Solving Equations by Interchanging Numbers and Signs

The problem asks us to find which combination of interchanging numbers and mathematical signs will make the given equation true.

The given equation is:

\(5 \div 10 \times 12 + 14 - 7 = 88\)

We need to test each option by applying the suggested interchanges and then evaluating the resulting equation using the order of operations (BODMAS/PEMDAS: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Testing Option 1: Interchange 10 and 5, and - and \(\times\)

Let's apply the interchanges from option 1:

• Interchange 10 and 5: The number 10 becomes 5, and the number 5 becomes 10.
• Interchange - and \(\times\): The minus sign (-) becomes a multiplication sign (\(\times\)), and the multiplication sign (\(\times\)) becomes a minus sign (-).

Applying these changes to the original equation \(5 \div 10 \times 12 + 14 - 7 = 88\), we get a new equation:

\(10 \div 5 - 12 + 14 \times 7\)

Now, let's evaluate this new equation step-by-step using BODMAS:

1. Division: \(10 \div 5 = 2\). The equation becomes \(2 - 12 + 14 \times 7\).
2. Multiplication: \(14 \times 7 = 98\). The equation becomes \(2 - 12 + 98\).
3. Subtraction and Addition (from left to right):
   • \(2 - 12 = -10\). The equation becomes \(-10 + 98\).
   • \(-10 + 98 = 88\).

The result of the evaluation is 88. The original equation is required to equal 88. Since \(88 = 88\), the interchanges suggested in Option 1 make the equation correct.

Let's quickly summarize the changes and the result for Option 1 in a table:

New Equation: \(10 \div 5 - 12 + 14 \times 7\)

Evaluation: \(10 \div 5 - 12 + 14 \times 7 = 2 - 12 + 98 = -10 + 98 = 88\)

Result: \(88 = 88\). The equation is correct with these interchanges.

Since Option 1 makes the equation correct, we do not need to test the other options.

Revision Table: Number and Sign Interchange Problems

Additional Information on Equation Solving

Solving equations often involves isolating a variable or verifying if a given set of values or conditions (like interchanges) satisfies the equality. In this type of problem, we are not solving for a variable, but rather testing specific modifications to the equation structure itself.

• Always perform operations strictly following the order of operations (BODMAS/PEMDAS).
• Be careful with negative numbers and the order of subtraction and addition.
• When interchanging, ensure both directions of the swap are applied (e.g., if 5 and 10 are interchanged, every 5 becomes 10 and every 10 becomes 5).

These types of problems test your understanding of mathematical operations and your ability to follow instructions precisely to modify and evaluate an expression.