Answered

Question 1:

What did the duck say to the waiter about how he would pay for the meal?

[tex]\[
\overline{11} \overline{5} \quad \overline{36} \quad \overline{15} \quad \overline{36} \quad \overline{81} \quad \overline{41} \quad \overline{16} \quad \overline{38} \quad \overline{10} \quad \overline{15} \quad \overline{54} \quad \overline{54}
\][/tex]

Question 2:

Challenge: Place parentheses to make the equations true.

1. [tex]\(16 - 8 \div 4 \times 2 = 1\)[/tex]
2. [tex]\(24 - 18 - 12 \div 2 + 1 = 13\)[/tex]



Answer :

GinnyAnswer
Let's solve these problems step-by-step:

### 1) [tex]\(16 - 8 \div 4 \times 2 = 1\)[/tex]

To solve this equation, we need to place parentheses in such a way that the operations follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right)). The goal is to make the expression equal to 1.

Here's the detailed process:

1. Original Expression:
[tex]\[ 16 - 8 \div 4 \times 2 \][/tex]

2. Evaluate different placement of parentheses to achieve the desired result (1):

- Case 1: [tex]\((16 - 8) \div 4 \times 2\)[/tex]
[tex]\[ 16 - 8 = 8 \quad \text{then} \quad 8 \div 4 = 2 \quad \text{then} \quad 2 \times 2 = 4 \][/tex]
This does not equal 1.

- Case 2: [tex]\(16 - (8 \div 4) \times 2\)[/tex]
[tex]\[ 8 \div 4 = 2 \quad \text{then} \quad 2 \times 2 = 4 \quad \text{then} \quad 16 - 4 = 12 \][/tex]
This does not equal 1.

- Case 3: [tex]\(16 - 8 \div (4 \times 2)\)[/tex]
[tex]\[ 4 \times 2 = 8 \quad \text{then} \quad 8 \div 8 = 1 \quad \text{then} \quad 16 - 1 = 15 \][/tex]
This does not equal 1.

After analyzing different placements, we see that none of them make the expression equal to 1. Therefore, let's consider the parentheses and perform the calculations:

- Correct placement:

[tex]\[ 16 - (8 \div 4) \times 2 \][/tex]

[tex]\[ 8 \div 4 = 2 \][/tex]

[tex]\[ 2 \times 2 = 4 \][/tex]

[tex]\[ 16 - 4 = 12 \][/tex]

This configuration makes the equation not equal to 1.

### Final Answer for 1):
[tex]\( \boxed{12} \)[/tex]

### 2) [tex]\(24 - (18 - (12 \div 2)) + 1 = 13\)[/tex]

Next, let's solve the second equation by breaking down the operations inside the parentheses:

1. Original Expression:
[tex]\[ 24 - (18 - (12 \div 2)) + 1 \][/tex]

2. Evaluate the innermost parentheses first:
[tex]\[ 12 \div 2 = 6 \][/tex]

3. Evaluate the next level of parentheses:
[tex]\[ 18 - 6 = 12 \][/tex]

4. Evaluate the remaining expression:
[tex]\[ 24 - 12 + 1 = 13 \][/tex]

So, by following the steps, we see that the proper placement of parentheses makes the equation true as follows:

[tex]\[ 24 - (18 - (12 \div 2)) + 1 = 13 \][/tex]

### Final Answer for 2):
[tex]\( \boxed{13} \)[/tex]