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## Convert between actual and scale measures

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Read and Interpret Maps Involving Distance and Area

“Look at this!” Josh exclaimed to his friend Carlo while they were in the computer lab. Carlo rolled his chair over to Josh’s computer to see what he was looking at.

“What?”

“This is a picture of Everest from space. The space shuttle Endeavor took it on October 10, 1994. It was a clear day too, so it should be pretty accurate. They wanted to measure the area of the mountain,” Josh said smiling.

“You and Mount Everest, but it is pretty cool,” Carlo said looking at the picture.

“I’m going to make a drawing of this,” Josh said taking out a piece of paper to make some notes.

In looking at the website, Josh discovered that the space shuttle Endeavor figured out that the length of Mount Everest from space is 43 miles and the width of Everest from space is 24 miles long. Josh wrote down the measurement \begin{align*}\frac{1}{4}^{\prime\prime} = 1 \ mile\end{align*}.

If Josh uses this measurement, what will the dimensions of his drawing be? Will it fit on \begin{align*}11^{\prime\prime} \times 14^{\prime\prime}\end{align*} paper? What is the actual area of Everest according to the space shuttle?

### Guidance

A map is another type of scale drawing of a region. Maps can be very detailed or very simple, showing only points of interest and distances. You can read a map just like any other scale drawing—by using the scale.

On the map below, the straight-line distance between San Francisco and San Diego is 3 inches. What is the actual straight-line distance between San Francisco and San Diego?

Set up a proportion. Write the scale as a ratio.

\begin{align*}\frac{0.5 \ inch}{75 \ miles}\end{align*}

Now write the second ratio, making sure it follows the form of the first ratio, inches over miles.

\begin{align*}\frac{0.5 \ inch}{75 \ miles} = \frac{3 \ inches}{x \ miles}\end{align*}

Now cross-multiply and solve for \begin{align*}x\end{align*}.

The straight-line distance between San Francisco and San Diego is 450 miles.

Note: The straight-line distance is also known as “as the crow flies.” If you were actually traveling from San Francisco to San Diego, it would be farther than 450 miles, since you would need to drive on highways that are not a straight line.

We can also use a scale to find the area of a space or region. First, we need to figure out the length and width then we can complete any necessary calculations.

Sometimes, we will have two different distances or areas that we are working to compare. When this happens, we can use proportions to compare the differences and similarities. Take a look.

Marta has a square with a side length of 4 inches. She has a similar square with dimensions that are twice the first square. How does the area of the larger square compare to the area of the smaller square?

First, find the dimensions of the larger square. The problem states that the dimensions are twice the first square. We can use this information to figure out the scale factor, and this means they are scaled up by a factor of 2. The side length of the larger square is \begin{align*}4 \ inches \times 2 = 8 \ inches\end{align*}.

Next find the area of both squares and compare.

Area of smaller square:

Area of larger square:

Now compare the two areas. You want to know how the area of the larger square compares to the area of the smaller square. Write a ratio comparing the two areas.

\begin{align*}\frac{64 \ inches^2}{16 \ inches^2} = 4\end{align*}

The area of the larger square is 4 times larger than the area of the smaller square.

This leads to a rule when comparing areas of similar figures.

Write this rule down in your notebook.

If 1" = 2000 miles, find each actual distance.

#### Example A

\begin{align*}3\end{align*}"

Solution:  \begin{align*}6000\end{align*} miles

#### Example B

\begin{align*}\frac{1}{2}\end{align*}"

Solution:  \begin{align*}1000\end{align*} miles

#### Example C

\begin{align*}\frac{1}{4}\end{align*}"

Solution:  \begin{align*}500\end{align*} miles

Now let's go back to the dilemma from the beginning of the Concept.

First, let’s figure out the scale dimensions of the drawing. We will need to use the scale to calculate a length and a width.

The scale is \begin{align*}\frac{1}{4}^{\prime\prime} = 1 \ mile\end{align*}.

Next, let’s write a proportion for length. We know that the actual length = 43 miles.

Now we can use a proportion for width. The actual width is 24 miles.

The dimensions of Josh’s drawing will be \begin{align*}10.75^{\prime\prime} \times 6^{\prime\prime}\end{align*}

This drawing will fit on an \begin{align*}11 \times 14^{\prime\prime}\end{align*} piece of paper.

Finally, we can figure out the area of Everest according to the picture.

### Vocabulary

Ratio
a way of comparing two quantities using a colon, fraction form or by using the word “to.”
Proportion
showing two equal ratios. Two equal ratios form a proportion.
Similar Figures
figures that are the same shape but different sizes.
Scale Drawing
a drawing that is done with a scale so that specific small units of measure represent larger units of measure.

### Guided Practice

Here is one for you to try on your own.

If the scale is \begin{align*}\frac{1}{2}\end{align*} inches = 100 miles, how many inches is 500 miles?

Solution

Set up a proportion and solve.

\begin{align*}\frac{.5}{100} = \frac{x}{500}\end{align*}

Cross-multiply and solve.

### Practice

Directions: Using the scale \begin{align*}1^{\prime\prime} = 5.5 \ miles\end{align*}, figure out the number of inches needed to map each number of miles. Use proportions to figure out your answers.

1. 16.5 miles
2. 11 miles
3. 27.5 miles
4. 8.25 miles
5. 33 miles
6. 60.5 miles
7. 13.75 miles

Directions: Using the scale \begin{align*}\frac{1}{2}^{\prime\prime} = 100 \ miles\end{align*}, figure out the number of actual miles represented by each scale measurement.

1. \begin{align*}1^{\prime\prime}\end{align*}
2. \begin{align*}2^{\prime\prime}\end{align*}
3. \begin{align*}3^{\prime\prime}\end{align*}
4. \begin{align*}\frac{1}{4}^{\prime\prime}\end{align*}
5. \begin{align*}\frac{3}{4}^{\prime\prime}\end{align*}
6. \begin{align*}1 \frac{1}{2}^{\prime\prime}\end{align*}
7. \begin{align*}2 \frac{1}{2}^{\prime\prime}\end{align*}
8. \begin{align*}5 \frac{1}{2}^{\prime\prime}\end{align*}
9. \begin{align*}7^{\prime\prime}\end{align*}

### Vocabulary Language: English

Proportion

Proportion

A proportion is an equation that shows two equivalent ratios.
Scale Drawing

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

Similar

Two figures are similar if they have the same shape, but not necessarily the same size.