Making Math Equations Correct by Interchanging Signs

This question asks us to find which two mathematical signs, when swapped in the given equation, make the equation true. The original equation is:

63 ÷ 21 – 7 + 28 × 3 = 144

To solve this, we need to evaluate the equation after interchanging signs for each given option and check if the result equals 144. We will follow the order of operations (BODMAS/PEMDAS) when evaluating the expressions.

Understanding BODMAS/PEMDAS

BODMAS or PEMDAS is a rule to remember the order of operations in mathematical expressions:

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

Evaluating the Original Equation

Let's first evaluate the given equation with the original signs:

\(63 \div 21 - 7 + 28 \times 3\)

1. Division: \(63 \div 21 = 3\)
2. Multiplication: \(28 \times 3 = 84\)
3. The expression becomes: \(3 - 7 + 84\)
4. Subtraction/Addition (from left to right): \(3 - 7 = -4\)
5. Addition: \(-4 + 84 = 80\)

So, \(80 = 144\), which is false. The original equation is incorrect.

Testing the Options for Sign Interchange

We will now test each option by swapping the indicated signs and re-evaluating the equation.

Option 1: Interchange + and ÷

Original equation: \(63 \div 21 - 7 + 28 \times 3 = 144\)

After interchanging + and ÷:

\(63 + 21 - 7 \div 28 \times 3\)

1. Division: \(7 \div 28 = \frac{7}{28} = \frac{1}{4} = 0.25\)
2. Multiplication: \(0.25 \times 3 = 0.75\)
3. The expression becomes: \(63 + 21 - 0.75\)
4. Addition/Subtraction: \(63 + 21 = 84\)
5. Subtraction: \(84 - 0.75 = 83.25\)

\(83.25 \neq 144\). Option 1 is incorrect.

Option 2: Interchange × and +

Original equation: \(63 \div 21 - 7 + 28 \times 3 = 144\)

After interchanging × and +:

\(63 \div 21 - 7 \times 28 + 3\)

1. Division: \(63 \div 21 = 3\)
2. Multiplication: \(7 \times 28 = 196\)
3. The expression becomes: \(3 - 196 + 3\)
4. Subtraction/Addition: \(3 - 196 = -193\)
5. Addition: \(-193 + 3 = -190\)

\(-190 \neq 144\). Option 2 is incorrect.

Option 3: Interchange ÷ and –

Original equation: \(63 \div 21 - 7 + 28 \times 3 = 144\)

After interchanging ÷ and –:

\(63 - 21 \div 7 + 28 \times 3\)

1. Division: \(21 \div 7 = 3\)
2. Multiplication: \(28 \times 3 = 84\)
3. The expression becomes: \(63 - 3 + 84\)
4. Subtraction/Addition: \(63 - 3 = 60\)
5. Addition: \(60 + 84 = 144\)

\(144 = 144\). Option 3 makes the equation correct.

Option 4: Interchange + and –

Original equation: \(63 \div 21 - 7 + 28 \times 3 = 144\)

After interchanging + and –:

\(63 \div 21 + 7 - 28 \times 3\)

1. Division: \(63 \div 21 = 3\)
2. Multiplication: \(28 \times 3 = 84\)
3. The expression becomes: \(3 + 7 - 84\)
4. Addition/Subtraction: \(3 + 7 = 10\)
5. Subtraction: \(10 - 84 = -74\)

\(-74 \neq 144\). Option 4 is incorrect.

Conclusion

After testing all options, we found that interchanging the signs ÷ and – makes the equation \(63 - 21 \div 7 + 28 \times 3\) evaluate to 144, which matches the right side of the original equation.

Revision Table: Key Concepts for Sign Interchange Problems

Here's a quick review of what we used to solve this problem:

• Order of Operations: Always follow BODMAS/PEMDAS.
• Systematic Testing: Test each option by carefully interchanging the specified signs.
• Re-evaluation: Calculate the value of the expression after each interchange using the correct order of operations.
• Comparison: Compare the result with the target value (144 in this case) to determine if the equation is correct.

Additional Information: Arithmetic Operations

This problem involves the basic arithmetic operations:

• Addition (+): Combining numbers.
• Subtraction (-): Finding the difference between numbers.
• Multiplication (×): Repeated addition.
• Division (÷): Splitting into equal parts.

Understanding how these operations work and the correct order in which to perform them is crucial for solving mathematical equations and expressions correctly.