Final Answer:
The next date on which both Olga and Nadia will rest is January 21. They will both rest on this day because it is the least common multiple of 4 and 5, the respective rest intervals for Olga and Nadia.
Step-by-step explanation:
Olga rests every fourth day, and Nadia rests every fifth day. To find the next date on which both will rest, we need to determine the least common multiple (LCM) of 4 and 5.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ...
The common multiples are 20, 40, 60, ...
So, the next date they will both rest is on the 20th day. However, since they both rested on the first day, we need to add 20 days to January 1, giving us January 21 as the next date on which both Olga and Nadia will rest.
In conclusion, by finding the least common multiple of their rest intervals, we determined that Olga and Nadia will both rest on January 21, following their respective workout routines. This calculation ensures that we account for the day they both started their new routines and find the next occurrence of simultaneous rest days.