2.8: Scale Distances or Dimensions

Difficulty Level: At Grade Created by: CK-12
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Have you ever tried to make a map of something real? To do this successfully, you will need to use a scale and actual measurements.

Now that Alex has figured out what he wants the garden to look like, he wants to make a drawing of the plot that is accurate. What does this mean? It means that Alex wants to use a scale to draw his design. When you use a scale, you choose a unit of measure to represent the real thing. For example, if you want to draw a picture of a ship that is 100 feet long, it doesn’t make sense to actually draw it 100 feet long. You have to choose a unit of measurement like an inch to help you. Alex decides to use a 1” = 1 ft scale, but he is having a difficult time. He has two pieces of paper to choose from that he wants to draw the design on. One is \begin{align*}8\frac{1}{2}'' \times 11''\end{align*} and the other is \begin{align*}14\frac{1}{2}'' \times 11''\end{align*}. He starts using a 1 inch scale and begins to measure the garden plot onto the \begin{align*}8\frac{1}{2}'' \times 11''\end{align*} sheet of paper. At that moment, Tania comes in from outside. She looks over Alex’s shoulder and says, “That will never fit on there. You are going to need a smaller scale or a larger sheet of paper.” Alex is puzzled. He starts to rethink his work. He wonders if he should use a \begin{align*}\frac{1}{2}''\end{align*} scale.

If he uses a 1” scale, what will the measurements be? Does he have a piece of paper that will work? If he uses a \begin{align*}\frac{1}{2}''\end{align*} scale, what will the measurements be? Does he have a piece of paper that will work?

Use this Concept to learn all about scale dimensions, then you will be able to answer these questions at the end of the Concept.

Guidance

In the last Concept, you learned how to figure out actual dimensions or distances when you had been given a scale.

Now we are going to look at figuring out the scale given the actual dimensions.

To do this, we work in reverse. To make a map, for instance, we need to “shrink” actual distances down to a smaller size that we can show on a piece of paper. Again, we use the scale. Instead of solving for the actual distance, we solve for the map distance.

Suppose we are making a map of some nearby towns. We know that Trawley City and Oakton are 350 kilometers apart. We are using a scale of 1 cm : 10 km. How far apart do we draw the dots representing Trawley City and Oakton on our map?

We use the scale to write ratios that make a proportion. Then we fill in the information we know. This time we know the actual distance between the two towns, so we put that in and solve for the map distance.

\begin{align*}\frac{1 \ cm}{10 \ km}=\frac{x \ cm}{350 \ km}\end{align*} Next we cross multiply to find the number of centimeters that we would need to draw on the map.

\begin{align*}1(350) &= 10x\\ 350 &= 10x\\ 35 &= x\end{align*}

Using our scale, to draw a distance of 350 km on our map, we need to put Trawley City 35 centimeters away from Oakton.

We can figure out the scale using a model and an actual object too.

Jesse wants to build a model of a building. The building is 100 feet tall. If Jesse wants to use a scale of 1” to 25 feet, how tall will his model be?

Let’s start by looking at our scale and writing a proportion to show the measurements that we know. \begin{align*}\frac{1''}{25 \ ft}=\frac{x}{100 \ ft}\end{align*}

To solve this proportion we cross multiply.

\begin{align*}1(100) &= 25(x)\\ 100 &= 25x\\ 4 &= x\end{align*} Jesse’s model will be 4 inches tall.

Our answer is \begin{align*}4''\end{align*}.

Now let's practice. Use the scale 1" = 100 miles.

Example A

The distance from Kara's home to the family summer house is 150 miles. How many inches is that on the map?

Solution: 1.5 inches

Example B

The distance from Kara's home to her Grandmother's home is 2000 miles. How many inches is that on the map?

Solution: 20 inches

Example C

If the distance from Mark's home to his Grandmother's is half of Kara's, how many inches is that on the map?

Solution: 10 inches

Here is the original problem once again.

Now that Alex has figured out what he wants the garden to look like, he wants to make a drawing of the plot that is accurate.

What does this mean?

It means that Alex wants to use a scale to draw his design. When you use a scale, you choose a unit of measurement to represent the real thing. For example, if you want to draw a picture of a ship that is 100 feet long, it doesn’t make sense to actually make a drawing 100 feet long. You have to choose a unit of measurement like an inch to help you.

Alex’s decides to use a scale of 1” = 1 ft., but he is having a difficult time.

He has two pieces of paper to choose from that he wants to draw the design on. One is \begin{align*}8\frac{1}{2}'' \times 11''\end{align*} and the other is \begin{align*}14 \frac{1}{2}'' \times 11''\end{align*}. He starts using a 1 inch scale and begins to measure the garden plot onto the \begin{align*}8\frac{1}{2}'' \times 11''\end{align*} sheet of paper.

At that moment, Tania comes in from outside. She looks over Alex’s shoulder and says, “That will never fit on there. You are going to need a smaller scale or a larger sheet of paper.” Alex is puzzled. He starts to rethink his work. He wonders if he should a \begin{align*}\frac{1}{2}''\end{align*} scale.

