Erica noted that a train to Muizenberg took $8$ minutes to pass her. A train in the opposite direction to Cape Town took $12$ minutes to pass her. The trains took $9 $ minutes to pass each other. Assuming each train maintained a constant speed, and given that the train to Cape Town was $ 150$m long, what was the length of the train to Muizenberg?
My attempt:
let $x$ be the length of the train to Muizenberg.
let $s_1$ be the speed of the train to Muizenberg.
let $s_2$ be the speed of the train to Cape Town.
let $t_1 =$ the time it takes the train to Muizenberg to pass the train to Cape Town
let $t_2 =$ the time it takes the train to Cape Town to pass the train to Muizenberg
We have $x = 8 \times s_1$
$$s_2 = \frac{150}{12} = 12.5$$
Now by the third statement, if it takes $9$ minutes for the two trains to pass each other then $$t_1 = t_2$$ $$ \frac{150}{12.5} = \frac{x}{s_1} = 9$$ (Since $Time = \frac{distance}{Speed}$.)
But clearly there is something wrong in my solution. Where did I go wrong? Can someone explain to me what is my mistake? Thank you.
You mistake is in the fact that when the two trains pass each other, the relative speed is the sum of their speed, and the length is the sum of their lengths. Letting $s_1$ (resp. $s_2$) be the speed of the first train (resp the second) relative to Erica, you get $$ 9\cdot 60 = \frac{x+150}{s_1+s_2} $$ (the multiplication by 60 being to express the speeds in $m.s^{-1}$).