BOATING Magazine: Bad Math Returns

[jimh]

In a prior article I commented about the "certified" boat test in BOATING magazine, in which the data showed that a boat with a 300-HP VERADO engine was able to get a fuel economy of 54.5-MPG at full throttle. I recently came across another boat test in BOATING magazine that has some familiarly bad mathematics.

In the APRIL 2019 concern of BOATING on page 65 there is a "BOATING Certified Test" of a Four Winns VISTA 355. The fuel consumption, engine speed, and boat speed data look quite reasonable, but the author of the article, Kevin Falvey, seems a bit confused about basic mathematics. Kevin is also the magazine's Editor-in-Chief, so perhaps no one else looks at his copy before it goes to print. He writes:

The boat's most economical cruising speed comes at 5,000 rpm and 33 mph, where you can net 1.1 mpg. If sea conditions allow, bump it up to 37 mph: you'll only give up one-tenth of 1 percent in efficiency...

With real math, that means that the MPG would decrease from 1.1-MPG to something less by a factor of "one-tenth of 1 percent." We just need to multiply 1.1 by 0.001. That means a decrease of 0.0011. So we would expect the MPG at the higher speed to then be 1.0989 MPG. However, in the table of data, the decrease in MPG is to 1.0 from 1.1, which is a factor of -0.1. To express -0.1 as a percentage of 1.1, we would call it a 9-percent decrease. To call a 9-percent (0.09) decrease a decrease of only 1/10th of 1-percent (0.001) is an error by a factor of 0.09/0.001 or 90-to-1. Expressed as a percentage that is an error of about 900-percent.

When I read published statements expressing mathematical relationships that are in error by a factor of 90-to-1, I have to wonder what else in the accompanying article might be improperly stated.

[jimh]

Here is a little math problem. It is a bit harder than it looks at first glance.

A person wants to rent a car. The rental company has two similar cars, A and B.
Their fuel economy is:

Car A = 17-MPG and is bigger and roomier
Car B = 22-MPG and is smaller and not so roomy

Interestingly the rental rates for the period of the rental are as follows:

Car A is $4 cheaper than Car B.

This seems odd because Car A is is bigger and roomier, but apparently people like Car B because it gets better fuel economy, so the rental company charges more for Car B.

Assuming: that the two cars get their rated MPG, that Gasoline costs $3.80 per gallon, and that you rented Car B, how many miles do you have to drive before you save $4 in gasoline compared to your fuel costs if you had rented Car A?

[rtk]

Car A costs $.224 per mile for gas. ($3.80 / 17)

Car B costs $.173 per mile for gas. ($3.80 / 22)

Car B costs $.051 per mile less for gas to operate.

$4.00 / $.051 dollars = 78.431 miles.

Rich

[jimh]

RICH--a very good solution.

Here is another approach to the problem, a bit more complicated:

First we convert $4 into the number of gallons that must be saved:

$4 x 1-gallon/$3.80 = 1.053 gallons.

To solve for the miles needed to create a difference in fuel consumption of 1.053 gallons, we let x be the number of miles driven.

When Car A drives x mile it will consume x/17 gallons
When Car B drives x miles it will consume x/22 gallons

The goal is to find x such that Car A's fuel minus Car B's fuel =1.053 gallon. This is said mathematically

(1) x/17 - x/22 = 1.053/1

We have a fractional relationship with three different denominators, 17, 22, and 1. We factor them looking for the least common denominator: 17, 11, 2, 1. The least common denominator is 17 x 11 x 2 x 1 = 374. Now we multiply both sides of equation (1) by 374:

(2) (374x/17) - (374x/22) = 374(1.053)
(3) 22x -17x = 394
(4) 5x = 394
(5) x = 78.77 miles

CHECK:

(78.77/17) - (78.77/22) = 1.053
1.053 = 1.053

[rtk]

Very nice work Jim.

I also cringe when I see data/percentages used in the way they were used in your referenced article. In order to even come up with a measurement of one tenth of one percent relative fuel economy with the data that is available via the engine's fuel consumption calculator is suspect. I'm not quite sure the measurement is that precise.

Percent of change/relative difference is typically always for lack of a better term mismanaged. The decimal equivalent of a percentage figure is not well understood for some reason.

Here is some basic fun with percentages.

A 90 horsepower engine has 50% greater horsepower than a 60 horsepower engine. A 60 horsepower engine has 33% less horsepower than a 90 horsepower engine.

