The problem requires finding which pair of numbers to interchange in the given equation to make it true.

Analyzing the Equation and Options

The initial equation is: $14 \times 8 \div 12 + 64 - 2 = 108$ Let's evaluate the left side using the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

1. Multiplication: $14 \times 8 = 112$
2. Division: $112 \div 12 = \frac{112}{12} = \frac{28}{3}$
3. Addition: $\frac{28}{3} + 64 = \frac{28}{3} + \frac{192}{3} = \frac{220}{3}$
4. Subtraction: $\frac{220}{3} - 2 = \frac{220}{3} - \frac{6}{3} = \frac{214}{3} \approx 71.33$

Since $\frac{214}{3} \neq 108$, the equation is incorrect. We need to test swapping pairs of numbers.

Testing Number Swaps

Swapping 2 and 12 (Option C)

If we swap the numbers 2 and 12, the equation becomes:

Let's evaluate this new equation:

1. Multiplication: $14 \times 8 = 112$
2. Division: $112 \div 2 = 56$
3. Addition: $56 + 64 = 120$
4. Subtraction: $120 - 12 = 108$

The result is 108, which matches the right side of the original equation. Therefore, swapping 2 and 12 makes the equation true.

Verifying Other Options (Briefly)

Testing other swaps confirms they do not result in 108:

• Swap 2, 8: $14 \times 2 \div 12 + 64 - 8 = 28 \div 12 + 56 = \frac{7}{3} + 56 = \frac{175}{3} \neq 108$
• Swap 8, 14: $8 \times 14 \div 12 + 64 - 2 = 112 \div 12 + 62 = \frac{28}{3} + 62 = \frac{214}{3} \neq 108$
• Swap 12, 14: $12 \times 8 \div 14 + 64 - 2 = 96 \div 14 + 62 = \frac{48}{7} + 62 = \frac{482}{7} \neq 108$

Conclusion

Swapping the numbers 2 and 12 balances the equation.