# How many trains do not stop at either of these stations on that day?

## Oaklands and Brighton are two busy train stations on the same train line. On one particular day: 1/5th of the trains do not stop at Oaklands 45 trains do not stop at Brighton 60 trains stop at both Brighton and Oaklands 60 trains stop at only Brighton or Oaklands (not both) How many trains do not stop at either of these stations on that day?

Aug 9, 2018

5 trains do not stop at either station

#### Explanation:

Start by assigning letters to all the different possibilities we can think of (draw a diagram if it helps).
It won't matter if we don't use them all. At this point they are unknowns, so lets use a letter to represent them.

$T$ = total number of trains
$O$ = number of trains stopping only at Oaklands
$B$ = number of trains stopping only at Brighton
$N$ = number of trains stopping at neither
$U$ = number of trains stopping at both

$\overline{O}$ = number of trains not stopping at Oaklands
$\overline{B}$ = number of trains not stopping at Brighton
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List what we are told:

$U = 60$

$O + B = 60$

$\overline{O} = \frac{T}{5}$

$\overline{B} = 45$

$T = N + O + B + U$
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Write equations for things we know. It doesn't matter if they look a bit complicated at first. Try different ideas. Put the values in later, after rearranging the equations.
We are told that $O + B = 60$ is so lets try finding what they are in terms of other unknowns (so we can maybe add them... )

$\overline{O} = N + B = \frac{T}{5}$ ....(i.e. trains stopping at Neither + only B )

$N + B = \frac{1}{5} \left(N + O + B + U\right)$

$N + B = \frac{1}{5} \left(N + 60 + 60\right)$

$N + B = \frac{N}{5} + \frac{120}{5}$
$B = 24 - \frac{4}{5} N$ ..........................................(1)
...................................................

And:

$\overline{B} = N + O = 45$ ....(trains stopping at neither + only O )

$O = 45 - N$ ................................................(2)
....................................................

But $B + O = 60$, so

$24 - \frac{4}{5} N + 45 - N = 60$ .........adding (1) and (2)

$- \frac{4}{5} N - N = 60 - 24 - 45$

$- 1.8 N = - 9$

$N = \frac{- 9}{-} 1.8$

$N = 5$

Note that it is now easy to find B (=20 ) and O (=40 ) by using $N = 5$ in eqns (1) and (2)