Car A gets 17 MPG. Car B gets 22 MPG. Car A fuel economy (MPG) is .227 or 22.7% less than Car B. Car B fuel economy (MPG) is .294 or 29.4% greater than Car A.

It is helpful to know the math behind the statistic before relying on the result as a relative point of reference or a relative point of central tendency.

Rich

[jimh]

Another problem with the comparison given in the BOATING article is the metric: "efficiency." The data is just about miles-per-gallon. The notion that this is equivalent to "efficiency" is a bit complicated. There is a variable, gallons, and a basis of measurement, miles. But there are two processes involved.

The gallons are a measurement of fuel. The fuel has to be converted into useful power, usually measured in horsepower. A better measurement of the efficiency of conversion of fuel to power is brake specific fuel consumption. Every engine has its own curve of brake specific fuel consumption, and we can't really tell where this process is most efficient from the MPG data.

Once the engine develops power, it has to be converted to thrust to propel the boat. That brings the propeller and the hull form into the process. The hull form efficiency can change with hull speed, and the propeller efficiency can change with the speed it travels through the water.

A fundamental problem with the data we have is there is no data about power. All we know is engine speed, and we don't really know how much power is being developed at each engine speed.

With regard to boats, we already know that pushing a boat faster is not a linear relationship to propeller shaft power, at least not with moderate V-hull planing hulls. We know if the power of the engine is doubled the boat speed will not be doubled. As a general case, pushing the boat through the water at higher speeds always requires proportionally more horsepower. In the boat test data we see that the speed increases to 37-MPH from 33-MPH, a change of plus 4-MPH, or expressed as a percentage of 33-MPH, an increase in speed of 12-percent.

In the data in the BOATING test we see that the MPG decreases only 9-percent while the boat speed increases 12-percent. (We do not see the the boat speed increase with only "one-tenth of 1-percent" decrease in MPG.) This seems a bit paradoxical, because we expect the hull form to become less efficient at being pushed through the water as speed increases.

There are several things that may be occurring. The engine may be operating at a more efficient brake specific fuel consumption, that is, the engine is getting more power out of the fuel at the higher engine speed. The propeller may also be getting more efficient as the speed it travels through the water while turning increases. And the hull might become more efficient at a certain boat speed.

Another efficiency may be in the time domain. By going faster the boat may get to its destination sooner; that may be an important efficiency, too. But there is no standard way to measure saved time. For some people, it might be hardly important at all.

[rtk]

Efficiency is a very broad term in the context of the referenced article. A motor vehicle user could be more concerned with time efficiencies (time saved travelling over a fixed distance) vs. fuel efficiencies (fuel saved travelling over a fixed distance).

https://www.dictionary.com/browse/efficiency

"...the ratio of the work done or energy developed by a machine, engine, etc., to the energy supplied to it, usually expressed as a percentage."

I am more accustomed to collecting fuel usage data to determine fuel efficiency- fuel consumed over a fixed distance at a certain speeds. Sea conditions and the amount of weight on a boat most certainly has an impact on the amount of fuel necessary to produce the engine horsepower needed to achieve a desired speed over water.

The thing I noticed most was the boat speeds that the author uses to opine on efficiency. Operating speeds of 33-37 miles per hour. It is very rare that one can run a boat of any size consistently at 33-37 miles per hour for any meaningful distance or time due to sea and boat traffic conditions. I have spent a fair amount of time on 30 foot boats fishing offshore in the Atlantic Ocean. Even on "bluebird" perfect weather days with light winds there is always some type of swell on the ocean. 30 miles per hour is a good rate of speed on the ocean. 25 miles per hour is more typical unless sea conditions get bad. That's when fuel consumption over a distance becomes a concern with regard to having enough fuel and also the cost of fuel. When 30 or more gallons per hour at cruise are being consumed over 50 to 100 miles fuel costs can get pricey.

I guess the moral of my story is fuel usage data at 34-37 miles per hour is not useful nor relevant to me because I rarely travel at those boat speeds for any appreciable amount of time or distance. And the ability to travel 37 miles in an hour vs. 34 miles in an hour is not relevant to me unless we are trying to make a fish weigh in by a certain time. If that is the case then who cares how much fuel is burned just go as fast as one can and take the "beating" of the uncomfortably fast ride.

Rich