Enough for everyone

Dinner for four: one big flatbread pizza cut into eight slices, so two slices each. Each person gets one quarter of the pizza.

Dinner for $4$: one pizza in $8$ slices, so $8÷4=2$ slices each, which is $\frac{1}{4}$ of the pizza.

Pizza of $8$ slices, people $n=4$: share $\frac{1}{n}=\frac{1}{4}$, that is $\frac{8}{4}=2$ slices.

Then two guests knocked at the door. Six people, one pizza. Everyone should get one sixth of it, and one sixth of eight slices is eight sixths, which is one and a third slices.

Two guests arrive: $4+2=6$ people. A fair share is $\frac{1}{6}$ of the pizza, which is $\frac{8}{6}=\frac{4}{3}$ slices.

For $n=6$ the share is $\frac{1}{6}$, that is $\frac{8}{6}=\frac{4}{3}$ slices.

One and a third is awkward, so we cut every slice into three small pieces. Now there were twenty-four pieces, and twenty-four divided by six is four pieces each.

Awkward, so cut each slice in three: $8\times 3=24$ pieces, and $24÷6=4$ pieces each.

Each slice in three: $8\times 3=24$ and $\frac{24}{6}=4$ pieces.

The four of us gave up only two pieces each, one twelfth of the pizza, and the two guests received four pieces each. Eight pieces given, eight pieces received: the sharing was exact.

We four gave up $6-4=2$ pieces each, which is $\frac{1}{4}-\frac{1}{6}=\frac{1}{12}$ of the pizza. The guests got $2\times 4=8$ pieces and we gave $4\times 2=8$. Exact.

Given: $4\times 2=8$; received: $2\times 4=8$; each gave up $\frac{1}{4}-\frac{1}{6}=\frac{1}{12}$.