4.7: Use Scale Factor When Problem Solving
Have you ever applied scale factor to a real-world dilemma? Take a look at this one.
A driveway has a length of 24 feet. If the scale is 2 inches : 4 feet, what is the scale factor? In a diagram, how many inches would be drawn to represent the driveway?
Pay attention and you will know how to figure this out by the end of the Concept.
Guidance
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.
The scale that we use can help us with scale dimensions or actual dimensions. This scale is key in problem solving.
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*}.
Take a look at this situation where we can use scale factor.
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.
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.
Take a look at this one.
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.
Use the scale factor of \begin{align*}\frac{1}{4}\end{align*}":\begin{align*}4\end{align*}' to find the actual dimensions in each example.
Example A
\begin{align*}8\end{align*}"
Solution: \begin{align*}\frac{1}{2}\end{align*} inch
Example B
\begin{align*}12\end{align*}"
Solution: \begin{align*}\frac{3}{4}\end{align*} inch
Example C
\begin{align*}16\end{align*}"
Solution: \begin{align*}1\end{align*} inch
Now let's go back to the dilemma from the beginning of the Concept.
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.
Vocabulary
- 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.
Guided Practice
Here is one for you to try on your own.
Find the missing actual dimension if the scale factor is 2" : 3' and the scale measurement is 6".
Solution
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*}
The actual dimension is 9 feet.
Video Review
Practice
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*}
- \begin{align*}\frac{18 \ inches}{36 \ feet}\end{align*}
- \begin{align*}\frac{6 \ inches}{48 \ feet}\end{align*}
- \begin{align*}\frac{6 \ inches}{12 \ 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?
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Actual Dimension
The actual dimensions are the real–life measures of the object or building.Proportion
A proportion is an equation that shows two equivalent ratios.Scale Dimension
A scale dimension is the measurement used to represent actual dimensions in a drawing or on a map.Image Attributions
Here you'll use scale factor when problem solving.