Balancing Mathematical Equation with Signs

The problem asks us to find the correct sequence of mathematical signs to replace the '%' symbols in the expression \(35 \% 4 \% 12 \% 3 \% 6 \% 5 \% 5\) so that the equation balances. We are given four options, each providing a sequence of six signs. We need to test each option to see which one results in a true mathematical statement.

The expression has six '%' symbols. This means the sequence of signs provided in each option must replace the symbols from left to right.

Testing Options to Balance the Equation

We will test each option by substituting the signs into the expression and evaluating the resulting equation. Remember to follow the order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Testing Option 1: +, ̶ , \(\div\), \(\times\), =, +

Substituting these signs into the expression gives the equation:

\(35 + 4 - 12 \div 3 \times 6 = 5 + 5\)

Let's evaluate both sides of the equation:

• Left Hand Side (LHS): \(35 + 4 - 12 \div 3 \times 6\)
• First, perform division: \(12 \div 3 = 4\)
• Equation becomes: \(35 + 4 - 4 \times 6\)
• Next, perform multiplication: \(4 \times 6 = 24\)
• Equation becomes: \(35 + 4 - 24\)
• Perform addition and subtraction from left to right: \(35 + 4 = 39\)
• Equation becomes: \(39 - 24\)
• Finally, \(39 - 24 = 15\)

So, LHS = \(15\)

• Right Hand Side (RHS): \(5 + 5\)
• Perform addition: \(5 + 5 = 10\)

So, RHS = \(10\)

Comparing LHS and RHS: \(15 \neq 10\). Option 1 does not balance the equation.

Testing Option 2: - , +, \(\times\), =, \(\div\), +

Substituting these signs into the expression gives the equation:

\(35 - 4 + 12 \times 3 = 6 \div 5 + 5\)

Let's evaluate both sides of the equation:

• Left Hand Side (LHS): \(35 - 4 + 12 \times 3\)
• First, perform multiplication: \(12 \times 3 = 36\)
• Equation becomes: \(35 - 4 + 36\)
• Perform subtraction and addition from left to right: \(35 - 4 = 31\)
• Equation becomes: \(31 + 36\)
• Finally, \(31 + 36 = 67\)

So, LHS = \(67\)

• Right Hand Side (RHS): \(6 \div 5 + 5\)
• First, perform division: \(6 \div 5 = 1.2\)
• Equation becomes: \(1.2 + 5\)
• Finally, \(1.2 + 5 = 6.2\)

So, RHS = \(6.2\)

Comparing LHS and RHS: \(67 \neq 6.2\). Option 2 does not balance the equation.

Testing Option 3: - , +, \(\div\), \(\times\), =, +

Substituting these signs into the expression gives the equation:

\(35 - 4 + 12 \div 3 \times 6 = 5 + 5\)

Let's evaluate both sides of the equation:

• Left Hand Side (LHS): \(35 - 4 + 12 \div 3 \times 6\)
• First, perform division: \(12 \div 3 = 4\)
• Equation becomes: \(35 - 4 + 4 \times 6\)
• Next, perform multiplication: \(4 \times 6 = 24\)
• Equation becomes: \(35 - 4 + 24\)
• Perform subtraction and addition from left to right: \(35 - 4 = 31\)
• Equation becomes: \(31 + 24\)
• Finally, \(31 + 24 = 55\)

So, LHS = \(55\)

• Right Hand Side (RHS): \(5 + 5\)
• Perform addition: \(5 + 5 = 10\)

So, RHS = \(10\)

Comparing LHS and RHS: \(55 \neq 10\). Option 3 does not balance the equation.

Testing Option 4: - , +, \(\div\), =, \(\times\), +

Substituting these signs into the expression gives the equation:

\(35 - 4 + 12 \div 3 = 6 \times 5 + 5\)

Let's evaluate both sides of the equation:

• Left Hand Side (LHS): \(35 - 4 + 12 \div 3\)
• First, perform division: \(12 \div 3 = 4\)
• Equation becomes: \(35 - 4 + 4\)
• Perform subtraction and addition from left to right: \(35 - 4 = 31\)
• Equation becomes: \(31 + 4\)
• Finally, \(31 + 4 = 35\)

So, LHS = \(35\)

• Right Hand Side (RHS): \(6 \times 5 + 5\)
• First, perform multiplication: \(6 \times 5 = 30\)
• Equation becomes: \(30 + 5\)
• Finally, \(30 + 5 = 35\)

So, RHS = \(35\)

Comparing LHS and RHS: \(35 = 35\). Option 4 balances the equation.

Therefore, the correct combination of signs is - , +, \(\div\), =, \(\times\), +.

Revision Table: Key Concepts for Equation Balancing

Additional Information on Mathematical Operations

When working with mathematical expressions, especially those involving multiple operations, correctly applying the order of operations is vital. This ensures that everyone gets the same result when evaluating the same expression.

The common acronyms BODMAS or PEMDAS help remember the order:

• Brackets (Parentheses)
• Orders (Exponents, square roots, etc.) / Exponents
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In this problem, we used division, multiplication, addition, and subtraction. Division and multiplication are done before addition and subtraction. If both division and multiplication (or addition and subtraction) are present, you perform them in the order they appear from left to right.

An equation is a statement that two mathematical expressions are equal. The symbol '=' signifies this equality. To 'balance' an equation means to find the values or conditions (like the correct signs in this problem) that make the statement of equality true.