5.7: Scale Factor
Introduction
The Reading Corner
Mrs. Henderson is thrilled with the progress that her students are making in the reading challenge. As a reward, she has decided to make a small reading corner in the back of her room. She has gotten a few bean bag chairs donated and one small couch. When the students arrived on Monday morning, they were thrilled.
“Wow! This is awesome!” they exclaimed as they took turns trying out the furniture.
“Next, we need a small rug,” Mrs. Henderson said.
On the board she drew a rectangle.
“This rug has a scale factor of \begin{align*}\frac{1}{10}\end{align*}
“Mrs. Henderson, this isn’t math,” Jessica said smiling.
“A little math never hurt anyone, let’s get started.”
While the students work on this problem, it is your turn to learn about scale factor.
What You Will Learn
In this lesson, you will learn how to demonstrate the following skills:
- Recognize scale factor as the ratio of scale dimensions to actual dimensions, without regard to units.
- Use scale factor to find dimensions of scale models, drawings or maps given actual dimensions.
- Use scale factor to find actual dimensions from scale models, drawings or maps.
- Solve real-world problems involving scale models, drawings and maps using scale factors and unit conversion.
Teaching Time
I. Recognize Scale Factor as the Ratio of Scale Dimensions to Actual Dimensions, Without Regard to Units
You already know that scale drawings and scale models allow us to represent objects accurately that would otherwise be too large or too small to represent at the correct size.
The relationship between the dimensions on a scale drawing or model and the actual dimensions of the object being represented is described by a ratio. One type of ratio that could be used is called a unit scale, in which units of measure are used to relate scale dimensions to actual dimensions. For example, a unit scale for a scale drawing might be: 1 centimeter = 2 meters.
Another ratio that could be used is called a scale factor. Like a unit scale, a scale factor relates the scale dimensions to actual dimensions however it does so without regard to specific units.
For example, a scale factor for a scale drawing might be \begin{align*}\frac{1}{200}\end{align*}
If you think about the unit scale that we just introduced, 1 centimeter to 2 meters, the scale factor would be 1 to 2.
II. Use Scale Factor to Find Dimensions of Scale Models, Drawings or Maps Given Actual Dimensions
Let’s look at how we can work with scale factors when given actual dimensions.
Suppose you wanted to make a scale model or a scale drawing. You would need to know the dimensions of the object you wanted to represent. Then, you could choose a scale factor for your model or drawing. You would then use that scale factor to create the drawing.
Since a scale factor disregards units of measure, you will need to convert units of length to solve problems involving scale drawings and models. Knowing the unit conversions in this table can help you convert units of length.
Customary Units of Length | Metric Units of Length |
---|---|
1 foot (ft) = 12 inches (in.) | 1 meter (m) = 100 centimeters (cm) = 100 millimeters (mm) |
1 yard (yd) = 3 feet (ft) | |
1 mile (mi) = 1760 yards (yd) = 5280 feet (ft) | 1 kilometer (km) = 1000 meters (m) |
Write these conversions down in your notebook and then continue. This way you can refer back to them as you work.
Example
A small airplane has a wingspan of 16 feet. Finn wants to make a scale model of the airplane. The scale factor for his model will be \begin{align*}\frac{1}{48}\end{align*}
The scale factor compares the dimensions of the scale model to the dimensions of the actual airplane.
\begin{align*}\frac{scale}{actual} = \frac{1}{48}\end{align*}
The actual wingspan of the airplane is 16 feet. Use \begin{align*}w\end{align*}
\begin{align*}\frac{scale}{actual} = \frac{w}{16}\end{align*}
Set up and solve a proportion to find \begin{align*}w\end{align*}
\begin{align*}\frac{1}{48} &= \frac{w}{16}\\
48 \cdot w &= 1 \cdot 16\\
48w &= 16\\
\frac{48w}{48} &= \frac{16}{48}\\
w &= \frac{16}{48} = \frac{16 \div 16}{48 \div 16} = \frac{1}{3}\end{align*}
The wingspan of the model airplane will be \begin{align*}\frac{1}{3}\end{align*} foot.
