Three cloud servers, X, Y, and Z, working simultaneously can process a large database migration in $13\frac{1}{3}$ hours, whereas servers X and Z operating together can complete the same migration in 16 hours. How many hours would server Y take working alone to process 75% of the same database migration?

The correct option is 60 hours.

Step-by-step Explanation:

Step 1: Calculate the combined work rate of servers X, Y, and Z working together.
The three servers X, Y, and Z operating together can complete the database migration in
$13\frac{1}{3}$
hours, which converts to an improper fraction as
$\frac{40}{3}$
hours.
The combined work rate of (X + Y + Z) per hour is:
$\text{Rate of (X + Y + Z)}=\frac{1}{40/3}=\frac{3}{40}$
of the total migration work per hour.

Step 2: Calculate the combined work rate of servers X and Z working together.
Servers X and Z operating together can complete the database migration in 16 hours.
The combined work rate of (X + Z) per hour is:
$\text{Rate of (X + Z)}=\frac{1}{16}$
of the total migration work per hour.

Step 3: Determine the work rate of server Y operating alone.
The work rate of server Y is the difference between the combined rate of (X + Y + Z) and the combined rate of (X + Z):
$\text{Rate of Y}=\text{Rate of (X + Y + Z)}-\text{Rate of (X + Z)}$
$\text{Rate of Y}=\frac{3}{40}-\frac{1}{16}$
Taking the least common multiple of 40 and 16, which is 80:
$\text{Rate of Y}=\frac{6}{80}-\frac{5}{80}=\frac{1}{80}$
Server Y processes
$\frac{1}{80}$
of the database migration per hour.

Step 4: Find the total time server Y takes to complete 100% of the migration alone.
Time required for server Y to complete the full migration =
$\frac{1}{1/80}=80$
hours.

Step 5: Calculate the time server Y takes to process 75% of the database migration.
75% of the total migration is equal to:
$75\%=\frac{75}{100}=\frac{3}{4}$
of the total work.
Time taken by Y to process 75% of the migration =
$80\times \frac{3}{4}=60$
hours.

Thus, server Y working alone would take 60 hours to process 75% of the database migration.