Question

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.

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.

The new Scale Factor is *blank 1* *blank 2* *blank 3* *blank 4*

Answer 1

To determine the scale factor that relates Maya's two scale drawings (1:2 and 1:6), we need to compare these two scale factors.


1. The scale factor from the actual cage to the first drawing is \(1:2\).
2. The scale factor from the actual cage to the second drawing is \(1:6\).

To find the scale factor that relates the two drawings to each other, we can set up a proportion based on the scale factors.

Since we want to relate the first scale drawing (1:2) to the second scale drawing (1:6), we can find the factor by dividing the first scale factor by the second scale factor:

\[
\text{Scale Factor} = \frac{1:2}{1:6}
\]

This can be interpreted as:

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

Therefore, the scale factor that relates the two scale drawings to each other is **3**.

Next, to find the length and width of the smaller scale drawing with the scale factor of \(1:6\), we need the dimensions of the first scale drawing. However, you did not provide specific dimensions for the first drawing, so let's assume the dimensions for the first scale drawing (1:2) are as follows:

- Length = \(L_1\)
- Width = \(W_1\)

Then the dimensions of the actual cage can be calculated as:

\[
\text{Actual Length} = L_1 \times 2
\]
\[
\text{Actual Width} = W_1 \times 2
\]

To convert these dimensions to the new smaller scale drawing (1:6), the dimensions would be calculated as:

\[
\text{Length of Smaller Drawing} = \frac{\text{Actual Length}}{6} = \frac{L_1 \times 2}{6} = \frac{L_1}{3}
\]
\[
\text{Width of Smaller Drawing} = \frac{\text{Actual Width}}{6} = \frac{W_1 \times 2}{6} = \frac{W_1}{3}
\]

In summary, the scale factor relating the two drawings is **3**, and the length and width of the smaller scale drawing would be \(L_1/3\) and \(W_1/3\), respectively, based on the dimensions of the first drawing \(L_1\) and \(W_1\).

To fill the blanks for “The new Scale Factor is *blank 1* *blank 2* *blank 3* *blank 4*”, it would be:

- *blank 1*: 3
- *blank 2*: (leave blank or add ":", depending on format)
- *blank 3*: (leave blank or add "and", depending on format)
- *blank 4*: (leave blank or add "between drawings", depending on format)

If you have specific dimensions, you can replace \(L_1\) and \(W_1\) with those values to calculate the exact length and width.