Solving Equations by Interchanging Mathematical Signs

The question asks us to find which pair of mathematical signs, when interchanged in the given equation, makes the equation correct. The initial equation is:

\(25 + 21 \div 3 - 35 \times 5 = 81\)

First, let's evaluate the original equation using the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

\(25 + (21 \div 3) - (35 \times 5)\)

\(25 + 7 - 175\)

\(32 - 175\)

\(-143\)

Since \(-143 \neq 81\), the original equation is incorrect. We need to test each option by interchanging the specified signs and re-evaluating the equation.

Testing Option 1: Interchanging + and -

If we interchange + and - signs, the equation becomes:

\(25 - 21 \div 3 + 35 \times 5 = 81\)

Evaluate using BODMAS:

\(25 - (21 \div 3) + (35 \times 5)\)

\(25 - 7 + 175\)

\(18 + 175\)

\(193\)

Since \(193 \neq 81\), interchanging + and - is not the correct solution.

Testing Option 2: Interchanging ÷ and ×

If we interchange ÷ and × signs, the equation becomes:

\(25 + 21 \times 3 - 35 \div 5 = 81\)

Evaluate using BODMAS:

\(25 + (21 \times 3) - (35 \div 5)\)

\(25 + 63 - 7\)

\(88 - 7\)

\(81\)

Since \(81 = 81\), interchanging ÷ and × makes the equation correct. This is the required solution.

Testing Option 3: Interchanging + and ×

If we interchange + and × signs, the equation becomes:

\(25 \times 21 \div 3 - 35 + 5 = 81\)

Evaluate using BODMAS:

\(25 \times (21 \div 3) - 35 + 5\)

\(25 \times 7 - 35 + 5\)

\(175 - 35 + 5\)

\(140 + 5\)

\(145\)

Since \(145 \neq 81\), interchanging + and × is not the correct solution.

Testing Option 4: Interchanging × and -

If we interchange × and - signs, the equation becomes:

\(25 + 21 \div 3 \times 35 - 5 = 81\)

Evaluate using BODMAS:

\(25 + (21 \div 3) \times 35 - 5\)

\(25 + 7 \times 35 - 5\)

\(25 + 245 - 5\)

\(270 - 5\)

\(265\)

Since \(265 \neq 81\), interchanging × and - is not the correct solution.

Based on the evaluation of each option, interchanging the ÷ and × signs makes the equation correct.

Revision Table: Evaluating Sign Interchanges

Additional Information: Order of Operations (BODMAS/PEMDAS)

To correctly evaluate mathematical expressions, we follow a specific order of operations. This is often remembered using acronyms like BODMAS or PEMDAS.

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

Both acronyms represent the same order. Division and Multiplication have the same priority and should be performed from left to right as they appear. Similarly, Addition and Subtraction have the same priority and are performed from left to right.

In this problem, correctly applying BODMAS/PEMDAS after interchanging signs was crucial to determining which interchange resulted in the correct equation.