9.

# Proportionality

Equations are useful things when you want to solve for one of the variables in them, but they also tell you something about the relationship between these variables.Consider the equation $a={bc^2\over d}$.

- Because $a$ and $b$ are in the numerator on opposite sides, $a$ is directly proportional to $b$. That means that if $b$ gets bigger, and
*all the other variables stay the same*, then $a$ will get bigger too. - The variables $a$ and $d$ are inversely proportional, because one is in the numerator on one side, and the other is in the denominator on the other side. That means that if $d$ gets bigger, and all the other variables stay the same, then $a$ will get
*smaller*. - The variables $b$ and $d$ are directly proportional, even though one is in the numerator and one in the denominator, because they are on the
*same side*of the equation. If we solved the equation for $d$, we would see that $d={bc^2\over a}$, and the proportional relationship would be more obvious. - The variables $a$ and $c^2$ are also directly proportional. Furthermore, because $c$ is squared, $a$ depends more strongly on $c$ than it does on $b$ or $d$. The variable $a$ is doubled if $b$ is doubled, but it is
*quadrupled*if $c$ is doubled.

$$v=\sqrt{k\over m}d$$

Thus, equations tell us how different quantities depend on one another. For instance, the equation shown here is the speed of a ball that is launched from a spring: it depends on the stiffness $k$ of the spring, the mass $m$ of the ball, and the distance $d$ the spring is pulled back. We see that we can make the ball go faster by - increasing $d$ (i.e. compressing the spring more),
- increasing $k$ (i.e. using a stiffer spring), or
*decreasing*$m$ (i.e. using a less massive ball). However, the

*radius*of the ball doesn’t appear in the equation, so the speed doesn’t depend on that (at least according to this equation).