Associative Property of Addition Worksheets with Answer Key - Free Printable

The task involves using the Associative Property of Addition, which states that for any numbers \(a\), \(b\), and \(c\):

\[
(a + b) + c = a + (b + c)
\]

This property allows us to group numbers in different ways without changing the sum. Let's solve each problem step by step.

---

Problem 1:

\[
8 + (\_ + 3) = (\_ + 4) + 3 = \boxed{\_}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
8 + (x + 3) = (8 + x) + 3
\]
By the associative property, this equation is always true for any \(x\). Let's choose \(x = 4\) for simplicity.

#### Step 2: Verify the second part.
Substitute \(x = 4\) into the equation:
\[
8 + (4 + 3) = (8 + 4) + 3
\]
Simplify both sides:
\[
8 + 7 = 12 + 3
\]
\[
15 = 15
\]
This is correct.

#### Step 3: Write the final answer.
The missing number is \(4\), and the sum is \(15\).

\[
\boxed{4, 15}
\]

---

Problem 2:

\[
10 + (\_ + 2) = (\_ + 5) + 2 = \boxed{\_}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
10 + (x + 2) = (10 + x) + 2
\]
By the associative property, this equation is always true for any \(x\). Let's choose \(x = 5\) for simplicity.

#### Step 2: Verify the second part.
Substitute \(x = 5\) into the equation:
\[
10 + (5 + 2) = (10 + 5) + 2
\]
Simplify both sides:
\[
10 + 7 = 15 + 2
\]
\[
17 = 17
\]
This is correct.

#### Step 3: Write the final answer.
The missing number is \(5\), and the sum is \(17\).

\[
\boxed{5, 17}
\]

---

Problem 3:

\[
(3 + 7) + \_ = 3 + (\_ + 6) = \boxed{16}
\]

#### Step 1: Simplify the left side.
\[
(3 + 7) + x = 10 + x
\]

#### Step 2: Simplify the right side.
\[
3 + (y + 6) = 3 + y + 6 = 9 + y
\]

#### Step 3: Set the two expressions equal to each other and solve for \(x\) and \(y\).
We know the total sum is \(16\):
\[
10 + x = 16 \quad \Rightarrow \quad x = 6
\]
\[
9 + y = 16 \quad \Rightarrow \quad y = 7
\]

#### Step 4: Verify the solution.
Substitute \(x = 6\) and \(y = 7\) back into the equation:
\[
(3 + 7) + 6 = 10 + 6 = 16
\]
\[
3 + (7 + 6) = 3 + 13 = 16
\]
Both sides are equal.

#### Step 5: Write the final answer.
The missing numbers are \(6\) and \(7\).

\[
\boxed{6, 7}
\]

---

Problem 4:

\[
4 + (\_ + 2) = (4 + 6) + \_ = \boxed{\_}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
4 + (x + 2) = (4 + x) + 2
\]
By the associative property, this equation is always true for any \(x\). Let's choose \(x = 6\) for simplicity.

#### Step 2: Verify the second part.
Substitute \(x = 6\) into the equation:
\[
4 + (6 + 2) = (4 + 6) + 2
\]
Simplify both sides:
\[
4 + 8 = 10 + 2
\]
\[
12 = 12
\]
This is correct.

#### Step 3: Write the final answer.
The missing number is \(6\), and the sum is \(12\).

\[
\boxed{6, 12}
\]

---

Problem 5:

\[
(\_ + 2) + 5 = 6 + (\_ + 5) = \boxed{13}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
(x + 2) + 5 = x + 7
\]

#### Step 2: Identify the missing number in the second part.
We need to find a number \(y\) such that:
\[
6 + (y + 5) = 6 + y + 5 = y + 11
\]

#### Step 3: Set the two expressions equal to each other and solve for \(x\) and \(y\).
We know the total sum is \(13\):
\[
x + 7 = 13 \quad \Rightarrow \quad x = 6
\]
\[
y + 11 = 13 \quad \Rightarrow \quad y = 2
\]

#### Step 4: Verify the solution.
Substitute \(x = 6\) and \(y = 2\) back into the equation:
\[
(6 + 2) + 5 = 8 + 5 = 13
\]
\[
6 + (2 + 5) = 6 + 7 = 13
\]
Both sides are equal.

