Finding the Correct Mathematical Signs to Balance an Equation

This problem requires us to identify the correct sequence of mathematical operators (+, -, ×, ÷) to substitute for the hash symbols (#) in the given expression, ultimately balancing the equation. The expression provided is:

[{(44 # 38) # (2 # 5)} # (4 # 2)] # 5 # 10

Our goal is to determine which combination of signs correctly fits into the '#' placeholders.

Step-by-Step Substitution and Evaluation

To solve this, we first substitute the operators suggested by the options into the expression. Then, we evaluate the resulting mathematical expression using the standard order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication/Division from left to right, Addition/Subtraction from left to right).

Based on the analysis of the options, the correct sequence of operators is: –, +, ×, ÷, ×, ×

Let's substitute these signs into the original expression:

[{(44 - 38) + (2 \times 5)} \div (4 \times 2)] \times 5 \times 10

Now, we proceed with the step-by-step evaluation:

Calculation Breakdown

• Step 1: Evaluate the innermost parentheses/brackets first.

  First, calculate the expression inside the innermost parentheses:

  (44 - 38) evaluates to 6.

  (2 \times 5) evaluates to 10.

  (4 \times 2) evaluates to 8.

  Substituting these results back into the expression, we get:

  [{(6) + (10)} \div (8)] \times 5 \times 10

• Step 2: Evaluate the next level of parentheses/brackets.

  Next, we calculate the sum within the curly braces:

  {6 + 10} evaluates to 16.

  The expression simplifies to:

  [16 \div 8] \times 5 \times 10

• Step 3: Evaluate the expression within the square brackets.

  Now, we perform the division within the square brackets:

  16 \div 8 evaluates to 2.

  The expression becomes:

  2 \times 5 \times 10

• Step 4: Perform the final multiplications.

  Finally, we perform the multiplications from left to right:

  2 \times 5 evaluates to 10.

  10 \times 10 evaluates to 100.

The expression evaluates to 100. Therefore, the sequence of signs –, +, ×, ÷, ×, × successfully balances the equation.