1. A rectangle with side lengths of $$ \frac{1}{3} $$ unit and $$ \frac{1}{6} $$ unit is shown. Use the rectangle to complete parts (a) (c). a. Create a unit square. Partition the unit square into equal parts. b. How many equal parts did you need to create a unit square? c. What is the area of the rectangular tile with side lengths of $$ \frac{1}{3} $$ unit and $$ \frac{1}{6} $$ unit?

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UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Part (a): Create a unit square and partition it into equal parts.

To create a unit square, we need to consider a square with side lengths of 1 unit. The area of this unit square is:

$$
A = 1 \times 1 = 1 \text{ square unit}
$$

Next, we need to partition this unit square into equal parts that can accommodate the rectangle with side lengths of $$ \frac{1}{3} $$ unit and $$ \frac{1}{6} $$ unit.

### Part (b): How many equal parts did you need to create a unit square?

To find out how many equal parts we need, we can determine the area of the rectangle and then see how many of these rectangles fit into the unit square.

### Part (c): What is the area of the rectangular tile with side lengths of $$ \frac{1}{3} $$ unit and $$ \frac{1}{6} $$ unit?

The area $$ A_r $$ of the rectangle can be calculated as follows:

$$
A_r = \text{length} \times \text{width} = \frac{1}{3} \times \frac{1}{6}
$$

Now, let's calculate the area of the rectangle.

$$
A_r = \frac{1}{3} \times \frac{1}{6} = \frac{1 \times 1}{3 \times 6} = \frac{1}{18} \text{ square units}
$$

Now, we can find out how many of these rectangles fit into the unit square:

$$
\text{Number of rectangles} = \frac{\text{Area of unit square}}{\text{Area of rectangle}} = \frac{1}{\frac{1}{18}} = 18
$$

### Summary of Answers:
- **(a)** A unit square can be created with side lengths of 1 unit.
- **(b)** We need to create **18 equal parts** to fill the unit square with rectangles of size $$ \frac{1}{3} $$ unit by $$ \frac{1}{6} $$ unit.
- **(c)** The area of the rectangular tile is $$ \frac{1}{18} $$ square units.
To create a unit square, partition it into 18 equal parts. Each rectangle has an area of $$ \frac{1}{18} $$ square units.

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