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# Distances or Dimensions Given Scale Measurements

## Identify actual dimensions given scale dimensions

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Practice Distances or Dimensions Given Scale Measurements

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Read and Interpret Scale Drawings and Floor Plans

The local community is planning to put a playground on a block of unused land. This is the drawing that they have created. You can see that the width of the playground is 7 inches wide in the drawing. If this is the case, what is the actual width of the playground?

In this concept, you will learn to interpret scale drawings and floor plans.

### Scale Drawing

A scale drawing or floor plan is a representation of an actual object or space drawn in two-dimensions. For a floor plan, you can imagine that you are directly above the building looking down. The lines represent the walls of the building, and the space in between the lines represents the floor.

In order to find the actual dimensions from a floor plan, you can set up and solve a proportion. The scale given in the drawing is the first ratio. The unknown length and the scale length is the second ratio.

Let’s look at an example.

The floor plan below shows several classrooms at Craig’s school. The length of Classroom 2 in the floor plan is 2 inches. What is the actual length, in feet, of Classroom 2?

First, set up a proportion. The scale in the drawing says that 12=3\begin{align*}{\frac{1}{2}}^{{\prime}{\prime}} = 3^{\prime}\end{align*}, therefore the proportion is:

0.5 inches3 feet\begin{align*}\frac{0.5 \text{ inches}}{3 \text{ feet}}\end{align*}

Next, write the second ratio. You know the scale length is 2 inches. The unknown length is x\begin{align*}x\end{align*}. Make sure that the second ratio follows the form of the first ratio: inches over feet.

0.5 inches3 feet=2 inchesx feet\begin{align*}\frac{0.5 \text{ inches}}{3 \text{ feet}}=\frac{2 \text{ inches}}{x \text{ feet}}\end{align*}Then, cross-multiply.

0.530.5x0.5x===2x2×36\begin{align*}\begin{array}{rcl} \frac{0.5}{3} &=& \frac{2}{x} \\ 0.5 x &=& 2 \times 3 \\ 0.5 x &=& 6 \end{array}\end{align*}Then, divide both sides by 0.5 to solve for x\begin{align*}x\end{align*}.

0.5x0.5x0.5x===660.512\begin{align*}\begin{array}{rcl} 0.5 x &=& 6\\ \frac{0.5 x}{0.5} &=& \frac{6}{0.5} \\ x &=& 12 \end{array}\end{align*}The answer is 12.

The actual length of the classroom is 12 feet.

### Examples

#### Example 1

Remember the playground to scale?

Consider the earlier drawing of the playground’s scale: 1=20\begin{align*}1^{{\prime}{\prime}}= 20^{\prime}\end{align*}.

First, set up a proportion. The scale in the drawing says that 1=20\begin{align*}1^{{\prime}{\prime}}= 20^{\prime}\end{align*}, therefore the proportion is:

1 inch20 feet\begin{align*}\frac{1 \text{ inch}}{20 \text{ feet}}\end{align*}

Next, write the second ratio. You know the scale length is 7\begin{align*}7^{{\prime}{\prime}}\end{align*} inches. The unknown length is x\begin{align*}x\end{align*}.

1 inch20 feet=7 inchesx feet\begin{align*}\frac{1 \text{ inch}}{20 \text{ feet}}=\frac{7 \text{ inches}}{x \text{ feet}}\end{align*}Then, cross-multiply to solve for x\begin{align*}x\end{align*}.

1201xx===7x20×7140\begin{align*}\begin{array}{rcl} \frac{1}{20}&=&\frac{7}{x} \\ 1x &=& 20 \times 7 \\ x &=& 140 \end{array}\end{align*}

The actual width of the playground is 140 feet.

#### Example 2

The image below shows the fountain in front of a hotel. The diameter of the fountain is 4 centimeters. What is the actual diameter of the fountain?

First, set up a proportion. The scale for the image says that 1 cm=0.5 m\begin{align*}1 \ cm = 0.5 \ m\end{align*}, therefore the proportion is:

1 cm0.5 m\begin{align*}\frac{1 \ cm}{0.5 \ m}\end{align*}

Next, write the second ratio. You know the scale length is 4 cm. The unknown length is x\begin{align*}x\end{align*}.

