Which two signs should be interchanged to make the following equation correct?
+ and ÷
Swapping Signs to Correct the Equation
The problem asks us to identify which two mathematical signs in the given equation need to be swapped so that the equation becomes true. The original equation is:
\(24 \div 12 - 6 \times 6 + 2 = 18\)
Currently, let's evaluate the original equation using the BODMAS/PEMDAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction):
- Division: \(24 \div 12 = 2\)
- Multiplication: \(6 \times 6 = 36\)
- Now substitute back: \(2 - 36 + 2\)
- Addition/Subtraction (from left to right): \(2 - 36 = -34\), then \(-34 + 2 = -32\)
So, the original equation evaluates to \(-32\), which is not \(18\).
We need to test each option by swapping the suggested signs and re-evaluating the equation.
Testing the Options for Sign Interchange
Option 1: Swapping \(\div\) and \(\times\)
Original equation: \(24 \div 12 - 6 \times 6 + 2 = 18\)
Swap \(\div\) and \(\times\): \(24 \times 12 - 6 \div 6 + 2\)
Evaluate:
- Division: \(6 \div 6 = 1\)
- Multiplication: \(24 \times 12 = 288\)
- Substitute back: \(288 - 1 + 2\)
- Addition/Subtraction: \(288 - 1 = 287\), then \(287 + 2 = 289\)
\(289 \neq 18\). This option is incorrect.
Option 2: Swapping \(+\) and \(\div\)
Original equation: \(24 \div 12 - 6 \times 6 + 2 = 18\)
Swap \(+\) and \(\div\): \(24 + 12 - 6 \times 6 \div 2\)
Evaluate using BODMAS/PEMDAS:
- Division: \(6 \div 2 = 3\)
- Multiplication: \(6 \times 3 = 18\)
- Substitute back: \(24 + 12 - 18\)
- Addition/Subtraction: \(24 + 12 = 36\), then \(36 - 18 = 18\)
\(18 = 18\). This makes the equation correct.
Option 3: Swapping \(+\) and \(-\)
Original equation: \(24 \div 12 - 6 \times 6 + 2 = 18\)
Swap \(+\) and \(-\): \(24 \div 12 + 6 \times 6 - 2\)
Evaluate:
- Division: \(24 \div 12 = 2\)
- Multiplication: \(6 \times 6 = 36\)
- Substitute back: \(2 + 36 - 2\)
- Addition/Subtraction: \(2 + 36 = 38\), then \(38 - 2 = 36\)
\(36 \neq 18\). This option is incorrect.
Option 4: Swapping \(\times\) and \(+\)
Original equation: \(24 \div 12 - 6 \times 6 + 2 = 18\)
Swap \(\times\) and \(+\): \(24 \div 12 - 6 + 6 \times 2\)
Evaluate:
- Division: \(24 \div 12 = 2\)
- Multiplication: \(6 \times 2 = 12\)
- Substitute back: \(2 - 6 + 12\)
- Addition/Subtraction: \(2 - 6 = -4\), then \(-4 + 12 = 8\)
\(8 \neq 18\). This option is incorrect.
Based on the evaluation of each option, swapping the \(+\) and \(\div\) signs makes the equation correct.
Summary of Evaluation
| Signs Swapped | New Equation | Evaluated Result | Correct? |
|---|---|---|---|
| \(\div\) and \(\times\) | \(24 \times 12 - 6 \div 6 + 2\) | \(289\) | No |
| \(+\) and \(\div\) | \(24 + 12 - 6 \times 6 \div 2\) | \(18\) | Yes |
| \(+\) and \(-\) | \(24 \div 12 + 6 \times 6 - 2\) | \(36\) | No |
| \(\times\) and \(+\) | \(24 \div 12 - 6 + 6 \times 2\) | \(8\) | No |
Revision Table: Key Concepts
| Concept | Description | Importance in Problem Solving |
|---|---|---|
| Order of Operations | Rules (like BODMAS/PEMDAS) that dictate the sequence for evaluating mathematical expressions (Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). | Ensures consistent and correct evaluation of mathematical expressions. Crucial for verifying if the swapped signs make the equation correct. |
| Sign Interchange | Swapping the positions of two mathematical operation signs (\(+, -, \times, \div\)) in an equation. | The core operation required by the problem. Each potential swap creates a new equation to test. |
Additional Information: Applying BODMAS/PEMDAS
The BODMAS or PEMDAS rule is fundamental when evaluating expressions with multiple operations. It helps avoid ambiguity and ensures everyone arrives at the same answer. Remember that division and multiplication have the same priority, as do addition and subtraction. When they appear together, you perform them from left to right.
Let's re-examine the correct case (\(24 + 12 - 6 \times 6 \div 2\)) with strict BODMAS:
- No Brackets or Orders (powers, square roots).
- Division and Multiplication (left to right):
- \(6 \times 6 = 36\). Equation becomes: \(24 + 12 - 36 \div 2\)
- \(36 \div 2 = 18\). Equation becomes: \(24 + 12 - 18\)
- Addition and Subtraction (left to right):
- \(24 + 12 = 36\). Equation becomes: \(36 - 18\)
- \(36 - 18 = 18\).
The final result is 18, matching the required value.
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