, etc are helpful, they aren’t necessary for figuring out a missing dimension as long as you know the scale factor.
IV. Use Scale Factor to Find Actual Dimensions or Scale Dimensions, Given Necessary Information
Now we can look at applying scale factor to our work when we do know the units. To use scale factor to find actual dimensions or scale dimensions, we will need to know a few things.
Necessary Information:
- Scale Factor
- One other dimension either the actual or the scale dimension must be given
So, if we have three parts of the proportion, we can solve for the last missing part. Let’s look at an example.
Example
The plans for a flower garden show that it is 6 inches wide on the plan. If the scale for the flower garden is 1 : 12, what is the actual width of the flower garden?
To work on this problem, we first need to write two ratios that form a proportion. We have the scale factor and we have the scale measurement. We are missing the actual measurement. Let’s figure out the actual measurement of the garden.
\begin{align*}1 : 12 = 6 : x\end{align*}
Now we have two ratios that form a proportion. Let’s write them both in fraction form so that we can work easily in solving for the missing measurement.
\begin{align*}\frac{1}{3} = \frac{12}{x}\end{align*}
Now we can cross multiply or solve it by using equal ratios.
\begin{align*}1 \times 12 = 12\end{align*}
\begin{align*}3 \times 12 = 36\end{align*}
The measurement of the garden is 36 inches, which is the same as three feet.
Example
A driveway has a length of 24 feet. If the scale is 2 inches : 4 feet, what is the scale factor? How many inches will be drawn to represent the driveway?
Notice that there are two parts to this problem. First, we have to identify the scale factor.
\begin{align*}\frac{2}{4} = \frac{1}{2}\end{align*}
The scale factor is 1 : 2.
Next, we need to figure out how many inches will be drawn to represent the driveway. To do this, we write a proportion.
\begin{align*}\frac{2}{4} = \frac{x}{24}\end{align*}
We can cross multiply and divide or use equal ratios to solve this. Let’s use equal ratios. We work with the denominators.
\begin{align*}4 \times 6 = 24\end{align*}
\begin{align*}2 \times 6 = 12\end{align*}
The driveway will be represented by 12 inches or 1 foot.
Now let’s go back to the problem in the introduction and work on solving it with what we have learned.
Real-Life Example Completed
A Model of Everest
Here is the original problem once again. First, reread it. Then write a ratio to represent the scale that Josh has decided to use for the model. After that, figure out how many feet one inch will represent. Then use that scale to figure out the actual dimensions of Josh’s model using proportions. Notice that there are three parts to your answer.
Josh is so excited about Mount Everest that he has decided to create a model of the mountain for his geography class. Mr. Watkins approved the project and so Josh is putting together all of the information. During lunch, instead of talking with his friends, he took out his notebook and began figuring out how to measure the mountain. His friend Sasha spotted him and came over.
“Hi Josh, what are you working on?” she asked.
“Well, I am going to build a model of Mount Everest for my geography project. I am reading this really cool book about it. So I’m trying to figure out what scale to use for the model,” he explained grinning from ear to ear.
“What do you mean “scale”?”
“If you are going to build a model, you have to use a scale. You shrink down the real dimensions so that you can actually build a model that everyone will be able to see, think about it, Everest is 29,035 feet high, you can’t build that using the real dimensions,” Josh said.
“Oh I get it, so you use \begin{align*}1^{\prime\prime}\end{align*} for 1 foot.”
“That’s the right idea, but \begin{align*}1^{\prime\prime}\end{align*} is definitely too big. If I used \begin{align*}1^{\prime\prime}\end{align*} equals 1 foot for the scale, I would have a model that is 29,035 feet high. I need to use a smaller unit of measurement,” Josh said.
“What about \begin{align*}\frac{1}{4}\end{align*}?” Sasha suggested.
“That’s possible, but it still will probably be too big,” he stopped talking to scribble some numbers down in his notebook. “I think that \begin{align*}\frac{1}{8}^{\prime\prime} = 2000 \ feet\end{align*} will be perfect.”
Sasha looked puzzled.
Now figure out the three parts of the answer. Notice that there are three parts to your answer.
