## Thursday, April 21, 2011

### There are two types of drivers

This puzzle celebrates our master puzzle solver: Bean.

A holiday table conversation at Bean's household this week was prompted by one of the guests getting stuck on the way to the dinner without gas, which led to musing about two types of drivers. There are those who prefer to live on a safe side, possibly because they have nightmarish memories (or nightmarish fantasies) of their car stopping in the worst place and at the worst time. They refill as soon as the gas meter is showing less than half a tank. And there are those drivers, like our expert puzzle solver Bean, who enjoy taking a bit of risk here and there, rationalizing that to get to know this world you need to explore its boundaries. They zip to the gas station only after the tank is 90% empty and the low gas light turns on.

Do you think that by taking some risk this second group of people can save some time and/or money?
Bean was arguing that she can save a few hours per year and enjoy an extra family movie by pushing refills from 40% full tank to 5% full tank. Is she right? Assume a 17 gallon tank that runs empty in exactly a week, and that it takes 7 min to stop at the gas station.

Answers accepted all day long on Friday April 22nd and Saturday April 23rd, on our Family Puzzle Marathon. They will be hidden until Sunday morning (EST) and everyone who solved it will get a puzzle point. Please, explain your answer.

Donna said...

Bean is correct, if my calculations are correct. By waiting til his tank is almost empty, he fills up once a week. I assumed he spends about \$51 to fill up a 17 gallon tank (gas costing \$3/gallon, unrealistic, I know) which when extended over a period of a year, would cost him about \$2,652.00. Compare this with the non-Bean person who fills up whenever his tank is down by 60%, which would take only about 4 days to occur. This person would visit the gas station 91 times in a year (as opposed to Bean's 52 times), and would spend approximately .6 x \$51 each time, or \$30.60. When extended over a period of a year, this would cost about \$2,784.60 (\$30.60 x 91)

Now, looking at time spent, Bean would spend 7 minutes at the gas station every week for an entire year, which would amount to 364 minutes total (52x7). Non-Bean would spend about .6x7 min each visit, amounting to about 4.2 min. Extended over an entire year, with Non-Bean making 91 trips to the gas station, this would result in a total of about 382 min (91x4.2).

So, in conclusion, Bean would save about \$132.60 and about 18 minutes over Non-Bean.

Donna said...

Oops! Just re-read the problem, and realized Bean is a she....sorry Bean!

anne-marie said...

I consider that Bean does not travel long distances with her car and then, she does not have to rest every two or three hours.
Pushing refills from 40% to 5% means that every times that 16.15 gallons have been used, Bean is refillinig the tank instead of every times 10.2 gallons have been used.
16.5/10.2=1.6 and 1.6*7=11mn per week if I consider that 1 year is around 48 weeks, Bean would save around 9 hours.
Now, if there is no gas station near by, you can also lose so much time without considering the stress...I am not sure that it is worth it...

Tom said...

She MIGHT save some time by visiting the gas station less often. Barely. Saves no money at all I think.

There is increased risk of running out of fuel, however, and when that happens, a driver can lose a great deal of time and perhaps money. That's going to make y'all late for the movie.

And there is definitely increased risk of damage to the expensive fuel pump if it runs to empty (just on NPR this Friday morning). The pump runs very hot if not cooled by fuel flow. And the last mouthful of gas is nasty, dirty, etc.

The decisions should be made based on convenience, spare time, distance to the gas station if is not on the usual way to work or home, etc. Running OUT is not smart and can be dangerous in traffic, and even expensive. Many places it is illegal, punishable by a fine.

Playing with the given arithmetic...given that Bean is burning 15-16 gallons a week (about 90-95% of the 17 gallon tank), certainly she'd better stop for fuel once a week or oftener. The alternative presented, 40% full and 60% burned, would mean refueling approximately "half again oftener," or 3 times each two weeks. Spending an extra 7 minutes every two weeks at this chore is probably wise. "Saving up" that 7 minutes toward some imagined family activity seems silly; we might even say that such "time thrift" is compulsive in some way and might need to be discussed.

