Bob just filled his car's gas tank..

[DeepthiS561]

Student Question: How would you solve this problem using the teeter totter method that's in the All the Quant Guides? I understand how to do this problem algebraically, but I usually prefer teeter totter method and don't understand how to utilize that method in this problem.

Thanks!

Problem:

Bob just filled his car's gas tank with 20 gallons of gasohol, a mixture consisting of 5% ethanol and 95% gasoline. If his car runs best on a mixture consisting of 10% ethanol and 90% gasoline, how many gallons of ethanol must he add into the gas tank for his car to achieve optimum performance?


A) 9/10


B) 1


C) 10/9


D) 20/19


E) 2

[esledge]

For the teeter-totter, put the two things you are mixing at the ends of the teeter-totter:
(1) the gasohol with 5% ethanol and (2) straight ethanol with 100% ethanol.
Label one end with 5% and the other 100%. (It's probably also a good idea to write the word "ethanol," too, to help you keep track.)

Then, the "weight" is how much of those two things you have:
--At the 5% end, the "weight" is 20 gallons, so write "20 gallons" there.
--At the 100% end, the "weight" is what you are trying to find, so give that a variable and write "y gallons" there.

Then I'd draw a mark on the teeter-totter (or the fulcrum of the teeter-totter) for where the resulting mixture should fall. Since "his car runs best on a mixture consisting of 10% ethanol and 90% gasoline," the resulting mixture should be 10% ethanol. Try to locate this mark accurately and to scale: the teeter-totter goes from 5% to 100%, so 10% is much closer to the 5% end than to the 100%.

In fact, the fulcrum is (10-5)/(100-5) = 5/95 = 1/19 of the distance from 5% to 100%.
So, the 20 gallons at the 5% end is 18/19 of the total weight: 20 gallons = 18/19 * total
And the x gallons at the 100% end is 1/19 of the total weight: x gallons= 1/19 * total

This can be solved algebraically for x. Or just note that the expression for x is 1/18th the expression that is equivalent to 20 gallons, so x = 20/18 = 10/9 gallons.

Just a note: as an alternative to the teeter-totter, you can do the same setup but think of it as a tug-of-war instead (in this problem the 5% end is "winning," by 18/19 of the total), if that suits your thought process better.