Understanding the Equation and the Problem

The problem asks us to identify which pair of mathematical signs, when swapped in the given equation, makes the equation true. The original equation is:

\(588 \div 28 \times 32 + 72 - 160 = 760\)

To solve this, we need to follow the order of operations, commonly known as BODMAS or PEDMAS. This rule dictates the sequence for performing operations: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

Let's first evaluate the original equation to see if it's already correct or what value it yields:

• Division: \(588 \div 28 = 21\)
• Multiplication: \(21 \times 32 = 672\)
• Addition and Subtraction (from left to right): \(672 + 72 - 160 = 744 - 160 = 584\)

So, the original equation is \(584 = 760\), which is false. We need to swap signs according to the options to make the equation evaluate to 760.

Testing Sign Interchange Options

We will test each option by swapping the specified signs and then applying the BODMAS rule to the modified equation.

Detailed Calculation: Option 1 (– and +)

Interchange the subtraction (–) and addition (+) signs in the original equation.

Original: \(588 \div 28 \times 32 + 72 - 160 = 760\)

After swapping – and +:

\(588 \div 28 \times 32 - 72 + 160 = ?\)

Now, evaluate this new equation using BODMAS:

• Division: \(588 \div 28 = 21\)
• Multiplication: \(21 \times 32 = 672\)
• Subtraction and Addition (from left to right): \(672 - 72 + 160 = 600 + 160 = 760\)

The result is 760, which matches the right side of the equation. So, interchanging the – and + signs makes the equation correct.

Detailed Calculation: Option 2 (÷ and –)

Interchange the division (÷) and subtraction (–) signs.

Original: \(588 \div 28 \times 32 + 72 - 160 = 760\)

After swapping ÷ and –:

\(588 - 28 \times 32 + 72 \div 160 = ?\)

Evaluate using BODMAS:

• Division: \(72 \div 160 = \frac{72}{160} = \frac{9}{20} = 0.45\)
• Multiplication: \(28 \times 32 = 896\)
• Subtraction and Addition (from left to right): \(588 - 896 + 0.45 = -308 + 0.45 = -307.55\)

The result is -307.55, which is not 760. This option is incorrect.

Detailed Calculation: Option 3 (+ and ×)

Interchange the addition (+) and multiplication (×) signs.

Original: \(588 \div 28 \times 32 + 72 - 160 = 760\)

After swapping + and ×:

\(588 \div 28 + 32 \times 72 - 160 = ?\)

Evaluate using BODMAS:

• Division: \(588 \div 28 = 21\)
• Multiplication: \(32 \times 72 = 2304\)
• Addition and Subtraction (from left to right): \(21 + 2304 - 160 = 2325 - 160 = 2165\)

The result is 2165, which is not 760. This option is incorrect.

Detailed Calculation: Option 4 (÷ and +)

Interchange the division (÷) and addition (+) signs.

Original: \(588 \div 28 \times 32 + 72 - 160 = 760\)

After swapping ÷ and +:

\(588 + 28 \times 32 \div 72 - 160 = ?\)

Evaluate using BODMAS:

• Division: \(32 \div 72 = \frac{32}{72} = \frac{4}{9}\)
• Multiplication: \(28 \times \frac{4}{9} = \frac{112}{9} \approx 12.44\)
• Addition and Subtraction (from left to right): \(588 + \frac{112}{9} - 160 = 588 - 160 + \frac{112}{9} = 428 + \frac{112}{9} = \frac{3852 + 112}{9} = \frac{3964}{9} \approx 440.44\)

The result is approximately 440.44, which is not 760. This option is incorrect.

Conclusion on Sign Interchange

Based on the calculations, only interchanging the subtraction (–) and addition (+) signs makes the equation correct.

Revision Table: Key Concepts

Additional Information: BODMAS Rule Explained

The BODMAS or PEDMAS rule is crucial for solving mathematical expressions unambiguously. Let's break down the steps:

• B/P: Brackets/Parentheses - Solve any expressions inside brackets first.
• O/E: Orders/Exponents - Calculate any powers or square roots.
• DM: Division and Multiplication - Perform these operations from left to right in the expression.
• AS: Addition and Subtraction - Perform these operations from left to right in the expression.

When you swap signs in an equation, you create a new equation that must be evaluated carefully following this order. This type of question checks your ability to apply these fundamental rules consistently under modified conditions.