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In real-world situations, quantities are accompanied by appropriate units so that the situations can be easily understood and interpreted. Before analyzing situations involving too large or too small quantities, their models are created by keeping the ratio between corresponding lengths constant. In this lesson, this ratio will be named and used to analyze real-life situations.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

## Scaling the Solar System

In the model of the Solar System below, the initial labeled distance corresponds to miles. Zoom in or out to see how the labeled distance is affected.
Think about the following questions:
• Is it possible to make a model of the Solar System using the actual sizes of planets and distances between planets?
• Is it possible to draw a model of the Solar System on a sheet of paper? If so, how many miles should be represented by inch?
Discussion

## Different Presentations in Modeling

When modeling a real-life situation, it is sometimes impossible to represent an object or scenario using the original dimensions. In these situations, it is more convenient to work with manageable units while still being able to maintain the same properties as the original. This can be done by making a scale drawing.

Concept

## Scale Drawing

A scale drawing is a two-dimensional drawing that is similar to an actual object or place. In a scale drawing, the ratio of any length on the drawing to the actual length is always the same.
Possible examples of scale drawing are floor plans, blueprints, and maps.

In the case of the original real-life situation involving a three-dimensional object, making a model is more useful than a drawing. The idea behind a model is the same as a scale drawing, but the model has three dimensions instead of two.

Concept

## Scale Model

A scale model is a three-dimensional model that is similar to a three-dimensional object. The ratio of a linear measurement of a model to the corresponding linear measurement of the actual object is always the same.
Here is an example scale model of a building.
In any case, it is important to define appropriate quantities to represent the distances of the original object or situation. This relationship between the new quantities used and the actual distances is known as the scale.
Discussion

## Scale

The scale of a model or drawing is the ratio between any length on the model or drawing and its corresponding length on the actual object or place.

Suppose a drawing has a scale of This means that inch on the drawing represents feet on the actual object. Apart from the colon notation, a scale can be expressed using an equals sign or as a fraction, as it is a ratio.

Denoting a Scale
Ratio
Equals Sign
Fraction

When a scale is written without specifying the units, it is understood that both numbers have the same units of measure. For example, a scale of means that the actual object is twice the size of the model. A scale of means that the actual object is half the size of the model.

External credits: Derek Quinn
Knowing the scale is helpful in different scenarios, like when reading a map. The scale allows the reader to compute actual distances.
Example

## Using the Scale to Find Actual Dimensions

The Sweden Solar System is the world's largest model of the planetary system. In this model, the Globe Arena in Stockholm represents the Sun, while the planets align to the north. Click on the dots below, representing the planets, to show the dimensions, location, and distance to the Globe Arena.
The scale used to make this model is million.
a What is the actual diameter of the Earth? Give the answer in kilometers.
b What is the actual distance from Pluto to the Sun?

### Hint

a Note that the scale does not specify the units used. This means that both numbers share the same units of length.
b To find the actual dimension, multiply both numbers in the scale by the matching dimension on the model.

### Solution

a Since the scale does not specify the units, it means that both numbers have the same units. Then, to find the actual diameter of the Earth, multiply both numbers by the diameter of the Earth in the model, centimeters.
This shows that centimeters on the model corresponds to centimeters in real life. This means that the actual diameter of the Earth is centimeters. A more appropriate unit to represent this is kilometers, so next, use a conversion factor to convert the centimeters to kilometers.
Simplify
b Similarly, to find the actual distance from Pluto to the Sun, multiply both numbers in the scale by
In conclusion, the distance from Pluto to the Sun is billion kilometers.
When the scale is not given, it can be found using a dimension on the model and the corresponding dimension on the actual object.
Example

## Finding the Scale of a 3D Model

Tom made a model of the Empire State Building for a school project. The model has a height of centimeters.
The actual height of the Empire State Building is meters, including its antenna.
a According to the scale Tom used, how many meters correspond to centimeter on the model?
b What is the actual width of the Empire State Building in meters?

### Hint

a The scale compares a dimension on the model to the corresponding dimension on the original object.
b Multiply each side of the scale by the width of the model.

### Solution

a By definition, the scale of a model compares a dimension on the model to the corresponding dimension on the actual object. In this case, compare the height of the model and the actual height of the Empire State Building in order to find the scale.
According to this scale, centimeters correspond to meters. The scale can be simplified by dividing both numbers by
In the scale used by Tom, centimeter corresponds to meters.
b Although the scale was found using the height, the same scale is used for the width as well. Therefore, the actual width can be calculated by multiplying each side of the scale by the width of the model, which is centimeters. This time the scale in the form of an equation will be used.
Therefore, the Empire State Building is meters wide.
A scale can also be seen as a rate. For example, consider inch : miles. This rate can then be written as a ratio.
In a rate, however, the numerator and denominator do not have to represent units of distance. For example, when the numerator represents a population and the denominator represents an area, this unit of measurement is called population density.
Example

## Calculating Population Density

The United States of America has an area of square kilometers. In mid the total population was people.

