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Proportions to Find Dimensions

Use proportions to find scale measurements or actual measurements.

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Proportions to Find Dimensions
License: CC BY-NC 3.0

Sarah is at a miniature exhibit. There is a model of a skyscraper. Underneath the model was a ratio. It said:

\begin{align*}1 \text{ inch} = 30 \text{ feet}.\end{align*}1 inch=30 feet.

If the model is 38 inches tall, how Sarah use this information to find the actual height of the skyscraper?

In this concept, you will learn how proportions are used to find actual and scale dimensions using a scale ratio.

Using Proportions to Find Dimensions

A scale is a ratio that shows the relationship between the representation of an object and the real measurement of an object. Toy makers use scale ratios to make models of cars and airplanes that are proportional to the real thing. Architects use scale drawings to plan the design of buildings.

A scale drawing is a drawing that is used to represent and object that is too large to be drawn in its actual dimensions. If a drawing is to scale, you can use proportions to determine the actual dimensions of the scale drawing.

Let’s look at a scale drawing of a sculpture. In the drawing, the sculpture is 4 inches tall. The scale is 1 inch : 4 feet. It scale can be written as

\begin{align*}\frac{1 \text{ in}}{4 \text{ ft}}\end{align*}1 in4 ft

(Note: Inch is abbreviated “in.” or with the double prime mark, ″. Foot/feet is abbreviated “ft” or with the single prime mark, ′.)

A drawing with this scale tells you that 1 inch on paper is equal to 4 feet in real life. If the drawing is 4 inches tall, use the scale to find the actual height of the sculpture.

First, create a proportion using equivalent ratios. Remember to write the corresponding units in the numerator and in the denominator. The actual height of the sculpture is represented by the variable \begin{align*}x\end{align*}x.

\begin{align*}\frac{1 \text{ in.}}{4 \text{ ft.}} = \frac{4 \text{ in.}}{x \text{ ft.}}\end{align*}1 in.4 ft.=4 in.x ft.

Then, cross multiply and simplify the equation to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} x & = & 4(4)\\ x & = & 16 \end{array}\end{align*}xx==4(4)16

The sculpture will be 16 feet tall.

Let’s make a scale drawing. Use the scale \begin{align*}1^{\prime\prime} = 2^{\prime}\end{align*}1=2. Draw a room that is \begin{align*}8^{\prime} \times 12^{\prime}\end{align*}8×12.

First, write a proportion to find the measurement of the width.

\begin{align*}\frac{1^{\prime\prime}}{2^{\prime}}=\frac{x^{\prime\prime}}{8^{\prime}}\end{align*}12=x8

Next, cross multiply and simplify the equation to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} 2x & = & 1(8)\\ 2x & = & 8\\ x & = & 4 \end{array}\end{align*}2x2xx===1(8)84

The drawing will be 4 inches wide.

Then, write a proportion to find the measurement of the length.

\begin{align*}\frac{1^{\prime\prime}}{2^{\prime}}=\frac{x^{\prime\prime}}{12^{\prime}}\end{align*}12=x12

Finally, cross multiply and simply the equation to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} 2x & = & 12\\ x & = & 6 \end{array}\end{align*}2xx==126

The drawing will be 6 inches long

In the drawing room will be \begin{align*}4 \text{ inches} \times 6 \text{ inches}\end{align*}4 inches×6 inches. If one unit on the drawing is equal to one inch, here is the room.

License: CC BY-NC 3.0

Scale dimensions are also used to figure out the actual dimensions of something.

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

First, write a proportion to find the actual measurement of the flower bed.

\begin{align*}\frac{1 \text{ in.}}{5 \text{ ft.}} = \frac{6 \text{ in.}}{x \text{ ft.}}\end{align*}1 in.5 ft.=6 in.x ft.

Then, cross multiply and simplify the equation to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} x & = & 5(6)\\ x & = & 30 \end{array}\end{align*}xx==5(6)30

The actual flower bed is 30 feet wide.

Examples

Example 1

Earlier, you were given a problem about Sarah at the miniature exhibit.

The model is 38 inches tall and it uses the scale \begin{align*}1 \text{ inch}= 30 \text{ feet}\end{align*}1 inch=30 feet. To find the actual height, Sarah can use a proportion.

