Evaluating Equations After Interchanging Numbers and Signs

The problem asks us to find the values of two given equations after interchanging specific numbers and signs. We need to interchange the number 5 and the number 4, and also interchange the addition sign (+) and the multiplication sign (×).

Let's apply these interchanges to each equation one by one and then evaluate them using the order of operations (BODMAS/PEMDAS).

Step-by-Step Evaluation of Equation (I)

The original equation (I) is:

\( 4 + 6 \times 5 - 7 \div 1 \)

Now, let's apply the interchanges:

• Interchange 5 and 4. The equation becomes: \( 5 + 6 \times 4 - 7 \div 1 \)
• Interchange + and ×. The equation becomes: \( 5 \times 6 + 4 - 7 \div 1 \)

The modified equation (I) is:

\( 5 \times 6 + 4 - 7 \div 1 \)

Now, let's evaluate this using the BODMAS rule (Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

1. Division: Evaluate \( 7 \div 1 \).
   \( 5 \times 6 + 4 - 7 \)
2. Multiplication: Evaluate \( 5 \times 6 \).
   \( 30 + 4 - 7 \)
3. Addition: Evaluate \( 30 + 4 \).
   \( 34 - 7 \)
4. Subtraction: Evaluate \( 34 - 7 \).
   \( 27 \)

So, the value of equation (I) after the interchanges is 27.

Step-by-Step Evaluation of Equation (II)

The original equation (II) is:

\( 5 \times 3 - 4 + 8 \div 2 \)

Now, let's apply the interchanges:

• Interchange 5 and 4. The equation becomes: \( 4 \times 3 - 5 + 8 \div 2 \)
• Interchange + and ×. The equation becomes: \( 4 + 3 - 5 \times 8 \div 2 \)

The modified equation (II) is:

\( 4 + 3 - 5 \times 8 \div 2 \)

Now, let's evaluate this using the BODMAS rule.

1. Multiplication and Division (from left to right): First, \( 5 \times 8 \).
   \( 4 + 3 - 40 \div 2 \)
   Next, \( 40 \div 2 \).
   \( 4 + 3 - 20 \)
2. Addition: Evaluate \( 4 + 3 \).
   \( 7 - 20 \)
3. Subtraction: Evaluate \( 7 - 20 \).
   \( -13 \)

So, the value of equation (II) after the interchanges is -13.

Summary of Equation Values

After interchanging the number 5 with 4, and the sign + with ×, the values of the equations are:

• Equation (I): 27
• Equation (II): -13

The values of equation (I) and (II) respectively are 27 and -13.

Revision Table: Key Concepts

Additional Information: Understanding BODMAS/PEMDAS

The order of operations is essential for consistently evaluating mathematical expressions. Let's look at BODMAS and PEMDAS:

• BODMAS: Brackets, Orders (powers/roots), 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).

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

In this problem, we only had Division, Multiplication, Addition, and Subtraction, so the order followed was Division/Multiplication first (left-to-right), then Addition/Subtraction first (left-to-right).