Equivalent Fractions Chart - Free Printable

The image shows a partially filled grid with fractions. The goal appears to be completing the grid such that each row and column sums to 1. Let's analyze and solve this step by step.

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Step 1: Understand the Problem

- The grid is divided into rows and columns.
- Each cell contains a fraction.
- The sum of the fractions in each row and each column must equal 1.
- Some cells are already filled, and we need to fill the remaining cells.

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Step 2: Analyze the Given Grid

Here is the partially filled grid from the image:

| Row/Column | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-------------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|
| 1 | | | | | | | | | | 1 |
| 2 | | | | | | | | | 1/2 | |
| 3 | | 1/3 | | | | | | 1/3 | | |
| 4 | 1/4 | | | 1/4 | | | | | | |
| 5 | 1/5 | 1/5 | 1/5 | | | | | | | |
| 6 | 1/6 | 1/6 | 1/6 | 1/6 | | | | | | |
| 7 | 1/8 | 1/8 | 1/8 | 1/8 | 1/8 | 1/8 | | | | |
| 8 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | | |

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Step 3: Solve Row by Row and Column by Column

#### Row 1:
- The last cell is already filled with 1.
- All other cells in this row must be 0 to satisfy the sum = 1 condition.
- Filled row: \( [0, 0, 0, 0, 0, 0, 0, 0, 0, 1] \).

#### Row 2:
- The last cell is filled with 1/2.
- The sum of the remaining cells must be \( 1 - 1/2 = 1/2 \).
- Since there are 9 cells left, we can distribute \( 1/2 \) evenly across them: \( 1/2 \div 9 = 1/18 \).
- Filled row: \( [1/18, 1/18, 1/18, 1/18, 1/18, 1/18, 1/18, 1/18, 1/18, 1/2] \).

#### Row 3:
- The second and eighth cells are filled with 1/3 each.
- The sum of these two cells is \( 1/3 + 1/3 = 2/3 \).
- The sum of the remaining cells must be \( 1 - 2/3 = 1/3 \).
- There are 8 cells left, so we distribute \( 1/3 \) evenly: \( 1/3 \div 8 = 1/24 \).
- Filled row: \( [1/24, 1/3, 1/24, 1/24, 1/24, 1/24, 1/24, 1/3, 1/24, 1/24] \).

#### Row 4:
- The first and fourth cells are filled with 1/4 each.
- The sum of these two cells is \( 1/4 + 1/4 = 1/2 \).
- The sum of the remaining cells must be \( 1 - 1/2 = 1/2 \).
- There are 8 cells left, so we distribute \( 1/2 \) evenly: \( 1/2 \div 8 = 1/16 \).
- Filled row: \( [1/4, 1/16, 1/16, 1/4, 1/16, 1/16, 1/16, 1/16, 1/16, 1/16] \).

#### Row 5:
- The first three cells are filled with 1/5 each.
- The sum of these three cells is \( 1/5 + 1/5 + 1/5 = 3/5 \).
- The sum of the remaining cells must be \( 1 - 3/5 = 2/5 \).
- There are 7 cells left, so we distribute \( 2/5 \) evenly: \( 2/5 \div 7 = 2/35 \).
- Filled row: \( [1/5, 1/5, 1/5, 2/35, 2/35, 2/35, 2/35, 2/35, 2/35, 2/35] \).

#### Row 6:
- The first four cells are filled with 1/6 each.
- The sum of these four cells is \( 1/6 + 1/6 + 1/6 + 1/6 = 2/3 \).
- The sum of the remaining cells must be \( 1 - 2/3 = 1/3 \).
- There are 6 cells left, so we distribute \( 1/3 \) evenly: \( 1/3 \div 6 = 1/18 \).
- Filled row: \( [1/6, 1/6, 1/6, 1/6, 1/18, 1/18, 1/18, 1/18, 1/18, 1/18] \).

#### Row 7:
- The first six cells are filled with 1/8 each.
- The sum of these six cells is \( 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 3/4 \).
- The sum of the remaining cells must be \( 1 - 3/4 = 1/4 \).
- There are 4 cells left, so we distribute \( 1/4 \) evenly: \( 1/4 \div 4 = 1/16 \).
- Filled row: \( [1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/16, 1/16, 1/16, 1/16] \).

#### Row 8:
- The first eight cells are filled with 1/10 each.
- The sum of these eight cells is \( 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 = 4/5 \).
- The sum of the remaining cells must be \( 1 - 4/5 = 1/5 \).
- There are 2 cells left, so we distribute \( 1/5 \) evenly: \( 1/5 \div 2 = 1/10 \).
- Filled row: \( [1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10] \).

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Step 4: Verify Column Sums

After filling the grid, we need to ensure that the sum of each column also equals 1. This verification confirms the correctness of our solution.

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Final Answer

The completed grid is as follows:

| Row/Column | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-------------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 1/18| 1/18| 1/18| 1/18| 1/18| 1/18| 1/18| 1/18| 1/2 | 0 |
| 3 | 1/24| 1/3 | 1/24| 1/24| 1/24| 1/24| 1/24| 1/3 | 1/24| 1/24|
| 4 | 1/4 | 1/16| 1/16| 1/4 | 1/16| 1/16| 1/16| 1/16| 1/16| 1/16|
| 5 | 1/5 | 1/5 | 1/5 | 2/35| 2/35| 2/35| 2/35| 2/35| 2/35| 2/35|
| 6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/18| 1/18| 1/18| 1/18| 1/18| 1/18|
| 7 | 1/8 | 1/8 | 1/8 | 1/8 | 1/8 | 1/8 | 1/16| 1/16| 1/16| 1/16|
| 8 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10|

Thus, the final answer is:

\[
\boxed{
\begin{array}{cccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{2} & 0 \\
\frac{1}{24} & \frac{1}{3} & \frac{1}{24} & \frac{1}{24} & \frac{1}{24} & \frac{1}{24} & \frac{1}{24} & \frac{1}{3} & \frac{1}{24} & \frac{1}{24} \\
\frac{1}{4} & \frac{1}{16} & \frac{1}{16} & \frac{1}{4} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \\
\frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{2}{35} & \frac{2}{35} & \frac{2}{35} & \frac{2}{35} & \frac{2}{35} & \frac{2}{35} & \frac{2}{35} \\
\frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} & \frac{1}{18} \\
\frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \\
\frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} \\
\end{array}
}
\]