Solving the Equation by Swapping Numbers

The problem asks us to find which pair of numbers in the given equation, \(104 \div 4 - 6 + 13 \times 5 = 22\), should be interchanged to make the equation mathematically correct.

First, let's evaluate the original equation using the order of operations (BODMAS/PEMDAS):

\(104 \div 4 - 6 + 13 \times 5\)

• Division: \(104 \div 4 = 26\)
• Multiplication: \(13 \times 5 = 65\)

Substitute these values back into the equation:

\(26 - 6 + 65\)

• Subtraction: \(26 - 6 = 20\)
• Addition: \(20 + 65 = 85\)

So, the original equation evaluates to \(85 = 22\), which is false. We need to swap numbers to achieve the result 22.

Testing the Options for Swapping Numbers

We will test each given option by interchanging the specified numbers and then evaluating the new equation.

Option 1: Interchange 13 and 4

If we swap 13 and 4, the equation becomes:

\(104 \div 13 - 6 + 4 \times 5\)

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

• Division: \(104 \div 13 = 8\)
• Multiplication: \(4 \times 5 = 20\)

Substitute back into the equation:

\(8 - 6 + 20\)

• Subtraction: \(8 - 6 = 2\)
• Addition: \(2 + 20 = 22\)

The result is 22. So, \(22 = 22\). This option makes the equation correct.

Option 2: Interchange 6 and 4

If we swap 6 and 4, the equation becomes:

\(104 \div 6 - 4 + 13 \times 5\)

Evaluating this equation:

• Division: \(104 \div 6\) (This does not result in a whole number, which suggests this might not be the correct option if we expect integer results)
• Multiplication: \(13 \times 5 = 65\)

The equation is approximately \(17.33 - 4 + 65 = 13.33 + 65 = 78.33\). This is not 22.

Option 3: Interchange 6 and 5

If we swap 6 and 5, the equation becomes:

\(104 \div 4 - 5 + 13 \times 6\)

Evaluating this equation:

• Division: \(104 \div 4 = 26\)
• Multiplication: \(13 \times 6 = 78\)

Substitute back into the equation:

\(26 - 5 + 78\)

• Subtraction: \(26 - 5 = 21\)
• Addition: \(21 + 78 = 99\)

The result is 99. This is not 22.

Option 4: Interchange 13 and 6

If we swap 13 and 6, the equation becomes:

\(104 \div 4 - 13 + 6 \times 5\)

Evaluating this equation:

• Division: \(104 \div 4 = 26\)
• Multiplication: \(6 \times 5 = 30\)

Substitute back into the equation:

\(26 - 13 + 30\)

• Subtraction: \(26 - 13 = 13\)
• Addition: \(13 + 30 = 43\)

The result is 43. This is not 22.

Based on testing all options, interchanging 13 and 4 is the only swap that results in the equation being mathematically correct.

Revision Table: Key Concepts

Additional Information: Understanding Mathematical Equations and Operations

A mathematical equation is a statement that two expressions are equal. It uses an equals sign (=). The goal is often to find values that make the statement true or, as in this case, to manipulate the equation so it becomes true.

The operations in the equation are division (\(\div\)), subtraction (\(-\)), addition (\(+\)), and multiplication (\(\times\)). The order in which these operations are performed is critical and is governed by the order of operations rule (BODMAS or PEMDAS).

• BODMAS: Brackets, Orders (powers, square roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).

Applying this rule ensures that everyone gets the same answer when evaluating an expression. In this problem, by swapping numbers, we change the operands for these operations, thus changing the final result of the expression on the left side of the equation. We needed to find the swap that makes the left side equal to the right side (22).

This type of problem tests your understanding of basic arithmetic operations and the importance of the correct order of operations when solving mathematical expressions.