Scale Factor

Example 1:

A hand drawn painting has a scale of 4 in. : 5 ft. What is the scale factor of this painting?

Solution: 

Here, we have to ensure that the scale is in the same units. So, we will convert feet into inches. 

As we know, 1 ft. = 12 in. 

Hence, 5 ft. = 5 x 12 = 60 in.

Now, we have 4 in. : 60 in. and we will divide both quantities by 4 to determine the scale factor.

4 in. : 60 in.

1 in. : 15 in.

So, the scale factor is \(\frac{1}{15}\).

Example 2: 

The scale model of a toy soldier is 25 inches tall. The original toy is 50 inches tall. Find the scale factor of the toy.

Solution:

Here, the ratio of the toy’s height to the original height is 25 in. : 50 in.

We will divide both sides by 25 to ensure 1 inch in the model toy.

So, the scale factor will be 1 in. : 2 in. that can be written as, \(\frac{1}{2}\).

Example 3:

The scale of a landscape painting is 5 mm : 30 cm. What is the scale factor of the painting?

Solution: 

In order to have the scale with the same units, we will convert the units. Here, 1 cm = 10 mm

So, 30 cm = 30 x 10 mm = 300 mm

Now, the scale is 5 mm : 300 mm where we will divide both sides by 5 to represent 1 mm in the model.

Thus, the scale factor will be 1 mm: 60 mm or we can write it as \(\frac{1}{60}\).

Example 4:

The following image shows two similar triangles. The area of the bigger triangle is 40 square units and the area of the smaller triangle is 10 square units. Find the scale factor used to create the smaller triangle.

Solution:  

Here, the given details are 40 square units for the bigger triangle and 10 square units for the smaller triangle.

So, scale factor = \(\frac{\text{Dimension of new object}}{\text{Dimension of actual object}}\)

= \(\frac{10}{40}\)

= \(\frac{1}{4}\) 

Thus, the scale factor is \(\frac{1}{4}\).