Question

use fractions to complete the sentences. the side length of each small tile is \\(\square\\) units. the area of each small tile is \\(\square\\) square units. the area of the shaded rectangle is \\(\square\\) square units.

Explanation:

Step1: Determine side length of small tile

The large square has a side length of 1 unit and is divided into 7 columns and 7 rows? Wait, no, looking at the grid: the horizontal side (length) is divided into 7 parts? Wait, no, the grid: let's count the number of columns and rows. Wait, the large square is 1 unit by 1 unit. Let's see the number of small squares along one side. Let's count the columns: from left to right, how many small squares? Let's see the grid: the first part (unshaded) has 3 columns, then shaded has 4? Wait, no, total columns: 3 + 4 = 7? Wait, no, the vertical side (height) has how many rows? Let's see: the shaded part has 6 rows? Wait, no, the bottom row is unshaded. Wait, the large square is 1 unit by 1 unit. Let's count the number of small squares along the horizontal (length) and vertical (width) directions.

Wait, the horizontal side (length) of the large square is 1 unit, and it's divided into 7 equal - sized small squares? No, wait, let's check the number of columns. Let's count the small squares in one row: the first row has 3 unshaded and 4 shaded, then the next 5 rows (up to the 6th row) have shaded, and the 7th row (bottom) is unshaded? Wait, no, maybe I miscounted. Let's do it properly.

The large square has a side length of 1 unit. Let's look at the number of small squares along the horizontal (length) direction: let's count the number of columns. From the diagram, the horizontal side is divided into 7 equal - width small rectangles (since the top - bottom is 1 unit, and the horizontal division: let's see, the number of columns: 3 (unshaded) + 4 (shaded) = 7. So each small square has a side length of $\frac{1}{7}$ units? Wait, no, wait the vertical direction: the number of rows. Let's see, the shaded part has 6 rows? Wait, the bottom row is unshaded, so total rows: 7. So each small square has a side length of $\frac{1}{7}$ units? Wait, no, maybe the horizontal is divided into 7 parts and vertical into 7 parts? Wait, the large square is 1x1, so if it's divided into a grid with 7 columns and 7 rows, then each small square has a side length of $\frac{1}{7}$ units.

Wait, let's re - examine the first question: "The side length of each small tile is [ ] units."

Since the large square has a side length of 1 unit and is divided into 7 equal - length segments (columns) and 7 equal - length segments (rows), each small tile (square) has a side length of $\frac{1}{7}$ units. Wait, no, maybe the horizontal is 7 and vertical is 7? Wait, the large square is 1 unit by 1 unit. So if we have $n$ small squares along one side, the side length of each small square is $\frac{1}{n}$. Let's count the number of small squares along the horizontal side: looking at the grid, in one row, the number of small squares is 7 (3 unshaded + 4 shaded). Similarly, along the vertical side, the number of small squares is 7 (6 shaded rows + 1 unshaded row at the bottom). So each small square has a side length of $\frac{1}{7}$ units.

Step2: Area of each small tile

The area of a square is side length squared. Since the side length of each small tile is $\frac{1}{7}$ units, the area of each small tile is $(\frac{1}{7})\times(\frac{1}{7})=\frac{1}{49}$ square units.

Step3: Area of the shaded rectangle

First, find the length and width of the shaded rectangle.

Length of the shaded rectangle: the number of columns in the shaded part is 4, and each column has a width of $\frac{1}{7}$ units, so the length of the shaded rectangle is $4\times\frac{1}{7}=\frac{4}{7}$ units.

Width of the shaded rectangle: the number of rows in the shaded part is 6, and each row has a height of $\frac{1}{7}$ units, so the width of the shaded rectangle is $6\times\frac{1}{7}=\frac{6}{7}$ units.

The area of a rectangle is length times width, so the area of the shaded rectangle is $\frac{4}{7}\times\frac{6}{7}=\frac{24}{49}$ square units.

Wait, but let's check again. Maybe the horizontal division is 7 columns (so each column width is $\frac{1}{7}$) and vertical division is 7 rows (each row height is $\frac{1}{7}$).

1. **Side length of each small tile**:
The large square has a side length of 1 unit and is divided into 7 equal - length segments along each side (since there are 7 small squares along one side). So the side length of each small tile is $\frac{1}{7}$ units.

2. **Area of each small tile**:
The area of a square is $s\times s$, where $s$ is the side length. So $A = (\frac{1}{7})\times(\frac{1}{7})=\frac{1}{49}$ square units.

3. **Area of the shaded rectangle**:
First, find the number of columns (length - wise) and rows (width - wise) of the shaded part.
- Number of columns (length) of the shaded rectangle: 4 (since the shaded part has 4 columns).
- Number of rows (width) of the shaded rectangle: 6 (since the shaded part has 6 rows).
- The length of the shaded rectangle is $4\times\frac{1}{7}=\frac{4}{7}$ units (because each column has a width of $\frac{1}{7}$ units).
- The width of the shaded rectangle is $6\times\frac{1}{7}=\frac{6}{7}$ units (because each row has a height of $\frac{1}{7}$ units).
- The area of a rectangle is $A = l\times w$, so $A=\frac{4}{7}\times\frac{6}{7}=\frac{24}{49}$ square units.

Answers:

- The side length of each small tile is $\boldsymbol{\frac{1}{7}}$ units.
- The area of each small tile is $\boldsymbol{\frac{1}{49}}$ square units.
- The area of the shaded rectangle is $\boldsymbol{\frac{24}{49}}$ square units.

Answer:

s:

• The side length of each small tile is $\boldsymbol{\frac{1}{7}}$ units.
• The area of each small tile is $\boldsymbol{\frac{1}{49}}$ square units.
• The area of the shaded rectangle is $\boldsymbol{\frac{24}{49}}$ square units.