Problem Solving

An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month?

A $4000
B $13,000
C $9000
D $5000
E $11,000

OA: E

Musicat wrote:An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month?

A $4000
B $13,000
C $9000
D $5000
E $11,000

An analysis of the monthly incentives received by 5 salesmen: The mean and median of the incentives is $7000.
Sum of the 5 incentives = (number of incentives)(average of the incentives) = (5)(7000) = 35000.

The only mode among the observations is $12,000.
Since 3*12,000 = 36,000 -- and the sum of all 5 incentives = 35000 -- it is not possible that 3 of the incentives = 12000.
Thus, a mode of 12000 implies that exactly 2 incentives = 12000.

In ascending order, the 5 incentives are as follows:
a, b, 7000, 12000, 12000.
Since the sum of the 5 incentives = 35000, we get:
a+b = 35000 - (7000+12000+12000) = 4000.

Incentives paid to each salesman were in full thousands.
Since the set has only one mode, a≠b.
Thus, a+b = 4000 implies that a=1000 and b=3000.

Resulting set:
1000, 3000, 7000, 12000, 12000.
What is the difference between the highest and the lowest incentive?
highest - lowest = 12000-1000 = 11000.

The correct answer is E.

Musicat wrote:An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month?

A $4000
B $13,000
C $9000
D $5000
E $11,000

OA: E

Start with what we know. For an odd set of numbers, the median will simply be the middle number. So here we have the median (that is, the third number in the set) equal to $7,000. So we know there are two values less than this and two values greater than this amount. We are told that the only mode is $12,000. The mode is the number appearing the most times in the set. Since only two of the numbers in the set can be greater than $7000, it must be the case that both of them are $12,000.

So we now have: x, y, 7000, 12000, 12000.

The mean is the average of all five numbers - that is the sum of the five numbers, divided by 5. So the mean is 7000 = (x + y + 7000 + 12000 + 12000)/5.

Multiply both sides by 5 and simplify the right side of the equation and we get: 35000 = x + y + 31000.

Subtract 31000 from both sides of the equation to get x + y = 4000.

The only possible positive values of x and y, such that x and y are both whole multiples of 1000, are x = 1000, y = 3000 ; x = 2000, y = 2000; and x = 3000, y = 1000.

However, if x and y were both $2000, it would be another mode of the set. But we were already told that 12000 was the only mode. Therefore, it must be the case that one of the values is 1000 and the other is 3000.

Therefore, the difference between the least and greatest value is 12000 - 1000 = 11000, which is answer choice E.

Since the mean is $7000, the total must be $7000 * 5, or $35,000.

Since the mode is $12,000, we must have more than one of these in the set. We can't have three of them, though, or we'd have 3*$12,000, which is greater than the actual total of $35,000. So we have exactly two of these.

Since we have 5 terms in the set, the middle one is the median, so we have $7000 in the set. That means we have two salaries (call them x and y) which are each ≤ $7000.

At this point, we have

{x, y, $7000, $12000, $12000}

Since the sum is $35000, we have x + y + 7000 + 2*12,000 = 35,000, or x + y = $4000.

x and y are both full thousands, so we must have $1000 + $3000 or $2000 + $2000. But $2000 + $2000 is out, because we only have one mode, so $1000 + $3000 it is.

That gives us x = $1000, and a range of $12,000 - $1000, or $11,000.