Understanding the Equation and Number Interchange

The problem asks us to find which two numbers, when swapped in the given equation, make the equation mathematically correct. The original equation is:

\(119 + 11 \times 5 - 153 \div 17 = 201\)

To solve this, we need to test each option by interchanging the specified pair of numbers and then evaluate the modified equation using the order of operations (BODMAS/PEMDAS).

Evaluating the Original Equation

Before swapping any numbers, let's evaluate the original equation to see if it is already correct. We follow the BODMAS/PEMDAS rule:

1. Brackets/Parentheses
2. Orders/Exponents
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Applying this to \(119 + 11 \times 5 - 153 \div 17\):

• First, Division: \(153 \div 17 = 9\)
• Next, Multiplication: \(11 \times 5 = 55\)
• The equation becomes: \(119 + 55 - 9\)
• Now, Addition: \(119 + 55 = 174\)
• Finally, Subtraction: \(174 - 9 = 165\)

So, the original equation evaluates to \(165 = 201\), which is incorrect.

Testing the Interchange Options

We will now test each option by swapping the given pair of numbers in the original equation \(119 + 11 \times 5 - 153 \div 17 = 201\) and re-evaluating.

Option 1: Interchange 17 and 5

The equation becomes: \(119 + 11 \times 17 - 153 \div 5\)

• Division: \(153 \div 5\) is not an integer. This is likely not the correct swap if the result is an integer (201). \(153 \div 5 = 30.6\)
• Multiplication: \(11 \times 17 = 187\)
• The equation becomes: \(119 + 187 - 30.6\)
• Addition: \(119 + 187 = 306\)
• Subtraction: \(306 - 30.6 = 275.4\)

\(275.4 \neq 201\). So, this option is incorrect.

Option 2: Interchange 119 and 17

The equation becomes: \(17 + 11 \times 5 - 153 \div 119\)

• Division: \(153 \div 119\) is not an integer. \(153 \div 119 \approx 1.28\)
• Multiplication: \(11 \times 5 = 55\)
• The equation becomes: \(17 + 55 - 153/119\)
• Addition: \(17 + 55 = 72\)
• Subtraction: \(72 - 153/119 \neq 201\)

So, this option is incorrect.

Option 3: Interchange 119 and 153

The equation becomes: \(153 + 11 \times 5 - 119 \div 17\)

• Division: \(119 \div 17 = 7\)
• Multiplication: \(11 \times 5 = 55\)
• The equation becomes: \(153 + 55 - 7\)
• Addition: \(153 + 55 = 208\)
• Subtraction: \(208 - 7 = 201\)

\(201 = 201\). This matches the right side of the original equation. So, this option makes the equation correct.

Option 4: Interchange 11 and 17

The equation becomes: \(119 + 17 \times 5 - 153 \div 11\)

• Division: \(153 \div 11\) is not an integer. \(153 \div 11 \approx 13.9\)
• Multiplication: \(17 \times 5 = 85\)
• The equation becomes: \(119 + 85 - 153/11\)
• Addition: \(119 + 85 = 204\)
• Subtraction: \(204 - 153/11 \neq 201\)

So, this option is incorrect.

Based on the evaluation of each option, interchanging 119 and 153 makes the equation correct.

Revision Table: Order of Operations (BODMAS/PEMDAS)

Additional Information: Solving Equations by Number Interchange

Problems involving interchanging numbers or mathematical operators require careful application of the order of operations. The goal is to find a rearrangement that satisfies the equality. This often involves trial and error, systematically checking each possibility provided in the options.

• Always evaluate the original expression/equation first to understand the starting point.
• For each interchange option, rewrite the equation clearly with the numbers swapped.
• Strictly follow the order of operations when evaluating the modified equation.
• Perform calculations step by step to avoid errors.
• Compare the result of the evaluation with the target value on the other side of the equality sign.
• The option that results in a true equality is the correct answer.