Understanding Sign Interchanges in Equations

The problem asks us to find which interchange of two mathematical signs in the given equation \(20 \div 4 \times 8 + 16 - 15 = 11\) will make the equation correct. To solve this, we need to test each given option by replacing the specified signs and then evaluating the resulting expression using the order of operations (BODMAS/PEMDAS).

Evaluating Sign Interchange Options

We will examine each option systematically.

Option 1: Interchanging '-' and '÷'

Original Equation: \(20 \div 4 \times 8 + 16 - 15 = 11\)

After interchanging '-' and '÷', the equation becomes:

\(20 - 4 \times 8 + 16 \div 15\)

Let's evaluate this expression following BODMAS/PEMDAS:

• First, Division: \(16 \div 15 \approx 1.0667\)
• Next, Multiplication: \(4 \times 8 = 32\)
• Then, Addition and Subtraction from left to right: \(20 - 32 + 1.0667\)
• \(20 - 32 = -12\)
• \(-12 + 1.0667 = -10.9333\)

The result is approximately \(-10.9333\), which is not equal to \(11\). So, Option 1 is incorrect.

Option 2: Interchanging '+' and '-'

Original Equation: \(20 \div 4 \times 8 + 16 - 15 = 11\)

After interchanging '+' and '-', the equation becomes:

\(20 \div 4 \times 8 - 16 + 15\)

Let's evaluate this expression:

• First, Division: \(20 \div 4 = 5\)
• Next, Multiplication: \(5 \times 8 = 40\)
• Then, Subtraction and Addition from left to right: \(40 - 16 + 15\)
• \(40 - 16 = 24\)
• \(24 + 15 = 39\)

The result is \(39\), which is not equal to \(11\). So, Option 2 is incorrect.

Option 3: Interchanging '×' and '÷'

Original Equation: \(20 \div 4 \times 8 + 16 - 15 = 11\)

After interchanging '×' and '÷', the equation becomes:

\(20 \times 4 \div 8 + 16 - 15\)

Let's evaluate this expression:

• First, Multiplication and Division from left to right: \(20 \times 4 = 80\)
• Then, \(80 \div 8 = 10\)
• Next, Addition and Subtraction from left to right: \(10 + 16 - 15\)
• \(10 + 16 = 26\)
• \(26 - 15 = 11\)

The result is \(11\), which is equal to the right side of the original equation. So, Option 3 makes the equation correct.

Option 4: Interchanging '+' and '×'

Original Equation: \(20 \div 4 \times 8 + 16 - 15 = 11\)

After interchanging '+' and '×', the equation becomes:

\(20 \div 4 + 8 \times 16 - 15\)

Let's evaluate this expression:

• First, Division: \(20 \div 4 = 5\)
• Next, Multiplication: \(8 \times 16 = 128\)
• Then, Addition and Subtraction from left to right: \(5 + 128 - 15\)
• \(5 + 128 = 133\)
• \(133 - 15 = 118\)

The result is \(118\), which is not equal to \(11\). So, Option 4 is incorrect.

Based on the evaluation, interchanging the '×' and '÷' signs makes the equation correct.

Revision Table: Key Concepts

Additional Information: BODMAS/PEMDAS Explained

The order of operations is crucial for solving mathematical expressions correctly. Without a standard order, the same expression could yield multiple different results. BODMAS and PEMDAS are acronyms to help remember the sequence:

• BODMAS:
  • Brackets
  • Orders (powers, 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 acronyms represent the same order of priority. Division and Multiplication have equal priority and are performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have equal priority and are performed from left to right.

In the given problem, applying this order after interchanging the signs is the key to determining which interchange makes the equation true.