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A train started from Mumbai at $6.00$ A.M. On the next (second) station $1/3$ passengers got down and $96$ got in. On the next (third) station, $1/2$ of the total passengers present in the train, got down and $12$ came in. Now there were $248$ passengers in the train, when the train started from Mumbai, the number of passengers was:

- $435$
- $564$
- $654$
- $736$

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Let the number of passenger in train(when train started from mumbai) $ = x$

From second station: The number of passenager $ = x – \dfrac{x}{3} + 96 = \dfrac{2x + 288}{3}$

From second station: The number of passenager $ = \dfrac{2x + 288}{3} – \left(\dfrac{ \dfrac{2x + 288}{3} }{2}\right) + 12$

Now, $\dfrac{2x + 288}{3} – \left(\dfrac{ \dfrac{2x + 288}{3} }{2}\right) + 12 = 248$

$\implies \dfrac{2x + 288}{3} – \left(\dfrac{x + 144}{3}\right) + 12 = 248$

$\implies \dfrac{2x + 288 – x – 144 + 36}{3} = 248$

$\implies x + 180 = 744$

$\implies x = 564.$

So, the correct answer is $(B).$

From second station: The number of passenager $ = x – \dfrac{x}{3} + 96 = \dfrac{2x + 288}{3}$

From second station: The number of passenager $ = \dfrac{2x + 288}{3} – \left(\dfrac{ \dfrac{2x + 288}{3} }{2}\right) + 12$

Now, $\dfrac{2x + 288}{3} – \left(\dfrac{ \dfrac{2x + 288}{3} }{2}\right) + 12 = 248$

$\implies \dfrac{2x + 288}{3} – \left(\dfrac{x + 144}{3}\right) + 12 = 248$

$\implies \dfrac{2x + 288 – x – 144 + 36}{3} = 248$

$\implies x + 180 = 744$

$\implies x = 564.$

So, the correct answer is $(B).$