# Expressing using algebra- a wordy question

I am currently practising UKMT questions and this question was from the 1998 IMC. I am struggling to picture how this works. Does the $$x$$ amount of people behind Wallace equal the $$y$$ amount of places in front of Gromit? I don't really see the differences between $$x,y$$ and $$n$$. Aren't they all the same? I know which one's the answer but it would really help if someone explained how this works with maybe a diagram? Thank you.

Draw the queue like this, where $$W$$ is wallace and $$G$$ is gromit:

$$\begin{array}{ccccc} \cdots & G & \cdots & W & \cdots \end{array}$$

Let's call $$a,b,c$$ the number of people in line in each of the segments labelled "$$\cdots$$''. So there are $$a$$ people behind Gromit, $$b$$ people in between them, and $$c$$ people after Wallace. Let's think about what each statement in the original problem says:

1. "There are $$x$$ people behind Wallace". This means $$a+b+1 = x$$
2. "There are $$n$$ people in front of Gromit". This means $$b+c+1 = n$$.
3. "Wallace is $$y$$ places in front of Gromit". This means $$b=y-1$$.

You should be able to use these equations to solve for $$a$$ and $$c$$. Finally, the total number of people in line is $$a+b+c+2$$.

Alternate Way: Add all the people behind Wallace plus all the people in front of Gromit. This gives $$x+n$$. But this double-counts all the people in between, of which there are $$y-1$$. So the total is $$x+n-(y-1)$$.

• Sorry, i'm confused. If there are c people after Wallace then doesn't c=x?
– yt.
Nov 7 '18 at 19:01
• Okay i think i understand the alternative way. But its quite confusing with the diagram ...G...W... above- are we assuming the queue is starting from the right so that Gromit is behind Wallace? I think that is where the confusion is coming from.
– yt.
Nov 7 '18 at 19:06
• I meant the "front" of the line is at the right and the "back" is the left. So for example, $c$ is the number of people "in front of Wallace" and $a$ is the number of people "behind" Gromit.
– Nick
Nov 7 '18 at 20:05