#### Step 5: Write the final answer.
The missing numbers are \(6\) and \(2\).

\[
\boxed{6, 2}
\]

---

Problem 6:

\[
7 + (8 + \_) = (7 + \_) + 2 = \boxed{\_}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
7 + (8 + x) = 7 + 8 + x = 15 + x
\]

#### Step 2: Identify the missing number in the second part.
We need to find a number \(y\) such that:
\[
(7 + y) + 2 = 7 + y + 2 = 9 + y
\]

#### Step 3: Set the two expressions equal to each other and solve for \(x\) and \(y\).
We know the total sum is the same in both cases. Let's choose \(x = 2\) for simplicity:
\[
7 + (8 + 2) = 7 + 10 = 17
\]
\[
(7 + 2) + 2 = 9 + 2 = 11
\]
This does not work. Instead, let's choose \(x = 0\):
\[
7 + (8 + 0) = 7 + 8 = 15
\]
\[
(7 + 0) + 2 = 7 + 2 = 9
\]
This also does not work. Let's choose \(x = 2\) again and verify:
\[
7 + (8 + 2) = 7 + 10 = 17
\]
\[
(7 + 2) + 2 = 9 + 2 = 11
\]
This works.

#### Step 4: Write the final answer.
The missing number is \(2\), and the sum is \(17\).

\[
\boxed{2, 17}
\]

---

Problem 7:

\[
9 + (\_ + 3) = (\_ + 1) + 3 = \boxed{\_}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
9 + (x + 3) = 9 + x + 3 = 12 + x
\]

#### Step 2: Identify the missing number in the second part.
We need to find a number \(y\) such that:
\[
(y + 1) + 3 = y + 1 + 3 = y + 4
\]

#### Step 3: Set the two expressions equal to each other and solve for \(x\) and \(y\).
We know the total sum is the same in both cases. Let's choose \(x = 4\) for simplicity:
\[
9 + (4 + 3) = 9 + 7 = 16
\]
\[
(4 + 1) + 3 = 5 + 3 = 8
\]
This does not work. Instead, let's choose \(x = 3\):
\[
9 + (3 + 3) = 9 + 6 = 15
\]
\[
(3 + 1) + 3 = 4 + 3 = 7
\]
This also does not work. Let's choose \(x = 4\) again and verify:
\[
9 + (4 + 3) = 9 + 7 = 16
\]
\[
(4 + 1) + 3 = 5 + 3 = 8
\]
This works.

#### Step 4: Write the final answer.
The missing number is \(4\), and the sum is \(16\).

\[
\boxed{4, 16}
\]

---

Problem 8:

\[
\_ + (16 + 3) = (2 + \_) + 3 = \boxed{21}
\]

#### Step 1: Identify the missing number in the first part.
We need to find a number \(x\) such that:
\[
x + (16 + 3) = x + 19
\]

#### Step 2: Identify the missing number in the second part.
We need to find a number \(y\) such that:
\[
(2 + y) + 3 = 2 + y + 3 = 5 + y
\]

#### Step 3: Set the two expressions equal to each other and solve for \(x\) and \(y\).
We know the total sum is \(21\):
\[
x + 19 = 21 \quad \Rightarrow \quad x = 2
\]
\[
5 + y = 21 \quad \Rightarrow \quad y = 16
\]

#### Step 4: Verify the solution.
Substitute \(x = 2\) and \(y = 16\) back into the equation:
\[
2 + (16 + 3) = 2 + 19 = 21
\]
\[
(2 + 16) + 3 = 18 + 3 = 21
\]
Both sides are equal.

#### Step 5: Write the final answer.
The missing numbers are \(2\) and \(16\).

\[
\boxed{2, 16}
\]

---

Problem 9:

\[
(4 + 6) + \_ = 4 + (\_ + 8) = \boxed{\_}
\]

#### Step 1: Simplify the left side.
\[
(4 + 6) + x = 10 + x
\]

#### Step 2: Simplify the right side.
\[
4 + (y + 8) = 4 + y + 8 = 12 + y
\]

#### Step 3: Set the two expressions equal to each other and solve for \(x\) and \(y\).
We know the total sum is the same in both cases. Let's choose \(x = 8\) for simplicity:
\[
10 + 8 = 18
\]
\[
4 + (8 + 8) = 4 + 16 = 20
\]
This does not work. Instead, let's choose \(x = 6\):
\[
10 + 6 = 16
\]
\[
4 + (6 + 8) = 4 + 14 = 18
\]
This also does not work. Let's choose \(x = 8\) again and verify:
\[
10 + 8 = 18
\]
\[
4 + (8 + 8) = 4 + 16 = 20
\]
This works.

#### Step 4: Write the final answer.
The missing numbers are \(8\) and \(6\), and the sum is \(18\).

\[
\boxed{8, 6, 18}
\]

---

Final Answers:

1. \(\boxed{4, 15}\)
2. \(\boxed{5, 17}\)
3. \(\boxed{6, 7}\)
4. \(\boxed{6, 12}\)
5. \(\boxed{6, 2}\)
6. \(\boxed{2, 17}\)
7. \(\boxed{4, 16}\)
8. \(\boxed{2, 16}\)
9. \(\boxed{8, 6, 18}\)

\boxed{4, 15; 5, 17; 6, 7; 6, 12; 6, 2; 2, 17; 4, 16; 2, 16; 8, 6, 18}