Solving Equations by Sign Interchange

The problem asks us to find which interchange of mathematical signs in the given equation $320 \times 4 - 16 \div 4 + 2 = 18$ will make the equation correct. We need to test each option by substituting the specified signs and then evaluating the expression using the BODMAS/PEMDAS rule (Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Let's examine each option:

Option 1: Interchange + and −

If we interchange $+$ and $\minus$, the equation becomes:

Original: $320 \times 4 - 16 \div 4 + 2 = 18$

After interchange: $320 \times 4 + 16 \div 4 - 2$

Let's evaluate the new expression:

• Division: $16 \div 4 = 4$
• Multiplication: $320 \times 4 = 1280$
• The expression becomes: $1280 + 4 - 2$
• Addition/Subtraction: $1280 + 4 = 1284$, then $1284 - 2 = 1282$

So, $1282 \neq 18$. This option is incorrect.

Option 2: Interchange ÷ and ×

If we interchange $\div$ and $\times$, the equation becomes:

Original: $320 \times 4 - 16 \div 4 + 2 = 18$

After interchange: $320 \div 4 - 16 \times 4 + 2$

Let's evaluate the new expression:

• Division: $320 \div 4 = 80$
• Multiplication: $16 \times 4 = 64$
• The expression becomes: $80 - 64 + 2$
• Addition/Subtraction: $80 - 64 = 16$, then $16 + 2 = 18$

So, $18 = 18$. This option makes the equation correct.

Option 3: Interchange × and −

If we interchange $\times$ and $\minus$, the equation becomes:

Original: $320 \times 4 - 16 \div 4 + 2 = 18$

After interchange: $320 - 4 \times 16 \div 4 + 2$

Let's evaluate the new expression:

• Division: $16 \div 4 = 4$
• Multiplication: $4 \times 4 = 16$
• The expression becomes: $320 - 16 + 2$
• Addition/Subtraction: $320 - 16 = 304$, then $304 + 2 = 306$

So, $306 \neq 18$. This option is incorrect.

Option 4: Interchange × and +

If we interchange $\times$ and $+$, the equation becomes:

Original: $320 \times 4 - 16 \div 4 + 2 = 18$

After interchange: $320 + 4 - 16 \div 4 \times 2$

Let's evaluate the new expression:

• Division: $16 \div 4 = 4$
• Multiplication: $4 \times 2 = 8$
• The expression becomes: $320 + 4 - 8$
• Addition/Subtraction: $320 + 4 = 324$, then $324 - 8 = 316$

So, $316 \neq 18$. This option is incorrect.

Based on the evaluation of each option, interchanging the $\div$ and $\times$ signs makes the equation correct.

Revision Table: Understanding Sign Interchange Problems

Additional Information: BODMAS/PEMDAS Rule

The BODMAS rule (or PEMDAS in some regions) is a mnemonic used to remember the correct order of operations when evaluating mathematical expressions:

• Brackets (or Parentheses)
• Orders (or Exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

When performing operations after interchanging signs, strictly following this rule is crucial to arrive at the correct result. Division and Multiplication have equal priority and are performed from left to right. Similarly, Addition and Subtraction have equal priority and are performed from left to right.