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# 8.4: Proportions and Scale Drawings

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Supermarket Display

Jessica is in charge of creating a new project display at the supermarket. Her manager noticed her doodling in a notebook and was impressed with her artistic ability.

He called Jessica into his office and handed her a design on a sheet of paper.

“We want to create a special poster to focus on a key product each week. This is the sketch of the design. That product will be on sale and hopefully this new display will help people notice it and buy it,” Jessica’s manager explains.

The design on the page is 8” \begin{align*}\times\end{align*} 5”. It is a rectangle and the scale at the bottom of the page says 1” = 6”.

Jessica takes out some foamcore and an exacto knife. She knows that she needs the scale to figure out the exact measurements of the display. The problem is that she can’t remember how to do the math.

“If it were 1” to 1 ft,” she thinks, “Then the poster would be 8 ft \begin{align*}\times\end{align*} 5 ft, but that isn’t it. 6" is only 1/2 of 1 ft, so this scale is 1" to 1/2 ft.”

Jessica is puzzled. What are the dimensions for the display?

This lesson works on scale and proportions. You can use a scale to figure out the change in measurements from a picture to the actual thing.

Be ready so that you can help Jessica at the end of the lesson.

What You Will Learn

In this lesson you will learn how to do the following skills.

• Given scale ratios, find actual or scale dimensions using proportions.
• Given actual and scale dimensions, find scale ratios using proportions.
• Solve real-world problems involving scale drawings and maps using proportions.

Teaching Time

I. Given Scale Ratios, Find Actual or Scale Dimensions Using Proportions

In our last lesson we talked about how you can find proportions in everyday life. This lesson focuses on scale and scale drawings. This is a key place where we use proportions in everyday life.

What is a scale drawing?

A scale drawing is a drawing that is used to represent an object that is too large to be drawn in its actual dimensions. For example, if you had a very tall building, you couldn’t make a drawing of the building that is as large as the building itself. Think about how many sheets of paper you would need to draw a 25 foot tall building.

This is an example where a scale drawing would be very useful. We can use a scale to represent measurements and then draw the building in a size that makes more sense.

What is a scale?

When we talk about scale, we aren’t talking about the object that we use to weigh things.

The scale that we are talking about is a fraction that shows the relationship between the measurement in a drawing and the measurement of a real object.

Let’s look at an example.

\begin{align*}\frac{1\ inch}{4\ feet}\end{align*}

This scale says that we would use one inch to represent every four feet. The top number is the scale that we would use in the drawing. The bottom number represents the measurement of the actual building.

Let’s say that we wanted to draw a building that is sixteen feet tall using this scale. We could set up a proportion to solve for the number of inches that we would need to draw.

\begin{align*}\frac{1\ inch}{4\ feet} = \frac{x\ inches}{16\ feet}\end{align*}

Now we can use what we learned in our last lesson about cross-multiplying to solve proportions. This will help us to figure out the scale dimension of the building.

\begin{align*}16 = 4x\end{align*}

Our answer is four inches. The two ratios now form a proportion.

What about if we have more than one dimension?

Let’s say that we have a room that is 8’ \begin{align*}\times\end{align*} 12’ and we want to use a scale of 1” = 2 feet. How many inches long and wide would this drawing be?

First, let’s look at the width of the room. It is eight feet wide. We can set up a proportion using the scale and the actual width to figure out the measurement of the drawing.

\begin{align*}\frac{1\ in}{2\ ft} = \frac{x}{8\ ft}\end{align*}

Next, we solve the proportion.

\begin{align*}2x & = 8 \\ x & = 4\ inches\end{align*}

On the drawing, the width will be four inches.

Now we need to look at the length. The room is 12 feet long. We can set up a proportion using the scale and the actual length to figure out the length of the drawing.

\begin{align*}\frac{1\ in}{2\ ft} = \frac{x}{12\ ft}\end{align*}

Next, we solve the proportion by cross-multiplying.

\begin{align*}2x & = 12 \\ x & = 6\end{align*}

On the drawing, the length of the room will be six inches.

Next, we can do a scale drawing of this room. If one unit on the drawing is equal to one inch, here is our room.

We can also use scale dimensions to figure out the actual dimensions of something. We will use proportions to do this as well. Here is an example.

Example

The flower bed design shows that the width of the garden on the drawing is six inches. If the scale is 1” = 5 feet, how wide is the actual flower garden?

To solve this problem, we need to set up a proportion. Let’s start by writing the scale in the form of a ratio.

\begin{align*}\frac{1"}{5\ ft}\end{align*}

Next, we can write the actual dimensions that we know with a variable as our unknown and make this the second ratio in this proportion.

\begin{align*}\frac{1"}{5’} = \frac{6"}{x}\end{align*}

The drawing of our flower bed is six inches. We can solve the proportion for the actual dimensions of the flower bed by cross-multiplying.

\begin{align*}1x & = 30 \\ x & = 30\ ft\end{align*}

The actual flower bed is 30 feet wide.

Practice a few of these on your own. Solve each proportion for the scale measurement or the actual measurement.

1. \begin{align*}\frac{1"}{3\ ft} = \frac{x}{21\ ft}\end{align*}
2. \begin{align*}\frac{3\ in}{6\ ft} = \frac{9\ in}{x}\end{align*}
3. \begin{align*}\frac{2\ in}{10\ ft} = \frac{x}{120\ ft}\end{align*}

Check the accuracy of your work with a friend. Did you remember to label your work with the appropriate units?

II. Given Actual and Scale Dimensions, Find Scale Ratios Using Proportions

Now that you have figured out how to use proportions to figure out actual and scale dimensions, we can look at figuring out scale factors.

