X does 1/2 as much work as Y, let's break it right here. So that means X does equal amount of work as Y in twice the amount of time mentioned (for half the work done) which is 2 * 1/3 of Y = 2/3 of Y.
So now we know X will do same amount of work as Y in 2/3 the time. if it takes Y 60 minutes to do something, it takes X 60*2/3 = 40 minutes to do the same job.
now we know they're doing equal amounts of work in different amounts of time, so lets see what proportion of work each can do working together for a given amount of time:
let's say the time is 1 hour (the number of hours/days here doesn't matter, we could've just started with 20 days, but goes easier if it's unit time). out of 1 hour, if Y finishes 1 job, X will have already finished it in 40 minutes, as mentioned above, nonetheless X keeps working for the full hour. How much work will X accumulate in another 20 minutes needed to complete the hour?
well, 40 minutes for a job means 20 minutes for half a job, so that's a total of 1.5 jobs per hour for X, while it is 1 job per hour for Y. How much total work was done by both X and Y combined? 1 + 1.5 = 2.5 jobs in 1 hour(or unit time).
now break down each person's share of jobs accomplished, as a fraction of total jobs accomplished by both X and Y combined.
X's proportion: 1.5/2.5 = 3/5 = 60% of the work
Y's proportion: 1/2.5 = 2/5 = 40% of the work
so Y can accomplish 40% of the work in 20 days, while X puts in the other 60%.
we want to know how long Y takes working alone. it should be intuitive that if both X and Y work together and accomplish the job in 20 days, it should take X or Y individually longer to accomplish the same task working alone.
Y completes 40% of the job in 20 days, so how many days does is take Y to complete a 100% of the task?
so we have 40%/20days = 100%/?days
cross multiply to get:
40 * ? = 100 * 20
which is 40 * ? = 2000
now divide both sides by 40 to find the question mark
2000/40 = 50
it takes Y 50 days to complete the task alone.