Solving Mathematical Equation Sign Interchange Problems

The problem asks us to identify which pair of mathematical signs, when swapped in the given equation $60 \times 4 + 9 \div 3 - 8 = 34$, will make the equation correct. We need to test each option by performing the interchange and then evaluating the resulting expression according to the order of operations (BODMAS/PEMDAS).

Understanding the Order of Operations (BODMAS/PEMDAS)

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

• Brackets (Parentheses)
• Orders (Exponents, Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Testing the Options for Sign Interchange

Let's test each option by swapping the specified signs in the original equation: $60 \times 4 + 9 \div 3 - 8 = 34$.

Option 1: Swapping + and -

The new equation becomes: $60 \times 4 - 9 \div 3 + 8$

Evaluate using BODMAS:

• Multiplication: $60 \times 4 = 240$
• Division: $9 \div 3 = 3$
• The expression is now: $240 - 3 + 8$
• Subtraction: $240 - 3 = 237$
• Addition: $237 + 8 = 245$

The result is $245$. Since $245 \neq 34$, this option is incorrect.

Option 2: Swapping ÷ and +

The new equation becomes: $60 \times 4 \div 9 + 3 - 8$

Evaluate using BODMAS:

• Multiplication: $60 \times 4 = 240$
• Division: $240 \div 9 = \frac{240}{9} = \frac{80}{3}$ (This results in a fraction, which is unlikely to yield the integer 34. Let's continue to confirm).
• The expression is now: $\frac{80}{3} + 3 - 8$
• Convert to common denominator: $\frac{80}{3} + \frac{9}{3} - \frac{24}{3} = \frac{80 + 9 - 24}{3} = \frac{89 - 24}{3} = \frac{65}{3}$

The result is $\frac{65}{3}$. Since $\frac{65}{3} \neq 34$, this option is incorrect.

Option 3: Swapping + and ×

The new equation becomes: $60 + 4 \times 9 \div 3 - 8$

Evaluate using BODMAS:

• Multiplication: $4 \times 9 = 36$
• Division: $36 \div 3 = 12$
• The expression is now: $60 + 12 - 8$
• Addition: $60 + 12 = 72$
• Subtraction: $72 - 8 = 64$

The result is $64$. Since $64 \neq 34$, this option is incorrect.

Option 4: Swapping ÷ and ×

The new equation becomes: $60 \div 4 + 9 \times 3 - 8$

Evaluate using BODMAS:

• Division: $60 \div 4 = 15$
• Multiplication: $9 \times 3 = 27$
• The expression is now: $15 + 27 - 8$
• Addition: $15 + 27 = 42$
• Subtraction: $42 - 8 = 34$

The result is $34$. Since $34 = 34$, this option makes the equation correct.

Conclusion

After testing all the options, we found that swapping the signs ÷ and × makes the given equation correct.

Revision Table: Equation Sign Swapping Concepts

Additional Information: Solving Techniques for Equation Problems

When tackling mathematical equation problems involving sign or number interchanges, consider these strategies:

• Understand the Target: Always keep the right-hand side of the equation in mind. If the initial calculation is very far off, consider swaps that involve multiplication or division first, as they tend to have a larger impact on the result.
• Apply BODMAS Methodically: Strictly follow the order of operations for each potential new equation. Do one step at a time to avoid errors.
• Estimate: Before doing the full calculation, sometimes you can quickly estimate the result of a potential swap. For example, in the original equation $60 \times 4$ is 240, which is much larger than 34. This suggests that the $\times$ sign might need to be replaced by an operation that yields a smaller number, like $\div$ or $+$.
• Check All Options (if necessary): While estimation can guide you, for accuracy, it's often best to verify each given option unless you are completely certain.
• Practice: Solving more problems of this type helps build intuition about which swaps are likely to work.