Solving Equations by Interchanging Signs and Numbers

This question requires us to find which pair of numbers and which pair of signs, when swapped in the given equation, make the equation mathematically correct. The given equation is:

\(24 - (9 \div 12) + (84 \times 4) + 28 = 23\)

We need to test each option by performing the proposed interchanges and then evaluating the resulting equation using the BODMAS/PEMDAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Analysing the Original Equation

Let's first evaluate the original equation to confirm it is incorrect:

\(24 - (9 \div 12) + (84 \times 4) + 28\)

Following BODMAS:

• Brackets: \(9 \div 12 = 0.75\), \(84 \times 4 = 336\)
• Equation becomes: \(24 - 0.75 + 336 + 28\)
• Addition/Subtraction (from left to right): \(24 - 0.75 = 23.25\)
• \(23.25 + 336 = 359.25\)
• \(359.25 + 28 = 387.25\)

So, \(387.25 = 23\), which is False.

Testing the Interchange Options

Option 1: Interchange 24 and 28, - and ÷

Original equation: \(24 - (9 \div 12) + (84 \times 4) + 28 = 23\)

After interchanging 24 with 28 and - with ÷:

\(28 \div (9 - 12) + (84 \times 4) + 24 = 23\)

Evaluate:

• Brackets: \((9 - 12) = -3\), \((84 \times 4) = 336\)
• Equation becomes: \(28 \div (-3) + 336 + 24\)
• Division: \(28 \div (-3) \approx -9.33\)
• Equation becomes: \(-9.33 + 336 + 24\)
• Addition/Subtraction: \(-9.33 + 336 = 326.67\), \(326.67 + 24 = 350.67\)

\(350.67 = 23\), which is False.

Option 2: Interchange 24 and 28, × and ÷

Original equation: \(24 - (9 \div 12) + (84 \times 4) + 28 = 23\)

After interchanging 24 with 28 and × with ÷:

\(28 - (9 \times 12) + (84 \div 4) + 24 = 23\)

Evaluate:

• Brackets: \((9 \times 12) = 108\), \((84 \div 4) = 21\)
• Equation becomes: \(28 - 108 + 21 + 24\)
• Addition/Subtraction: \(28 - 108 = -80\), \(-80 + 21 = -59\), \(-59 + 24 = -35\)

\(-35 = 23\), which is False.

Option 3: Interchange 9 and 4, ÷ and ×

Original equation: \(24 - (9 \div 12) + (84 \times 4) + 28 = 23\)

After interchanging 9 with 4 and ÷ with ×:

\(24 - (4 \times 12) + (84 \div 9) + 28 = 23\)

Evaluate:

• Brackets: \((4 \times 12) = 48\), \((84 \div 9) \approx 9.33\)
• Equation becomes: \(24 - 48 + 9.33 + 28\)
• Addition/Subtraction: \(24 - 48 = -24\), \(-24 + 9.33 = -14.67\), \(-14.67 + 28 = 13.33\)

\(13.33 = 23\), which is False.

Option 4: Interchange 12 and 4, ÷ and ×

Original equation: \(24 - (9 \div 12) + (84 \times 4) + 28 = 23\)

After interchanging 12 with 4 and ÷ with ×:

\(24 - (9 \times 4) + (84 \div 12) + 28 = 23\)

Evaluate using BODMAS:

• Brackets: \((9 \times 4) = 36\), \((84 \div 12) = 7\)
• Equation becomes: \(24 - 36 + 7 + 28\)
• Addition/Subtraction (from left to right): \(24 - 36 = -12\)
• \(-12 + 7 = -5\)
• \(-5 + 28 = 23\)

The equation becomes \(23 = 23\), which is True.

Thus, interchanging the numbers 12 and 4, and the signs ÷ and × makes the equation correct.

Revision Table: Key Concepts

Additional Information: Solving Operator and Number Interchange Problems

Problems involving the interchange of operators and numbers test your understanding of mathematical operations and the order of operations. Here are some tips for solving such problems efficiently:

• Always follow the BODMAS/PEMDAS rule strictly after performing the interchanges.
• Test each option methodically. It's easy to make calculation errors, so double-check each step.
• Pay close attention to the signs (positive and negative) and the order of operations, especially with division and subtraction.
• Sometimes, you can eliminate options by quickly estimating the result after interchange, if the numbers are large or the required target value is significantly different from the original calculation.
• Practice with different types of equations and interchanges to become faster and more accurate.

These types of problems are common in logical reasoning and quantitative aptitude tests.