Understand Ratios with “Oomph” and “Often”

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Ratios summarize a scenario with a number, such as “income per day”. Unfortunately, this hides the explanation for how the result came about.

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} }}

With Oomph and Often, I visualize two distinct levers to increase. 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 }}

Easier, I think. What could Oomph and Often mean?

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?).

In the same minute, suppose Frank lifted 100lbs ten times, while Annie lifted 1000lbs once. From the equation, they have the same power (though to be honest, I’m more frightened by Annie.)

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 imagine something (a mini-engine?) is moving the Oomph around inside the wire. 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: find analogies where 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!)

The ratios think every type of power reduces to a generic Work/Time calculation. 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 a new parameter to 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:

  • Instead of asking what happens to the observer, can we change parameters and ask what the mover sees? (Degrees vs. radians.)

  • Can we see a giant function as being parameterized by smaller ones? (See the chain rule.)

  • Can we express probabilities as odds, instead of percentages? (It makes Bayes Theorem easier.)

Adjusting parameters is a way to morph an idea that doesn’t click into one that does. Since I don’t naturally think with inverted units, I’ve made it easier on myself: deal with two multiplications, instead of a division.

Happy math.

Other Posts In This Series

  1. Rethinking Arithmetic: A Visual Guide
  2. Techniques for Adding the Numbers 1 to 100
  3. Understand Ratios with "Oomph" and "Often"
  4. Mental Math Shortcuts
  5. How to Develop a Sense of Scale
  6. Easy Permutations and Combinations
  7. How To Understand Combinations Using Multiplication
  8. Surprising Patterns in the Square Numbers (1, 4, 9, 16…)
  9. Navigate a Grid Using Combinations And Permutations
  10. A Quick Intuition For Parametric Equations
  11. Understanding Algebra: Why do we factor equations?
  12. How To Learn Trigonometry Intuitively
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22 Comments

  1. Kalid– Brilliant thinking as usual. I will use these ideas in my Physics and AP physics classes. Giving credit where it is due of course. I have been pushing your internet book all year. Hope they have listened.

    –david

  2. Awesome, glad you liked it [and thanks for the support]. Yes, feel free to steal/remix/reword any part of the article that’s useful! One of the tricky things for me (with physics) was going from the formal Work/Time definitions to a more intuitive understanding that I could apply.

  3. I always take time to read your emails. I never finished your 7 day intuitive limits course but I will get caught up! Reading your articles are never painful and always intriguing! Thanks

  4. Thanks Sean, glad you enjoyed it! (And no worries on the courses, go through things at your own pace :)).

  5. Hi Kalid,

    you have touched a great topic. derivatives, probability, interest rate etc are ratios as well, and so is any fraction at the basic levels. if things are explained in the way you explain, all the advanced math (calculus, probability etc. can be explained to even third grade students :-)).

  6. @Kumar: Really appreciate it! I think the basic intuition behind most math (calculus, algebra, etc.) can be understood by young students. I hope to cover more topics like this down the road.

    @Pintu: Thanks

    @Mike: Very glad it helped, I strive to make things practical and I’m glad to hear that resonated with you.

  7. As always, Kalid, a new and enlightening slant on an old topic. So now I understand how I passed my physics courses – Oomph was the little bit I learned each time I studied, and Often was how frequently I studied during the term. Passing just required tinkering with the ratios.

  8. Dear Kalid,

    I wish you could write books on computer programming like c++ or java in such a lucid fashion that anyone can understand.
    I still remember the fortran classes long time ago where fortran was taught in such a gruesome fashion. Later I mastered a few languages in my own way.
    I believe java will be easier if somebody rewrites the books in your style.

    A million thanks.

  9. You have an even greater opportunity for understanding mechanical power if you break the equation down further and recombine it different ways. Usually Power is explained as how quickly you can do Work. But since Work = Force * Distance then Power = Force * Distance / Time; expressing Distance / Time as Velocity leads to Power = Force * Velocity. Now you can see that Power is also the amount of Force you can muster even when already moving quickly. For example, can you still put as much force into a crank while keeping it spinning at 60 rpm as you did at 1 rpm? If you are at your power limit, the answer is no. If you can, you will be producing more power at the same force as the speed increases.

  10. Kalid,
    Thank you for this website. I wish you were around back when I studied Calc years ago, it would have made sense. You may have inspired me to go back and revisit the subject!

  11. Dear Kalid!

    Thnaks fro this great insight!
    I’m (too) lacking a good intuition fro elctricity. But the \displaystyle{Oomph*Often} analogy makes a bit easier to deal with.

