swerve wrote:Last year, Company X paid out a total of $1,050,000 in salaries to its 21 employees. If no employee earned a salary that is more than 20% greater than any other employee, what is the lowest possible salary that any one employee earned?
(A) $40,000
(B) $41,667
(C) $42,000
(D) $50,000
(E) $60,000
We can PLUG IN THE ANSWERS, which represent the smallest possible salary.
To MINIMIZE the smallest salary, we must MAXIMIZE the other 20 salaries.
Thus, each of the other 20 salaries must be the maximum allowed value: 20% greater than the smallest salary.
Since the sum of the salaries is 1,050,000 -- a multiple of 1,000 -- the correct answer choice is almost certainly a multiple of 1,000.
Eliminate B.
When the correct answer is plugged in, the sum of all 21 salaries will be 1,050,000.
Answer choice D: Smallest salary = 50,000.
Maximum value of each of the other 20 salaries = 50,000 + 0.2(50,000) = 60,000.
Sum of the 21 salaries = 50,000 + 20(60,000) = 50,000 + 1,200,000 = 1,250,000.
The sum is too big.
Eliminate D and E.
A: Smallest salary = 40,000
Maximum value of each of the other 20 salaries = 40,000 + 0.2(40,000) = 48,000.
Sum of the 21 salaries = 40,000 + 20(48,000) = 40,000 + 960,000 = 1,000,000.
The sum is too small.
Eliminate A.
The correct answer is
C.
Algebraically:
Let x = the smallest salary.
As noted above, to MINIMIZE the smallest salary, we must MAXIMIZE the other 20 salaries.
Thus, each of the other 20 salaries must be the maximum allowed value:
20% greater than the smallest salary = 1.2x.
Sum the 20 greatest salaries = 20(1.2x) = 24x.
Since the sum of all 21 salaries is equal to 1,050,000, we get:
x + 24x = 1,050,000
25x = 1,050,000
x = 42,000.
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