Understanding the Equation Problem

The question asks us to find which specific interchange of two numbers and two mathematical signs will correct the given equation. The original equation is:$$13 - 65 \div 45 \times 21 + 2 = 82$$Our goal is to determine which proposed interchange makes the left side of the equation equal to 82.

Applying the Order of Operations (BODMAS/PEMDAS)

To evaluate the equation after each interchange, we must follow the standard order of mathematical operations:

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

We will test each option provided.

Testing Each Interchange Option

Option 1: Interchange 65 and 13, $\times$ and $+$

Original equation: $13 - 65 \div 45 \times 21 + 2 = 82$
Interchange 65 and 13: $65 - 13 \div 45 \times 21 + 2$
Interchange $\times$ and $+$: $65 - 13 \div 45 + 21 \times 2$

Let's evaluate the expression:$$65 - 13 \div 45 + 21 \times 2$$First, perform Division and Multiplication:$$65 - \frac{13}{45} + 42$$Then, perform Addition and Subtraction:$$107 - \frac{13}{45}$$This result is not an integer and definitely not 82. So, Option 1 is incorrect.

Option 2: Interchange 45 and 13, $\times$ and $+$

Original equation: $13 - 65 \div 45 \times 21 + 2 = 82$
Interchange 45 and 13: $45 - 65 \div 13 \times 21 + 2$
Interchange $\times$ and $+$: $45 - 65 \div 13 + 21 \times 2$

Let's evaluate the expression:$$45 - 65 \div 13 + 21 \times 2$$First, perform Division and Multiplication:$$45 - (65 \div 13) + (21 \times 2)$$$$45 - 5 + 42$$Then, perform Addition and Subtraction from left to right:$$(45 - 5) + 42$$$$40 + 42$$$$82$$The result is 82, which matches the right side of the original equation. So, Option 2 makes the equation correct.

Option 3: Interchange 45 and 21, $\times$ and $+$

Original equation: $13 - 65 \div 45 \times 21 + 2 = 82$
Interchange 45 and 21: $13 - 65 \div 21 \times 45 + 2$
Interchange $\times$ and $+$: $13 - 65 \div 21 + 45 \times 2$

Let's evaluate the expression:$$13 - 65 \div 21 + 45 \times 2$$First, perform Division and Multiplication:$$13 - \frac{65}{21} + 90$$Then, perform Addition and Subtraction:$$103 - \frac{65}{21}$$This result is not an integer and definitely not 82. So, Option 3 is incorrect.

Option 4: Interchange 45 and 13, $\minus$ and $+$

Original equation: $13 - 65 \div 45 \times 21 + 2 = 82$
Interchange 45 and 13: $45 - 65 \div 13 \times 21 + 2$
Interchange $\minus$ and $+$: $45 + 65 \div 13 \times 21 - 2$

Let's evaluate the expression:$$45 + 65 \div 13 \times 21 - 2$$First, perform Division and Multiplication from left to right:$$45 + (65 \div 13) \times 21 - 2$$$$45 + 5 \times 21 - 2$$$$45 + (5 \times 21) - 2$$$$45 + 105 - 2$$Then, perform Addition and Subtraction from left to right:$$(45 + 105) - 2$$$$150 - 2$$$$148$$The result is 148, which is not 82. So, Option 4 is incorrect.

Conclusion

After testing all options and performing the calculations according to the order of operations, we find that only the interchange of 45 and 13, and the interchange of $\times$ and $+$ correctly solves the equation.$$45 - 65 \div 13 + 21 \times 2 = 82$$Therefore, the correct option is the one that suggests interchanging 45 and 13, and $\times$ and $+$.

The correct interchange is 45 and 13, $\times$ and +.

Revision Table: Solving Equations by Interchange

Additional Information: Quantitative Aptitude Puzzles

Questions involving interchanging numbers and signs to balance an equation are common in quantitative aptitude sections of various exams. These puzzles test logical reasoning and arithmetic skills. Here are some tips for approaching them:

• Always check the options systematically. Do not guess.
• Clearly write down the modified equation after applying the interchange.
• Strictly follow the order of operations (BODMAS/PEMDAS) at every step of the calculation.
• Pay attention to signs and their precedence.
• Practice different types of interchange problems to become familiar with potential pitfalls.