Understanding the Problem: Sign Interchange in Mathematical Expressions

The question asks us to evaluate a given mathematical expression after interchanging two specific signs: multiplication (×) and division (÷). We need to find the numerical value of the expression after performing this sign swap and then applying the correct order of operations.

The original expression is: 20 + 5 ÷ 6 × 2 – 5

We are required to interchange the '×' and '÷' signs.

Step-by-Step Evaluation After Interchanging Signs

First, let's rewrite the expression with the interchanged signs:

• Replace '÷' with '×'.
• Replace '×' with '÷'.

The new expression becomes: 20 + 5 × 6 ÷ 2 – 5

Now, we evaluate this new expression following the order of operations (BODMAS/PEMDAS).

BODMAS stands for:

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

Applying the BODMAS rule to 20 + 5 × 6 ÷ 2 – 5:

1. Division and Multiplication: We perform these operations from left to right.
   • First, perform the multiplication: $\qquad 5 \times 6 = 30$
   • The expression is now: $\qquad 20 + 30 \div 2 – 5$
   • Next, perform the division: $\qquad 30 \div 2 = 15$
   • The expression is now: $\qquad 20 + 15 – 5$
2. Addition and Subtraction: We perform these operations from left to right.
   • First, perform the addition: $\qquad 20 + 15 = 35$
   • The expression is now: $\qquad 35 – 5$
   • Finally, perform the subtraction: $\qquad 35 – 5 = 30$

The final value of the expression after interchanging the signs is 30.

Summary of Calculation

Original Expression: $20 + 5 \div 6 \times 2 – 5$
Signs to Interchange: $\times$ and $\div$
New Expression: $20 + 5 \times 6 \div 2 – 5$

Calculation:

$\qquad 20 + 5 \times 6 \div 2 – 5$
$\qquad 20 + 30 \div 2 – 5$ (Multiplication first)
$\qquad 20 + 15 – 5$ (Division next)
$\qquad 35 – 5$ (Addition next)
$\qquad 30$ (Subtraction last)

The value of the expression after interchanging the given two signs is 30.

Revision Table: Mathematical Operators and Order

Additional Information: Importance of Order of Operations

The order of operations is crucial in mathematics to ensure that expressions are evaluated consistently, leading to a unique result. If the operations are performed in a different order, the result would likely be incorrect. For example, in the expression $20 + 5 \times 3$, if we added first ($20+5=25$) and then multiplied ($25 \times 3 = 75$), we would get a different result than multiplying first ($5 \times 3 = 15$) and then adding ($20 + 15 = 35$). The BODMAS/PEMDAS rule provides a standard convention for this.

Problems involving interchanging signs test your understanding of both rewriting expressions based on specific rules and correctly applying the order of operations to the modified expression. Always remember to evaluate expressions step-by-step, dealing with operations according to their priority.