Analyzing the Equation and Sign Interchange

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

\(15 \div 3 + 2 \times 10 - 6 = 14\)

Currently, let's evaluate the left side using the order of operations (BODMAS/PEMDAS):

• Division: \(15 \div 3 = 5\)
• Multiplication: \(2 \times 10 = 20\)
• Addition and Subtraction (from left to right): \(5 + 20 - 6 = 25 - 6 = 19\)

So, \(15 \div 3 + 2 \times 10 - 6 = 19\). This is not equal to 14, so the original equation is incorrect as written. We need to test each option by swapping the specified signs and re-evaluating the equation.

Testing Option 1: Interchanging ÷ and -

If we interchange the division (÷) and subtraction (-) signs, the equation becomes:

\(15 - 3 + 2 \times 10 \div 6 = ?\)

Let's evaluate this new equation using BODMAS/PEMDAS:

• Division: \(10 \div 6 = \frac{10}{6} = \frac{5}{3}\)
• Multiplication: \(2 \times \frac{5}{3} = \frac{10}{3}\)
• Addition and Subtraction (from left to right): \(15 - 3 + \frac{10}{3} = 12 + \frac{10}{3} = \frac{36}{3} + \frac{10}{3} = \frac{46}{3}\)

Since \(\frac{46}{3} \neq 14\), interchanging ÷ and - does not correct the equation.

Testing Option 2: Interchanging ÷ and +

If we interchange the division (÷) and addition (+) signs, the equation becomes:

\(15 + 3 \div 2 \times 10 - 6 = ?\)

Let's evaluate this new equation using BODMAS/PEMDAS:

• Division: \(3 \div 2 = \frac{3}{2}\)
• Multiplication: \(\frac{3}{2} \times 10 = \frac{30}{2} = 15\)
• Addition and Subtraction (from left to right): \(15 + 15 - 6 = 30 - 6 = 24\)

Since \(24 \neq 14\), interchanging ÷ and + does not correct the equation.

Testing Option 3: Interchanging + and ×

If we interchange the addition (+) and multiplication (×) signs, the equation becomes:

\(15 \div 3 \times 2 + 10 - 6 = ?\)

Let's evaluate this new equation using BODMAS/PEMDAS:

• Division: \(15 \div 3 = 5\)
• Multiplication: \(5 \times 2 = 10\)
• Addition and Subtraction (from left to right): \(10 + 10 - 6 = 20 - 6 = 14\)

Since \(14 = 14\), interchanging + and × corrects the equation.

Testing Option 4: Interchanging + and -

If we interchange the addition (+) and subtraction (-) signs, the equation becomes:

\(15 \div 3 - 2 \times 10 + 6 = ?\)

Let's evaluate this new equation using BODMAS/PEMDAS:

• Division: \(15 \div 3 = 5\)
• Multiplication: \(2 \times 10 = 20\)
• Addition and Subtraction (from left to right): \(5 - 20 + 6 = -15 + 6 = -9\)

Since \(-9 \neq 14\), interchanging + and - does not correct the equation.

Based on our analysis, interchanging the + and × signs makes the equation correct.

Revision Table: Key Concepts for Equation Problems

Additional Information: Solving Sign Interchange Problems

Sign interchange problems require careful application of the order of operations. It's crucial to recalculate the entire expression after swapping the signs, following BODMAS or PEMDAS strictly.

• BODMAS: Brackets, Of (orders/powers), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and Multiplication have the same priority and are performed from left to right. Similarly, Addition and Subtraction have the same priority and are performed from left to right.

For problems like this, the most straightforward approach is to test each given option systematically until the equation balances (LHS = RHS). Keep your calculations neat to avoid errors.