Solving the Mathematical Sign Interchange Problem

The question asks us to find which two signs in the given mathematical expression, when interchanged, will make the expression evaluate to the value '37'. The expression is:

27 + 3 ÷ 81 × 93 – 57

To solve this problem, we need to test each given option. For each option, we will interchange the specified signs in the expression and then evaluate the new expression following the order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Step-by-Step Evaluation of Options

Option 1: Interchange + and ÷

The expression becomes: 27 ÷ 3 + 81 × 93 – 57

Let's evaluate:

\(27 \div 3 + 81 \times 93 - 57\)

Perform Division first:

\(9 + 81 \times 93 - 57\)

Perform Multiplication:

\(9 + 7533 - 57\)

Perform Addition:

\(7542 - 57\)

Perform Subtraction:

\(7485\)

This value (7485) is not 37.

Option 2: Interchange – and +

The expression becomes: 27 – 3 ÷ 81 × 93 + 57

Let's evaluate:

\(27 - 3 \div 81 \times 93 + 57\)

Perform Division first:

\(27 - \frac{3}{81} \times 93 + 57\)

\(27 - \frac{1}{27} \times 93 + 57\)

Perform Multiplication:

\(27 - \frac{93}{27} + 57\)

\(27 - \frac{31}{9} + 57\)

Perform Addition/Subtraction from left to right:

\(84 - \frac{31}{9}\)

\(\frac{84 \times 9 - 31}{9}\)

\(\frac{756 - 31}{9}\)

\(\frac{725}{9}\)

This value (\(\frac{725}{9}\)) is not 37.

Option 3: Interchange × and +

The expression becomes: 27 × 3 ÷ 81 + 93 – 57

Let's evaluate:

\(27 \times 3 \div 81 + 93 - 57\)

Perform Multiplication and Division from left to right:

\(81 \div 81 + 93 - 57\)

\(1 + 93 - 57\)

Perform Addition and Subtraction from left to right:

\(94 - 57\)

\(37\)

This value (37) matches the required value.

Option 4: Interchange ÷ and –

The expression becomes: 27 + 3 – 81 × 93 ÷ 57

Let's evaluate:

\(27 + 3 - 81 \times 93 \div 57\)

Perform Multiplication and Division from left to right:

\(27 + 3 - 81 \times \frac{93}{57}\)

\(27 + 3 - 81 \times \frac{31}{19}\)

\(27 + 3 - \frac{81 \times 31}{19}\)

\(27 + 3 - \frac{2511}{19}\)

Perform Addition/Subtraction from left to right:

\(30 - \frac{2511}{19}\)

\(\frac{30 \times 19 - 2511}{19}\)

\(\frac{570 - 2511}{19}\)

\(\frac{-1941}{19}\)

This value (\(\frac{-1941}{19}\)) is not 37.

Based on the evaluation of each option, interchanging the '×' and '+' signs results in the expression evaluating to 37.

Revision Table: Key Concepts in Expression Evaluation

Additional Information: Understanding Order of Operations

The order of operations, often remembered by acronyms like BODMAS or PEMDAS, is crucial for consistently evaluating mathematical expressions. Without a standard order, an expression could have multiple different values depending on which operation is performed first. Here's a breakdown:

• Brackets/Parentheses: Operations inside brackets are always performed first. If there are nested brackets, work from the innermost ones outwards.
• Orders/Exponents: Next, evaluate anything involving powers or roots.
• Division and Multiplication: These operations are performed next. They have equal priority. If both appear in an expression, work from left to right.
• Addition and Subtraction: These are the last operations to be performed. They also have equal priority. If both appear, work from left to right.

Following this specific order ensures that everyone arrives at the same correct answer for any given mathematical expression. In problems involving interchanging signs, applying this order correctly after swapping the signs is key to finding the solution.