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August 31

MPG Word Problem

I added a word problem to a test and out of 22 students in the class, I received 22 radically different answers ranging from 3 to 156k. When I did it in my head, I got 56,351 (which none of the students got). So, I would like to know if the users here can come to a consensus about the answer.

I used to get 20.889 miles per gallon in my car. I paid $550 for an upgrade. Now, I get 23.261 miles per gallon. Assume that gas is constantly priced at $2 per gallon. How many miles do I have to drive to save $550 (paying off the upgrade)? 209.149.113.5 (talk) 16:26, 31 August 2017 (UTC)[reply]

If one goes x miles at 23.261 miles per gallon then that requires x/23.261 gallons

For x miles the saving is $2 * (x/20.889 - x/23.261) and if one save $550 going x miles that gives

x = 550/(2*(1/20.889-1/23.261))

Sticking that into Google as I can only do 2 decimal places in my head gives 56333.1504954 say 56333. So I guess you're only good to about 3 decimal places. :) Dmcq (talk) 16:42, 31 August 2017 (UTC)[reply]

I confirm Dmcq's result. Rounding to 56333 is the way to go since the given mpg values have 5 significant figures. -- Meni Rosenfeld (talk) 16:47, 31 August 2017 (UTC)[reply]

Well actually only four significant figures at best because of the subtraction 23.261-20.889. And my best guess at his original figures is 564 miles on 27 gallons before and 535 miles on 23 gallons afterwards. Supposing we're really generous and say they measured gallons accurately that would be 3 significant digits at a stretch. Dmcq (talk) 06:55, 1 September 2017 (UTC)[reply]

Sounds fair, and consistent with the OP's result only matching yours to 3 sf. -- Meni Rosenfeld (talk) 15:20, 1 September 2017 (UTC)[reply]

Thanks I got the same equation: 550 / (2/20.889 - 2/23.261). In MatLab, the answer was 5.6351e+04. It is highly likely that since I used the variables for MPG from previous calculations, it was using more digits for MPG than I saw when I did the MPG calculations. 209.149.113.5 (talk) 17:08, 31 August 2017 (UTC)[reply]

How old are your students? I hope they are not from a university? Ruslik_Zero 20:39, 31 August 2017 (UTC)[reply]

A student might take 23.261 mpg - 20.889 mpg = 2.372 mpg, and then wonder why their answer of 2.372 mpg * $550 / ($2.00/gal) = 652.3 miles is incorrect. The factor-label method is an important tool, but here is a case where dimensional analysis alone does not do the job as it doesn't differentiate between the erroneous solution above and the correct solution they would have found by using a fuel economy in terms of fuel consumed per distance instead of its reciprocal, distance traveled per unit of fuel: (1/20.889) gal/mi - (1/23.261) gal/mi = 0.0048817 gal/mi, and $550 / (0.0048817 gal/mi * $2.00/gal) = 56,333 miles.

While a student can be excused for coming up with the wrong answer, there is no excuse for not checking the result and discovering that it was incorrect. (Do you give partial credit -- or even full credit, in cases were strong understand and reasoning is exhibited -- to a student who comes up with the wrong answer, but is able to explain why they know it is wrong, even if they don't know where they went wrong?) This is a perfect example where you can teach your students that checking their result can lead to them finding the right way to do the problem.

Take the incorrect solution above and check it. How much money was saved on gas driving 652.3 miles after the upgrade? (652.3 miles / 20.889 mpg) * ($2.00/gal) - (652.3 miles / 23.261 mpg) * ($2.00/gal) = $6.3686, well short of the $550 goal. At this point the student: #1, knows that their original answer was wrong. #2, knows how far they were off -- that the correct answer was ($550/$6.3686) * 652.3 miles = 56,333 miles. And, #3, knows how they should have gone about the problem in the first place, by replacing their incorrect value 652.3 miles with a variable x in the equation they used to check their answer: (x / 20.889 mpg) * ($2.00/gal) - (x / 23.261 mpg) * ($2.00/gal) = $550, and solving for x, giving your original solution. -- ToE 21:40, 31 August 2017 (UTC)[reply]