Why variables in directly proportinality are multipiled

Question

Why variables (RHS) in directly proportionality are always multiplied. Suppose the Newton's 2nd law

$$F \propto m$$ $$F \propto a$$

$$F \propto m*a$$

Please don't give a rigorous proof. I just want to understand it intuitively.

Answer 1

Suppose you first increase mass $m$ by twice, that is $m\rightarrow2m$ thaen $F∝m$ implies that the force should also increase twice $F \rightarrow 2F$. After this let's increase the $a$ twice- this implies that the force shoudl again become twice, that is $2F\rightarrow 4F$.
If we simultaneously increase mass and acceleration to double then the force should increase 4 times. Symbolically this can be written as:
$$F∝m \ \ and\ \ \ F∝a \implies F∝m∗a$$

Answer 2

The first proportionality indicates that $F=m \cdot C$ for some value of $C$. Let $C=a$ and you get $F=ma$.

In the same way, the second proportionality gives you $F = a \cdot D$ for some value of $D$, then let $D=m$ and you again get $F = ma$.

Answer 3

pro·por·tion·al (prə-pôr′shə-nəl, -pōr′-) adj.

1. Forming a relationship with other parts or quantities; being in proportion.
2. Properly related in size, degree, or other measurable characteristics; corresponding: Punishment ought to be proportional to the crime.
3. Mathematics Having the same or a constant ratio.

The number of eggs that I break is directly proportional to the number of houses that don't give me candy. I don't necessarily need to throw just one egg at every house, I could throw 3 if I wanted to, in that case I would break 3 eggs for every cheap house. $$eggs = 3 \times houses$$