1 cm0.5 m=4 cmx m\begin{align*}\frac{1 \ cm}{0.5 \ m}=\frac{4 \ cm}{x \ m}\end{align*}Then, cross-multiply to solve for x\begin{align*}x\end{align*}.

10.51xx===4x0.5×42\begin{align*}\begin{array}{ rcl} \frac{1}{0.5} &=& \frac{4}{x} \\ 1x &=& 0.5\times4\\ x &=& 2 \end{array}\end{align*}The answer is 2.

The actual diameter of the fountain is 2 m.

#### Example 3

If the scale said 12=4 feet\begin{align*}\frac{1}{2^{\prime \prime}}= 4 \text{ feet}\end{align*}, what would be the inches drawn for a room 8×12\begin{align*}8^\prime \times 12^\prime\end{align*}?

First, set up a proportion. The scale in the drawing says that 12=4\begin{align*}\frac{1}{2}^{{\prime}{\prime}}= 4^{\prime}\end{align*}, therefore the proportion is:

0.5 inches4 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}\end{align*}

Next, write the second ratio. You know the scale length is 8 inches\begin{align*}8^\prime \text{ inches}\end{align*}. The unknown length is x\begin{align*}x\end{align*}.

0.5 inches4 feet=x inches8 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{8 \text{ feet}}\end{align*}

Then use the same ratio to find the other unknown length.

0.5 inches4 feet=x inches12 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}} = \frac{x \text{ inches}}{12 \text{ feet}}\end{align*}

Then, cross-multiply for both ratios.

0.544x4x===x80.54=x120.5×84x=0.5×124  4x=6\begin{align*}\begin{array}{rcl} \frac{0.5}{4} & = & \frac{x}{8} \qquad \quad \frac{0.5}{4}=\frac{x}{12} \\ 4x & = & 0.5 \times 8 \quad \quad 4x = 0.5 \times 12 \\ 4x & = & 4 \qquad \qquad \ \ 4x = 6 \end{array}\end{align*}

Then, divide both sides to solve for x\begin{align*}x\end{align*}.

4x4x4x===44x=6444x4=64 1 x=1.5\begin{align*}\begin{array}{rcl} 4x&=&4 \quad \quad \quad 4x = 6\\ \frac{4x}{4} &= &\frac{4}{4} \qquad \frac{4x}{4} =\frac{6}{4} \\ x &=& \ 1 \quad \qquad \ x = 1.5 \end{array}\end{align*}

The answers are 1 and 1.5.

The scale dimensions are 1×1.5\begin{align*}1 ^{{\prime}{\prime}} \times 1.5^{{\prime}{\prime}}\end{align*}.

#### Example 4

If the scale said 12=4 feet\begin{align*}\frac{1}{2^{{\prime}{\prime}}}=4 \text{ feet}\end{align*}, what would be the inches drawn for a room 20×28\begin{align*}20^{\prime} \times 28^{\prime}\end{align*}?

First, set up a proportion. The scale in the drawing says that  12=4\begin{align*}{\frac{1}{2}}^{{\prime}{\prime}} = 4^{\prime}\end{align*}, therefore the proportion is:

0.5 inches4 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}\end{align*}

Next, write the second ratio. You know the scale length is 8\begin{align*}8^{\prime}\end{align*} inches. The unknown length is x\begin{align*}x\end{align*}.

0.5 inches4 feet=x inches20 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{20 \text{ feet}}\end{align*}

Then use the same ratio to find the other unknown length.

0.5 inches4 feet=x inches28 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{28 \text{ feet}}\end{align*}

Then, cross-multiply for both ratios.

0.544x4x===x200.54=x280.5×20   4x=0.5×28104x=14\begin{align*}\begin{array}{rcl} \frac{0.5}{4}&=& \frac{x}{20} \qquad \qquad \frac{0.5}{4}=\frac{x}{28} \\ 4x &= & 0.5 \times 20 \qquad \ \ \ 4x = 0.5 \times 28 \\ 4x &=& 10 \qquad \quad \qquad 4x =14 \end{array}\end{align*}

Then, divide both sides to solve for x\begin{align*}x\end{align*}.

