Balancing the Mathematical Equation: Step-by-Step

The question asks us to find the correct sequence of mathematical signs to replace the asterisks (*) in the given expression, making the equation true. The expression is:

76 * 34 * 23 * 12 * 6 * 21

We need to test the options provided, substituting the operators in the order they appear and evaluating the expression using the standard order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right). The final operator in each option is '=', indicating the result the left side should equal.

Evaluating Operator Combinations

Let's examine each option to see which combination correctly balances the equation.

Testing Option 1: +, -, +, ÷, =

Substituting these signs: 76 + 34 - 23 + 12 ÷ 6 = 21

• First, perform the division: \( 12 \div 6 = 2 \)
• The equation becomes: \( 76 + 34 - 23 + 2 = 21 \)
• Next, perform addition and subtraction from left to right:
  • \( 76 + 34 = 110 \)
  • \( 110 - 23 = 87 \)
  • \( 87 + 2 = 89 \)
• The final result is \( 89 \), which is not equal to \( 21 \). Therefore, Option 1 is incorrect.

Testing Option 2: -, -, +, +, =

Substituting these signs: 76 - 34 - 23 + 12 + 6 = 21

• Perform subtraction and addition from left to right:
  • \( 76 - 34 = 42 \)
  • \( 42 - 23 = 19 \)
  • \( 19 + 12 = 31 \)
  • \( 31 + 6 = 37 \)
• The final result is \( 37 \), which is not equal to \( 21 \). Therefore, Option 2 is incorrect.

Testing Option 3: ×, -, +, ÷, =

Substituting these signs: 76 × 34 - 23 + 12 ÷ 6 = 21

• Perform multiplication and division first (from left to right):
  • \( 76 \times 34 = 2584 \)
  • \( 12 \div 6 = 2 \)
• The equation becomes: \( 2584 - 23 + 2 = 21 \)
• Perform subtraction and addition from left to right:
  • \( 2584 - 23 = 2561 \)
  • \( 2561 + 2 = 2563 \)
• The final result is \( 2563 \), which is not equal to \( 21 \). Therefore, Option 3 is incorrect.

Testing Option 4: -, -, +, ÷, =

Substituting these signs: 76 - 34 - 23 + 12 ÷ 6 = 21

• First, perform the division: \( 12 \div 6 = 2 \)
• The equation becomes: \( 76 - 34 - 23 + 2 = 21 \)
• Next, perform subtraction and addition from left to right:
  • \( 76 - 34 = 42 \)
  • \( 42 - 23 = 19 \)
  • \( 19 + 2 = 21 \)
• The final result is \( 21 \), which is equal to \( 21 \). This combination balances the equation.

Conclusion on Balancing Signs

By systematically testing each option and applying the correct order of operations, we found that Option 4, which uses the signs -, -, +, ÷, and =, is the correct combination that successfully balances the given mathematical equation.