A new hospital in Silvergrove starts out with 8 junior residents and 6 senior residents on its staff. Management plans to hire additional personnel at a rate of 1 junior resident per month and 3 senior residents per month. Eventually, there will be an equal number of each on the hospital staff. How long will that take? How many of each type will there be?
Let's assume it will take "x" months to have an equal number of junior and senior residents on the hospital staff.
After "x" months, the number of junior residents will be 8 + 1*x = 8 + x.
After "x" months, the number of senior residents will be 6 + 3*x = 6 + 3x.
To have an equal number of each type on the hospital staff, we set the two equations equal to each other:
8 + x = 6 + 3x.
Subtracting x from both sides, we get:
8 = 6 + 2x.
Subtracting 6 from both sides, we get:
2 = 2x.
Dividing both sides by 2, we get:
x = 1.
So it will take 1 month to have an equal number of junior and senior residents on the hospital staff.
After 1 month, the number of junior residents will be 8 + 1*1 = 8 + 1 = 9.
After 1 month, the number of senior residents will be 6 + 3*1 = 6 + 3 = 9.
Therefore, after 1 month, there will be 9 junior residents and 9 senior residents on the hospital staff.