The question asks for the wingspan in inches. So, convert \begin{align*}\frac{1}{3}\end{align*} foot to inches.
We know that 1 foot = 12 inches. One way to find the number of inches in \begin{align*}\frac{1}{3}\end{align*} foot is to multiply \begin{align*}\frac{1}{3}\end{align*} by 12.
\begin{align*}\frac{1}{3} \ foot = \frac{1}{3} \times 12 = \frac{1}{3} \times \frac{12}{1} = \frac{12}{3} = 4 \ inches\end{align*}
The wingspan of the model airplane will be 4 inches, or \begin{align*}\frac{1}{3}\end{align*} foot.
Example
The longest side of a triangular flower bed is 5.5 meters long. Leah wants to make a scale drawing of the flower bed. The scale factor for her drawing will be \begin{align*}\frac{1}{200}\end{align*}. What will be the length of the longest side of the flower bed in her drawing?
The scale factor compares the dimensions of the scale drawing to the dimensions of the actual flower bed.
\begin{align*}\frac{scale}{actual} = \frac{1}{200}\end{align*}
The actual length of the flower bed is 5.5 meters. Use \begin{align*}l\end{align*} to represent the unknown length of the flower bed in the scale drawing.
\begin{align*}\frac{scale}{actual} = \frac{l}{5.5}\end{align*}
Set up and solve a proportion to find the scale length, \begin{align*}l\end{align*}.
\begin{align*}\frac{1}{200} &= \frac{l}{5.5}\\ 200 \cdot l &= 1 \cdot 5.5\\ 200 l &= 5.5\\ \frac{200 l}{200} &= \frac{5.5}{200}\\ l &= 0.0275\end{align*} If you were unsure of how to divide 5.5 by 200, here is how you could have done it:
\begin{align*}&5.5 = 5.5000, \ \text{so} && \overset{ \quad \ 0.0275}{200 \overline{ ) { \ 5.5000 \;}}}\\ &&& \quad \ \ \ \underline{-400\;\;}\\ &&& \qquad \ 1500\\ &&& \qquad \ \underline{-1400\;\;}\\ &&& \qquad \quad 1000\\ &&& \qquad \ \underline{-1000\;\;}\\ &&& \qquad \qquad \ \ 0 \end{align*}
Since the actual length was 5.5 meters, the length of the flower bed in the scale drawing is 0.0275 meters.
Convert that scale length to centimeters. Write a ratio for the unit conversion: 1 meter = 100 centimeters. Then write a second ratio to compare \begin{align*}x\end{align*}, the unknown scale length, in centimeters, to the known scale length in meters.
\begin{align*}\frac{meters}{centimeters} = \frac{1}{100} && \frac{meters}{centimeters} = \frac{0.0275}{x}\end{align*} Now, set up and solve a proportion to find the scale length in centimeters.
\begin{align*}\frac{1}{100}& = \frac{0.0275}{x}\\ 100 \cdot 0.0275 &= 1 \cdot x\\ 2.75 &= x\end{align*} The length of the longest side of the flower bed in the scale drawing will be 0.0275 meter or 2.75 centimeters.
5M. Lesson Exercises
Using a scale factor of \begin{align*}\frac{1}{6}\end{align*}, find the scale dimensions given the actual dimensions.
- 18 feet
- 24 inches
- 30 feet
Take a few minutes to check your answers with a friend.
III. Use Scale Factor to Find Actual Dimensions from Scale Models, Drawings or Maps
You can also use scale factor to find the actual dimensions of an object that is represented in a scale drawing or scale model.
Example
An ant that Alison observed was too small to draw at its actual size. So, Alison made the scale drawing shown below. The scale factor for the drawing is 5. Find the actual length of the ant Alison observed.
