Pancakes for twelve

Batter for four people uses two eggs, two hundred grams of flour and five hundred millilitres of milk. It makes eight pancakes.

Batter for $4$ people: $2$ eggs, $200$ g flour, $500$ ml milk, making $8$ pancakes.

For $4$ people: eggs $E$, flour $F$, milk $M$: $(E,F,M)=(2,200,500)$, pancakes $8$.

We were twelve, and everyone wanted three pancakes. That is thirty-six pancakes, four and a half times the recipe. So: nine eggs, nine hundred grams of flour and two thousand two hundred and fifty millilitres of milk.

We were $12$, each wanting $3$: $12\times 3=36$ pancakes. Factor: $36÷8=4.5$. So $2\times 4.5=9$ eggs, $200\times 4.5=900$ g flour and $500\times 4.5=2250$ ml milk.

$12\times 3=36$ pancakes, so $k=\frac{36}{8}=4.5$ and $(E,F,M)\times k=(9,900,2250)$.

Frying does not scale so easily. A pan fries two pancakes in three minutes. One pan would need eighteen rounds of three minutes: fifty-four minutes, and the first pancakes would be cold. With two pans it takes nine rounds, which is twenty-seven minutes.

Frying does not scale like batter: $1$ pan makes $2$ pancakes per $3$ min, so $36÷2=18$ rounds and $18\times 3=54$ min. With $2$ pans: $36÷4=9$ rounds and $9\times 3=27$ min.

Pans $p$, time $t$: $t=3\times \frac{36}{2p}$. For $p=1$ it is $t=54$, for $p=2$ it is $t=27$.

The surprise: the batter was multiplied by four and a half, but the time was only halved, by using two pans. The pancakes were gone in ten minutes.

The batter grew $4.5$ times, but the time only fell from $54$ to $54÷2=27$ min with two pans. The pancakes were gone in $10$ min.

Batter: $k=4.5$. Time: $\frac{54}{27}=2$.