Gimme a Break… It’s Already Time to Rake?? - The Solutions

October 22, 2001

If they are working together, then they will both be working for the same amount of time, x. Since we want our answer in minutes, let’s change the amounts of time it will take them to do the entire job on their own to 150 minutes and 240 minutes. This means the Leif will get x/150 of the job done in x minutes, while Autumn will get x/240 of the job done in x minutes. However, together, their partial jobs should add up to the entire job, so: x/150 + x/240 = 1. Getting a common denominator of 1200, we see this is the same as (8x)/1200 + (5x)/1200 = 1; which is the same as (13x)/1200 = 1. Multiplying both sides by 1200/13, we see that x = 92 minutes… the length of time it will take for the two of them to rake the yard if working together.

If there are 6820 leaves with any yellow and 5550 leaves with any red, we can’t just add these numbers. We need to remember that 4370 of these leaves were counted twice because they had both yellow and red on them. Therefore, there were 6820 + 5550 – 4370 = 8000 leaves.

The most accurate way to do this would be to realize that there are 8000 leaves in the pile and 6820 of them will give Autumn the outcome she desires; a leaf with some yellow on it. Therefore, the probability of pulling out three “good” leaves in a row is (6820)/(8000) × (6819)/(7999) × (6818)/(7998) = .62, to the nearest hundredth. That’s a pretty good bet for Autumn.

Since there are so many leaves involved, we would get a pretty accurate answer if we just figured out the probability of success one time: (6820)/(8000) = .8525 and used that same probability for all three picks. (Removing one leaf is not going to change this probability by that much!) So we could find the probability by calculating .85253, which is a lot fewer calculator strokes and we still get .62!