In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and

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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 (arithmetic mean) annual salary of the 3 people that year7

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

Answer: B

Source: GMAT Prep

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Let Jim's annual salary = j
Let Mary's annual salary = m
Let Kate's annual salary = k
m - j = 2 (m-k)
Target question: what was the average (arithmetic mean) annual salary of the 3 people that year7
Note: Mary's annual salary is the highest

Statement 1: Jim's annual salary was $30,000 that year.
j = 30,000
From question stem; m - j = 2(m-k)
m - j = 2m - 2k
2m - m = 2k - j
m = 2k - j
$$Average\ of\ the\ 3=\frac{m+k+j}{3};\ where\ m=2k-j$$
$$Average=\frac{2k-j+k+j}{3}=\frac{3k}{3}=k$$
The information provided is not related to the value of k, hence, statement 1 is NOT SUFFICIENT.

Statement 2: Kate's annual salary was $40,000 that year.
k = $40000
From question stem, m - j = 2 (m-k)
m - j = 2m - 2k
2k - j = 2m - m
m = 2k - j
$$Average=\frac{m+k+j}{3}\ where\ m=2k-j$$
$$Average=\frac{2k-j+k+j}{3}=\frac{3k}{3}=k$$
Since k=40000, average salary is also $40000. Therefore, statement 2 alone is SUFFICIENT.
Answer = option B.

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Gmat_mission wrote:
Thu Sep 24, 2020 2:38 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 (arithmetic mean) annual salary of the 3 people that year7

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

Answer: B

Source: GMAT Prep
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.

For proof that 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
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 salary, we now know that the average salary must be $40,000
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B
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