What is a Ratio?
A ratio is a comparison of quantities having the same unit. A ratio helps us indicate how big or how small a quantity is, when compared to another quantity.
For example, “a : b” compares the value of a with b.
What is a Proportion?
Proportion is a concept that is closely interlinked with ratios and fractions. A proportion is an equation that states that two ratios or two fractions are equivalent. That is, two ratios are said to be proportional when they are equal.
For example, if \(\frac{5}{6} = \frac{10}{12}\). This indicates that 5 : 6 and 10 : 12 are equivalent ratios.
What is a Scale drawing?
Do architects draw sketches that are as big as the structures they have in mind? This is usually not the case. Similarly, can we draw the cells of the human body according to their actual size? It’s not always possible to draw real-life objects as per their actual dimensions. That’s where scale drawings can come in handy.
Scale drawings make it easy to make 2-D representations of real objects on paper. Scale drawings may depict a reduced or an enlarged size of a real-life object, depending on the size of the object. The interesting thing about scale drawings is that every part of the drawing gets magnified or reduced with the same factor called scale. Maps are examples of scale drawings that we use in everyday life.
Similarly, a scale model is a proportional 3-D model of a real object.
Scale and scale factor
As we have already learned, the dimensions of scale drawings and models are proportional to the dimensions of real objects. The scale gives us the ratio which compares the dimensions of the drawings or models with the size of real objects.
A ratio is usually expressed as a : b. That means it can also be expressed as \(1:\frac{b}{a}\). When writing the scale of a drawing or a model, we usually keep the antecedent, or the first unit, as 1.
When you change the size of the shape by making it larger or smaller, the degree to which you change the size is measured by its scale factor. A linear scale factor is simply the size of the enlargement or reduction of the shape.
For example, if the scale factor is \(3\), this means that the new shape is thrice the size of the original shape. Similarly, the scale factor of \(\frac{1}{5}\) tells us the new shape is one-fifth the size of the original. When the units are the same, the scale can be written without units. If y = Kx, where y and x represent dimensions on two different scale drawings, \(K\) is the scale factor of the two drawings.
Solved Examples
Example 1: The actual wingspan of an aircraft is 54 feet. Find the scale of the models if the wingspan of the small aircraft model is 1 foot and the wingspan of the bigger model is two feet.
Solution:
Wingspan of actual aircraft = 54 feet
The wingspan of the smaller model = 1 foot
The scale factor of the smaller aircraft model = 1 : 54
The wingspan of the smaller aircraft model is twice as long as the wingspan of the bigger aircraft model.
The wingspan of the larger model = 2 feet
Since the wingspan of the larger model is twice as long as the wingspan of the smaller model, the scale factor of the larger model
= 2 : 54
We can simplify this ratio by dividing both sides by 2.
= \(\frac{2}{2}:\frac{54}{2}\) Divide both sides of the ratio by 2
= \(1:\frac{54}{2}\) Simplify
= 1 : 27
Example 2: A model of the Empire State Building is 14.54 inches tall. The height of the actual building is 1454 feet tall. What does one inch in the model represent in real life? What is the scale factor of the model?
Solution:
The height of the model is 14.54 inches and the height of the real tower is 1454 feet.
14.54 inch : 1454 feet
To find what 1 inch represents, i.e. to find the scale, we need to divide both the sides by 14.54
\(\frac{14.54~\text{inch}}{14.54~\text{inch}}:\frac{1454~\text{feet}}{14.45~\text{inch}}\)
⇒ 1:100\(\frac{\text{feet}}{\text{inch}}\)
Hence, an inch of the model represents 100 feet in real life.
To find the scale factor, the units have to be the same. That is, we need to convert feet into inches.
1 feet = 12 inches
⇒ \(1:100\frac{\text{feet}}{\text{inch}}=1:100\times 12 = 1 : 1200\frac{\text{inch}}{\text{inch}}\)
⇒ 1 : 1200
So, the scale factor of the model is \(\frac{1}{1200} \).
Example 3: The scale of a map of Texas is 1 cm : 100 miles. If the distance between Dallas and Austin on the map is 1.84 cm, find the actual distance between the two cities.
Solution:
The distance between Austin and Dallas on the map is 1.84 cm.
The scale of the map is 1 cm : 100 miles.
Use the scale 1 cm : 100 miles and the ratio 1.84 cm : d miles to write and solve a proportion.
Therefore,\(\frac{1~\text{cm}}{100~miles} = \frac{1.84~\text{cm}}{d~\text{miles}}\), where “\(d\)” is the actual distance between Dallas and Austin.
\(d = 1.84\times 100\) Cross product
= 184 miles Multiply
The actual distance between Austin and Dallas = 184 miles
Example 4: Howard made a scale model of the solar system whose scale is 1 inch : 3 million miles. If the distance between the center of the Earth and the center of the Sun on the scale model is 31 inches, find the actual distance between Earth and the Sun.
Solution:
The distance between Earth and the Sun on the scale model = 31 inches
Scale factor of the model = 1 inch : 3 million miles
Assume that the ratio between the distance on the scale model and the actual distance is 31 inches : d million miles, where d is the actual distance between Earth and the Sun (in million miles).
We can use the scale 1 inch : 3 million miles and the ratio 31 inches:d million miles to form a proportion.
\(\frac{1~\text{inch}}{100~\text{million miles}} = \frac{31~\text{inch}}{d~\text{million miles}}\)
\(d = 31\times 3\) Cross product
\(d = 93\) Multiply
Since d is the distance between Earth and the Sun in million miles, we got the actual distance as 93 million miles.
The ratio that compares the dimensions of drawings or models with real-life objects is known as scale. A scale factor is a number with which you multiply the dimension of a drawing or a model to get the dimension of the real-life object.
If a model has more than one scale factor, it means that the model is not proportional to the real-life object.
To find the dimensions of the real-life object, we just need to multiply the dimensions of the drawing or model with the scale factor.