Finding the Correct Sign Interchange for Equations

The problem asks us to determine which pair of mathematical operation signs, when swapped in the given equation, makes the equation mathematically correct. The initial equation is:

\(18 – 2 × 7 ÷ 6 + 10 = 67\)

We need to test each option by interchanging the specified signs and then evaluating the resulting equation using the order of operations (BODMAS/PEMDAS).

The order of operations is:

• Brackets (Parentheses)
• Orders (Exponents, Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Testing Option 1: Interchange – and ÷

If we interchange the – and ÷ signs, the equation becomes:

\(18 ÷ 2 × 7 – 6 + 10\)

Now, let's evaluate this step-by-step using BODMAS:

1. Division/Multiplication (from left to right):
2. \(18 ÷ 2 = 9\)
3. The equation is now: \(9 × 7 – 6 + 10\)
4. Next multiplication: \(9 × 7 = 63\)
5. The equation is now: \(63 – 6 + 10\)
6. Addition/Subtraction (from left to right):
7. \(63 – 6 = 57\)
8. The equation is now: \(57 + 10\)
9. \(57 + 10 = 67\)

The result is 67, which matches the right side of the original equation. So, interchanging – and ÷ makes the equation correct.

Testing Option 2: Interchange × and ÷

If we interchange the × and ÷ signs, the equation becomes:

\(18 – 2 ÷ 7 × 6 + 10\)

Evaluate using BODMAS:

1. Division/Multiplication (from left to right):
2. \(2 ÷ 7 = \frac{2}{7}\)
3. The equation is now: \(18 – \frac{2}{7} × 6 + 10\)
4. Next multiplication: \(\frac{2}{7} × 6 = \frac{12}{7}\)
5. The equation is now: \(18 – \frac{12}{7} + 10\)
6. Addition/Subtraction (from left to right):
7. \(18 – \frac{12}{7} = \frac{18 \times 7 - 12}{7} = \frac{126 - 12}{7} = \frac{114}{7}\)
8. The equation is now: \(\frac{114}{7} + 10\)
9. \(\frac{114}{7} + 10 = \frac{114 + 10 \times 7}{7} = \frac{114 + 70}{7} = \frac{184}{7}\)

The result is \(\frac{184}{7}\), which is not equal to 67.

Testing Option 3: Interchange + and ×

If we interchange the + and × signs, the equation becomes:

\(18 – 2 + 7 ÷ 6 × 10\)

Evaluate using BODMAS:

1. Division/Multiplication (from left to right):
2. \(7 ÷ 6 = \frac{7}{6}\)
3. The equation is now: \(18 – 2 + \frac{7}{6} × 10\)
4. Next multiplication: \(\frac{7}{6} × 10 = \frac{70}{6} = \frac{35}{3}\)
5. The equation is now: \(18 – 2 + \frac{35}{3}\)
6. Addition/Subtraction (from left to right):
7. \(18 – 2 = 16\)
8. The equation is now: \(16 + \frac{35}{3}\)
9. \(16 + \frac{35}{3} = \frac{16 \times 3 + 35}{3} = \frac{48 + 35}{3} = \frac{83}{3}\)

The result is \(\frac{83}{3}\), which is not equal to 67.

Testing Option 4: Interchange + and -

If we interchange the + and - signs, the equation becomes:

\(18 + 2 × 7 ÷ 6 – 10\)

Evaluate using BODMAS:

1. Division/Multiplication (from left to right):
2. \(2 × 7 = 14\)
3. The equation is now: \(18 + 14 ÷ 6 – 10\)
4. Next division: \(14 ÷ 6 = \frac{14}{6} = \frac{7}{3}\)
5. The equation is now: \(18 + \frac{7}{3} – 10\)
6. Addition/Subtraction (from left to right):
7. \(18 + \frac{7}{3} = \frac{18 \times 3 + 7}{3} = \frac{54 + 7}{3} = \frac{61}{3}\)
8. The equation is now: \(\frac{61}{3} – 10\)
9. \(\frac{61}{3} – 10 = \frac{61 - 10 \times 3}{3} = \frac{61 - 30}{3} = \frac{31}{3}\)

The result is \(\frac{31}{3}\), which is not equal to 67.

Based on the evaluation of each option, interchanging the – and ÷ signs is the only option that makes the equation correct.

Revision Table: Checking Sign Interchange Options

Additional Information: Understanding BODMAS/PEMDAS

The BODMAS (or PEMDAS) rule is crucial when evaluating mathematical expressions with multiple operations. It establishes a standard order to ensure everyone gets the same result for a given calculation.

• B/P: First, evaluate anything inside Brackets or Parentheses.
• O/E: Next, calculate Orders, which include exponents (powers, indices) and roots.
• DM: Then, perform Division and Multiplication. These two operations have equal priority. You should perform them from left to right as they appear in the expression.
• AS: Finally, perform Addition and Subtraction. These also have equal priority and should be performed from left to right as they appear.

Following this order is essential to correctly solve equations like the one in this problem where signs are interchanged.