First, write a proportion.

\begin{align*}\frac{1 \text{ in.}}{30 \text{ ft.}} = \frac{38 \text{ in.}}{x \text{ ft.}}\end{align*}1 in.30 ft.=38 in.x ft.

Next, cross multiply simplify to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} x & = & 30(38)\\ x & = & 1140 \end{array}\end{align*}xx==30(38)1140

The actual height of the building is 1,140 feet.

Example 2

Solve the proportion: \begin{align*}\frac{7 \text{ in.}}{70 \text{ ft.}} = \frac{x \text{ in.}}{140 \text{ ft.}}\end{align*}7 in.70 ft.=x in.140 ft..

First, cross multiply and simplify to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} 70x & = & 7(140)\\ 70x & = & 980\\ x & = & 14 \end{array}\end{align*}70x70xx===7(140)98014

Or, use mental math to solve for \begin{align*}x\end{align*}x.

First, look at the relationship between the two given denominators. Think, “The second denominator is double the first denominator.”

Then, take the given numerator and multiply by 2. Think, “7 times 2 is 14.”

The answer is 14 inches.

Example 3

Solve the proportion: \begin{align*}\frac{1 \text{ in.}}{3 \text{ ft.}} = \frac{x \text{ in.}}{21 \text{ ft.}}\end{align*}1 in.3 ft.=x in.21 ft..

First, cross multiply and simplify to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} 3x & = & 21\\ x & = &7 \end{array}\end{align*}3xx==217

The answer is 7 inches.

Example 4

Solve the proportion: \begin{align*}\frac{3 \text{ in.}}{6 \text{ ft.}} = \frac{9 \text{ in.}}{x \text{ ft.}}\end{align*}3 in.6 ft.=9 in.x ft..

First, cross multiply and simplify to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} 3x & = & 6(9)\\ 3x & = & 54\\ x & = & 18 \end{array}\end{align*}3x3xx===6(9)5418

The answer is 18 feet.

Example 5

Find the proportion: \begin{align*}\frac{2 \text{ in.}}{10 \text{ ft.}} = \frac{x \text{ in.}}{120 \text{ ft.}}\end{align*}2 in.10 ft.=x in.120 ft..

First, cross multiply and simplify to find the value of \begin{align*}x\end{align*}x.

\begin{align*}\begin{array}{rcl} 10x & = & 2(120)\\ 10x & = & 240\\ x & = & 24 \end{array}\end{align*}10x10xx===2(120)24024

The answer is 24 inches.

Review

Find the actual dimension.

  1. The scale of the drawing shows that 1  inch = 5 feet. If the drawing shows the height of the building as 5 inches, how tall is the actual building?
  2. Given this scale, a drawing of a building is 7 inches how tall is the actual building?
  3. Given this scale, how tall is a building that has a drawing that is 15 inches?
  4. The scale of the drawing shows that 2 inches = 10 feet. If the drawing shows the height of the building as 8 inches, how tall is the actual building?
  5. The scale of the drawing shows that 1 inch = 3 feet. If the drawing shows the height of the tree as 9 inches, how tall is the tree?
  6. The scale of the drawing shows that 2 inches = 7 feet. If the drawing shows that the height of the tree is 6 inches, how tall is the tree?
  7. The scale of the drawing shows that 1 inch = 3 feet. If the drawing shows that the height of the tree house is 3 inches, how high is the actual tree house?

Find the scale dimension.

  1. The scale of the map shows that 1 inch = 50 miles. If the map shows that there is 5 inches between the two cities, what is the actual distance?
  2. The scale of the map shows that 2 inches = 100 km. If the map shows that there are 3 inches between the two cities, what is the actual distance between them?
  3. The scale of the map shows that 4 inches = 200 km. If the map shows that there are 5 inches between the two cities, what is the actual distance between them?
  4. The scale of the garden design shows that 2 inch = 3 feet. How big is the garden if the rectangular plot is 4″ × 6″?
  5. The scale of the room design shows that 1 inch = 2 feet. How big is the actual room if the design shows a square that is 5 inches wide?
  6. The scale of the room design shows 2 inches = 4 feet. How big is the actual room if the design shows a square that is 10 inches wide?
  7. Using this same scale, how big is the actual room if the design shows a square that is 15 inches wide?
  8. Using this same scale, how tall is a building if the drawing is 12 inches tall?

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.8. 

Resources

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Vocabulary

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.

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0

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