Solve Equation by Interchanging Signs

The task is to find which pair of mathematical signs needs to be interchanged in the equation 18 + 3 × 27 − 9 ÷ 3 = 78 to make it true.

We will test the options by swapping the signs and checking if the equation holds true using the order of operations (PEMDAS/BODMAS: Multiplication and Division from left to right, then Addition and Subtraction from left to right).

Checking Original Equation

Evaluate the expression 18 + 3 × 27 − 9 ÷ 3:

• Perform division: \( 9 \div 3 = 3 \)
• Perform multiplication: \( 3 \times 27 = 81 \)
• The expression becomes: \( 18 + 81 - 3 \)
• Perform addition: \( 18 + 81 = 99 \)
• Perform subtraction: \( 99 - 3 = 96 \)

The original equation evaluates to 96, not 78. Thus, sign interchange is necessary.

Testing Option: '+' and '×' Interchange

Let's interchange the addition (+) sign and the multiplication (×) sign as suggested by Option 1.

The equation becomes: 18 × 3 + 27 - 9 ÷ 3 = 78

Now, evaluate this new expression:

• Perform multiplication: \( 18 \times 3 = 54 \)
• Perform division: \( 9 \div 3 = 3 \)
• The expression becomes: \( 54 + 27 - 3 \)
• Perform addition: \( 54 + 27 = 81 \)
• Perform subtraction: \( 81 - 3 = 78 \)

The result is 78, which matches the equation's right-hand side.

Final Conclusion

Interchanging the addition (+) and multiplication (×) signs makes the equation correct.