Sebastian, Abe, and Rajeesh have taken up geocaching. The boys have use of a GPS and can see the general location of the cache on a map. If the scale is 1" = 80 meters, and the map shows the cache as being 2 inches from the boys' current location, how far do they have to walk to reach it?

In this concept, you will learn about scale distances.

### Scale Distances or Dimensions

A **ratio** compares two quantities. Ratios can be written as fractions, with a colon or with the word “to.”

A **proportion** is created when two ratios are found to be equivalent or equal.

is a proportion.

A **unit rate** is a comparison of two measurements, one of which has a value of 1.

55 miles per hour, \begin{align*}\frac{55}{1}\frac{miles}{hour}\end{align*}is a unit rate.

Somewhat like a unit rate, a unit scale is a ratio that compares two measurements.

A **unit scale** is a ratio that compares the dimensions of an actual object to the dimensions of a scale drawing or model that represents the actual object. Neither value in a unit scale has to equal 1.

The unit scale on a map may read 1 inch = 100 feet.

** **The ratio would be written \begin{align*}\frac{1 \ inch} {100 \ ft}\end{align*}.

To represent a line 500 feet long, the unit scale would be used to draw a line 5 inches long. A line 8 inches long would represent an actual line of 800 feet.

Unit scales and proportions can be used to calculate actual distances from maps, drawings, or models. Actual distances can be represented on maps, drawings, or models by using unit scales.

### Examples

#### Example 1

Earlier, you were given a problem about the three boys who went geocaching.

The scale on their GPS unit is 2.5 cm = 80 meters, and the map shows the cache as being 5 cm from the boys' current location, how far do they have to walk to reach it?

First, write the unit scale as a ratio

Next, keeping the units consistent, write a ratio that compares the scale distance to the actual distance from the cache.

Next, set this ratio equal to the unit scale to form a proportion.

Then, cross multiply.

The answer is d = 160 meters. The cache is 160 meters away.

#### Example 2

Rano has a rectangular backyard. Its actual dimensions are 50 feet by 30 feet. Rano wants to make a scale drawing of his backyard and has chose a scale of \begin{align*}\frac{1}{2} \ in. = 5 \ ft\end{align*}.

Find the dimensions of the backyard for the scale drawing.

First, write the unit scale as a ratio.

**\begin{align*}\frac{1}{2} \ in. = 5 \ ft\end{align*}**

\begin{align*}\frac{inches}{feet} = \frac{0.5}{5}\end{align*}

Next, keeping the units consistent, write ratios that compare the scale distance to the actual distance for both the length and width of the backyard.

** \begin{align*}Length=\frac{scale}{actual} \ in \ \frac{inches}{feet}=\frac{1}{50}\end{align*}**

** \begin{align*}Width=\frac{scale}{actual} \ in \ \frac{inches}{feet}=\frac{w}{30}\end{align*}**

Then, set each of these ratios equal to the unit scale to form two proportions, one for length and one for width.

\begin{align*}Length=\frac{l}{50} = \frac{0.5}{5}\end{align*}

\begin{align*}Length=\frac{w}{30}= \frac{0.5}{5}\end{align*}

Next, solve each proportion by cross multiplication.

\begin{align*}Length\\ 5l&=0.5\times 50\\ 5l&=25\\ l&=5 \ inches\end{align*} \begin{align*}Width\\ 5w&=0.5\times 30\\ 5w&=15\\ w&=3 \ inches\end{align*}

The answer is

.Rano's scale drawing of his backyard should be 5 inches long and 3 inches wide.

#### Example 3

A rectangular fish pond is 10 feet by 20 feet. Using the scale \begin{align*}\frac{1}{2}\end{align*}in = 5 ft , determine the dimensions for a scale model to be used in a miniature display.

First, write the unit scale as a ratio. \begin{align*}\frac{1}{2}\end{align*} in = 5 ft or \begin{align*}\frac{1}{2}\end{align*} inches per 5 feet.

\begin{align*}\frac{inches}{feet}=\frac{0.5}{5}\end{align*}

Next, keeping the units consistent, write ratios that compare the scale distance to the actual distance for both the length and width of the pond.

\begin{align*}Length = \frac{scale}{actual} in \frac{inches}{feet}=\frac{1}{20}\end{align*}

Then, set each of these ratios equal to the unit scale to form two proportions, one for length and one for width.

\begin{align*}Length = \frac{l}{20}=\frac{0.5}{5}\end{align*}

\begin{align*}Width=\frac{w}{10}=\frac{0.5}{5}\end{align*}

Next, solve each proportion by cross multiplication.

\begin{align*}Length\\ 5l &= 0.5×20\\ 5l &= 10\\ l &= 2 \ inches\end{align*}

The answer is *l* = 2 inches, and *w* = 1 inch. The miniature fish pond should be 2 inches by 1 inch.

