Understanding the Problem: Interchanging Signs in Equations

The question asks us to identify which pair of mathematical signs, when swapped in the given equation, makes the equation correct. The given equation is:

225 + 5 × 3 ÷ 5 - 7 = 133

Our goal is to test each option by interchanging the specified signs and then evaluating the resulting equation using the standard order of operations (BODMAS/PEMDAS) to see if the result is 133.

Applying the Order of Operations (BODMAS/PEMDAS)

To correctly evaluate mathematical expressions, we follow a specific order:

1. Brackets (Parentheses)
2. Orders (Exponents/Powers and Square Roots)
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Let's apply this rule while testing each option.

Analysing Sign Interchange Options

Option 1: Interchanging + and ×

If we interchange the '+' and '×' signs in the original equation 225 + 5 × 3 ÷ 5 - 7 = 133, the new equation becomes:

225 × 5 + 3 ÷ 5 - 7

Now, let's evaluate this using BODMAS:

• First, Division and Multiplication from left to right:
  • \(225 \times 5 = 1125\)
  • \(3 \div 5 = 0.6\)
• The equation is now: 1125 + 0.6 - 7
• Next, Addition and Subtraction from left to right:
  • \(1125 + 0.6 = 1125.6\)
  • \(1125.6 - 7 = 1118.6\)

The result is 1118.6, which is not equal to 133.

Option 2: Interchanging + and ÷

If we interchange the '+' and '÷' signs in the original equation 225 + 5 × 3 ÷ 5 - 7 = 133, the new equation becomes:

225 ÷ 5 × 3 + 5 - 7

Now, let's evaluate this using BODMAS:

• First, Division and Multiplication from left to right:
  • \(225 \div 5 = 45\)
  • \(45 \times 3 = 135\)
• The equation is now: 135 + 5 - 7
• Next, Addition and Subtraction from left to right:
  • \(135 + 5 = 140\)
  • \(140 - 7 = 133\)

The result is 133, which is equal to the target value.

Option 3: Interchanging × and -

If we interchange the '×' and '-' signs in the original equation 225 + 5 × 3 ÷ 5 - 7 = 133, the new equation becomes:

225 + 5 - 3 ÷ 5 × 7

Now, let's evaluate this using BODMAS:

• First, Division and Multiplication from left to right:
  • \(3 \div 5 = 0.6\)
  • \(0.6 \times 7 = 4.2\)
• The equation is now: 225 + 5 - 4.2
• Next, Addition and Subtraction from left to right:
  • \(225 + 5 = 230\)
  • \(230 - 4.2 = 225.8\)

The result is 225.8, which is not equal to 133.

Option 4: Interchanging × and ÷

If we interchange the '×' and '÷' signs in the original equation 225 + 5 × 3 ÷ 5 - 7 = 133, the new equation becomes:

225 + 5 ÷ 3 × 5 - 7

Now, let's evaluate this using BODMAS:

• First, Division and Multiplication from left to right:
  • \(5 \div 3 \approx 1.666...\)
  • \(1.666... \times 5 \approx 8.333...\)
• The equation is now: 225 + 8.333... - 7
• Next, Addition and Subtraction from left to right:
  • \(225 + 8.333... \approx 233.333...\)
  • \(233.333... - 7 \approx 226.333...\)

The result is approximately 226.333..., which is not equal to 133.

Conclusion

After testing each option by interchanging the given signs and evaluating the resulting equation using the BODMAS rule, we found that interchanging the '+' and '÷' signs makes the equation correct.

Revision Table: Checking Sign Interchange

Additional Information: Mathematical Operator Properties

Understanding how mathematical operators work and their properties is crucial for solving such problems. Key points include:

• Commutative Property: For addition (\(a+b = b+a\)) and multiplication (\(a \times b = b \times a\)). Subtraction and division are NOT commutative.
• Associative Property: For addition (\((a+b)+c = a+(b+c)\)) and multiplication (\((a \times b) \times c = a \times (b \times c)\)). Subtraction and division are NOT associative.
• Distributive Property: Multiplication distributes over addition and subtraction (\(a \times (b+c) = a \times b + a \times c\)).
• Inverse Operations: Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

These properties, along with the order of operations, ensure consistent results when evaluating mathematical expressions.