Why are we seemingly hard-wired to get the above problem incorrect? Two reasons.

In their paper on using cognitive illusion to improve quantitative literacy:

Numeracy Infusion Course for Higher Education (NICHE), 1: Teaching Faculty How to Improve Students’ Quantitative Reasoning Skills through Cognitive Illusions *Wang, F., & Wilder, E. I. (2015). *Numeracy*, 8(2), 6.*

the authors describe the following scenario, originally presented in a *Science* magazine article and summarized in the above slide from my Grow Up, Branch Out interactive video.

Consider a family that has an SUV that gets 10 MPG and a sedan that gets 25 MPG. Both are driven equal distances in a year. Is the family better off replacing the SUV with a minivan that gets 20 MPG or replacing the sedan with a hybrid that gets 50 MPG?

This problem is described as a cognitive illusion because many of us, thinking heuristically (that is, without stepping back and working out the details), are drawn to either conclude that:

- The 25-to-50-mpg upgrade saves more fuel, since 50 mpg is qualitatively much more efficient. (It’s a “big number” in this context.)
- Or, the two upgrades save equal amounts of fuel, since each represents the same doubling of fuel economy in miles per gallon. (Our preferred, and very often quite useful, proportional-reasoning heuristic.)

However, **neither is true.** Doubling the miles-per-gallon economy of the *lesser* efficient vehicle results in greater fuel savings. Why is this the case? We’ll look at how arithmetic exposes this to be an issue of denominator neglect, and how the way Americans think about fuel economy is essentially shaped by cultural choices.

### The Arithmetic

The key observation here is about unit conversion. Converting a quantity from one set of units into another is always a matter of either multiplying or dividing by the number 1, by representing 1 as a clever fraction whose numerator and denominator are equal measurements using unequal units.

Fuel economy, in this view, can be thought of as a conversion factor representing an equality between an amount of fuel consumed (in gallons) and a distance driven (in miles). The SUV making 10 miles per gallon, for instance, can be represented as the “equation”

[latex] 10 {\rm \; miles} = 1 {\rm \; gallon} [/latex]

This can be made into a conversion factor, then, by dividing this equation by either of its sides:

[latex] \frac{10 {\rm \; miles}}{1 {\rm\; gallon}} = 1 = \frac{1 {\rm \;gallon}}{10 {\rm \; miles}} [/latex]

Where we get tripped up, when we’re thinking heuristically, is in figuring out **which of these two conversion factors is appropriate to the question. **That is, are we more worried about multiplying something by 10? Or dividing it by 10?

Asking a crucial quantitative reasoning question — what’s changing and what’s not? — reveals that the number of miles the vehicles are being driven each year is remaining constant. So we will have the same number of miles for the SUV as for the minivan; the question is, what happens to the amount of fuel consumed? We therefore need to convert from miles into gallons, meaning we must “cancel the miles and introduce the gallons.” **This means using the [latex]\frac{1 {\rm \;gallon}}{10 {\rm \; miles}}[/latex] factor, or dividing by the MPG rating.**

A little additional thought should persuade us that the *actual number* of miles driven per year will not affect our answer, so we’ll choose a convenient, round number: say, 10,000 miles. Here, then, are the results of *dividing* that number by each of the MPG figures to determine the amount of fuel each of these four vehicles would need to drive that far.

SUV (10 mpg):
[latex]10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{10 {\rm \; miles}} = 1\, 000 {\rm \; gallons}[/latex] |
Sedan (25 mpg):
[latex]10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{25 {\rm \; miles}} = 400 {\rm \; gallons}[/latex] |

Minivan (20 mpg):
[latex]10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{20 {\rm \; miles}} = 500 {\rm \; gallons}[/latex] |
Hybrid (50 mpg):
[latex]10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{50 {\rm \; miles}} =200 {\rm \; gallons}[/latex] |

This upgrade saves: [latex]1\, 000 – 500 = 500[/latex] gallons |
This upgrade saves: [latex]400 – 200 = 200[/latex] gallons |

So, the mere fact that the numbers we’re seeing on the screen (the MPG ratings) belong *in the denominator* of the conversion not only thwarts our proportional reasoning, it inverts it. Denominators, as I often tell my students, are Bizarro World: up is down, bigger is smaller, and “nothing could mean anything.”

### The Gravamen of Culture

Quantitative reasoning is a human expression of mathematical thought. As such, QR is inescapably socially constructed and culturally informed.

In this example, the cultural information that obscures our reasoning is the distinctly American habit of *reporting fuel economy with the fuel in the denominator* (miles *per gallon*). Cognitively, this has several effects.

