Number System — One fourth of a number subtracted from one third of the same number equals 12. Find the original number.

Introduction / Context:
This problem involves forming and solving a linear equation with fractional coefficients. Careful handling of fractions and finding a common denominator streamlines the solution. The task assesses basic algebraic manipulation and fraction operations.

Given Data / Assumptions:

• Let the number be x.
• Equation from words: (1/3)*x − (1/4)*x = 12.
• We need the exact value of x.

Concept / Approach:
Combine like terms with fractional coefficients by finding a common denominator (12). The expression (1/3 − 1/4) simplifies to 1/12. Then solve the resulting simple linear equation. This avoids mistakes from prematurely converting to mixed numbers or decimals.

Step-by-Step Solution:
1) Start: (1/3)*x − (1/4)*x = 12.2) Compute the coefficient: 1/3 − 1/4 = (4 − 3) / 12 = 1/12.3) So (1/12)*x = 12.4) Multiply both sides by 12: x = 12 * 12 = 144.

Verification / Alternative check:
Plug back: one third of 144 is 48; one fourth is 36; 48 − 36 = 12. The condition is satisfied exactly.

Why Other Options Are Wrong:
120 gives (40 − 30) = 10; 72 gives (24 − 18) = 6; 63 gives (21 − 15.75) = 5.25; 96 gives (32 − 24) = 8. None produce 12.

Common Pitfalls:
Adding fractions instead of subtracting; computing 1/3 − 1/4 as 1/7; converting to decimals and rounding prematurely; forgetting to multiply both sides by the common denominator.

Final Answer:
144