Going on an outing

If we let $n$ be the number of people and $c$ the cost per person, we get two equations: $nc = 8000 \\ (n+4)(c-100) = 8000$ But, if we rewrite the first one as $c=\frac{8000}{n}$ , we can rewrite the second as $(n+4)\left(\frac{8000}{n} - 100\right) = 8000$ .

Expanding yields: $\frac{32000}{n} - 100n - 400 = 0$

Tidying up: $n² + 4n - 320 = 0$

Factorising: $(n + 20)(n - 16) = 0$

Having $-20$ people in a group is absurd, so the original number of people is $\boxed{16}$