What is a scale factor?

A scale factor is another name for a scale ratio. When looking for a scale factor, you can look at the relationship between the scale measurement and the actual measurement to determine what scale was used. This scale is called the scale factor.

Let’s look at an example.

Example

A fence is actually 16 feet long. If the fence is drawn as four inches, what is the scale factor?

To figure this out, we need to write a ratio to compare the drawing of the fence to the actual measurement.

\begin{align*}\frac{4"}{16\ ft}\end{align*}

Now we want to figure out the scale factor. To do this, we simplify the ratio using the greatest common factor.

The greatest common factor of 4 and 16 is 4.

\begin{align*}4 \div 4 & = 1 \\ 16 \div 4 & = 4\end{align*}

The scale factor is \begin{align*}\frac{1"}{4\ ft}\end{align*}.

Use this information to simplify and find the following scale factors.

1. \begin{align*}\frac{5”}{25’}\end{align*}
2. \begin{align*}\frac{2”}{50’}\end{align*}
3. \begin{align*}\frac{12.5”}{25’}\end{align*}

Check your work with a partner. Did you figure out the scale factor of the lst problem?

## Real Life Example Completed

The Supermarket Display

Now it is time to use what you have learned to help Jessica. Here is the problem again. Begin by underlining all of the important information.

Jessica is in charge of creating a new project display at the supermarket. Her manager noticed her doodling in a notebook and was impressed with her artistic ability.

He called Jessica into his office and handed her a design on a sheet of paper.

“We want to create a special poster to focus on a key product each week. This is the sketch of the design. The key product will be on sale and hopefully this new display will help people notice it and buy it,” Jessica’s manager explains.

The design on the page is 8” \begin{align*}\times\end{align*} 5”. It is a rectangle and the scale at the bottom of the page says 1” = 6”.

Jessica takes out some foamcore and an exacto knife. She knows that she needs the scale to figure out the exact measurements of the display. The problem is that she can’t remember how to do the math.

“If it were 1” to 1 ft,” she thinks, “Then the poster would be 8 ft \begin{align*}\times\end{align*} 5 ft, but that isn’t it. 6" is only 1/2 of 1 ft, so this scale is 1" to 1/2 ft.”

Now that you have underlined the important information, write the scale factor for the poster.

Scale factor is 1” : 6”

Or \begin{align*}\frac{1}{6}\end{align*}

The drawing shows that the length of the rectangular sign is 8” and the width is 5”. If the one inch is equal to one half of a foot, then the length is 4 ft and width is 2.5 ft.

Jessica is amazed at how simple it actually was to figure that out once she knew how to use the scale factor. Her poster is 4 ft \begin{align*}\times\end{align*} 2.5 ft. She gets right to work on the poster and design!

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Proportion
two equal ratios
Scale Drawing
a drawing used when a real life object is too big to draw with its actual dimensions.
Scale
the relationship of the size of a drawing to the size of the real object
Scale Factor
the relationship between the measurement of the drawing and the measurement of the real object.

## Technology Integration

Other Videos:

http://www.teachertube.com/members/viewVideo.php?video_id=169091&title=Finding_dimensions_using_scale&vpkey – This is a blackboard video on how to use a small scale to determine dimensions. You'll need to register at the site to view this video.

http://www.teachertube.com/members/viewVideo.php?video_id=169086&title=Scale_drawings This is a basic blackboard video on how to determine measurements using a scale. You'll need to register at the site to view this video.

## Time to Practice

Directions: Find actual dimensions using proportions.

1. The scale of the drawing shows that 1” = 5 feet. If the drawing shows the height of the building as 5 inches, how tall is the actual building?

2. The scale of the drawing shows that 2” = 10 feet. If the drawing shows the height of the building as 8 inches, how tall is the actual building?

3. The scale of the drawing shows that 1” = 3 feet. If the drawing shows the height of the tree as 9 inches, how tall is the tree?

4. The scale of the drawing shows that 2” = 7 feet. If the drawing shows that the height of the tree is 6 inches, how tall is the tree?

5. The scale of the drawing shows that 1” = 3 feet. If the drawing shows that the height of the tree house is 3”, how high is the actual tree house?

Directions: Find the scale dimensions using proportions.

6. The scale of the map shows that 1” = 50 miles. If the map shows that there is 5” between the two cities, what is the actual distance?

7. The scale of the map shows that 2” = 100 km. If the map shows that there are 3” between the two cities, what is the actual distance between them?

8. The scale of the map shows that 4” = 200 km. If the map shows that there are 5 inches between the two cities, what is the actual distance between them?

9. The scale of the garden design shows that 2” = 3 feet. How big is the garden if the rectangular plot is 4” \begin{align*}\times\end{align*} 6”?

10. The scale of the room design shows that 1” = 2 feet. How big is the actual room if the design shows a square that is 5 inches wide?

Directions: Simplify each ratio to find the scale factor.

11. \begin{align*}\frac{4”}{6\ ft}\end{align*}

12. \begin{align*}\frac{12”}{24\ ft}\end{align*}

13. \begin{align*}\frac{6”}{18\ ft}\end{align*}

14. \begin{align*}\frac{9”}{27\ ft}\end{align*}

15. \begin{align*}\frac{4”}{16\ ft}\end{align*}

16. \begin{align*}\frac{5”}{30\ ft}\end{align*}

17. \begin{align*}\frac{3”}{30\ ft}\end{align*}

18. \begin{align*}\frac{3”}{60\ miles}\end{align*}

19. \begin{align*}\frac{4”}{100\ miles}\end{align*}

20. \begin{align*}\frac{5”}{1000\ km}\end{align*}

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