Which number gives the same result when added to $1 \frac{1}{2}$ and when multiplied by $1 \frac{1}{2}$?

### Concept & Formula This problem requires creating a direct equivalence between an addition equation and a multiplication equation using an unknown value. $$x + k = x \cdot k$$ ### Step-by-Step Solution * **Convert the mixed number to an improper fraction:** $$1 \frac{1}{2} = \frac{3}{2}$$ * **Set up the equation:** Let the required number be $x$. $$x + \frac{3}{2} = \frac{3}{2}x$$ * **Solve for x:** Subtract $x$ from both sides to gather terms: $$\frac{3}{2} = \frac{3}{2}x - x$$ $$\frac{3}{2} = x\left(\frac{3}{2} - 1\right)$$ $$\frac{3}{2} = \frac{1}{2}x$$ Multiply both sides by 2: $$x = 3$$ ### Exam Strategy & Shortcut Use option testing for instant verification. Test option (b) 3: Addition: $3 + 1.5 = 4.5$ Multiplication: $3 \times 1.5 = 4.5$ Both operations yield identical results, making 3 the immediate solution. ### Common Pitfall Students often confuse mixed fraction representations ($1 \frac{1}{2}$) with simple whole numbers, or make minor errors while collecting like fractional variables on one side of an algebraic identity. ### Final Answer **Therefore, the correct answer is 3.**