The scale factor compares the dimensions of the scale drawing to the dimensions of the actual ant. Since \begin{align*}5 = \frac{5}{1}\end{align*}, we can represent the ratio as, \begin{align*}\frac{5}{1}\end{align*}.
\begin{align*}\frac{scale}{actual} = \frac{5}{1}\end{align*}
The scale length of the ant in the drawing is 2.75 centimeters. Use \begin{align*}l\end{align*} to represent the unknown length of the actual ant.
\begin{align*}\frac{scale}{actual} = \frac{2.75}{l}\end{align*}
Set up and solve a proportion to find the actual length, \begin{align*}l\end{align*}.
\begin{align*}\frac{5}{1} &= \frac{2.75}{l}\\ 1 \cdot 2.75 &= 5 \cdot l\\ 2.75 &= 5l\\ \frac{2.75}{5} &= \frac{5l}{5}\\ 0.55 &= l\end{align*} Since the scale length you used was 2.75 centimeters, the length of the actual ant was 0.55 centimeter.
Convert that scale length to millimeters. Use the unit conversion, 1 centimeter = 10 millimeters, to write a ratio.
\begin{align*}\frac{centimeters}{millimeters} = \frac{1}{10}\end{align*}
Then write a second ratio to compare \begin{align*}x\end{align*}, the unknown scale length, in millimeters, to the known scale length in centimeters.
\begin{align*}\frac{centimeters}{millimeters} = \frac{0.55}{x}\end{align*}
Now, set up and solve a proportion to find the scale length in centimeters.
\begin{align*}\frac{1}{10} &= \frac{0.55}{x}\\ 10 \cdot 0.55 &= 1 \cdot x\\ 5.5 &= x\end{align*}
The actual length of the ant was 0.55 centimeter or 5.5 millimeters.
5N. Lesson Exercises
Use the scale 3 to 1 to figure out the actual dimensions given the following scale dimensions.
- 9 inches
- 12 mm
- 15 inches
Take a few minutes to check your work with a peer.
IV. Solve Real-World Problems Involving Scale Models, Drawings and Maps Using Scale Factors and Unit Conversion
We can use all of this information to solve more complex problems. These are the types of problems that you will often see in real-life.
Example
The scale drawing below shows a rectangular carpet. The scale factor for the drawing is \begin{align*}\frac{1}{24}\end{align*}. What is the area, in square feet, of the actual carpet?
You can find the area of a rectangle by multiplying the length by the width: \begin{align*}A = lw\end{align*}.
Before you can find the actual carpet's area, you must first find the length and width of the actual carpet.
The scale factor compares the dimensions of the scale drawing to the dimensions of the actual carpet.
\begin{align*}\frac{scale}{actual} = \frac{1}{24}\end{align*}
The scale length in the drawing is 4 inches and the scale width is 2 inches. Use \begin{align*}l\end{align*} to represent the unknown actual length and use \begin{align*}w\end{align*} to represent the unknown actual width.
\begin{align*}\frac{scale}{actual} = \frac{4}{l} \qquad \quad \frac{scale}{actual} = \frac{2}{w}\end{align*}
Set up and solve proportions to find the actual length, \begin{align*}l\end{align*}, and the actual width, \begin{align*}w\end{align*}.
\begin{align*}&\frac{1}{24} = \frac{4}{l} && \frac{1}{24} = \frac{2}{w}\\ &24 \cdot 4 = 1 \cdot l && 24 \cdot 2 = 1 \cdot w\\ &96 =l && 48 = w\end{align*} Since the scale dimensions you used were given in inches, the actual length is 96 inches and the actual width is 48 inches.
