Mario's watch runs fast . in 1 day , it gains an hour. so in 12 days it gains 12 hours and is correct again . julio's watch also runs fast. in 1 day , it gains 20 minutes. suppose they both set their 12 hour watches correctly at 9: 00 a. m on monday. when will their watches next show the correct time together?

Tuesday 09:00AM

In this case, the watch is 12 hour so you just need to complete 12 hours cycle not 24 hours. If seen at 12 hour watch, Mario's watch is moving 13 hours and Julio's watch moving 12+1/3 hour. First, you need to know how much days needed to make the watch show correct time for each of them.
For Mario's watch : 12 hour/1 hour= 12 days.
For Julio's watch: 12 hour/(1/3) hour= 36 days.

After that, you need to find the least common multiple. It's when both of their watches showing correct time together. The least common multiple of them is 36 days or 5 weeks +1 day

So the answer is Monday 9:00AM + 5 weeks + 1 day= Tuesday 09:00AM