Jane alone can do the work in 15 days and John alone can do it in 10 days. How many days are needed if Jane and John work together?

Jane rate    1 job / 15 days      John rate =  1 job/10 days

 

together     1 job / ( 1/15 + 1/10)   = 1/ ( 2/30 + 3/30)   = 1/ ( 5/30) = 30 / 5 = 6 days