Checking Equations with Sign and Number Interchange

The problem requires us to determine which of the given equations becomes valid after applying two specific interchanges: the sign '+' with '–', and the number '6' with '3'. This means:

• Every '+' sign should be replaced by a '–' sign, and every '–' sign should be replaced by a '+' sign.
• Every number '6' should be replaced by a '3', and every number '3' should be replaced by a '6'.

All other signs and numbers remain unchanged. We will apply these changes to each equation provided in the options and then evaluate the resulting mathematical expression using the standard order of operations (BODMAS/PEMDAS) to see if the left side equals the right side.

Checking Option 1 Equation After Swaps

Original Equation: \(8 – 3 \times 4 \div 6 + 9 = 18\)

Applying the interchanges (+ <–> –, 6 <–> 3):

The new equation becomes: \(8 + 6 \times 4 \div 3 – 9\)

Let's evaluate the left side:

1. First, perform the division: \(4 \div 3 = \frac{4}{3}\).
2. Next, perform the multiplication: \(6 \times \frac{4}{3} = \frac{24}{3} = 8\).
3. Finally, perform addition and subtraction from left to right: \(8 + 8 – 9 = 16 – 9 = 7\).

So, the equation is \(7 = 18\), which is incorrect.

Checking Option 2 Equation After Swaps

Original Equation: \(8 + 4 \times 6 \div 3 – 9 = 15\)

Applying the interchanges (+ <–> –, 6 <–> 3):

The new equation becomes: \(8 – 4 \times 3 \div 6 + 9\)

Let's evaluate the left side using the order of operations:

1. Perform the division: \(3 \div 6 = \frac{3}{6} = \frac{1}{2}\).
2. Perform the multiplication: \(4 \times \frac{1}{2} = 2\).
3. Perform addition and subtraction from left to right: \(8 – 2 + 9 = 6 + 9 = 15\).

So, the equation becomes \(15 = 15\), which is correct.

Checking Option 3 Equation After Swaps

Original Equation: \(9 \div 3 \times 4 – 6 + 7 = 10\)

Applying the interchanges (+ <–> –, 6 <–> 3):

The new equation becomes: \(9 \div 6 \times 4 + 3 – 7\)

Let's evaluate the left side:

1. Perform the division: \(9 \div 6 = \frac{9}{6} = \frac{3}{2} = 1.5\).
2. Perform the multiplication: \(1.5 \times 4 = 6\).
3. Perform addition and subtraction from left to right: \(6 + 3 – 7 = 9 – 7 = 2\).

So, the equation is \(2 = 10\), which is incorrect.

Checking Option 4 Equation After Swaps

Original Equation: \(6 \div 2 \times 8 + 3 – 1 = 20\)

Applying the interchanges (+ <–> –, 6 <–> 3):

The new equation becomes: \(3 \div 2 \times 8 – 6 + 1\)

Let's evaluate the left side:

1. Perform the division: \(3 \div 2 = \frac{3}{2} = 1.5\).
2. Perform the multiplication: \(1.5 \times 8 = 12\).
3. Perform addition and subtraction from left to right: \(12 – 6 + 1 = 6 + 1 = 7\).

So, the equation is \(7 = 20\), which is incorrect.

After interchanging the signs '+' and '–', and the numbers '6' and '3', only the equation in Option 2 holds true.

Revision Table: Summary of Interchanges

Additional Information: Understanding Order of Operations

The order of operations is crucial for solving mathematical expressions unambiguously. The widely used rule is BODMAS or PEMDAS:

• BODMAS: Brackets, Orders (powers, roots), Division and Multiplication (done from left to right), Addition and Subtraction (done from left to right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (done from left to right), Addition and Subtraction (done from left to right).

In this problem, we applied division and multiplication first, in the order they appeared from left to right, followed by addition and subtraction, also from left to right. This systematic approach helps in correctly evaluating expressions after applying the required interchanges.