A recipe written for four will happily feed six — every ingredient just rides the same escalator: $Q_2 = Q_1 \cdot \dfrac{S_2}{S_1}$ (read aloud: Q-two equals Q-one times S-two over S-one). Meet the four symbols before they go to work: $Q$ is a quantity of one ingredient — grams of flour, cups of milk — and $S$ is the number of servings. Subscript 1 means the recipe as written, subscript 2 means tonight. So $Q_1$ is the flour the card calls for, $S_1$ the mouths it was written for, $S_2$ the mouths you actually have, and $Q_2$ the flour you need — the one you are solving for.

The fraction $\dfrac{S_2}{S_1}$ is the scale factor — going from 4 servings to 6 makes it $\dfrac{6}{4} = 1.5$, and every gram, cup and pinch gets multiplied by exactly that.

The sanity check is free and instant: cooking for MORE people, the factor sits above 1 and every amount grows; cooking for fewer, it sits below 1 and everything shrinks. If your scaled flour moved the wrong way, the fraction is upside down — flip it before the oven finds out.