Math Addition Puzzles For Building Math And Logic Skills The Fun Way. - Free Printable

Problem Analysis

The task involves completing a grid-based arithmetic puzzle. The grid consists of rows and columns with numbers and operators (+). The goal is to fill in the missing values (gray cells) such that the sums of each row and column are consistent with the given totals.

#### Grid Structure:
- The grid has 3 rows and 4 columns.
- Each row and column contains numbers and addition operators.
- The totals for each row and column are provided at the ends of the respective rows and columns.

#### Known Information:
1. Row Totals:
- Row 1: \(8 + 5 + 10 = \text{?}\)
- Row 2: \(9 + 7 + 10 = \text{?}\)
- Row 3: \(10 + 10 + 9 = \text{?}\)

2. Column Totals:
- Column 1: \(8 + 9 + 10 = \text{?}\)
- Column 2: \(5 + 7 + 10 = \text{?}\)
- Column 3: \(10 + 10 + 9 = \text{?}\)
- Column 4: The totals for the gray cells need to be calculated.

#### Objective:
- Fill in the gray cells such that the sums of all rows and columns match the given totals.

---

Step-by-Step Solution

#### Step 1: Calculate Row Totals
First, calculate the totals for each row based on the given numbers.

1. Row 1:
\[
8 + 5 + 10 = 23
\]

2. Row 2:
\[
9 + 7 + 10 = 26
\]

3. Row 3:
\[
10 + 10 + 9 = 29
\]

So, the row totals are:
- Row 1: 23
- Row 2: 26
- Row 3: 29

#### Step 2: Calculate Column Totals
Next, calculate the totals for each column based on the given numbers.

1. Column 1:
\[
8 + 9 + 10 = 27
\]

2. Column 2:
\[
5 + 7 + 10 = 22
\]

3. Column 3:
\[
10 + 10 + 9 = 29
\]

4. Column 4:
The gray cells in Column 4 need to be determined. Let the gray cells be denoted as \(a\), \(b\), and \(c\) from top to bottom. The total for Column 4 will be:
\[
a + b + c
\]

#### Step 3: Fill in the Gray Cells
Now, we need to determine the values of the gray cells (\(a\), \(b\), and \(c\)) such that the row and column totals are satisfied.

##### Row 1:
The sum of Row 1 is 23. The gray cell in Row 1 is part of this sum. Let the gray cell in Row 1 be \(a\). Then:
\[
8 + 5 + 10 + a = 23
\]
Solving for \(a\):
\[
23 - (8 + 5 + 10) = 23 - 23 = 0
\]
So, \(a = 0\).

##### Row 2:
The sum of Row 2 is 26. The gray cell in Row 2 is part of this sum. Let the gray cell in Row 2 be \(b\). Then:
\[
9 + 7 + 10 + b = 26
\]
Solving for \(b\):
\[
26 - (9 + 7 + 10) = 26 - 26 = 0
\]
So, \(b = 0\).

##### Row 3:
The sum of Row 3 is 29. The gray cell in Row 3 is part of this sum. Let the gray cell in Row 3 be \(c\). Then:
\[
10 + 10 + 9 + c = 29
\]
Solving for \(c\):
\[
29 - (10 + 10 + 9) = 29 - 29 = 0
\]
So, \(c = 0\).

#### Step 4: Verify Column Totals
Now, verify that the column totals are consistent with the calculated values of \(a\), \(b\), and \(c\).

1. Column 1:
\[
8 + 9 + 10 = 27 \quad \text{(matches the given total)}
\]

2. Column 2:
\[
5 + 7 + 10 = 22 \quad \text{(matches the given total)}
\]

3. Column 3:
\[
10 + 10 + 9 = 29 \quad \text{(matches the given total)}
\]

4. Column 4:
\[
a + b + c = 0 + 0 + 0 = 0
\]

#### Final Answer:
The gray cells are all 0. The completed grid is:

\[
\begin{array}{c|c|c|c|c}
8 & + & 5 & + & 10 & = & 23 \\
+ & 0 & + & 0 & + & 0 & + \\
9 & + & 7 & + & 10 & = & 26 \\
+ & 0 & + & 0 & + & 0 & + \\
10 & + & 10 & + & 9 & = & 29 \\
= & 27 & = & 22 & = & 29 & = & 0 \\
\end{array}
\]

Thus, the final answer is:
\[
\boxed{0, 0, 0}
\]