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by GMATGuruNY » Fri Jan 02, 2015 1:03 am
In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. If Mary's annual salary was the highest of the 3 people, what was the average annual salary of the 3 people that year?

1. Jim's annual salary was $30,000 that year.
2. Kate's annual salary was $40,000 that year.
In ascending order, the salaries look like this:
J-------------K-------------M

According to the question stem:
M-K = x.
M-J = 2x.
Thus, the salaries look like this:
J------x------K------x------M

The number line above implies that the salaries are EVENLY SPACED.
When values are evenly spaced, AVERAGE = MEDIAN.
The median salary here is the value of K.

Question rephrased: What is the value of K?

Statement 1: J=30,000.
Since different values of K are possible, INSUFFICIENT.

Statement 2: K=40,000
SUFFICIENT.

The correct answer is B.

Algebraic approach:

The average of the 3 salaries = (M+J+K)/3.

Since the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries, we get:
M-J = 2(M-K)
M-J = 2M-2K
2K = M+J.

Substituting M+J = 2K into (M+J+K)/3, we can rephrase the question stem as follows:
Average of the 3 salaries = (M+J+K)/3 = (2K + K)/3 = 3K/3 = K.

Question rephrased: What is the value of K?
From here, we can proceed as we did above.
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by Brent@GMATPrepNow » Fri Jan 02, 2015 8:28 am
In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. If Mary's annual salary was the highest of the 3 people, what was the average annual salary of the 3 people that year?

1) Jim's annual salary was $30,000 that year.
2) Kate's annual salary was $40,000 that year.

Let's first deal with the given information.
Let J = Jim's salary
Let M = Mary's salary
Let K = Kate's salary

Notice that the salaries (in ascending order) must be J, K, M
Also, if the difference between Mary's and Jim's annual salaries equals twice the difference between Mary's and Kate's annual salaries, then we can conclude that the 3 salaries are equally spaced.

Target question: What was the average annual salary of the 3 people that year?

Statement 1: Jim's annual salary was $30,000 that year.
In other words, J = 30,000
So, the three salaries, arranged in ascending order are: 30,000, K, M
Plus we know that the 3 salaries are equally spaced.
Do we now have enough information to answer the target question?
NO.

To show that we don't have enough information, consider these 2 cases:
Case a: J=30,000, K=30,001, M=30,002, in which case the average salary is $30,001
Case b: J=30,000, K=30,002, M=30,004, in which case the average salary is $30,002
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Kate's annual salary was $40,000 that year.
In other words, K = 40,000
So, the three salaries, arranged in ascending order are: J, 40000, M
Perfect!
Since the 3 salaries are equally spaced, we can use a nice rule that says, "If the numbers in a set are equally spaced, then the mean and median of that set are equal"
Since Kate's salary must be the median (middle) salary, we now know that the average (mean) salary must be $40,000
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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