I’ve been an avid follower of Seth Godin ever since I watched his “why marketing is too important to be left to the marketing department” talk at the Business of Software conference. (If you haven’t made the time to see this, you really should.)

This morning his blog featured a simple quiz, which I must admit had me stumped too, despite my Bachelor of Mathematics degree:

A simple quiz for smart marketers:

Let's say your goal is to reduce gasoline consumption.

And let's say there are only two kinds of cars in the world. Half of them are Suburbans that get 10 miles to the gallon and half are Priuses that get 50.

If we assume that all the cars drive the same number of miles, which would be a better investment:

Get new tires for all the Suburbans and increase their mileage a bit to 13 miles per gallon.Replace all the Priuses and rewire them to get 100 miles per gallon (doublingtheir average!)

Trick question aside, the answer is the first one. (In fact, it's more than twice as good a move).

We're not wired for arithmetic. It confuses us, stresses us out and more often than not, is used to deceive.

Surely, there’s a trick, I thought; I immediately started reading too deeply into the subtleties of the implicit consumption associated with new Suburbans tires vs. replacing Priuses outright – this completely missed the point!

Frustrated, I opened up Notepad and wrote it all down:

- Let
*m*be number of miles driven by a car... - Let
*s*be the gas consumption (in gallons) for Suburbans (= m/10) - Let
*p*be the gas consumption (in gallons) for Priuses (= m/50) - Let T be the total consumption (in gallons) (= s +
*p*= m/10 + m/50 = 6m/50 = 0.12m)

So in Scenario #1, we have T = m/13 + m/50 = 50m+13m/650 = 63m/650 = 0.097m

And in Scenario #2, we have T = m/10 + m/100 = 11m/100 = 0.11m

Scenario #1 reduced consumption by 0.12-0.097 = 0.023; Scenario #2 only by 0.01; **Scenario #1 is 2.3x more efficient**! Sure, it all makes sense when it’s drawn out for you [1]:

This very interesting article inScience, “The MPG Illusion” by Richard P. Larrick and Jack B. Soll at the Fuqua School of Business in Duke University (Vol 320, June 20, 2008, p. 1593), points out the mathematically obvious truth thatgas used per mile is inversely proportional to miles per gallon, which means that you have a steeper slope at lower MPG ratings, and diminishing returns at higher MPG ratings.

Now, try to think about how this applies to your daily life and where you spend your time, particularly as an application developer.

More on this next time…

## 19 comments:

This is an interesting problem, but it speaks to the difference of how numbers grow - arithmetically vs. exponentially.

This problem plays on the fact that we tend to look at numbers in terms of relationships tied to addition/subtraction (A/S) and multiplication/division (M/D). While these are the same, M/D is really the same as A/S.

In this scenario, the growth is merely arithmetic - we are adding 3 MPG to the suburban and multiplying by two the Prius' MPG. Functionally, these are the same. We just believe that multiplying the Prius' MPG makes it more.

The further tricky part to this problem is that the difference between the numbers is still arithmetic. The prius gets 5 times more MPG then the surburban at the start. That is arithmetic.

Your last equation:

--T = 0.12/m

is classic hyperbolic reciprocal function which explains why the bang for our buck is high with the suburban. However, it also explains why we will never have 100% efficiency in gas engines.

Nice chart--that's a really good aid.

Yes, interesting as usual "Mischievous Marketing Wisdom" from Seth Godin and his Godinettes.

Once again Seth shows us his "Five and Dime -- Facts and Fiction".

I prefer your suggestion of people not spend 80% of their time on issues that impact a given environment by 20%.

Seth's artificial fabrication of a scenario to draw his own conclusion is as usual............zzzzzzzzzz..oh excuse me, I must have nodded off there.

The heart of the matter is the inherent tension between large numbers with small percentages and small numbers with large percentages. As this example illustrates sometimes small percentages translate to big results when the numbers they are acting on are large.

Why not convert both? Regardless of maths, numbers, data, graphs and explanations isn't the real problem overall gas consumption. Do both, simple. From a marketing perspective Toyota should run a campaign to get Prius drivers to convert their cars. Then for every 10 converted Prius' Toyota will replace the tires on one Suburban? Everyone wins?

My Housemate wrote me this after I placed the question to him -

i got bored reading papers and came back to your problem and extrapolated it further. i don't really know how blogs work but can you post stuff in them? if you can, cut and paste this and john nash the shit out of them.

firstly, when car sales are 50/50, option 1 is better. i think of it as driving 100 miles. under your conditions i would have to drive 50 miles in the suburban and 50 miles in the prius. under the control conditions it would take me 5 gallons to the suburban 50 miles and 1 gallon to travel the prius 50 miles totalling 6 gallons.

under option 1 conditions it would take me 3.836 (50/13) gallons to travel the suburban miles and 1 gallon to travel the prius miles totalling 4.846 gallons.

under option 2 conditions it would take me 5 gallons to travel the suburban miles and 0.5 (100/50) gallons to travel the prius miles totalling 5.5 gallons.

therefore option 1 is the more efficient choice for the IMMEDIATE FUTURE. however, i then thought, at what point would the percentage of distance travelled in the prius cancel out the benefits of making the suburban 3m/g more efficient. for example, if car sales were 80/20 in favour of prius (i.e. i drive 80 miles in the prius and 20 in the suburban) would option 1 still be the better choice?