You need to find the area in square feet. So, convert each scale length to feet before you find the area. Use the unit conversion, 1 foot = 12 inches.
\begin{align*}\frac{feet}{inches} = \frac{1}{12}\end{align*}
Then write two more ratios. One should compare \begin{align*}x\end{align*}, the actual length, in feet, to the known scale length in inches. The other should compare \begin{align*}y\end{align*}, the actual width, in feet, to the known scale width in inches,
\begin{align*}\text{ratio for length} = \frac{feet}{inches} = \frac{x}{96} && \text{ratio for width:}\ \frac{feet}{inches} = \frac{y}{48}\end{align*}
Now, set up and solve proportions to convert the dimensions to feet.
\begin{align*}\frac{1}{12} &= \frac{x}{96} && \quad \frac{1}{12} = \frac{y}{48}\\ 12 \cdot x &= 1 \cdot 96 && 12 \cdot y = 1 \cdot 48\\ 12x &= 96 && \ \ 12y = 48\\ \frac{12x}{12} &= \frac{96}{12} && \ \ \frac{12y}{12} = \frac{48}{12}\\ x &= 8 && \quad \ \ x = 4\end{align*}
Now, calculate the area in square feet.
\begin{align*}A = lw = 8 \ ft \cdot 4 \ ft = 32 \ ft^2\end{align*}
The area of the carpet is 32 square feet.
You may wonder why we converted the dimensions from inches to feet before finding the area. You might wonder if we could have simply found the area, in square inches, and then converted the area to square feet. We could have, but if we did, we would have needed to be careful to convert from square inches to square feet.
The area in square inches is:
\begin{align*}A = lw = 96 \ in. \cdot 48 \ in. = 4608 \ in.^2\end{align*}
How do we convert from square inches to square feet?
We know that 1 foot = 12 inches and \begin{align*}1 \ ft^2 = 1 \ ft \cdot 1 \ ft\end{align*}. We can use that information to find out how many square inches are in 1 square foot:
\begin{align*}1 \ ft^2 = 1 \ ft \times 1 \ ft = 12 \ in. \times 12 \ in. = 144 \ in.^2\end{align*}
So, we could set up and solve a proportion to convert 4,608 square inches to square feet, using the ratio: 1 square foot = 144 square inches.
\begin{align*}\frac{1}{144} & = \frac{a}{4608}\\ 144 \cdot a &= 1 \cdot 4608\\ 144a &= 4608\\ \frac{144a}{144} &= \frac{4608}{144}\\ a &= 32\end{align*} We would get the same answer, 32 square feet, for the area of the carpet using this method. If you choose to solve problems this way, just be sure to remember that even though 1 foot = 12 inches, 1 square foot = 144 square inches, not 12 square inches.
Now let’s go back and apply what we have learned to our introductory problem.
Real Life Example Completed
The Reading Corner
Here is the original problem once again. Reread it and underline any important information.
Mrs. Henderson is thrilled with the progress that her students are making in the reading challenge. As a reward, she has decided to make a small reading corner in the back of her room. She has gotten a few bean bag chairs donated and one small couch. When the students arrived on Monday morning, they were thrilled.
“Wow! This is awesome!” they exclaimed as they took turns trying out the furniture.
“Next, we need a small rug,” Mrs. Henderson said.
On the board she drew a rectangle.
“This rug has a scale factor of \begin{align*}\frac{1}{10}\end{align*}. If the length of the rug is 6” in this drawing and the width is 3”, then what is the area of the rug?”
“Mrs. Henderson, this isn’t math,” Jessica said smiling.
“A little math never hurt anyone, let’s get started.”
First, we need to figure out the actual length of the rug and the actual width of the rug given the drawing and the scale factor. Here is our proportion for the length.
\begin{align*}\frac{1}{10} = \frac{6}{l}\end{align*}
Notice that we use “\begin{align*}l\end{align*}” for the missing length. We use cross products and find that the length is 60 inches.
Next, we find the width.
\begin{align*}\frac{1}{10} = \frac{3}{w}\end{align*}
We use cross products to find that the width is 30 inches.
