Understand Ratios with “Oomph” and “Often”

Ratios summarize a scenario with a number, such as “income per day”. Unfortunately, this hides the explanation forhow结果来了。

For example, look at two businesses:

Annie’s Art Gallery sells a single, \$1000 piece every day
Frank’s Fish Emporium sells 250 trout at \$4/each every day

By the numbers, they’re identical \$1000/day operations, right? Hah.

Here’s how each business actually behaves:

$\displaystyle{\mathit{\frac{Dollars}{Day} = \frac{Dollars}{Transaction} \cdot \frac{Transactions}{Day} }}$

Transactions are the workhorse that drive income, but they’re lost in the dollars/day description. When studying an idea, separate the results into Oomph and Often:

$\displaystyle{\mathit{ Result = Oomph \cdot Often = \frac{Dollars}{Transaction} \cdot \frac{Transactions}{Day} }}$

在“Oomph”和“Often”中，我看到了两个需要增加的不同杠杆。A ratio like dollars/day makes me stumble through thoughts like:“For better results, I need 1/day to improve… which means the day gets shorter… How’s that possible? Oh, that must be the portion of the day used for each transaction…”.

Why make it difficult? Rewrite the ratio to include the root case: What’s the Oomph, and how Often does it happen?

Horsepower, Torque, RPM

In physics, we define everyday concepts like “power” with a formal ratio:

$\displaystyle{\mathit{ Power = \frac{Work}{Time} }}$

Ok. Power can be explained by a ratio, but we’re already in inverted-thinking mode. Just another hassle when exploring an already-tricky concept.

How about this:

$\displaystyle{\mathit{ Power = Oomph \cdot Often }}$

容易,我想。“魅力”和“经常”意味着什么?

Well, Oomph is probably the work we do (such as moving a weight) and Often is how frequently we do it (how many reps did you put in?).

在同一分钟内，假设弗兰克举起100磅10次，安妮举起1000磅1次。从等式来看，他们拥有同样的力量(不过说实话，我更害怕安妮)。

An engine mechanic might internalize power like this:

$\displaystyle{\mathit{ Power = \frac{Work}{Revolution} \cdot \frac{Revolutions}{Time} }}$

$\displaystyle{\mathit{ Horsepower = Torque \cdot RPM }}$

What does that mean?

Torque is the Oomph, or how much weight (and how far) can be moved by a turn of the engine (i.e., moving 500lbs by 1 foot)
RPM (revolutions per minute) is how frequently the engine turns

A motorcycle engine is designed for reps, i.e. spinning the wheels quickly. It doesn’t need much torque — just enough to pull itself and a few passengers — but it needs to send that to the wheels again and again.

A bulldozer is designed for “Oomph”, such as knocking over a wall. We don’t need to tap into that work very frequently, as one destroyed wall per minute is great, thanks.

I’m not a physicist or car guy, but I can at least conceptualize the tradeoffs with the Oomph/Often metaphor.

Gears can change the tradeoff between Oomph and Often in a given engine. If you’re going uphill, fighting gravity, what do you want more of? If you’re cruising on a highway? Trying to start from a standstill? Driving over slippery snow? Lost the brakes and need to slow down the car?

Oomph/Often gets me thinking intuitively, Work/Time does not.

Variation: Electric Power

Electric power has the same ratio as mechanical power:

$\displaystyle{\mathit{ Electric \ Power = \frac{Work}{Time} }}$

Yikes. It’s not clear what this means. How about:

$\displaystyle{\mathit{ Electric \ Power = Oomph \cdot Often }}$

It’s hard to have ideas out of the blue, but we might imaginesomething(一个迷你引擎?)正在铁丝内移动Oomph。If we call it a “charge” then we have:

$\displaystyle{\mathit{ Electric \ Power = \frac{Work}{Charge} \cdot \frac{Charges}{Time} }}$

And we can give those subparts formal names:

Voltage (Oomph): How much work each charge contributes
Current (Often): How quickly charges are moving through the wire

Now we get the familiar:

$\displaystyle{\mathit{ Electric \ Power = Voltage \cdot Current }}$

Boomshakala! I don’t have a good intuition for electricity, at least my goal is clear: findanalogieswhere voltage means Oomph, and current means Often.

And still, we can take a crack at intuitive thinking: when you get zapped by a doorknob in winter, was that Oomph or Often? What attribute should batteries maximize? What’s better for moving energy through stubborn power lines? (Vive la résistance!)

该比率认为，每一种类型的电力都可以简化为一个通用的工作/时间计算。The Oomph/Often metaphor gets us thinking about Torque/RPM in one scenario and Voltage/Current in another.

What’s Really Going On? Parameters, Baby.

The Oomph/Often viewpoint lets us think about the true cause of the ratio. Instead of dollars and days, we wonder how the actual transactions affect the outcome:

Can we increase the size of each transaction?
Can we increase the number each day?

In formal terms, we’ve introduced anew parameterto explain the interaction. To change a ratio from a/b to one parameterized by x, we can do:

$\displaystyle{\frac{a}{b} = \frac{(a/x)}{(b/x)} = (a/x) \cdot \frac{1}{(b/x)} = \frac{a}{x} \cdot \frac{x}{b} }$

We change our viewpoint to see x as the key component. In math, we often switch viewpoints to simplify problems:

与其问观察者发生了什么，我们能否改变参数，问移动者看到了什么?(Degrees vs.radians.)
Can we see a giant function as being parameterized by smaller ones? (See thechain rule.)
Can we express probabilities as odds, instead of percentages? (It makesBayes Theoremeasier.)

调整参数是一种将不合意的想法转变成合意的方法。因为我不习惯用倒单位来思考，所以我让自己更容易:处理两个乘法，而不是除法。

Happy math.

Horsepower, Torque, RPM

Variation: Electric Power

What’s Really Going On? Parameters, Baby.

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