Solving Mathematical Equations by Interchanging Numbers and Signs

The question asks us to find the new values of two given equations after interchanging specific numbers and signs. We need to swap the number 6 with 4, and the sign '–' with '×'. Then, we will evaluate the modified equations using the BODMAS rule (or PEMDAS).

Applying the Interchange Rules

The rules for interchange are:

• Number 6 becomes 4, and Number 4 becomes 6.
• Sign '–' becomes '×', and Sign '×' becomes '–'.

Evaluating Equation I

The original Equation I is: \(7 \times 6 + 8 \div 2 \ndash 4\)

Applying the interchange rules:

• The sign '×' becomes '–'.
• The number 6 becomes 4.
• The sign '–' becomes '×'.
• The number 4 becomes 6.

The modified Equation I becomes: \(7 \ndash 4 + 8 \div 2 \times 6\)

Now, we evaluate the modified Equation I using the BODMAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Step-by-step evaluation:

1. Division: \(8 \div 2 = 4\)
2. The equation is now: \(7 \ndash 4 + 4 \times 6\)
3. Multiplication: \(4 \times 6 = 24\)
4. The equation is now: \(7 \ndash 4 + 24\)
5. Subtraction/Addition (from left to right): \(7 \ndash 4 = 3\)
6. The equation is now: \(3 + 24\)
7. Addition: \(3 + 24 = 27\)

The value of Equation I after interchanging numbers and signs is 27.

Evaluating Equation II

The original Equation II is: \(4 \ndash 7 \times 6 + 8 \div 2\)

Applying the interchange rules:

• The number 4 becomes 6.
• The sign '–' becomes '×'.
• The sign '×' becomes '–'.
• The number 6 becomes 4.

The modified Equation II becomes: \(6 \times 7 \ndash 4 + 8 \div 2\)

Now, we evaluate the modified Equation II using the BODMAS rule.

Step-by-step evaluation:

1. Division: \(8 \div 2 = 4\)
2. The equation is now: \(6 \times 7 \ndash 4 + 4\)
3. Multiplication: \(6 \times 7 = 42\)
4. The equation is now: \(42 \ndash 4 + 4\)
5. Subtraction/Addition (from left to right): \(42 \ndash 4 = 38\)
6. The equation is now: \(38 + 4\)
7. Addition: \(38 + 4 = 42\)

The value of Equation II after interchanging numbers and signs is 42.

Summary of Results

After interchanging the numbers 6 and 4, and the signs – and ×, the values of the equations are:

• Equation I: 27
• Equation II: 42

Therefore, the values of equations (I) and (II) respectively are 27 and 42.

Revision Table: Interchanging Numbers and Signs

Additional Information: Understanding BODMAS/PEMDAS

The BODMAS rule is a mnemonic used to remember the order of operations in mathematical expressions. It stands for:

• Brackets
• Orders (powers, roots, etc.)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

PEMDAS is another common mnemonic used in some regions, standing for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Both rules essentially prescribe the same order of operations.

When solving mathematical expressions involving multiple operations, it is crucial to follow the order defined by BODMAS/PEMDAS to arrive at the correct result. Failing to follow this order can lead to incorrect answers.