Solving Equations by Swapping Signs

The question asks us to identify which two mathematical signs, when swapped in the given equation, will make the equation correct. The given equation is:

\(10 + 5 \div 10 \times 8 - 10 = 16\)

We need to test each option by swapping the specified signs and then evaluating the resulting equation using the BODMAS (or PEMDAS) rule, which dictates the order of operations: Brackets (Parentheses), Orders (Exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Checking Option 1: Swap × and +

If we swap the multiplication (×) and addition (+) signs, the equation becomes:

\(10 \times 5 \div 10 + 8 - 10\)

Let's evaluate this step-by-step using BODMAS:

• First, Division: \(5 \div 10 = 0.5\)
• The equation becomes: \(10 \times 0.5 + 8 - 10\)
• Next, Multiplication: \(10 \times 0.5 = 5\)
• The equation becomes: \(5 + 8 - 10\)
• Finally, Addition and Subtraction from left to right: \(5 + 8 = 13\)
• The equation becomes: \(13 - 10\)
• Subtraction: \(13 - 10 = 3\)

The result is 3. Since \(3 \neq 16\), swapping × and + does not make the equation correct.

Checking Option 2: Swap – and +

If we swap the subtraction (–) and addition (+) signs, the equation becomes:

\(10 - 5 \div 10 \times 8 + 10\)

Let's evaluate this step-by-step using BODMAS:

• First, Division: \(5 \div 10 = 0.5\)
• The equation becomes: \(10 - 0.5 \times 8 + 10\)
• Next, Multiplication: \(0.5 \times 8 = 4\)
• The equation becomes: \(10 - 4 + 10\)
• Finally, Addition and Subtraction from left to right: \(10 - 4 = 6\)
• The equation becomes: \(6 + 10\)
• Addition: \(6 + 10 = 16\)

The result is 16. Since \(16 = 16\), swapping – and + makes the equation correct.

Checking Option 3: Swap ÷ and ×

If we swap the division (÷) and multiplication (×) signs, the equation becomes:

\(10 + 5 \times 10 \div 8 - 10\)

Let's evaluate this step-by-step using BODMAS:

• First, Multiplication and Division from left to right: \(5 \times 10 = 50\)
• The equation becomes: \(10 + 50 \div 8 - 10\)
• Next, Division: \(50 \div 8 = 6.25\)
• The equation becomes: \(10 + 6.25 - 10\)
• Finally, Addition and Subtraction from left to right: \(10 + 6.25 = 16.25\)
• The equation becomes: \(16.25 - 10\)
• Subtraction: \(16.25 - 10 = 6.25\)

The result is 6.25. Since \(6.25 \neq 16\), swapping ÷ and × does not make the equation correct.

Checking Option 4: Swap + and ÷

If we swap the addition (+) and division (÷) signs, the equation becomes:

\(10 \div 5 + 10 \times 8 - 10\)

Let's evaluate this step-by-step using BODMAS:

• First, Division: \(10 \div 5 = 2\)
• The equation becomes: \(2 + 10 \times 8 - 10\)
• Next, Multiplication: \(10 \times 8 = 80\)
• The equation becomes: \(2 + 80 - 10\)
• Finally, Addition and Subtraction from left to right: \(2 + 80 = 82\)
• The equation becomes: \(82 - 10\)
• Subtraction: \(82 - 10 = 72\)

The result is 72. Since \(72 \neq 16\), swapping + and ÷ does not make the equation correct.

Based on the evaluation of each option, swapping the – and + signs results in the correct equation.

Thus, the signs that should be interchanged are – and +.

Revision Table for Mathematical Signs

Additional Information: BODMAS / PEMDAS Explained

The order of operations is crucial for solving mathematical expressions consistently. Two common mnemonics are used:

• BODMAS:
  • Brackets
  • Orders (powers/indices or roots)
  • Division and Multiplication (done from left to right)
  • Addition and Subtraction (done from left to right)
• PEMDAS:
  • Parentheses
  • Exponents (powers or roots)
  • Multiplication and Division (done from left to right)
  • Addition and Subtraction (done from left to right)

Both mnemonics represent the same order. It is important to remember that Division and Multiplication have the same priority and are performed from left to right as they appear. Similarly, Addition and Subtraction have the same priority and are performed from left to right.