Solution to Real – Life Example
First, let’s write a ratio to show the scale that Josh has selected.
\begin{align*}\frac{1}{8}^{\prime\prime} = 2000 \ feet\end{align*}
\begin{align*}\frac{\frac{1}{8}^{\prime\prime}}{2000 \ ft}\end{align*}
Next, we write a ratio to show how many feet are in \begin{align*}1^{\prime\prime}\end{align*}. We can use a proportion to complete this task.
\begin{align*}\frac{\frac{1}{8}^{\prime\prime}}{2000 \ ft} = \frac{1^{\prime\prime}}{x}\end{align*}
If we cross multiply and divide, we can see that \begin{align*}1^{\prime\prime} = 16,000 \ feet\end{align*}.
Now we can use this information to figure out the dimensions of the model. Once again, use a proportion.
\begin{align*}\frac{1^{\prime\prime}}{12,000 \ ft} = \frac{x}{29,035 \ ft}\end{align*}
We cross multiply and divide.
\begin{align*}12,000x &= 29,035\\ x &= 1.8 \ feet\end{align*}
Josh’s model will be 1.8 feet tall, a workable size for a model.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Ratio
- a comparison between two quantities. Ratios can be shown using a colon, a fraction bar or by using the word “to”.
- Unit Scale
- the scale of measurement used to represent actual dimensions in a model or drawing. The scale includes units of measurement such as inches, feet, meters.
- Scale Dimension
- the measurement used to represent actual dimensions in a drawing or on a map.
- Actual Dimension
- the real – life dimension of the object or building.
- Scale Factor
- the ratio of scale to actual dimension written in simplest form.
Time to Practice
Directions: Find the scale dimension given the scale and the actual dimension using proportions. Write a proportion and an answer for each problem. There are two answers for each problem.
- Scale is \begin{align*}1^{\prime\prime} = 2 \ ft\end{align*}, the actual dimension is 18 feet
- Scale is \begin{align*}1^{\prime\prime} = 5 \ feet\end{align*}, the actual dimension is 20 feet
- Scale is \begin{align*}\frac{1}{2}^{\prime\prime} = 2 \ feet\end{align*}, actual dimension is 10 feet
- Scale is \begin{align*}1^{\prime\prime} = 12 \ feet\end{align*}, actual dimension is 72 feet
- Scale is \begin{align*}3^{\prime\prime} = 4 \ feet\end{align*}, actual dimension is 16 feet
Directions: Using a scale of 1 to 2, figure out the actual dimensions given each scale.
- 4 to _____
- 6 to _____
- 9 to _____
- 12 to _____
- 14 to _____
Directions: Using a scale of 3 to 4, figure out the actual dimensions given each scale.
- 6 to _____
- 9 to _____
- 12 to _____
- 18 to _____
- 36 to _____
Directions: Figure out each scale factor.
- \begin{align*}\frac{2 \ inches}{8 \ feet}\end{align*}
- \begin{align*}\frac{3 \ inches}{12 \ feet}\end{align*}
- \begin{align*}\frac{6 \ inches}{24 \ feet}\end{align*}
- \begin{align*}\frac{11 \ inches}{33 \ feet}\end{align*}
- \begin{align*}\frac{16 \ inches}{32 \ feet}\end{align*}
Directions: Solve each problem.
- A rectangle has a width of 2 inches. A similar rectangle has a width of 9 inches. What scale factor could be used to convert the larger rectangle to the smaller rectangle?
- A drawing of a man is 4 inches high. The actual man is 64 inches tall. What is the scale factor for the drawing?
- A map has a scale of 1 inch = 4 feet. What is the scale factor of the map?
- A drawing of a box has dimensions that are 2 inches, 3 inches, and 5 inches. The dimensions of the actual box will be \begin{align*}3\frac{1}{4}\end{align*} times the dimensions in the drawing. What are the dimensions of the actual box?
- A room has a length of 10 feet. Hadley is drawing a scale drawing of the room, using the scale factor \begin{align*}\frac{1}{50}\end{align*}. How long will the room be in Hadley’s drawing?
- The distance from Anna’s room to the kitchen is 15 meters. Anna is making a diagram of her house using the scale factor of \begin{align*}\frac{1}{150}\end{align*}. What will be the distance on the diagram from Anna’s room to the kitchen?
- On a map of Cameron’s town, his house is 9 inches from his school. If the scale of the map is \begin{align*}\frac{1}{400}\end{align*}, what is the actual distance in feet from Cameron’s house to his school?