But it may be hard to change that habit of trying to run it to near-empty. The math will probably not be convincing, but a new fuel pump will get your attention.

Bean said...

Apparently Math Mom has been eavesdropping on the Bean household, where this discussion has occurred more than once. And yes, Bean is the defender of waiting for that orange light to come on. The one flight of fancy is that Bean only needs gas once a week. I wish!

Not only can she enjoy a movie, but she can even queue up one of those deadly long foreign films in the Netflix queue, or ten old 30 Rock episodes.

Anyway, 17 gallons x 52 weeks = 884 gallons/year. If you fill up at 40% full, you are filling every 10.2 gallons; at 5% full, you are filling every 16.16 gallons. Divide 884 by 10.2 and 16.16, respectively, and you get a choice between 86.67 fillups a year and 54.73. At 7 minutes a fillup, that's the difference between 606.69 minutes/year and 383.16 minutes/year. So Bean saves 223.53 minutes a year.

Now, if Bean runs out of gas and calls AAA, that would reduce the savings, as her husband has helpfully pointed out. On the other hand, seven minutes does not include the elaborate mapping problem involved in optimizing the getting to the Costco gas station/orange light balance.

Total Nerd said...

Yes I think it is possible.

Pat said...

Can Bean save time? yes. Money? yes, but I don;t think there's enough information here to determine how much.

52 weeks per year, 17 gals/week = 884 gals/year
60% * 17 gal. gas tank = 10.20 gals per fillup,
- meaning Bean would fill up 87 times/year
- requiring 10.1 hours
95% * 17 gal. gas tank = 16.15;
- meaning 55 times per year
- meaning 6.4 hours
So Bean would save 3.7 hours, enough time for even Avatar.

As to money, we can assume that time is money, also, the car is a little lighter (on average 3 gals * ~7.5 gals per pound), and thus should get slightly better mileage. Maybe enough to cover one additional (at home, on-demand) movie?

Pat said...

If you refill a 17 gal. tank when it is 40% full you will put 10.2 gal.in the tank at each fill-up. At 5% full you will add 16.15 gal. each time. 17 gal. a week times 52 weeks in a year is 884 gal.a year. To get 884 gal. at 10.2 gal. a fill-up you need to fill it 86.67 times at 7 min. each fill for a total time of approx. 606.67 min or 10 hrs. 6 min. 20 sec. At 16.15 gal. you fill-up 54.74 times for a total of 383.16 min. or about 6 hrs.23 min. 10 sec. You save 223.51 min. or about 3 hr.43 min. 31 sec. You can watch two movies of 111 min. or less. You make the popcorn, I'll bring the soft drinks.

thelittlebird said...

Happy Spring Ovoid Day!! At least we math lovers know that a 3 dimensional egg is not called a sphere.

So, to the puzzle. We're assuming that no matter how full the tank, it takes 7 minutes to fill it at the station. We also assume that we have 7 days of usage in a full tank.

If you fill your tank whenever it hits 40% full (or 60% empty), you will fill it every 4.2 days (.60 * 7 days). If you fill your tank every 4.2 days, you will fill it 86.9 times in a year (365/4.2). The time spent filling the tank in this scenario is 608.3 minutes per year (86.9 fills * 7 minutes per fill). [If you want to be logical and say that you have to go 87 times per year this would leave you with 609 minutes]

In the 5% full scenario (or 95% empty), the car will need filled every 6.65 days (.95 * 7 days). If you fill up every 6.65 days, you will visit the gas station 54.9 times per year (365/6.65). Fill time is 384.3 minutes per year (54.9 fills * 7 minutes per fill). [Again, rounding up to 55 times a year gives you 385 minutes of fill time]

The time saved by taking a risk and filling the tank at the 5% full mark instead of being safe and filling at 40% full is 224 minutes (608.3 - 384.3). This is more than enough time to watch the newly remastered director's cut of the 224 minute "Woodstock: 3 Days of Peace & Music: Ultimate Collector's Edition" and possibly even 2 short movies with your kids!! You also probably save enough money in additional grocery gas savings on the almost full tank to buy some popcorn and candy for the movie.