    Thanks for your effort sharing your ideas. I really appreciate it.

    Andy

  12. @Evan: Hah, I like the analogy :). I think many of us ended up squeezing all the Oomph into the evening before the test…

    @Prabu: Thanks!

    @Nandeesh: Appreciate the suggestion, thanks. I’d like to cover more programming topics down the road.

    @Walt: Really neat way to think about it — we can keep breaking up the parameters into even simpler terms. I like the notion of “already moving fast yet still able to push harder”, it helps illustrate the notion of having more power to spare. These are exactly the types of insights that don’t emerge from a plain Work/Time description.

    @Buddy: Happy to help :). You might like the calculus guide at http://betterexplained.com/calculus/lesson-1

    @Pedro: Oomph is probably closest to Fuerza. I’m not sure what Boomshakala would be (it seems I use a lot of slang as I translate my thoughts to paper…)

    @Edi: Thanks!

    @Andy: Appreciate the note. Electricity is a big area I want to explore more deeply, I think it’ll be about building intuition for one term at a time :).

  13. Hi Kalid,
    Just discovered your site and am an instant follower….thanks so much.

    I think I can help you with a translation of “boomshakalakala….”. I saw Sly and the Family Stone play (incongruously enough) at the 1969 Newport Jazz festival in RI. Well, due to Led Zeppelin also being there as the headliner group the festival was mobbed and the town fathers announced the gates were closed and everyone please go home. And (they lied) Led Zeppelin would not be appearing to close the show. A very large and unhappy crowd mostly without tickets gathered outside the fenced off area where the stage was and police started gathering atop the fence. It was getting pretty ugly fast. Bottles started being thrown over the fence from the outside but it wasn’t until someone inside the park threw one back one back that all hell broke loose. Really broke lose and the fence started being torn down and the police started clubbing everyone.
    It was then that Sly came on stage and ripped into “Wanna Take You Higher!” with the chorus singing “boomshakalalaka-boomshakalaalka!”…..and total magic happened…everyone started dancing and dancing and just like that the riot stopped. Completely stopped.
    So I think boomshakalakalacka” is like Euler’s “e” and pi and i= -1: it is a magical inverse force word that transforms negative energy into something much better.
    Led Zeppelin played the next day and we went home. Hope this helps!

  14. Wow, thanks Ernie — that’s an awesome story and analogy! Led Zeppelin is one of my favorite bands, you have no idea how badly I’ve wanted to see them in their heyday…

  15. another read, well worth it kalid. i’ve binge read most of your website recently as if it were breaking bad.

    that being said, i’ve been bugged by something. the idea that intuitive explanations are helpful when independently learning technical ideas. i’ve always viewed intuition as a sort of mirage to a logical understanding. so i’ve always taken the opposite approach. when learning math, physics, compsci from a textbook i pay attention to my logical insights doing my best to ignore my intuition until i feel like my intuition has gained credibility. your site shows how the creative nature of intuitive explanations are refreshing and helpful when being covertly grounded in logic, but should one apply this intuitive method when learning something new independently?

    sorry for being long-winded.

  16. Thanks Leon, if math articles can be in the same breath as Breaking Bad I’m pretty happy =).

    Great question on the role of intuition and raw logic. I view it as a spiral, where we develop one, check it with the other, and keep going. Logic needs to be checked by intuition (does this logical conclusion lead to something absurd?) and intuition needs to be checked by logic (does my initial thought make assumptions which are not true?).

    For example, there are things like the Birthday Paradox, which states that in a room of just 23 people, there’s a 50% chance of a shared birthday. People’s intuitions jump out and say “23 people! There’s not a 50% chance I’ll have a birthday with someone” but they’ve made a bad assumption, that it’s only their birthday being checked. In reality, we have (23*22)/2 birthday comparisons (every person in the room against everyone else), which is a lot more than the 23 comparisons you’re directly involved in.

    So, it’s a matter of recognizing that initial intuition, and training it when it leads to a logically incorrect result (when intuition is wrong, what assumption is being made that needs to be corrected?). Similarly, for logic, there must be some reasonably natural way to think about a scenario other than “We followed all the steps and got from proposition A to conclusion B”.

    I have a bit more here: http://betterexplained.com/articles/developing-your-intuition-for-math/

  17. great response. i suppose the culprit responsible for my different perspective is that i’m an ‘aspie.’ the didactic ‘spiral’ you use explains why i get so much out of your website: while I find the intuitive curves to be much harder to understand than the ‘raw logic’ i get the sense that your increasing intuitive ability.

    you’re doing great things for people who are not only neurotypical, but also for those with AS.

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