Solving Math Equations by Interchanging Signs

The problem asks us to identify which two mathematical signs in the given equation need to be swapped to make the equation correct. The original equation is:

\(23 + 84 \div 14 \times 8 - 3 = 5\)

Let's first evaluate the given equation using the standard order of operations (BODMAS/PEMDAS):

1. Division: \(84 \div 14 = 6\)
2. Multiplication: \(6 \times 8 = 48\)
3. Addition and Subtraction (from left to right): \(23 + 48 - 3 = 71 - 3 = 68\)

The result \(68\) is not equal to \(5\), so the original equation is incorrect.

Checking Options for Sign Interchange

We need to test each option by interchanging the specified signs and re-evaluating the equation.

Option 1: Interchanging ÷ and −

The new equation would be: \(23 + 84 - 14 \times 8 \div 3 = 5\)

Let's evaluate:

1. Multiplication: \(14 \times 8 = 112\)
2. Division: \(112 \div 3 = 37.33...\)

This does not seem to lead to an integer result like 5 easily. Let's check the option identified as correct.

Option 3: Interchanging − and ×

The original equation is: \(23 + 84 \div 14 \times 8 - 3 = 5\)

Swapping − and ×, the new equation becomes:

\(23 + 84 \div 14 - 8 \times 3 = 5\)

Now, let's evaluate this new equation following the order of operations:

1. Division: \(84 \div 14 = 6\)
2. Multiplication: \(8 \times 3 = 24\)
3. Addition and Subtraction (from left to right): \(23 + 6 - 24 = 29 - 24 = 5\)

The result \(5\) matches the right side of the equation. Therefore, interchanging the signs − and × makes the equation correct.

Verification of the Correct Interchange

By swapping the multiplication sign (×) and the subtraction sign (−), the equation transforms from:

\(23 + 84 \div 14 \times 8 - 3\)

to

\(23 + 84 \div 14 - 8 \times 3\)

Evaluating step-by-step:

\(23 + (84 \div 14) - (8 \times 3)\)

\(23 + 6 - 24\)

\(29 - 24\)

\(5\)

Since \(5 = 5\), the equation is correct after interchanging − and ×.

This confirms that interchanging the subtraction sign (−) and the multiplication sign (×) makes the equation correct.

Revision Table: Understanding Operator Precedence

Additional Information: Solving Sign Interchange Problems

Problems involving interchanging signs to correct an equation are common in logical reasoning and quantitative aptitude sections of various exams. The key steps are:

• Evaluate the original equation to see if it is correct.
• For each given option, swap the indicated signs in the original equation.
• Re-evaluate the new equation carefully, strictly following the order of operations (BODMAS/PEMDAS).
• Check if the result of the new equation matches the expected value on the right side of the equals sign.
• The option that yields the correct result is the answer.

It is important to perform the operations in the correct order, especially when division and multiplication or addition and subtraction appear in the same part of the equation; they are performed from left to right.