Next, we want to find the area of the rug. We want to find it in square feet. First, we need to convert the length and width to feet from inches.
\begin{align*}60 \div 12 &= 5 \ feet\\ 30 \div 12 &= 2.5 \ feet\end{align*}
The area is found by multiplying length times width.
\begin{align*}5 \times 2.5 = 12.5\end{align*} square feet
This is the area of the rug.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Unit Scale
- this ratio compares the scale dimensions of something to the actual dimensions. Unit scale includes units in the ratio.
- Scale Factor
- this ratio compares scale and actual dimensions without the units. Usually the answer to a problem with scale factor will include some kind of unit conversion.
Technology Integration
Khan Academy Scale and Indirect Measurement
Time to Practice
Directions: Solve each problem. Keep in mind that several problems may have more than one part.
1. Calvin drew a map of his neighborhood. The scale factor he used for his map was \begin{align*}\frac{1}{800}\end{align*}. The actual distance between Calvin's house and his best friend Frank's house is 80 meters. What should be the distance, in centimeters, between those two places on his scale drawing?
2. If the distance from Calvin’s house to the park is 40 meters, what would be the distance in centimeters?
3. If the distance from Calvin’s house to the corner store is double the distance from his house to Frank’s, what would be the distance on the map in centimeters?
4. If the distance from Calvin’s house to his Grandmother’s was halfway between his and Frank’s, what would be the distance on the map in centimeters?
5. Madeline built a model of a boat. The actual length of the boat is 24 feet. The scale factor she used for the model was \begin{align*}\frac{1}{36}\end{align*}. What should be the length, in inches, of the model boat?
6. If the scale factor was \begin{align*}\frac{1}{72}\end{align*}, what would the length be in inches of the model boat?
7. A metal pipe is 2.5 meters long. Josh wants to make a scale drawing of the pipe. The scale factor for his drawing will be \begin{align*}\frac{1}{100}\end{align*}. What will be the length, in centimeters, of the metal pipe in his drawing?
8. Sydra observed a housefly that was too small to draw at its actual size. So she made a scale drawing, using 10 as a scale factor. The actual length of the housefly was 8 millimeters. What is the length of the housefly in Sydra's drawing in millimeters?
9. What is the length of the housefly in Sydra's drawing in centimeters?
10. Luis made a scale model of the doghouse he is going to build. The scale factor he used for this model was \begin{align*}\frac{1}{24}\end{align*}. He wants the actual height of the doghouse to be 6 feet. What should be the height of the doghouse in his scale model?
11. Jean-Marc used a scale factor of 5 to make this scale drawing of a moth.
In the drawing, the wingspan of the moth measures 3 centimeters. What was the actual wingspan, in millimeters, of the moth Jean-Marc observed?
12. Below is a scale drawing of a swimming pool. In the scale drawing, the diameter of the pool measures \begin{align*}1 \frac{1}{2}\end{align*} inches. The scale factor for the drawing is \begin{align*}\frac{1}{72}\end{align*}. What is the actual diameter of the pool in feet?
13. Barbara made a scale model of the Washington Monument. The scale factor for her model is \begin{align*}\frac{1}{1332}\end{align*}. The height of her model is 5 inches. What is the actual height, in feet, of the Washington Monument?
14. Below is a map of a city park. This map was created using a scale factor of \begin{align*}\frac{1}{300}\end{align*}.
a. On the map, the distance between the sandbox and the swings is 2.5 centimeters. What is the actual distance between the sandbox and the swings in meters?
b. On the map, the distance between the sandbox and the jungle gym is 1.7 centimeters. What is the actual distance between the sandbox and the jungle gym in meters?
15. The scale drawing below shows the floor of Julian's bedroom.
The scale factor for the drawing is \begin{align*}\frac{1}{200}\end{align*}.
a. What are the dimensions, in meters, of the actual floor?
b. What is the area, in square meters, of the actual floor?