Solving Equation by Interchanging Signs and Numbers

The problem requires us to find which two signs and two numbers, when swapped in the given equation, make the equation mathematically correct.

The original equation is:

\(98 - 9 \times 21 \div 7 + 56 = 69\)

Let's evaluate the original equation using the BODMAS/PEDMAS rule to check if it is initially correct:

• Division first: \(21 \div 7 = 3\)
• Multiplication next: \(9 \times 3 = 27\)
• The equation becomes: \(98 - 27 + 56\)
• Addition/Subtraction from left to right: \(98 - 27 = 71\)
• Finally: \(71 + 56 = 127\)

So, the original equation is \(127 = 69\), which is false.

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

Testing Option 1: Interchange 56 and 9, + and ×

Interchange 56 with 9 and + with × in the original equation:

\(98 - \underline{9} \times 21 \div 7 \underline{+} \underline{56} = 69\)

New equation: \(98 \underline{-} \underline{56} \underline{+} 21 \div 7 \underline{\times} \underline{9} = 69\)

Let's evaluate the new equation using BODMAS/PEDMAS:

• Division first: \(21 \div 7 = 3\)
• Multiplication next: \(3 \times 9 = 27\)
• The equation becomes: \(98 - 56 + 27\)
• Addition/Subtraction from left to right: \(98 - 56 = 42\)
• Finally: \(42 + 27 = 69\)

So, the new equation is \(69 = 69\). This is correct.

Testing Option 2: Interchange 56 and 21, + and ×

Interchange 56 with 21 and + with ×:

Original: \(98 - 9 \times \underline{21} \div 7 \underline{+} \underline{56} = 69\)

New equation: \(98 - 9 \underline{+} \underline{56} \div 7 \underline{\times} \underline{21} = 69\)

Evaluate:

• Division: \(56 \div 7 = 8\)
• Multiplication: \(8 \times 21 = 168\)
• The equation becomes: \(98 - 9 + 168\)
• Subtraction: \(98 - 9 = 89\)
• Addition: \(89 + 168 = 257\)

So, \(257 = 69\), which is false.

Testing Option 3: Interchange 98 and 9, + and ×

Interchange 98 with 9 and + with ×:

Original: \(\underline{98} - \underline{9} \times 21 \div 7 \underline{+} 56 = 69\)

New equation: \(\underline{9} - \underline{98} \underline{\times} 21 \div 7 \underline{+} 56 = 69\)

Evaluate:

• Division: \(21 \div 7 = 3\)
• Multiplication: \(98 \times 3 = 294\)
• The equation becomes: \(9 - 294 + 56\)
• Subtraction: \(9 - 294 = -285\)
• Addition: \(-285 + 56 = -229\)

So, \(-229 = 69\), which is false. (Note: If multiplication and addition were swapped, the equation would be \(9 - 98 + 21 \div 7 \times 56\). Division: \(21 \div 7 = 3\). Multiplication: \(3 \times 56 = 168\). Equation: \(9 - 98 + 168\). Subtraction: \(9 - 98 = -89\). Addition: \(-89 + 168 = 79\). So, \(79 = 69\), still false.)

Testing Option 4: Interchange 21 and 7, + and −

Interchange 21 with 7 and + with −:

Original: \(98 \underline{-} 9 \times \underline{21} \div \underline{7} \underline{+} 56 = 69\)

New equation: \(98 \underline{+} 9 \times \underline{7} \div \underline{21} \underline{-} 56 = 69\)

Evaluate:

• Division: \(7 \div 21 = \frac{1}{3}\)
• Multiplication: \(9 \times \frac{1}{3} = 3\)
• The equation becomes: \(98 + 3 - 56\)
• Addition: \(98 + 3 = 101\)
• Subtraction: \(101 - 56 = 45\)

So, \(45 = 69\), which is false.

Based on the testing, interchanging 56 and 9, along with + and ×, makes the equation correct.

Revision Table: Equation Solving Concepts

Additional Information: Applying BODMAS/PEDMAS

The order of operations is crucial when evaluating expressions with multiple operations. Let's reiterate BODMAS/PEDMAS:

• B/P: Brackets or Parentheses - Solve expressions inside brackets first.
• O/E: Orders or Exponents - Calculate powers and square roots.
• D/M: Division and Multiplication - Perform these operations from left to right.
• A/S: Addition and Subtraction - Perform these operations from left to right.

When interchanging signs and numbers, always re-evaluate the modified expression strictly following this order to determine if the equation holds true.