Inverse proportionality

In an inverse relationship, the product of the two quantities never changes.
So if $a \times b = k$, making one bigger forces the other smaller.
They can never both be large at the same time.

The graph of an inverse relationship is a curve that drops steeply and then levels off.
As $x$ gets larger, $y$ gets smaller so that the product $xy$ stays the same.
The curve never touches either axis.

The key rule is that a change in one quantity is undone in the other.
If one is multiplied by $3$, the other is divided by $3$.
That keeps their product equal to the same constant $k$.

Sometimes the constant involves a power, like $k = s \times w^2$.
If $s$ is multiplied by $4$, then $w^2$ must be divided by $4$ to keep $k$ the same.
Since $w^2$ is divided by $4$, $w$ itself is divided by $2$, because $2^2 = 4$.

If $a$ and $b$ are inversely proportional with $a \times b = 36$, and $a$ is tripled, what happens to $b$?
To keep the product at $36$, $b$ must be divided by $3$.
So if $b$ was $12$, it becomes $4$.

The time to fill a pool is inversely proportional to the number of pumps. With $2$ pumps it takes $6$ hours. How long with $4$ pumps?
The product stays constant: $2 \times 6 = 12$.
With $4$ pumps, $4 \times \text{time} = 12$, so the time is $3$ hours.

In $k = p \times q^2$, if $p$ is multiplied by $9$, by what factor does $q$ change?
To keep $k$ constant, $q^2$ must be divided by $9$.
Since $q^2$ is divided by $9$, $q$ is divided by $3$, because $3^2 = 9$.