Hi, I was able to get access to this question, here's a response to your inquiry:

The Altius solution makes use of Ideal Gas Law but takes a bit of a shortcut so that you don't have to recalculate the volume with the length and radius change.

The law states: PV = nRT and you're asked for a ratio of n2 to n1. Since you know that R and T are not changing you can simplify to PV = n. The ratio reads: n2/n1 = P2V2/P1V1

So ask yourself, do P and V change?

The passage indicates that at max contraction the P = 500 000 Pa and the initial P = 100 000 Pa, this ia 5-fold increase. If P2 increases by 5, then n2 increases by 5 as well for the equation to balance.

Now let's look at V.

Since length and radius change, the V changes. V = (pi)(radius squared)(L) You don't have to fill in all of the details - think about this in terms of proportions. L decreases by a FACTOR of 1/2 and R increases by a FACTOR of 2. So V2 = (pi)(2 radius squared)(1/2L) = (pi)(2R)(1/2L) = 2(pi)RL. The overall effect is that V increases by a factor of 2.

Let's consider both P and V changes together: if P increases by 5 and V increases by 2, then P2V2 increases by 5 x 2 = 10. Therefore, the ratio is 10:1.

Hope this makes sense!