Solving the Equation by Interchanging Signs

The question asks us to identify which two mathematical signs in the given equation must be interchanged to make the equation correct. The equation is:

\(294 + 14 \times 4 \div 16 - 67 = 33\)

We need to test each option by swapping the specified signs and calculating the result using the correct order of operations (BODMAS/PEMDAS - Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Testing Option 1: Interchanging + and ×

If we interchange '+' and '×', the equation becomes:

\(294 \times 14 + 4 \div 16 - 67\)

Let's calculate step-by-step:

• First, calculate Division: \(4 \div 16 = 0.25\)
• Next, calculate Multiplication: \(294 \times 14 = 4116\)
• Now, perform Addition and Subtraction from left to right: \(4116 + 0.25 - 67 = 4116.25 - 67 = 4049.25\)

The result is \(4049.25\), which is not equal to \(33\). So, interchanging '+' and '×' does not make the equation correct.

Testing Option 2: Interchanging + and ÷

If we interchange '+' and '÷', the equation becomes:

\(294 \div 14 \times 4 + 16 - 67\)

Let's calculate step-by-step following the order of operations:

• First, calculate Division: \(294 \div 14 = 21\)
• Next, calculate Multiplication: \(21 \times 4 = 84\)
• Now, perform Addition and Subtraction from left to right: \(84 + 16 - 67 = 100 - 67 = 33\)

The result is \(33\), which is equal to the target value. So, interchanging '+' and '÷' makes the equation correct.

Testing Option 3: Interchanging × and ÷

If we interchange '×' and '÷', the equation becomes:

\(294 + 14 \div 4 \times 16 - 67\)

Let's calculate step-by-step:

• First, calculate Division: \(14 \div 4 = 3.5\)
• Next, calculate Multiplication: \(3.5 \times 16 = 56\)
• Now, perform Addition and Subtraction from left to right: \(294 + 56 - 67 = 350 - 67 = 283\)

The result is \(283\), which is not equal to \(33\). So, interchanging '×' and '÷' does not make the equation correct.

Testing Option 4: Interchanging + and -

If we interchange '+' and '-', the equation becomes:

\(294 - 14 \times 4 \div 16 + 67\)

Let's calculate step-by-step:

• First, calculate Multiplication: \(14 \times 4 = 56\)
• Next, calculate Division: \(56 \div 16 = 3.5\)
• Now, perform Addition and Subtraction from left to right: \(294 - 3.5 + 67 = 290.5 + 67 = 357.5\)

The result is \(357.5\), which is not equal to \(33\). So, interchanging '+' and '-' does not make the equation correct.

Based on the calculations, interchanging the signs '+' and '÷' correctly solves the equation.

Revision Table: Testing Sign Interchanges

Additional Information: Understanding Order of Operations

The order of operations is a rule that tells us the sequence in which to perform operations in a mathematical expression. A common acronym used is BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers and square roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and Multiplication have the same priority and should be performed from left to right as they appear. Similarly, Addition and Subtraction have the same priority and should be performed from left to right.

In this problem, applying the correct order of operations after interchanging the signs is crucial to arrive at the correct result.