A certain IT department of fewer than 15 people hires coders and systems administrators. Coders are paid $55,000 per year on average, while system administrators are paid an average yearly salary of $45,000. What is the ratio of coders to systems administrators?
(1) If two of the coders were made systems administrators instead, the yearly payroll for the IT department would be $535,000.
(2) If systems administrators' salaries were reduced by one-third, and coders' salaries were increased to $58,000, the department would save $57,000 in yearly payroll.
OA D
Source: Princeton Review
A certain IT department of fewer than 15 people hires coders
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Say there are c numbers of Coders and s numbers of systems administrators such that 2 ≤ c + s < 15BTGmoderatorDC wrote:A certain IT department of fewer than 15 people hires coders and systems administrators. Coders are paid $55,000 per year on average, while system administrators are paid an average yearly salary of $45,000. What is the ratio of coders to systems administrators?
(1) If two of the coders were made systems administrators instead, the yearly payroll for the IT department would be $535,000.
(2) If systems administrators' salaries were reduced by one-third, and coders' salaries were increased to $58,000, the department would save $57,000 in yearly payroll.
OA D
Source: Princeton Review
Given: Total payout = 55000c + 45000s
Question: What's the unique value of c/s?
Let's take each statement one by one.
(1) If two of the coders were made systems administrators instead, the yearly payroll for the IT department would be $535,000.
=> 55000(c - 2) + 45000(s + 2) = 535000
= 55(c - 2) + 45(s + 2) = 535
55c - 110 + 45s + 90 = 535
55c + 45s = 555
11c + 9s = 111
=> s = (111 - 11c)/9
s = 12 + (3 - 11c)/9
Since s and c are positive integers such that s + c < 15, we must plug-in few smart values to check the equation.
We see that (3 - 11c)/9 must be an integer so that s is a positive integer.
For c = 1 through 5, we do not get the integer value of s.
At c = 6, we have s = 12 + (3 - 11*6)/9 = 12 - 63/9 = 12 - 7 = 5; thus, c + s = 6 + 5 = 11 < 15 (Valid value)
From c = 7 through 13, we do not get the integer value of s.
Thus, the ratio of Coders to systems administrators = c/s = 6/5. A unique value. Sufficient.
(2) If systems administrators' salaries were reduced by one-third, and coders' salaries were increased to $58,000, the department would save $57,000 in yearly payroll.
=> 58000c + 2/3*45000s = (55000c + 45000s) - 57000
58c + 2/3*45s = 55c + 45s - 57
3c - 15s = -57
c - 5s = -19
c = 5s - 19
For c ≥ 1, we must have s ≥ 4.
At s = 4, we have c = 5*4 - 19 = 1 => c + s = 1 + 4 = 5 < 15 (Valid solution). Ratio of Coders to systems administrators = 1/4.
At s = 5, we have c = 5*5 - 19 = 6 => c + s = 6 + 5 = 11 < 15 (Valid solution). Ratio of Coders to systems administrators = 6/5.
No unique value of c/s. Insufficient.
The correct answer: A
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Let c = coders and s = system administrators.BTGmoderatorDC wrote:A certain IT department of fewer than 15 people hires coders and systems administrators. Coders are paid $55,000 per year on average, while system administrators are paid an average yearly salary of $45,000. What is the ratio of coders to systems administrators?
(1) If two of the coders were made systems administrators instead, the yearly payroll for the IT department would be $535,000.
(2) If systems administrators' salaries were reduced by one-third, and coders' salaries were increased to $58,000, the department would save $57,000 in yearly payroll.
Statement 1:
55,000(c-2) + 45,000(s+2) = 535,000
55(c-2) + 45(s+2) = 535
11(c-2) + 9(s+2) = 107.
Implication of the resulting equation:
(multiple of 11) + (multiple of 9) = 107.
(multiple of 9) = 107 - (multiple of 11).
Subtract multiples of 11 from 107.
The result must be a multiple of 9.
107-99=8.
107-88=19.
107-77=30.
107-66=41.
107-55=52.
107-44=63.
107-33=74.
107-22=85.
107-11=96.
Only the option in red yields a multiple of 9.
In this option, 11(c-2) = 44 and 9(s+2)=63, implying that c=6 and s=5.
Thus, c/s = 6/5.
SUFFICIENT.
Statement 2:
System administrator salaries reduced by 1/3 = (2/3)(45,000s) = 30,000s.
Coder salaries increased to 58,000 = 58,000c.
Total of new salaries = 58,000c + 30,000s.
Old total = 55,000c + 45,000s.
Since the new total is 57,000 less than the old total, we get:
55,000c + 45,000s - 57000 = 58,000c + 30,000s
55c + 45s - 57 = 58c + 30s
15s - 3c = 57
5s - c = 19
5s = c+19
s = (c+19)/5.
Case 1: c=1, s=4
In this case, c/s = 1/4.
Case 2: c=6, s=5
In this case, c/s = 6/5.
Since different ratios are possible, INSUFFICIENT.
The correct answer is A.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
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