What number multiplied by $48$ will give the same product as $173$ multiplied by $240$?

### Concept & Formula This is a direct application of algebraic equivalence. We are asked to set up a simple linear equation where two products are equal to each other. $$x \cdot A = B \cdot C$$ ### Step-by-Step Solution * **Given:** * Let the unknown number be $x$. * First product expression: $x \times 48$ * Second product expression: $173 \times 240$ * **Calculation / Deduction:** 1. Equate the two expressions according to the problem statement: $$x \times 48 = 173 \times 240$$ 2. Isolate $x$ by dividing both sides by $48$: $$x = \frac{173 \times 240}{48}$$ 3. Simplify the fraction before multiplying to save time. Notice that $240$ is a multiple of $48$. $240 / 48 = 5$ 4. Substitute this back into the equation: $$x = 173 \times 5$$ 5. Calculate the final multiplication: $$173 \times 5 = 865$$ ### Exam Strategy & Shortcut Instead of blindly multiplying $173$ by $240$ (which wastes time and introduces arithmetic error risk), always look for cancellation first. $240$ is exactly $5$ times $48$. So, the unknown number is just $5$ times $173$. Even faster: $173 \times 10 = 1730$. Half of $1730$ is $865$. ### Common Pitfall The worst approach here is calculating $173 \times 240 = 41520$ and then trying to do long division by $48$. While it yields the correct answer, it consumes excessive time on a timed competitive exam. Always simplify fractions first. ### Final Answer Therefore, the correct answer is 865.