Maya created several possible ways to rearrange her hamster's cage to give the hamster more room to run around. Her drawings were made with a scale factor of 1:2, but she wants to change the scale factor to 1:6 so she can compare them more easily.

Question

The diagram shows the lengths of the sides of her first scale drawing and the scale factors relating both drawings to the actual cage.

Identify the scale factor that relates the two scale drawings to each other. Then, find the length and width of the smaller scale drawing.

Scale Factor of 1:2 is
12 In top and bottom
6 left and right

Scale factor 1:6 unknown

The new scale factor is 1/2 ÷ 1/6 = 3

The widow the new rectangle is 3 ÷ 3 = 4
The width of the new rectangle is 2 inches

The height of the new rectangle *blank 1* ● *blank 2* = *blank 3*

The height of the new rectangle is *blank 4*

Options for blank 2

A 1/3
B 3
C 3/1

Answer 1

To find the scale factor that relates the two scale drawings to each other, let's analyze the given information carefully.

1. Scale Factors:

   • Original scale factor (Drawing 1) = 1:2
   • New scale factor (Drawing 2) = 1:6

2. Calculating the Relationship:

   • We want to determine how one scale drawing relates to the other. The scale factor of 1:2 is equivalent to a decimal of 1/2, and 1:6 is equivalent to 1/6.
   • To find the scale factor that relates the two drawings, we can take the ratio of the two scale factors:

   \[ \text{Scale factor relation} = \frac{1/2}{1/6} = \frac{1}{2} \times \frac{6}{1} = 3 \]

   This means that the first scale drawing is 3 times larger than the second scale drawing.

3. Finding the Dimensions of the Smaller Scale Drawing:

   • The dimensions of the first drawing are:
     • Length (top and bottom): 12 inches
     • Width (left and right): 6 inches
   • To find the dimensions of the smaller drawing (the one with the scale factor of 1:6), we can divide the original dimensions by the scale factor (3):
     • Smaller drawing length: \( \frac{12 \text{ inches}}{3} = 4 \text{ inches} \)
     • Smaller drawing width: \( \frac{6 \text{ inches}}{3} = 2 \text{ inches} \)

4. Filling the Blanks:

   • The height of the new rectangle is \( 4 \) inches (calculated) which corresponds to length.
   • For height (blank 1), the dimensions calculated would be:
     • Height of the new rectangle = \( 4 \) inches
     • Width of the new rectangle = \( 2 \) inches
     • Since the height is \( 4 \) inches and can be expressed with another multiplier, if we choose blank 2 to be the ratio, we should choose option \( A \) which is \( \frac{1}{3} \) to show the smaller drawing relation.

To summarize, the height of the new rectangle is \( 4 \) inches, and the completed equation for the rectangle dimensions can be written as:

Height of the new rectangle blank 1 \(\frac{1}{3}\)blank 2 = 2.

Answers:

• blank 1: 4
• blank 2: \( \frac{1}{3} \)
• blank 4: 4 inches