Balancing Mathematical Equations: Finding the Right Signs

The problem requires us to find the correct sequence of mathematical signs to replace the asterisks (*) in the equation 256 * 4 * 9 * 3 * 14 = 51 so that the equation holds true. We are given four options, each providing a different sequence of signs. To solve this, we need to test each sequence by substituting the signs into the equation and performing the calculations following the order of operations (BODMAS/PEMDAS).

The mathematical signs to consider are:

• Division ($\div$)
• Multiplication ($\times$)
• Subtraction ($-$)
• Addition ($+$)

Evaluating Each Option

Let's evaluate the expression for each given option:

Option 1: $\times, +, -, \div$

Replacing the * signs with this sequence, the equation becomes:

$\quad 256 \times 4 + 9 - 3 \div 14$

According to the order of operations, we perform multiplication and division first from left to right:

• $256 \times 4 = 1024$
• $3 \div 14 \approx 0.214$ (This indicates it's unlikely to be the correct option as the target is an integer 51)

Now substitute these values back:

$\quad 1024 + 9 - 0.214$

Perform addition and subtraction from left to right:

• $1024 + 9 = 1033$
• $1033 - 0.214 = 1032.786$

The result $1032.786$ does not equal $51$. So, Option 1 is incorrect.

Option 2: $\div, \times, -, +$

Replacing the * signs with this sequence, the equation becomes:

$\quad 256 \div 4 \times 9 - 3 + 14$

Perform division and multiplication from left to right:

• $256 \div 4 = 64$
• $64 \times 9 = 576$

Now substitute these values back:

$\quad 576 - 3 + 14$

Perform subtraction and addition from left to right:

• $576 - 3 = 573$
• $573 + 14 = 587$

The result $587$ does not equal $51$. So, Option 2 is incorrect.

Option 3: $\times, \div, +, -$

Replacing the * signs with this sequence, the equation becomes:

$\quad 256 \times 4 \div 9 + 3 - 14$

Perform multiplication and division from left to right:

• $256 \times 4 = 1024$
• $1024 \div 9 \approx 113.78$ (Again, a non-integer, unlikely to yield 51)

Substituting back:

$\quad 113.78 + 3 - 14$

This will not result in $51$. So, Option 3 is incorrect.

Option 4: $\div, -, \times, +$

Replacing the * signs with this sequence, the equation becomes:

$\quad 256 \div 4 - 9 \times 3 + 14$

Perform division and multiplication from left to right:

• $256 \div 4 = 64$
• $9 \times 3 = 27$

Now substitute these values back:

$\quad 64 - 27 + 14$

Perform subtraction and addition from left to right:

• $64 - 27 = 37$
• $37 + 14 = 51$

The result $51$ equals the right side of the equation. So, Option 4 is the correct combination.

Step-by-Step Calculation for the Correct Combination ($\div, -, \times, +$)

Let's show the calculation for Option 4 again clearly:

Equation: $256 * 4 * 9 * 3 * 14 = 51$

Substitute signs: $256 \div 4 - 9 \times 3 + 14 = 51$

1. First, perform division: $256 \div 4 = 64$
2. The equation becomes: $64 - 9 \times 3 + 14$
3. Next, perform multiplication: $9 \times 3 = 27$
4. The equation becomes: $64 - 27 + 14$
5. Now, perform subtraction from left to right: $64 - 27 = 37$
6. The equation becomes: $37 + 14$
7. Finally, perform addition: $37 + 14 = 51$

Since the result is $51$, the equation $256 \div 4 - 9 \times 3 + 14 = 51$ is balanced.

Conclusion: Identifying the Correct Signs

By testing each option and applying the correct order of operations, we found that the sequence of signs $\div, -, \times, +$ correctly balances the given equation $256 * 4 * 9 * 3 * 14 = 51$. This corresponds to Option 4.

Revision Table: Key Concepts in Equation Balancing

Additional Information: Understanding Order of Operations

The order of operations is a fundamental rule in mathematics that ensures everyone gets the same result when evaluating an expression with multiple operations. A common mnemonic for remembering the order is BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers and square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Both mnemonics represent the same order. Division and Multiplication are performed at the same level, from left to right as they appear. Similarly, Addition and Subtraction are performed at the same level, from left to right as they appear.

In our problem, when evaluating an option like $256 \div 4 - 9 \times 3 + 14$, we first handle the division ($256 \div 4$) and multiplication ($9 \times 3$) because they appear before addition and subtraction in the order. Since division appears before multiplication here, we do $256 \div 4$ first, then $9 \times 3$. After that, we are left with $64 - 27 + 14$. Now we handle subtraction and addition from left to right: $64 - 27 = 37$, then $37 + 14 = 51$. Following this specific order is key to solving these equation balancing problems correctly.