Math question below in the description.

Question

A new hospital in Lakeside starts out with 12 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?

Answer 1

Hi Josie,

The aim is to eventually have the same number of junior and senior residents.

Let X and Y be the number of new junior and senior hires respectively. Let t be the time it takes to meet this objective.

12 + X = 6 + Y           (Equal number of junior and senior residents)

 X = 1 junior resident/month * t = t.

 Y = 3 senior resident/month * t = 3t

You get

 

12 + t = 6 + 3t

       6 = 2t

        3 =t

X = 3, Y = 9

It will take 3 months to hire 3 Junior residents and 9 Senior residents

 

 

 

Answer 2

Let's assume that the hospital will have equal numbers of junior and senior residents in x months since when they had 12 junior residents and 6 senior residents on its staff

 

Thus, hiring 1 junior resident per month will give x junior residents after x months when total number of junior residents will become 12+x

 

And, hiring 3 senior residents per month will give 3x senior residents after x month when total number of senior residents will become 6+3x

 

Hence, 

 

6+3x=12+x

=>2x=6 ...............subtracting x and 6 from both sides

=> x=3 .............. dividing both sides by2

 

Thus in 3 months the number of Junior residents will be 12+3=15

And the number of Senior residents will be 6+9=15

 

Therefore, that will take 3 months. 15 of each type will be there.