Princeton--ration drill --salaries of an IT department

[prachich1987]

A certain IT department of fewer than 15 people hires coders & system 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 system administrators?

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.

2) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

[Night reader]

A certain IT department of fewer than 15 people hires coders & system 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 system administrators?

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.

2) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

[spoiler]
Answer C [/spoiler]

Useful data: call coders C; call system admins S; salary assigned to C=45,000; salary assigned to S=45,000
Not useful data: C+S<15
[spoiler]note the data not useful is to distract us from the right solution path it's quite difficult to apply this piece of data, for we are given one equation with two variables and one inequality with two variables - no other factors are given ... [/spoiler]

Find C/S-?

st(1) 55,000(C-2) + 45,000(S+2)=535,000 & C<15-S => Not Sufficient, test below ...
55,000*C=555,000-45,000*S => C=(555,000-45,000*S)/55,000 OR (555,000-45,000*S)/55,000<15-S =>
S<27 and C<-12 [spoiler]remember not useful data stuff happens with PR [/spoiler]

st(2) 30,000*S+58,000*C+57,000=45,000*S+55,000*C OR 3,000*C-15,000*S+57,000=0 => Not Sufficient
apply the inequality again...

Combining st(1&2) system of two equations
EQ1 { 3*[3,000*C-15,000*S+57,000]=0
EQ2 {__55,000*C+45,000*S-555,000=0 => add EQ1 to EQ2 => 64,000*C=384,000 OR C=6
find S => 3,000*6-15,000*S+57,000=0 => 3*6+57=15*S OR S=75/15, S=5

find the necessary C/S => 6/5 Sufficient.

[goyalsau]

A certain IT department of fewer than 15 people hires coders & system 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 system administrators?

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.

2) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

Though First i marked D, but when i saw the explanation of Night Reader I did my calculation again and got the answer as C

Let ----Coders are X & System Administrators are Y

X + Y < 15

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.
If two coders were not made system administrators then total pay roll must be 20 +

55 ---------------- 555------------45

I tried to find the Number for X and Y by the Help of this.

On maximum X can be 10 , Because 11 * 55 = 605

By Hit and Trial

X = 10 , Not possible 550 for total
X = 9 , Not possible 495 + 60 ( 60 is not a multiple of 45 )
X = 8 , Not possible 440 + 110 ( not a multiple of 45 )
X = 7 , Not possible 385 + 175 ( not a multiple of 45 )
X = 6 , Possible ...... 330 + 225 { multiple of 45 )

X = 5 , Not possible 275 + 275 { not a multiple of 45 ) 45 * 6 = 270
X = 4 Not possible 220 + 330 { Not a multiple of 45 ) 45 * 7 = 315
X = 3 , Not possible 165 + 385 { not multiple again ) 45 * 8 = 360
X = 2 , Not possible 110 + 440 ( clearly not a multiple )
X = 1 , possible 55 + 495 ...... 45 * 11 = 495

So two value for X and Y

{ 6 , 5 } & { 1 , 11 }

2 ) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

By this i realize there are two values again. X = 1 , Y = 4
Because 60 - 3 = 57

Second value X = 6 , Y = 5
75 - 18 = 57

by combining only one value...

X = 6 , Y = 5

sufficient
Answer is C

I posted my lengthy solution just for the curiosity to know, Whether any shortcut method is there to approach the values of X and Y , As there are two values in both the equation which satisfy the given condition,
If there is any shortcut method to get the different values from one equation. Please share that.......

[prachich1987]

A certain IT department of fewer than 15 people hires coders & system 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 system administrators?

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.

2) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

Though First i marked D, but when i saw the explanation of Night Reader I did my calculation again and got the answer as C

Let ----Coders are X & System Administrators are Y

X + Y < 15

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.
If two coders were not made system administrators then total pay roll must be 20 +

55 ---------------- 555------------45

I tried to find the Number for X and Y by the Help of this.

