### Scale and Indirect Measurement Applications

One place where ratios are often used is in making maps. The **scale** of a map describes the relationship between distances on a map and the corresponding distances on the earth's surface. These measurements are expressed as a fraction or a ratio.

So far we have only written ratios as fractions, but outside of mathematics books, ratios are often written as two numbers separated by a colon (:). For example, instead of \begin{align*}\frac{2}{3}\end{align*}, we would write 2:3.

Ratios written this way are used to express the relationship between a map and the area it represents. For example, a map with a scale of 1:1000 would be a map where one unit of measurement (such as a centimeter) on the map would represent 1000 of the same unit (1000 centimeters, or 10 meters) in real life.

#### Real-World Application: Maps

Anne is visiting a friend in London, and is using the map below to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where 1 cm on the map represents 1 km in real life. Using a ruler, she measures the distance on the map as 8.8 cm. How far is the real distance from the start of her journey to the end?

The scale is the ratio of distance on the map to the corresponding distance in real life. Written as a fraction, it is \begin{align*}\frac{1}{100000}\end{align*}. We can also write an equivalent ratio for the distance Anne measures on the map and the distance in real life that she is trying to find: \begin{align*}\frac{8.8}{x}\end{align*}. Setting these two ratios equal gives us our proportion: \begin{align*}\frac{1}{100000} = \frac{8.8}{x}\end{align*}. Then we can cross multiply to get \begin{align*}x = 880000\end{align*}.

That’s how many *centimeters* it is from Fleet Street to Borough Road; now we need to convert to kilometers. There are 100000 cm in a km, so we have to divide our answer by 100000.

\begin{align*}\frac{880000}{100000} = 8.8.\end{align*}

**The distance from Fleet Street to Borough Road is 8.8 km.**

In this problem, we could have just used our intuition: the \begin{align*}1 \ cm = 1 \ km\end{align*} scale tells us that any number of cm on the map is equal to the same number of km in real life. But not all maps have a scale this simple. You’ll usually need to refer to the map scale to convert between measurements on the map and distances in real life!

#### Real-World Application: Determining Scale

Antonio is drawing a map of his school for a project in math. He has drawn out the following map of the school buildings and the surrounding area

He is trying to determine the scale of his figure. He knows that the distance from the point marked A on the baseball diamond to the point marked B on the athletics track is 183 meters. Use the dimensions marked on the drawing to determine the scale of his map.

We know that the real-life distance is 183 m, and the scale is the ratio \begin{align*}\frac{\text{distance on map}}{\text{distance in real life}}\end{align*}.

To find the distance on the map, we use Pythagoras’ Theorem: \begin{align*}a^2 + b^2 = c^2\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are the horizontal and vertical lengths and \begin{align*}c\end{align*} is the diagonal between points \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

\begin{align*} 8^2 + 14^2 &= c^2\\ 64 + 196 &= c^2\\ 260 &= c^2\\ \sqrt{260} &= c\\ 16.12 & \approx c\end{align*}

So the distance on the map is about 16.12 cm. The distance in real life is 183 m, which is 18300 cm. Now we can divide:

\begin{align*}\text{Scale} = \frac{16.12}{18300} \approx \frac{1}{1135.23}\end{align*}

**The scale of Antonio’s map is approximately 1:1100.**

Another visual use of ratio and proportion is in **scale drawings**. Scale drawings (often called **plans**) are used extensively by architects. The equations governing scale are the same as for maps; the scale of a drawing is the ratio \begin{align*}\frac{\text{distance on diagram}}{\text{distance in real life}}\end{align*}.

#### Real-World Application: Drawing to Scale

Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 ft long. He would like his drawing to be at a 1:500 scale. How many inches long does his sheet of paper need to be?

We can reason intuitively that since the scale is 1:500, the paper must be \begin{align*}\frac{883}{500} = 1.766 \ feet\end{align*} long. Converting to inches means the length is \begin{align*}12(1.766) = 21.192 \ inches\end{align*}.

**Oscar’s paper should be at least 22 inches long.**

### Example

#### Example 1

The Rose Bowl stadium in Pasadena, California measures 880 feet from north to south and 695 feet from east to west. A scale diagram of the stadium is to be made. If 1 inch represents 100 feet, what would be the dimensions of the stadium drawn on a sheet of paper? Will it fit on a standard \begin{align*}8.5 \times 11\end{align*} inch sheet of paper?

Instead of using a proportion, we can simply use the following equation: (distance on diagram) = (distance in real life) \begin{align*}\times\end{align*} (scale). (We can derive this from the fact that \begin{align*}\text{scale} = \frac{\text{distance on diagram}}{\text{distance in real life}}\end{align*}.)

Plugging in, we get

\begin{align*}\text{height on paper} = 880 \ feet \times \frac{1 \ inch}{100 \ feet} = 8.8 \ inches\end{align*}

\begin{align*}\text{width on paper} = 695 \ feet \times \frac{1 \ inch}{100 \ feet} = 6.95 \ inches\end{align*}

**The scale diagram will be \begin{align*}8.8 \ in \times 6.95 \ in\end{align*}. It will fit on a standard sheet of paper.**

### Review

- A restaurant serves 100 people per day and takes in $908. If the restaurant were to serve 250 people per day, how much money would it take in?
- The highest mountain in Canada is Mount Yukon. It is \begin{align*}\frac{298}{67}\end{align*} the size of Ben Nevis, the highest peak in Scotland. Mount Elbert in Colorado is the highest peak in the Rocky Mountains. Mount Elbert is \begin{align*}\frac{220}{67}\end{align*} the height of Ben Nevis and \begin{align*}\frac{11}{12}\end{align*} the size of Mont Blanc in France. Mont Blanc is 4800 meters high. How high is Mount Yukon?
- At a large high school it is estimated that two out of every three students have a cell phone, and one in five of all students have a cell phone that is one year old or less. Out of the students who own a cell phone, what proportion owns a phone that is more than one year old?

For 4-6, suppose a map of Ratio City has a scale of 1:1,000,000, where 1 centimeter on the map represents 10 kilometers in real life. Use that scale to determine the real-life distances in kilometers.

- The distance on the map between city hall and high school is 1.2cm.
- The distance on the map between city hall and the main library is 0.6cm.
- The distance on the map between the main library and the high school is 0.4cm.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.12.