currently when you look at option 1 result and option 2 there is a discrepancy, with the lower result being the more efficient. but at what percentage would option 1 be identical to option 2 and result in 0 discrepancy?

for this we can follow the equation:

option 1, x/50 + (100-x)/13 = y (x = the percentage i am trying to find, x + (100-x) = 100%)

option 2, x/100 + (100-x)/10 = y

because i want option 1 = option 2 i can rearrange the equation to be:

x/50 + (100-x)/13 = x/100 + (100-x)/10

i'll convert to a common multiple of 1300 to calculate therefore becoming:

26x/1300 + (10,000-100x)/1300 = 13x/1300 + (13,000-130x)/1300

or simply, 26x + 10,000 - 100x = 13x + 13,000 - 130x

basic algerbra leaves a result of 43x = 3000 or x = 69.767

therefore, the improvements to suburban only outweigh the prius when car sales are below 69.767% prius. once prius sales are higher than this percentage, option 2 is the more efficient choice. the marketing should be aimed at increasing prius sales above this figure in order to counteract the short-term benefits of 3m/g increase in efficiency of suburban for a better long term solution.

My old '96 Suburban averages 16-18 on the hwy. In the last 30 years I've owned 20 Subs from 1965 to an '07 model and always have been happy with performance, room safety and mileage. So if my ol' 96 is getting that kind of mileage, what are the results now?

Mikey

6m/50 = .12m, not .12/m. The two are very different. You got it right where it mattered, though.

The real takeaway from this example is not, "Wow, let's make suburbans more fuel efficient!" but that when you artificially constrain the miles driven to NEVER CHANGE no matter what the gas mileage is, you can pull all kinds of clever stunts with numbers.

but with 2 gallons (each car given 1 gallon), the second scenario may reach further distance.

Scenario 1: 13 + 50 = 63 miles

Scenario 2: 10 + 100 = 110 miles

tricky...

I'm so not a mathematician and this problem has me stumped. Can someone explain it in non Math language? It's driving me crazy!

Check out this video from the original blog post...

http://charliepark.tumblr.com/post/169016492/in-seth-godins-post-this-morning-he-talks-about

Sure it's an artificial scenario - aimed at illustrating how our brains are wired rather than advocating a particular solution to fuel consumption. It's (if possible) even more counter-intuitive than it looks at first.

If the Sub improves from 10 -13 mpg it will beat

anyimprovement to the Prius - Even if the Prius is super-tuned to run a million miles per gallon.Take a simple example of 1 car of each type each doing 10,000 miles per year.

Tuning the Sub from 10-13 mpg results in a reduction from 1, gallons to 769 gallons a year - a saving of 231 gallons. The Prius only uses 200 gallons to start with and can never save more than this no matter how efficient it gets.

Well, I guess that's why in Europe fuel consumption is measured in litres per 100 kilometer ;)

> Well, I guess that's why in Europe fuel consumption is measured in litres per 100 kilometer ;)

Yes, in this case, it's by far more intuitive to understand:

10 miles per gallon = 23.5214583 litres per 100 km

13 miles per gallon = 18.0934295 litres per 100 km

-> 5.4280288 litres saved per 100 km

50 miles per gallon = 4.70429167 litres per 100 km

100 miles per gallon = 2.35214583 litres per 100 km

-> 2.35214584 litres saved per 100 km

It's really obvious. so I'm glad we use this more intuitive system in everyday life :)

Also really cool is that google is able to convert between these two systems so nicely, try googling "10 miles per gallon in litres per 100 km". Of cource, the opposite direction is also possible.

I think the key to this, and the bit that makes it feel "not intuitive" is the assumption that

both types of car drive the same number of milesNOT both types of cars are given the same amount of gas.If it was phrased, both cars have 1000 gallons of gas, then getting an extra 50 miles per gallon is far better than getting 3 extra miles per gallon.

I recently came accross your blog and have been reading along. I thought I would leave my first comment. I dont know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often.

Margaret

http://lotterymegamillions.net

i don't like it when one can already tell by the way a problem is posed that the answer is gonna be the one that seems less plausible at first sight. i cant help it but it always makes me think of a childish stupid little person desperately trying to get some attention... generally marketers make me feel that way, doing whatever possible to have you look at them/their product and then being all satisfied just because you took notice. and every possible critique is countered by pretending every reaction is already a win... that's just an all to easy way out. blah.

i'll add scenario three, one that would actually make some sense in the real world: replace 10% of the SUV with toyota priuses.

cheers

btw, your blog is great, i just dont like seth..

h4nne5's scenario is pretty interesting. Replace 10% of the SUVs with Priuses. Those SUV drivers would then be getting a 400% better gas mileage. So overall, the original SUV drivers would be getting 40% better gas mileage, or 14 mpg. Which is obviously an improvement over 13 mpg.

It's an interesting thought problem, but I doubt that replacing and properly inflating every SUVs tires will net a 30% increase in mpg. And we don't have a mass-produced 100mpg car. But we could replace clunkers with more fuel-efficient vehicles.... Wait. Where have I heard that idea before?

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