Sunday Numbers 2.0, Vol. 9: Reciprocal word problems.

Most word problems except the ones involving compound investment are solved with linear equations. They can look very different, like solving how much of two different alcohol solutions to combine x gallons of 25% alcohol solution and y gallons of 50% alcohol to make to make 10 gallons of 30% solution or how 19 dimes and quarters can add up to $2.50, but those kind of problems use linear methods. (The answers are 8 gallons 25% and 2 gallons 50%, and 15 dimes and 4 quarters. Figuring out how to set up the problems is left as an exercise to the reader.)

Let's consider something about the answers. With 19 coins that are either dimes or quarters, the lowest possible answer is $1.90 (all dimes) and the greatest total is $4.75 (all quarters). For the solution problem, the minimum percentage is 25% and the maximum percentage is 50%.

The answer must be between the two known extremes.

Consider the following question instead.

One drain pipe can empty a pool in 2 hours, while a smaller pipe can empty the same pool in 4 hours. How much time will it take if they work together, provided that they don't get in each others' way?

I always hate to use this phrase, bit it should be obvious the correct answers are not between 2 and 4 hours, but instead less than 2 hours.


This problem is solved using reciprocals. If a pipe can do the job in two hours, let's assume it finishes 1/2 the job every hour. (Depending on the physics, this assumption might not be accurate, but let's leave that alone for the moment.) Using this assumption, that means the smaller pipe which takes four hours finishes 1/4 of the job in an hour.

If we agree that they can work without getting in each others' way, in one hour they do 1/2 + 1/4 = 3/4 of the job. Once we add the reciprocals together, we reciprocate the sum to get the answer. The reciprocal of 3/4 is 4/3 or 1 1/3, so the two pipes working together finish the job in 1 hour and 20 minutes. This means opening the second pipe is only a 40 minute savings over doing the job with the first pipe alone.

This may very well be the trickiest word problem type around, though others involving rate, distance and time might also get the the award.

Some of those next week.