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Question 1210725: A high-tech research facility begins the year with a specific pool of scientists divided into three departments: AI, Biotech, and Quantum. 1/3 of the total scientists are in the AI department. The remaining scientists are split between Biotech and Quantum in the ratio of 5 : 3, respectively. Mid-Year. The Quantum department undergoes a major breakthrough and its staff size increases by 40%. The AI department faces budget cuts, and 1/5 of its scientists are reassigned to other non-research roles (leaving the total pool of research scientists). The Biotech department size remains unchanged. After these mid-year shifts, the Quantum department finishes the year with 210 scientists. How many scientists total did the facility start with at the beginning of the year, and what is the new size of the AI department?
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(54062)   (Show Source):
You can put this solution on YOUR website! .
A high-tech research facility begins the year with a specific pool of scientists divided into three departments:
AI, Biotech, and Quantum. 1/3 of the total scientists are in the AI department. The remaining scientists are split
between Biotech and Quantum in the ratio of 5 : 3, respectively.
Mid-Year, the Quantum department undergoes a major breakthrough, and its staff size increases by 40%.
The AI department faces budget cuts, and 1/5 of its scientists are reassigned to other non-research roles
(leaving the total pool of research scientists).
The Biotech department size remains unchanged.
After these mid-year shifts, the Quantum department finishes the year with 210 scientists.
(a) How many scientists total did the facility start with at the beginning of the year, and
(b) what is the new size of the AI department?
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Let x be the number of scientists the facility starts with at the beginning of the year.
Then the population in AI department is  ;
the remaining population is  ;
the Biotech population is  ;
the Quantum population is  .
After the mid-year changes
the Quantum population is  , or 210 scientists.
This gives us an equation
= 210.
From this equation, we find
x =  = 600.
It is the answer to question (a).
The new size of the AI department is  =  =  = 160.
It is the answer to question (b).
ANSWER. (a) 600; (b) 160.
Solved.
Answer by math_tutor2020(3852)  (Show Source):
You can put this solution on YOUR website!
q = original number of quantum researchers
1.4q = increase the previous amount by 40%
1.4q = 210
q = 210/1.4
q = 150
There were originally 150 quantum researchers at the start.
Originally the biotech and quantum departments have a staff ratio of 5:3
Meaning that for every 5 biotech workers, there were 3 quantum workers.
5/3 = bio/quantum
5/3 = b/q
5/3 = b/150
5*150 = 3b
b = 5*150/3
b = 250
There were 250 biotech researchers at the start.
At the start there were b+q = 250+150 = 400 biotech and quantum researchers.
This composes 2/3 of the original workforce
x = total number of researchers at first
(2/3)x = 400
x = 400*(3/2)
x = 600
There were 600 scientists at the start of the year.
That means there were x - (b+q) = 600-(250+150) = 200 researchers initially in the AI department.
1/5 of these people leave for non-research roles, so 4/5 of them remain
(4/5)*200 = 160 scientists is the new updated head count.
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Recap:
Question: How many scientists total did the facility start with at the beginning of the year?
Answer: 600
Question: What is the new size of the AI department?
Answer: 160
Answer by greenestamps(13391)  (Show Source):
You can put this solution on YOUR website!
So many different ways to set up a problem like this....
Given the ratio 5:3 of the scientists in Biotech and Quantum, that's where I would choose to start.
5x = Biotech scientists at the start of the year
3x = Quantum scientists at the start of the year
So the total number of Biotech and Quantum scientists at the beginning of the year is 8x. The AI scientists are 1/3 of the total at the beginning of the year, so the 8x Biotech and Quantum scientists are 2/3 of the total. So
4x = AI scientists at the start of the year
By the end of the year, the number of Quantum scientists has increased by 40%, the number of AI scientists has decreased by 20%, and the number of Biotech scientists has not changed.
(1.4)(3x) = 4.2x = Quantum scientists at end of year
(0.8)(4x) = 3.2x = AI scientists at end of year
5x = Biotech scientists at end of year
At the end of the year, the number of Quantum scientists is 210:
4.2x = 210
x = 210/4.2 = 50
Use that to find the numbers of scientists in each group at the beginning of the year and at the end of the year.
Beginning of year
Biotech: 5x = 250
Quantum: 3x = 150
AI: 4x = 200
End of year
Biotech: 5x = 250
Quantum: 4.2x = 210
AI: 3.2x = 160
Then answer the specific questions that were asked
ANSWERS:
(1) total number of scientists at the beginning of the year: 250+150+200 = 600
(2) number of AI scientists at the end of the year: 160
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