[FYI: 609-385 is also 224 minutes!]

anne-marie said...

I rectify my answer, I divide 10.2 by 16.15 which is 4.42 and I multiply it by 7*52 which is 229.9.
229.9/60 = around 4 hours per year.

Annie said...

Assume that both cars are empty at the 1st of the year and they both fill their tanks with 17 gallons of gas.

If they drove their cars until they were empty they would use 17 x 52 = 884 gal./yr.

The 40% full tank will fill up after using 10.2 gal. of gas (17 x .6 = 10.2) Therefore they will fill up (884 - 17) / 10.2 = 85 fillups plus the initial fill up = 86.

The 5% full tank will fill up after using 16.15 gallons. ( 884-17)/16.15 = 53.7 = 54 + initial fillup = 55.

86-55=31 fewer fillups for the 5%er. At 7 min/fill up, she saves 217 min/yr or 3 hrs and 37 min.

40% will buy 884 gal/yr ((10.2x85)+ 17) and 5% will buy 889.1 gal/yr((16.15x54)+17) By buying gas at 40% full, Bean will save the cost of 5.1 gal. of gas for that year. (But she'll be starting the following new year with some gas in her tank.)

I guess it'll come down to whether she'd rather have the time or the money.

Anonymous said...

OPTION A: A 17 gallon tank that is 40% full has 6.8 gallons of gas. That would mean each fill-up would require 10.2 gallons of gas.
OPTION BEAN: A 17 gallon tank that is 5% full has .85 gallon of gas. Each fill would require 16.15 gallons of gas.
If the 17 gallons of gas lasts one week, we need 884 gallons of gas per year.
Let's say that both cars started off with a full tank on January 1. The OPTION A driver then makes 85 trips to the gas station throughout the year, averaging 7 minutes each for a total of 595 minutes.
The OPTION BEAN driver makes 54 trips to the gas station for a total of 378 minutes.
Bean saves a whopping 217 minutes a year or 3 hours, 37 minutes.
I also looked at the average amount of gas each driver has in their tank.
The OPTION A driver has an average of just under 12 gallons of gas in his tank (fills to 17, runs down to 6.8 gallons).
Bean has an average of just under 9 gallons of gas (fills to 17, runs down to .85 gallon.)
Bean has an average of 3 gallons less than OPTION A which means about 18 lbs. less weight being driven around. I'm not sure how much savings that would be, but it would be something.
So, next time your significant other complains that you left the gas tank empty, just tell them that you're trying to save money! Mathmover

Maria said...

I am very proud of myself for matching Bean's personality (at least in refilling) without knowing anything about her. But I did miss on her commute length. The answers above, by the way, do reveal a lot about what type of driver each of us is and what movie we would watch if we have a choice.

I agree that she could save around 4 hours by waiting for the orange light and Bean herself provided the best explanation for the math.

As to the money - no money saving will be happening as the amount of gas spent in a year will be the same in both scenarios. The only tiny difference will be in having your gas tank full or empty at the end of the year.

A puzzle point for: Donna, anne-marie, Tom, Bean, Pat, thelittlebird, Annie and mathmover.
Total Nerd - welcome, you need to provide an explanation for your answer to get a puzzle point.

Have a great weekend and see you next Friday!

Maria said...

Mathmover - congratulations! You solved 5 puzzles and deserve a prize. Please email me your post address via contact link below and I will send you the prize.

Tom said...

Sunday comment from Tom (Happy Easter too). There is one other aspect we have overlooked; maybe two. (1) If a driver is able to choose to fill up at the cheapest station, seizing on an opportunity when it is found, then some cents can indeed be saved.

And (2) if gas prices keep climbing as they have been, then it may be worthwhile to fill the tank today instead of next week....inflation, see.