A train started from a station with a certain number of passengers. At the first halt, half of the passengers got down and $125$ passengers got in. At the second halt, half of the passengers left and $100$ entered. Then the train left for its destination with $250$ passengers. The number of passengers in the train at the start was

## Concept & Logic This problem requires reverse calculation (working backwards). Instead of starting with an unknown variable $x$ and building a complex equation forward, we can start with the final known number of passengers and reverse every operation step-by-step to find the initial count. ## Step-by-step Solution * **Final State:** * The train left for its destination with $250$ passengers. * **Reversing the Second Halt:** * Before $100$ passengers entered, the train had $250 - 100 = 150$ passengers. * These $150$ passengers represent the "half that remained" after the other half left. * Therefore, before half left, the train had $150 \times 2 = 300$ passengers arriving at the second halt. * **Reversing the First Halt:** * The train arrived at the second halt with $300$ passengers. * Before the $125$ passengers got in at the first halt, the train had $300 - 125 = 175$ passengers. * These $175$ passengers represent the "half that remained" after the first half got down. * Therefore, the initial number of passengers before anyone got down was $175 \times 2 = 350$. ## Exam Strategy & Shortcut Working backwards is the fastest shortcut here. Mentally invert the operations: Final = $250$ Subtract $100$ $\rightarrow 150$ Multiply by $2$ $\rightarrow 300$ Subtract $125$ $\rightarrow 175$ Multiply by $2$ $\rightarrow 350$. This takes less than $15$ seconds and requires no algebra. ## Common Pitfall If setting up an algebraic equation, failing to properly bracket the terms is a frequent mistake. E.g., writing $\frac{x}{2} + 125 \div 2 + 100 = 250$ instead of the correct $\frac{\frac{x}{2} + 125}{2} + 100 = 250$. The reverse method completely avoids this bracket confusion. ## Final Answer Therefore, the correct answer is **350**.