- On the plus side,
*it supports “more is better” cognition.*Except for trolls who roll coal, most of us consider more fuel-efficient vehicles to be more preferable when shopping for a car. Situating the fuel in the denominator means (again, Bizarro World) higher values mean less fuel: higher MPG rating means more fuel efficiency. - On the down side,
*it centers our cognition on miles instead of gallons**.*Not only is this the reason for our thwarted intuition in this problem, it also invites us — whether we realize it or not — to think of our fuel consumption as fixed, and imagine instead how much further we could drive on that same amount of fuel. By playing into our tendency toward denominator neglect, this framing also plays into the fossil-fuel industry’s hands because, for most drivers, annual mileage is more likely to remain constant when they purchase a new car. The real impact they are making when they upgrade is that they are (gasp!) consuming less fuel.

It’s just like the classic advertising story about A&W’s failed third-pound burger:

In the 1980s, A&W attempted to compete with McDonald's "Quarter Pounder" by introducing a third-pound burger.

However, it didn't sell because Americans thought 1/4th of a pound was larger.

— UberFacts (@UberFacts) February 7, 2019

Denominator neglect is the source of all manner of false cognitive illusions. So, why do I describe this as a cultural phenomenon in the U.S.?

Because European regulatory agencies have chosen to avoid it. *European agencies report fuel economy with fuel in the numerator instead of the denominator, *for instance, turning the dimensional “equation”

[latex]100\; {\rm kilometers} = 24 \; {\rm liters}[/latex]

into the conversion factor

[latex] \frac{24\; {\rm liters}}{100\; {\rm kilometers}} [/latex]

instead of the reciprocal. This approach sacrifices the “more is better” cognition, because a smaller figure — using less fuel over an equal distance — represents greater efficiency. But it avoids denominator neglect, it shines the spotlight on the amount of fuel consumed as the driver’s main independent variable, and it re-energizes our proportional reasoning.

In the EU, for example, this exercise would not create a cognitive illusion.

Consider a family that has an SUV that uses 24 L per 100 km and a sedan that uses 10 L per 100 km. Both are driven equal distances in a year. Is the family better off replacing the SUV with a minivan that uses 12 L per 100 km, or replacing the sedan with a hybrid that uses 5 L per 100 km?

Now, because the fuel usage is in the numerator instead of the denominator, and we are multiplying by these numbers in our conversion rather than dividing, our proportional reasoning should give us a more sensible insight. Let’s imagine 16,000 km of annual driving, which is approximately the same as 10,000 miles (though again, that specific amount will not affect the conclusion):

SUV (24 L / 100 km):
[latex]16\,000 {\rm \; km} \cdot \frac{24 {\rm \; L}}{100 {\rm \; km}} = 3\, 840 {\rm \; L}[/latex] |
Sedan (10 L / 100 km):
[latex]16\,000 {\rm \; km} \cdot \frac{10 {\rm \; L}}{100{\rm \; km}} = 1\, 600{\rm \; L}[/latex] |

Minivan (12 L / 100 km):
[latex]16\,000 {\rm \; km} \cdot \frac{12 {\rm \; L}}{100 {\rm \; km}} = 1\, 920 {\rm \; L}[/latex] |
Hybrid (5 L / 100 km):
[latex]16\,000 {\rm \; km} \cdot \frac{5 {\rm \; L}}{100 {\rm \; km}} = 800 {\rm \; L}[/latex] |

This upgrade saves: [latex]3\, 840- 1\, 920= 1\, 920 [/latex] L |
This upgrade saves: [latex]1\, 600 – 800= 800[/latex] L |

Shockingly, the answer is still the same as our MPG example! (In part, this is because the fuel economy figures I chose are approximately equivalent to the originals.) But the proportional reasoning “works” in that we can clearly see, in the last row, that halving the fuel consumption rates has indeed halved the amount of fuel consumed in either case. It’s just that the SUV is using *so much more* fuel in a year compared to the sedan, since both are being driven equal distances, so half of its total consumption is still significantly greater than half of the sedan’s. We’ve traded denominator neglect for a form of base rate neglect instead. But at least there’s a glimmer of valid proportional reasoning here.

### The Moral

Because of the precise nature with which numbers carry meaning, correct quantitative insights almost always require engaging our slower, analytical cognitive machinery — rather than relying on the quick, reflexive answers our relational minds provide by default.

But our “System 2” thinking requires significant effort to activate, and that means that the lazy, heuristic System 1 can occasionally catch out even more numerate people. Trying to power quantitative skills with heuristic thinking is like trying to build a campfire with only newspaper and lighter fluid: bound to generate more light than heat, and leave everyone out in the cold.

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