Which two signs need to be interchanged to make the following equation correct? 5 × 4 + 12 - 3 ÷ 6 = 18
÷ and -
Solving Equation by Interchanging Signs
The problem asks us to find which pair of signs, when swapped in the given equation, makes the equation correct. The original equation is:
\(5 \times 4 + 12 - 3 \div 6 = 18\)
Let's evaluate the original equation to see if it is already correct using the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction):
\(5 \times 4 + 12 - 3 \div 6\)
First, perform Multiplication and Division from left to right:
\((5 \times 4) + 12 - (3 \div 6)\)
\(20 + 12 - 0.5\)
Next, perform Addition and Subtraction from left to right:
\(32 - 0.5\)
\(31.5\)
Since \(31.5 \neq 18\), the original equation is not correct. We need to test each option by interchanging the specified signs and re-evaluating the equation.
Testing Option 1: Interchange + and -
If we swap the '+' and '-' signs, the equation becomes:
\(5 \times 4 - 12 + 3 \div 6\)
Evaluate using BODMAS:
Multiplication and Division:
\((5 \times 4) - 12 + (3 \div 6)\)
\(20 - 12 + 0.5\)
Addition and Subtraction:
\(8 + 0.5\)
\(8.5\)
Since \(8.5 \neq 18\), interchanging '+' and '-' does not make the equation correct.
Testing Option 2: Interchange ÷ and -
If we swap the '÷' and '-' signs, the equation becomes:
\(5 \times 4 + 12 \div 3 - 6\)
Evaluate using BODMAS:
Multiplication and Division:
\((5 \times 4) + (12 \div 3) - 6\)
\(20 + 4 - 6\)
Addition and Subtraction:
\(24 - 6\)
\(18\)
Since \(18 = 18\), interchanging '÷' and '-' makes the equation correct.
Testing Option 3: Interchange ÷ and ×
If we swap the '÷' and '×' signs, the equation becomes:
\(5 \div 4 + 12 - 3 \times 6\)
Evaluate using BODMAS:
Multiplication and Division:
\((5 \div 4) + 12 - (3 \times 6)\)
\(1.25 + 12 - 18\)
Addition and Subtraction:
\(13.25 - 18\)
\(-4.75\)
Since \(-4.75 \neq 18\), interchanging '÷' and '×' does not make the equation correct.
Testing Option 4: Interchange × and +
If we swap the '×' and '+' signs, the equation becomes:
\(5 + 4 \times 12 - 3 \div 6\)
Evaluate using BODMAS:
Multiplication and Division:
\(5 + (4 \times 12) - (3 \div 6)\)
\(5 + 48 - 0.5\)
Addition and Subtraction:
\(53 - 0.5\)
\(52.5\)
Since \(52.5 \neq 18\), interchanging '×' and '+' does not make the equation correct.
Based on the evaluation of each option, interchanging the '÷' and '-' signs is the correct choice to make the equation true.
The final answer is Option 2: ÷ and -.
Revision Table: Evaluating Sign Interchanges
| Option | Signs Interchanged | New Equation | Evaluation (Step-by-step) | Result | Correct? |
|---|---|---|---|---|---|
| Original | None | \(5 \times 4 + 12 - 3 \div 6\) | \(20 + 12 - 0.5 = 32 - 0.5 = 31.5\) | \(31.5\) | No |
| 1 | + and - | \(5 \times 4 - 12 + 3 \div 6\) | \(20 - 12 + 0.5 = 8 + 0.5 = 8.5\) | \(8.5\) | No |
| 2 | ÷ and - | \(5 \times 4 + 12 \div 3 - 6\) | \(20 + 4 - 6 = 24 - 6 = 18\) | \(18\) | Yes |
| 3 | ÷ and × | \(5 \div 4 + 12 - 3 \times 6\) | \(1.25 + 12 - 18 = 13.25 - 18 = -4.75\) | \(-4.75\) | No |
| 4 | × and + | \(5 + 4 \times 12 - 3 \div 6\) | \(5 + 48 - 0.5 = 53 - 0.5 = 52.5\) | \(52.5\) | No |
Additional Information: Order of Operations (BODMAS/PEMDAS)
When solving mathematical equations with multiple operations, it is crucial to follow a specific order to ensure the correct result. This order is commonly remembered using acronyms like BODMAS or PEMDAS.
- BODMAS:
- Brackets (Parentheses)
- Orders (Exponents, Roots)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
- PEMDAS:
- Parentheses
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Both acronyms represent the same order of operations. Division and Multiplication have the same priority, as do Addition and Subtraction. When operations of the same priority appear in an expression, you perform them from left to right.
In this problem, we applied the BODMAS/PEMDAS rule each time we evaluated the equation after interchanging the signs. This systematic approach is essential for solving such arithmetic puzzles accurately.
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