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If $m$ varies jointly as $n$ and $p$ is like this

$m = K(n)(p)$

is the $v$ varies jointly as $l$, $w$ and $h$ becomes like this

$v = k(l)(w)(h)$

or it becomes like this

$v = (l)(w)(h)$

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    Sorry. It seems that this is not about inequality. It's about variations. I can't find a tag about variations – 2017-01-05

1 Answers 1

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The amount of work that gets done varies jointly with the number of people working and the amount of time that is spent working.

$$W_{ork} = P_{eople} \times T_{ime} \times K$$

Where $K$ is a constant.

For example, if one person can paint $100$ square feet in one hour, then $K = 100$ square feet per hour per person, or $\frac {100 ft^2}{hr\times person}$.

So $K$ is really the rate. It also includes the units to make everything work out.

The point here—the point about "varies jointly"—is that if you double the number of people, you will double the amount of work that gets done. If you double the time that they work, you will double the amount of work that gets done.

Or, on the other side, if you quadruple the amount of work that needs to be done, you will need to quadruple the number of people working, OR quadruple the amount of time they have to work in, OR double the number of people and double the amount of time.

(There are other options that involve fractional multiples; for instance you might change the number of people from $10$ to $15$ and change the amount of time from $3$ hours to $8$. Together, these actions would quadruple the amount of work that would get done.)

Note that it's possible for $K$ to equal $1$. Volume of a rectangular prism is like that; the volume varies jointly with the length, width and height—but no conversion is needed. The product of the length, width and height is the volume.

If you relate the volume to something else—say you're buying styrofoam that's priced by volume—then $K$ will not equal $1$. The price of the styrofoam varies jointly with the length, width and height, but to calculate the actual price of a particular rectangular prism of styrofoam, you will have to multiply the computed volume by the "price per cubic foot" to get a dollar amount. So $$V=lwh$$ but $$P_{rice}=lwhK$$where $K$ might be measured in dollars per cubic foot.