The problem asks us to find which interchange of numbers and mathematical signs will make the given equation correct. The equation is:

\(72 \div 6 - 20 + 30 \times 5 = 70\)

Analyzing the Original Equation

First, let's evaluate the original equation using the order of operations (BODMAS/PEMDAS):

Division: \(72 \div 6 = 12\)
Multiplication: \(30 \times 5 = 150\)
The equation becomes: \(12 - 20 + 150\)
Addition and Subtraction (from left to right): \(12 - 20 = -8\)
\(-8 + 150 = 142\)

So, the original equation evaluates to 142, which is not equal to 70. We need to find an interchange that makes the left side equal to 70.

Testing the Interchange Options

We will test each option provided by performing the specified interchange and then evaluating the new equation.

Option 1: Interchange 5 and 20, and − and ×

Original equation: \(72 \div 6 - 20 + 30 \times 5 = 70\)
Interchange 5 with 20, and the sign − with the sign ×.

The new equation becomes:
\(72 \div 6 \times 5 + 30 - 20 = 70\)

Let's evaluate this new equation using the order of operations:

• Division: \(72 \div 6 = 12\)
• Multiplication: \(12 \times 5 = 60\)
• The equation is now: \(60 + 30 - 20\)
• Addition: \(60 + 30 = 90\)
• Subtraction: \(90 - 20 = 70\)

The result is 70. This matches the target value on the right side of the original equation. Therefore, this interchange makes the equation correct.

Option 2: Interchange 20 and 30, and × and ÷

Original equation: \(72 \div 6 - 20 + 30 \times 5 = 70\)
Interchange 20 with 30, and the sign × with the sign ÷.

The new equation becomes:
\(72 \times 6 - 30 + 20 \div 5 = 70\)

Let's evaluate this new equation:

• Division: \(20 \div 5 = 4\)
• Multiplication: \(72 \times 6 = 432\)
• The equation is now: \(432 - 30 + 4\)
• Subtraction: \(432 - 30 = 402\)
• Addition: \(402 + 4 = 406\)

The result is 406, which is not equal to 70. This interchange does not make the equation correct.

Option 3: Interchange 5 and 70, and + and −

Original equation: \(72 \div 6 - 20 + 30 \times 5 = 70\)
Interchange 5 with 70, and the sign + with the sign −. Note that 70 is the target value, not a number within the equation's left side. We will interpret this as swapping the number 5 on the left side with 70, and swapping the + sign with the - sign on the left side.

The new equation becomes:
\(72 \div 6 + 20 - 30 \times 70 = 70\)

Let's evaluate this new equation:

• Division: \(72 \div 6 = 12\)
• Multiplication: \(30 \times 70 = 2100\)
• The equation is now: \(12 + 20 - 2100\)
• Addition: \(12 + 20 = 32\)
• Subtraction: \(32 - 2100 = -2068\)

The result is -2068, which is not equal to 70. This interchange does not make the equation correct based on this interpretation.

Option 4: Interchange 6 and 20, and + and ÷

Original equation: \(72 \div 6 - 20 + 30 \times 5 = 70\)
Interchange 6 with 20, and the sign + with the sign ÷.

The new equation becomes:
\(72 \div 20 - 6 \div 30 \times 5 = 70\)

Let's evaluate this new equation:

• Division 1: \(72 \div 20 = 3.6\)
• Division 2: \(6 \div 30 = 0.2\)
• The equation is now: \(3.6 - 0.2 \times 5\)
• Multiplication: \(0.2 \times 5 = 1\)
• Subtraction: \(3.6 - 1 = 2.6\)

The result is 2.6, which is not equal to 70. This interchange does not make the equation correct.

Based on the evaluation of each option, the interchange specified in Option 1 is the only one that makes the given equation correct.

Revision Table: Equation Interchange

Additional Information on Order of Operations

The order of operations is a rule used in mathematics to clarify which procedures should be performed first in a given mathematical expression. Common mnemonics to remember this order are BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers and square 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).

Applying this order consistently is crucial when solving equations involving multiple operations, as demonstrated in the solution above where division and multiplication were performed before addition and subtraction in each step.