Solving Equation by Interchanging Signs

The question asks us to find which two mathematical signs, when swapped in the given equation, make the equation correct. The equation is: $306 \div 17 + 4 - 50 \times 8 = 30$.

Currently, let's evaluate the left side of the equation using the standard order of operations (BODMAS/PEMDAS):

Division: $306 \div 17 = 18$

Equation becomes: $18 + 4 - 50 \times 8$

Multiplication: $50 \times 8 = 400$

Equation becomes: $18 + 4 - 400$

Addition: $18 + 4 = 22$

Equation becomes: $22 - 400$

Subtraction: $22 - 400 = -378$

So, $306 \div 17 + 4 - 50 \times 8 = -378$, which is not equal to $30$. We need to interchange two signs to achieve the result $30$. Let's test each option provided.

Testing Sign Interchange Options

We will apply the sign interchanges suggested in each option and then evaluate the resulting expression.

Option 1: Interchange '+' and '-'

Swap '+' and '-' in the original equation.

Original: $306 \div 17 + 4 - 50 \times 8 = 30$

After swap: $306 \div 17 - 4 + 50 \times 8$

Now, evaluate the new expression:

• Division: $306 \div 17 = 18$
• The expression is now: $18 - 4 + 50 \times 8$
• Multiplication: $50 \times 8 = 400$
• The expression is now: $18 - 4 + 400$
• Subtraction/Addition (from left to right): $18 - 4 = 14$
• The expression is now: $14 + 400$
• Addition: $14 + 400 = 414$

Result: $414$. This does not equal $30$.

Option 2: Interchange '+' and '×'

Swap '+' and '×' in the original equation.

Original: $306 \div 17 + 4 - 50 \times 8 = 30$

After swap: $306 \div 17 \times 4 - 50 + 8$

Now, evaluate the new expression:

• Division: $306 \div 17 = 18$
• The expression is now: $18 \times 4 - 50 + 8$
• Multiplication: $18 \times 4 = 72$
• The expression is now: $72 - 50 + 8$
• Subtraction/Addition (from left to right): $72 - 50 = 22$
• The expression is now: $22 + 8$
• Addition: $22 + 8 = 30$

Result: $30$. This equals the right side of the original equation.

Option 3: Interchange '÷' and '×'

Swap '÷' and '×' in the original equation.

Original: $306 \div 17 + 4 - 50 \times 8 = 30$

After swap: $306 \times 17 + 4 - 50 \div 8$

Now, evaluate the new expression:

• Division: $50 \div 8 = 6.25$
• The expression is now: $306 \times 17 + 4 - 6.25$
• Multiplication: $306 \times 17 = 5202$
• The expression is now: $5202 + 4 - 6.25$
• Addition/Subtraction (from left to right): $5202 + 4 = 5206$
• The expression is now: $5206 - 6.25$
• Subtraction: $5206 - 6.25 = 5199.75$

Result: $5199.75$. This does not equal $30$.

Option 4: Interchange '×' and '-'

Swap '×' and '-' in the original equation.

Original: $306 \div 17 + 4 - 50 \times 8 = 30$

After swap: $306 \div 17 + 4 \times 50 - 8$

Now, evaluate the new expression:

• Division: $306 \div 17 = 18$
• The expression is now: $18 + 4 \times 50 - 8$
• Multiplication: $4 \times 50 = 200$
• The expression is now: $18 + 200 - 8$
• Addition/Subtraction (from left to right): $18 + 200 = 218$
• The expression is now: $218 - 8$
• Subtraction: $218 - 8 = 210$

Result: $210$. This does not equal $30$.

Conclusion

After testing all options, we found that interchanging the '+' and '×' signs makes the equation correct, resulting in $30$ on the left side, which matches the right side.

Therefore, the signs that should be interchanged are + and ×.

Revision Table: Equation Sign Interchange

Understanding how interchanging signs affects an equation is a key skill in mathematical reasoning. It tests your ability to apply the order of operations correctly after modifying the expression.

Additional Information: BODMAS/PEMDAS and Operator Precedence

The order of operations is crucial when evaluating mathematical expressions, especially when multiple operators are present. The acronyms BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) help remember the sequence.

• B/P: Brackets or Parentheses first. Evaluate expressions inside them.
• O/E: Orders or Exponents (like powers and square roots) next.
• DM: Division and Multiplication come next. These have equal precedence and should be performed from left to right as they appear in the expression.
• AS: Addition and Subtraction come last. These also have equal precedence and should be performed from left to right as they appear.

When you interchange signs, you create a new expression. You must apply BODMAS/PEMDAS to this new expression to see if it evaluates to the target value (in this case, 30).

For example, in the expression $18 \times 4 - 50 + 8$, Multiplication ($18 \times 4$) is done before Subtraction ($72 - 50$) and Addition ($22 + 8$) according to the rules.