#### Example 4

A drawing has a scale of 1" = 10 ft, find the actual dimensions of a building that measures 10 inches by 14 inches on the drawing.

First, write the unit scale as a ratio

Next, keeping the units consistent, write ratios that compare the scale distance to the actual distance for both the length and width of the building.

Then, set each of these ratios equal to the unit scale to form two proportions, one for length and one for width.

\begin{align*}Length = \frac{14}{l}=\frac{1}{10}\\ Width = \frac{10}{w}=\frac{1}{10}\end{align*}

Next, solve each proportion by cross multiplication.

\begin{align*}Length\\ l &= 10×14\\ l &= 140 \ feet\\ Width\\ w &= 10×10\\ w &= 100 \ feet\\ \end{align*}

The answer is *l* = 140 feet, and *w* = 100 feet. The actual dimensions of the building are 140 ft by 100 feet.

#### Example 5

Below is a map of Camp Skyview.

This morning, Mia walked directly from her bunk to the dining hall for breakfast. Then she walked directly to the Art & Craft Center. How many meters did she walk altogether?

First, add together the distances from the map that show how far Mia walked

4 cm + 2.25 cm = 6.25 cm

Next, write the unit scale as a ratio

Then, keeping the units consistent, write a ratio that compares the scale distance to the actual distance that Mia walked.\begin{align*}\frac{scale}{actual} in \frac{cm}{m}=\frac{6.25}{d}\end{align*}

Next, set this ratio equal to the unit scale to form a proportion.

Then, cross multiply.

\begin{align*}0.5d &= 6.25 \times 14\\ d &= 175 \ meters \end{align*}

The answer is d = 175 meters. Mia walked a total of 175 meters.

### Review

Solve each problem using scale and measurement.

- Haley made a scale model of her new school building. The scale she used for her model was 1 inch = 6 feet. The actual height of her school building is 30 feet. What was the height of the school building in her scale model?
- If the width of Haley’s school is 120 feet, what would be the width in the scale model?
- If the length of Haley’s school is 180 feet, what would be the length in the scale model?
- Eddie drew a map of Main Street in his hometown. The scale he used for his map was 1 centimeter = 8 meters. The actual distance between the post office and City Hall, both of which are on Main Street, is 56 meters. What is the distance between those two places on Eddie's map?
- If the distance from the post office to the library is twice the distance from the post office to the City Hall, what is the distance on Eddie’s map?
- If the distance from the library to the school is three times the distance as from the post office to the City Hall, what is the distance on Eddie’s map?
- Asharah built a model of a car. The actual length of the car is 12 feet. The scale of her model is \begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}. What is the length of her model car?
- Kenya built a model of the same car. His scale of the model is \begin{align*}\frac{1}{2}'' = 1 \ foot\end{align*}. What is the length of his model?
- Jonah observed a spider that was too small to draw at its actual size. So he made a scale drawing, using the scale 0.5 centimeter = 4 millimeters. The actual length of the spider's body, not including its legs, was 16 millimeters. What is the length of the spider's body, not including its legs, in Jonah's drawing?
- If Jonah made a drawing that is half the size of this one, what would length of the spider’s body be in the new drawing?
- Alyssa made a scale drawing of her rectangular classroom. She used the scale \begin{align*}\frac{1}{2} \ inch = 4 \ feet\end{align*}. Her actual classroom has dimensions of 32 feet by 28 feet. What are the dimensions of her classroom in the scale drawing?
- Below is a scale drawing of a circular fountain. In the scale drawing, the diameter of the fountain measures 3 centimeters. What is the actual diameter of the fountain?

- On a map, Brandon measured the straight-line distance between Los Angeles, California and San Francisco, California to be 2 inches. The scale on the map shows that \begin{align*}\frac{1}{4} \ inch = 43 \ miles\end{align*}. What is the actual straight-line distance between Los Angeles and San Francisco?
- A butterfly that Adriana observed was too small to draw at its actual size. So, she made this scale drawing.

In the drawing, the wingspan of the butterfly measures 4.5 centimeters. What was the actual wingspan of the butterfly Adriana observed?

- Jeremy made this scale model of Taipei 101, one of the tallest buildings in the world. The scale height of his model is \begin{align*}2\frac{1}{2}\end{align*} inches. Find the actual height of Taipei 101.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.11.