External credits: @rocketpixel
Find the population density in people per square mile. Round the answer to the nearest integer. Then, based on the units, write a few sentences to describe what population density measures.

In the United States of America, the population density is people per square mile. The population density measures the number of people per unit of area.

### Hint

People per square mile can be written as This implies that the population density is expressed as a ratio. Remember that square kilometer is equal to square mile.

### Solution

According to the problem, the population density needs to be given using the unit of people per square mile. This means that the population density is found by dividing the number of people by the area of a certain region.
Before calculating the quotient, note that the area given is in square kilometers. Therefore, a unit conversion from square kilometers to square miles should be performed. To do this, use the fact that square kilometer is equal to about square miles.
Simplify
The next step is to divide the population by the area, which gives the population density. Here, the result is rounded to the nearest integer.
Now that the population density is known, write the answer in a sentence form.

The population density of the United States of America is approximately per

This statement assumes that the total population is spread out evenly across the United States.

The population density measures the number of people per unit of area.
Example

## Practicing Converting a Scale

Mark wants to paint all of the walls of his bedroom except for the wall that contains the door. Each rectangular wall is meters high. The paint he will use is sold in liter cans — the price per liter is

After painting for a few minutes, Mark noticed that square meters can be covered with one liter of paint.

a What is the total area, in square meters, that Mark has to paint?
b What is the minimum number of liters of paint Mark needs?
c How many cans should Mark buy?
d How many dollars could Mark save if the paint was sold in liter cans?

### Hint

a Use the formula for the area of a rectangle to calculate the area of each wall.
b How many liters of paint are needed to cover one square meter of wall?
c Compare the number of liters needed to cover all three walls to the amount of paint in one can.
d Calculate the difference between the amount of money that Mark paid and the cost of liters of paint.

### Solution

a Start by writing the dimensions of the walls to be painted. Note that two walls have the same dimensions. This means that they also have the same area.
Dimensions
Wall
Wall
Wall
The area of a rectangular wall is found by multiplying the length by the height.
Remember that Walls and are the same size and have the same area. The area of Wall can be calculated similarly.
Now, to find the total area Mark has to paint, find the sum of the areas of all three walls.
b Saying that square meters can be covered with a liter of paint is equivalent to saying that of a liter of paint can cover square meter.
Multiplying by the total area gives the number of liters that Mark needs to paint the three walls.
c Mark needs liters of paint, but each can of paint contains liters. This means that Mark will actually have to buy cans to have enough paint to cover all three walls.
d The cost of the cans can be calculated by multiplying the number of liters in both cans by the price per liter. Each can contains liters, so in total there are liters of paint.
If the paint was sold in liter cans, Mark would need to buy cans. In this case, he would pay The amount of money that Mark could have saved can be found by calculating the difference between the price of liters of paint and liters of paint.
Note that using the appropriate units is critical when making decisions based on comparing two or more quantities.
Pop Quiz

## Practicing Finding or Using the Scale

On the applet, the model and the actual object are shown. Using the given information, find the scale or the size of either the model or the actual object.

• When entering the scale, write it in the form of a fraction with the numerator of
• When entering the size of a model or an actual objects, round to the closest integer.
Closure

## Calculating the Scale to Choose Better Phone Plan

Tadeo wants to buy a cheap phone plan for calling his friends and family. He asked Ali and Ramsha how much they pay per call.

• Ali said he paid for his last call.
• Ramsha said she paid for her last call.

Initially, Tadeo decided to take Ramsha's plan since she paid less. Later, he realized that this information is not helpful since he does not know the duration of each of the calls made by Ali and Ramsha.

Price per call is not an appropriate unit.

Therefore, Tadeo decided to ask Ali and Ramsha how long each phone call lasted.

• Ali said his call lasted minutes.
• Ramsha said her call lasted minutes.
Since the duration of the calls is different, Tadeo became confused and made the following diagram to think about the situation.
After this, Tadeo realized that dividing each call's cost by its duration will give him the price per minute, which is an appropriate unit to compare the plans.
Person Scale of the Plan
Ali per minute
Ramsha per minute
Consequently, Tadeo decided to take Ali's plan because the price per minute is cheaper. What are other possible situations where calculating the scale can help to make best decision?