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.

OA D

Source: Princeton Review

Say there are c numbers of Coders and s numbers of systems administrators such that 2 ≤ c + s < 15

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

Hope this helps!

-Jay
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