Keep in mind the measurements he figured out in the last Concept.

If he uses a 1” scale, what will the measurements be? Does he have a piece of paper that will work? If he uses a \begin{align*}\frac{1}{2}''\end{align*} scale, what will the measurements be? Does he have a piece of paper that will work?

First, let’s begin by underlining all of the important information in the problem. Next, let’s look at the dimensions given each scale, a 1” scale and a \begin{align*}\frac{1}{2}''\end{align*} scale. Let’s start with the 1" scale.

First, we start by figuring out the dimensions of the square. Here is our proportion.

\begin{align*}\frac{1''}{1 \ ft} &= \frac{x \ ft}{9 \ ft}\\ 9 &= x\end{align*} To draw the square on a piece of paper using this scale, the three matching sides would each be 9 inches. Next, we have the short side. It is one foot, so it would be 1” long on the paper. Now we can work with the rectangle. If the rectangle is 12 ft \begin{align*}\times\end{align*} 8” and every foot is measured with 1”, then the dimensions of the rectangle are 12” \begin{align*}\times\end{align*} 8”. You would think that this would fit on either piece of paper, but it won’t because remember that Alex decided to put the two garden plots next to each other. If one side of the square is 9” and the length of the rectangle is 12” that equals 21”. 21 inches will not fit on a piece of \begin{align*}8\frac{1}{2}'' \times 11''\end{align*} paper or \begin{align*}14\frac{1}{2}'' \times 11''\end{align*} paper.

Let’s see what happens if we use a \begin{align*}\frac{1}{2}'' = 1\end{align*} foot scale. We already figured out a lot of the dimensions here. We can use common sense and divide the measurements from the first example in half since \begin{align*}\frac{1}{2}''\end{align*} is half of 1”. The square would be 4.5” on each of the three matching sides. The short side of the square would be \begin{align*}\frac{1}{2}''\end{align*}. The length of the rectangle would be 6”. The width of the rectangle would be 4”. With the square and the rectangle side-by-side, the length of Alex's drawing would be 10.5". This will fit on either piece of paper.

Use your notebook to draw Alex’s garden design. Use a ruler and draw it to scale. The scale is \begin{align*}\frac{1}{2}'' = 1\end{align*} foot.

Vocabulary

Here are the vocabulary words used in this Concept.

Scale
a ratio that compares a small size to a larger actual size. One measurement represents another measurement in a scale.
Ratio
the comparison of two things
Proportion
a pair of equal ratios, we cross multiply to solve a proportion

Guided Practice

Here is one for you to try on your own.

Joaquin is going to build a model of a building that is 480 feet tall. If Joaquin decided to use a scale of \begin{align*}\frac{1}{2}'' = 1\end{align*} foot, what would the new height of the model be in inches?

How many feet tall will the model be? Would this scale work for a model?

To figure this out, we first have to look at the scale that Joaquin is using. If Joaquin had chosen 1" = 1 foot then the scale height of the model would be 480 feet. But Joaquin used one - half inch as his scale, so the model will be 240 inches tall.

That means that it will be 20 feet high. This is too big! Joaquin will need to use a smaller scale.

Video Review

Here are a few videos for review.

Other Videos

http://teachertube.com/viewVideo.php?video_id=79418&title=PSSA_Grade_7_Math_19_Map_Scale – You will need to register with this website. This is a video about solving a ratio and proportion problem.

Practice

Directions: Use the given scale to determine the scale measurement given the actual distance.

Given: Scale 2” = 150 miles

1. How many scale inches would 300 miles be?

2. How many scale inches would 450 miles be?

3. How many scale inches would 75 miles be?

4. How many scale inches would 600 miles be?

5. How many scale inches would 900 miles be?

Directions: Use the given scale to determine the scale measurement for the following dimensions.

Given: Scale 1” = 1 foot

6. What is the scale measurement for a room that is 8’ \begin{align*}\times\end{align*} 12’?

7. What is the scale measurement for a tree that is 1 yard high?

8. What is the scale measurement for a tower that is 36 feet high?

9. How many feet is that?

10. What is the scale measurement for a room that is \begin{align*}12' \times 16 \frac{1}{2}'\end{align*}?

Directions: Use what you have learned about scale and measurement to answer each of the following questions.

11. Joaquin is building the model of a tower. He is going to use a scale of 1” = 1 foot. How big will his tower be in inches if the actual tower if 480 feet tall?

12. How many feet high will the model be?

13. Is this a realistic scale for this model? Why or why not?

14. If Joaquin decided to use a scale that was \begin{align*}\frac{1}{4}''\end{align*} for every 1 foot, how many feet high would his model be?

15. What scale would Joaquin need to use if he wanted his model to be 5 feet tall?

16. How tall would the model be if Joaquin decided to use \begin{align*}\frac{1}{16}'' = 1\end{align*} foot?

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Proportion A proportion is an equation that shows two equivalent ratios.
Scale Scale is the relationship between the size of a drawing and the size of the real object.

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