On maximum X can be 10 , Because 11 * 55 = 605

By Hit and Trial

X = 10 , Not possible 550 for total
X = 9 , Not possible 495 + 60 ( 60 is not a multiple of 45 )
X = 8 , Not possible 440 + 110 ( not a multiple of 45 )
X = 7 , Not possible 385 + 175 ( not a multiple of 45 )
X = 6 , Possible ...... 330 + 225 { multiple of 45 )

X = 5 , Not possible 275 + 275 { not a multiple of 45 ) 45 * 6 = 270
X = 4 Not possible 220 + 330 { Not a multiple of 45 ) 45 * 7 = 315
X = 3 , Not possible 165 + 385 { not multiple again ) 45 * 8 = 360
X = 2 , Not possible 110 + 440 ( clearly not a multiple )
X = 1 , possible 55 + 495 ...... 45 * 11 = 495

So two value for X and Y

{ 6 , 5 } & { 1 , 11 }

2 ) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

By this i realize there are two values again. X = 1 , Y = 4
Because 60 - 3 = 57

Second value X = 6 , Y = 5
75 - 18 = 57

by combining only one value...

X = 6 , Y = 5

sufficient
Answer is C

I posted my lengthy solution just for the curiosity to know, Whether any shortcut method is there to approach the values of X and Y , As there are two values in both the equation which satisfy the given condition,
If there is any shortcut method to get the different values from one equation. Please share that.......

Thanks goyalsau for above explanation.
But according to Princeton the OA is D
Also I have difficulty in understanding x & y above.
You have assumed that "Coders are X & System Administrators are Y".
But then while doing calculations for statement 1, you should shift two coders to administrators & then do the calculations.

[goyalsau]

Thanks goyalsau for above explanation.
But according to Princeton the OA is D
Also I have difficulty in understanding x & y above.
You have assumed that "Coders are X & System Administrators are Y".
But then while doing calculations for statement 1, you should shift two coders to administrators & then do the calculations.

No Prachi I did not interchange X and Y.

X is for Codes that's why 55 * 6 = 330
Y is for System .. 45 * 5 = 225

[GMATGuruNY]

A certain IT department of fewer than 15 people hires coders & system 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 t system administrators?

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.

2) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

I received a PM asking me to comment.

Let c = coders and s = system administrators.

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.

[Night reader]

@Mitch, thanks for solution. It's apparent that this DS falls within plug and try rather only algebraic solution.

[prachich1987]

A certain IT department of fewer than 15 people hires coders & system 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 system administrators?

1) If two if the coders were made system administrators instead,the yearly payroll for the IT department would be $535,000.

2) If system administrator's salaries were reduced by one third and coders's salaries were increased to $58,000,the department would save $57000 dollar in yearly payroll.

I received a PM asking me to comment.

Let c = coders and s = system administrators. We know that c+s < 15 and that c and s are positive integers.

Statement 1:
55,000c + 45,000s = 535,000
55c + 45s = 535
11c + 9s = 107.
The total amount paid to the coders must be a multiple of 11. The total amount paid to the system adminstrators must be a multiple of 9.
Let's subtract multiples of 11 from 107 until we get a multiple of 9.
107-99=8.
107-88=19.
107-77=30.
107-66=41.
107-55=52.
107-44=63. This works: 63/9 = 7. So we could have 4 coders and 7 system administrators, because 11*4 + 9*7 = 107.
107-33=74.
107-22=85.
107-11=96.
The only combination that works is 4 coders and 7 system administrators.
Since in statement 1 two coders have been made system administrators, we started with 4+2 = 6 coders and 7-2 = 5 system administrators.
Sufficient.

Statement 2:
System administrator salaries reduced by 1/3 = 2/3*45,000c = 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.
Let's try the values that satisfied statement 1: s=5 and c=6.
5s - c = 5*5 - 6 = 25-6 = 19. This works!
Since c+s < 15, s=5 and c=6 is the only combination that will work.
Sufficient.

The correct answer is D.

Thanks Mitch
very nicely explained!

[shobhits]

Hi Mitch,

In your explanation, for statement 2, the final equation is:
5s - c = 19. Given that, c + s < 15.

Two set of values {s,c} satisfy this equation {4,1} and {5,6}. Because we cannot get unique value for the ratio c/s, shouldn't this statement be INSUFFICIENT.

Please correct me if I am wrong.

[GMATGuruNY]

Hi Mitch,

In your explanation, for statement 2, the final equation is:
5s - c = 19. Given that, c + s < 15.

Two set of values {s,c} satisfy this equation {4,1} and {5,6}. Because we cannot get unique value for the ratio c/s, shouldn't this statement be INSUFFICIENT.

Please correct me if I am wrong.

Good catch.
I've amended my solution accordingly.