Finding the Correct Mathematical Signs to Balance an Equation

The question asks us to find the sequence of mathematical signs from the given options that will make the equation 60 * 2 * 3 * 6 * 5 * 43 true when the signs replace the asterisks in order.

We need to test each option by substituting the signs into the equation and performing the calculations according to the order of operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Testing Option 1: ÷, ×, +, -, =

Let's substitute the signs into the equation:

\(60 \div 2 \times 3 + 6 - 5 = 43\)

Now, we perform the calculations:

• First, Division: \(60 \div 2 = 30\)
• The equation becomes: \(30 \times 3 + 6 - 5 = 43\)
• Next, Multiplication: \(30 \times 3 = 90\)
• The equation becomes: \(90 + 6 - 5 = 43\)
• Now, Addition and Subtraction from left to right: \(90 + 6 = 96\)
• The equation becomes: \(96 - 5 = 43\)
• Finally: \(91 = 43\)

Since \(91 \neq 43\), this option is incorrect.

Testing Option 2: ÷, +, ×, =, -

Let's substitute the signs into the equation:

\(60 \div 2 + 3 \times 6 = 5 - 43\)

We evaluate both sides of the equation:

Left side: \(60 \div 2 + 3 \times 6\)

• First, Division: \(60 \div 2 = 30\)
• The left side becomes: \(30 + 3 \times 6\)
• Next, Multiplication: \(3 \times 6 = 18\)
• The left side becomes: \(30 + 18 = 48\)

Right side: \(5 - 43\)

• Subtraction: \(5 - 43 = -38\)

Comparing both sides: \(48 = -38\). Since \(48 \neq -38\), this option is incorrect.

Testing Option 3: ÷, +, ×, -, =

Let's substitute the signs into the equation:

\(60 \div 2 + 3 \times 6 - 5 = 43\)

Now, we perform the calculations following the order of operations:

• First, Division: \(60 \div 2 = 30\)
• The equation becomes: \(30 + 3 \times 6 - 5 = 43\)
• Next, Multiplication: \(3 \times 6 = 18\)
• The equation becomes: \(30 + 18 - 5 = 43\)
• Now, Addition from left to right: \(30 + 18 = 48\)
• The equation becomes: \(48 - 5 = 43\)
• Finally, Subtraction: \(48 - 5 = 43\)

This gives us \(43 = 43\), which is a balanced equation.

Testing Option 4: +, ÷, ×, =, -

Let's substitute the signs into the equation:

\(60 + 2 \div 3 \times 6 = 5 - 43\)

We evaluate both sides of the equation:

Left side: \(60 + 2 \div 3 \times 6\)

• First, Division: \(2 \div 3 = \frac{2}{3}\)
• The left side becomes: \(60 + \frac{2}{3} \times 6\)
• Next, Multiplication: \(\frac{2}{3} \times 6 = \frac{12}{3} = 4\)
• The left side becomes: \(60 + 4 = 64\)

Right side: \(5 - 43\)

• Subtraction: \(5 - 43 = -38\)

Comparing both sides: \(64 = -38\). Since \(64 \neq -38\), this option is incorrect.

Conclusion

Only Option 3 results in a balanced equation. Therefore, the correct combination of mathematical signs to replace the * signs sequentially is ÷, +, ×, -, =.

Revision Table: Evaluating Options

Additional Information: Order of Operations (BODMAS/PEMDAS)

When solving mathematical expressions with multiple operations, it's crucial to follow a specific order to ensure a correct result. This order is commonly remembered using mnemonics like BODMAS or PEMDAS.

• BODMAS:
  • Brackets (Parentheses)
  • Orders (Exponents, square roots, etc.)
  • 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 have equal priority and are performed from left to right as they appear. Similarly, addition and subtraction have equal priority and are performed from left to right as they appear.

In the problem above, we used this order to evaluate the expressions after substituting the signs.