Question 14 Marks

A train starts from station $A$ with a certain number of passengers. At station $B$, the train drops one third of the passengers and takes in 96 more. At the next station $C$ one half of the passengers on board get down while 12 new passengers get on board. At station $D$, a quarter of the passengers get down and 20 new passengers board the train. It then reaches its final destination-station $E$-with 200 passengers.
(1) How many passengers were on board when the train left station $A$ ?
(a) 460$\quad$(b) 520$\quad$(c) 540$\quad$(d) 560
(2) How many passengers were on board when the train left station $B$ ?
(a) 428$\quad$(b) 436$\quad$(c) 444$\quad$(d) 456
(3) How many passengers were on board when the train left station $C$ ?
(a) 228$\quad$(b) 240$\quad$(c) 256$\quad$(d) 264
(4) Had the train started with 720 passengers from station $A$, how many passengers would have got down at station $E$ ?
(a) 228$\quad$(b) 236$\quad$(c) 245$\quad$(d) 256

Answer

Let the number of passengers on leaving station $A$ be $x$. Then, 
number of passengers on leaving station $B=\left(\frac{2 x}{3}+96\right)$ 
number of passengers on leaving station $C=\frac{1}{2}\left(\frac{2 x}{3}+96\right)+12=\left(\frac{x}{3}+48\right)+12=\left(\frac{x}{3}+60\right)$ 
number of passengers on leaving station $D=\frac{3}{4}\left(\frac{x}{3}+60\right)+20=\left(\frac{x}{4}+45\right)+20=\left(\frac{x}{4}+65\right)$
(1) (C) 540
$\frac{x}{4}+65=200 \Rightarrow \frac{x}{4}=135 \Rightarrow x=135 \times 4=540$
(2) (D) 456
Required number $=\left(\frac{2 x}{3}+96\right)=\left(\frac{2 \times 540}{3}+96\right)=360+96=456$.
(3) (B) 240
Required number $=\left(\frac{x}{3}+60\right)=\left(\frac{540}{3}+60\right)=180+60=240$.
(4) (C) 245
Required number $=\left(\frac{720}{4}+65\right)=180+65=245$.

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