4x4x4x===104x=141044x4=1442.5    x=3.5\begin{align*}\begin{array}{rcl} 4x&=&10 \qquad \quad \quad 4x= 14 \\ \frac{4x}{4} &= & \frac{10}{4} \qquad \quad \frac{4x}{4} = \frac{14}{4} \\ x &=& 2.5 \qquad \quad \ \ \ \ x = 3.5 \end{array}\end{align*}

The answers are 2.5 and 3.5. Therefore the scale dimensions are 2.5×3.5\begin{align*}2.5^{{\prime}{\prime}} \times 3.5^{{\prime}{\prime}}\end{align*}.

#### Example 5

If the scale said 12=4 feet\begin{align*}\frac{1}{ 2^{\prime \prime}}= 4 \text{ feet}\end{align*}, what would be the inches drawn for a room 16×24\begin{align*}16^\prime \times 24^\prime\end{align*}?

First, set up a proportion. The scale in the drawing says that 12=4\begin{align*}\frac{1}{ 2}^{\prime \prime}= 4^\prime\end{align*}, therefore the proportion is:

0.5 inches4 feet\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}\end{align*}

Next, write the second ratio. You know the scale length is 8 inches\begin{align*}8^\prime \text{ inches}\end{align*}. The unknown length is \begin{align*}x\end{align*}.

\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{16 \text{ feet}}\end{align*}

Then use the same ratio to find the other unknown length.

\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{24 \text{ feet}}\end{align*}Then, cross-multiply for both ratios.

\begin{align*}\begin{array}{rcl} \frac{0.5}{4}&=&\frac{x}{16} \qquad \quad \quad \frac{0.5}{4}=\frac{x}{24} \\ 4x &= &0.5 \times 16 \qquad \ \ \ 4x = 0.5 \times 24 \\ 4x &=& 8 \qquad \qquad \quad \ 4x =12 \end{array}\end{align*}

Then, divide both sides to solve for \begin{align*}x\end{align*}.
\begin{align*}\begin{array}{rcl} 4x&=&8 \quad \quad \quad 4x =12 \\ \frac{4x}{4} &= & \frac{8}{4} \qquad \frac{4x}{4} = \frac{12}{4} \\ x &=& 2 \qquad \quad \ \ x = 3 \end{array}\end{align*}
The answers are 2 and 3.

The scale dimensions are \begin{align*}2^{{\prime}{\prime}} \times 3^{{\prime}{\prime}}\end{align*}.

### Review

This floor plan shows Bonnie’s house. Use it to answer the following questions.

1. The width of the bedroom on the floor plan measures 1.5 inches. What is the actual width of the bedroom?

2. The length of the kitchen on the floor plan measures 3 inches. What is the actual length of the kitchen?

3. The study measures 2 inches by 1.5 inches on the floor plan. What is the actual area of the study?

4. The study measures 2 inches by 1.5 inches on the floor plan. What is the actual perimeter of the study?

5. The study measures 2 inches by 1.5 inches on the floor plan. What are the actual dimensions of the study?

6. If the length of the bedroom is the same as the study, what are the actual dimensions of the bedroom?

7. What is the area of the bedroom?

8. What is the actual length of the outside of the house?

9. A square measures 10 inches on each side. What is its area?

10. If the scale is \begin{align*}1^ {{\prime}{\prime}}= 50 \ ft\end{align*}, what is the actual side length of the same square?

11. What is its area?

12. A classroom measures 21 feet by 15 feet. A second classroom has a similar shape but the dimensions are \begin{align*}\frac{1}{5}\end{align*} as long. How does the area of the second classroom compare to the area of the first classroom?

13. What are the dimensions of the second classroom?

14. Yuri’s backyard has an area of 1,000 square feet. The dimensions of Kyle’s backyard are all \begin{align*}\frac{1}{5}\end{align*} the size of Yuri’s. What is the area of Kyle’s backyard?

15. Mary’s yard is double the area of Kyle’s backyard. What is the area of Mary’s yard?

what would be the inches drawn for a room

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### Vocabulary Language: English

TermDefinition
Inches An inch is a customary unit of measurement, measured best by a ruler.
Length Length is a measurement of how long something is. Examples of customary units of length are inches, feet, yards and miles.
Proportion A proportion is an equation that shows two equivalent ratios.
Scale Drawing A scale drawing is a drawing that is done with a scale so that specific small units of measure represent larger units of measure.
Similar Two figures are similar if they have the same shape, but not necessarily the same size.
Two – Dimensional A figure drawn in two dimensions is only drawn using length and width.