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By adding n into {3, 6, 9, 10}, the average is equal to the median. What is the sum of possible values of n?

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By adding n into {3, 6, 9, 10}, the average is equal to the median. What is the sum of possible values of n?

by Max@Math Revolution » Thu Feb 27, 2020 1:10 am

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[GMAT math practice question]

By adding n into {3, 6, 9, 10}, the average is equal to the median. What is the sum of possible values of n?

A. 24
B. 26
C. 28
D. 30
E. 32
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Re: By adding n into {3, 6, 9, 10}, the average is equal to the median. What is the sum of possible values of n?

by deloitte247 » Sun Mar 01, 2020 11:06 pm
Average = Median
The value of n can alter the portion of the median
$$If\ n\le6\ then\ median\ =\ 6\ $$
$$\frac{28+n}{5}=6$$
$$28+n=30$$
$$n=30-28=2$$

If median = n
$$\frac{3+6+9+10+n}{5}=n$$
$$\frac{28+n}{5}=n$$
$$28+n=5n$$
$$\frac{28}{5}=\frac{4n}{4}$$
$$n=7$$

$$If\ n\ge9,\ then\ median\ =9$$
$$\frac{3+6+9+10+n}{5}=9$$
$$\frac{28+n}{5}=9$$
$$28+n=45$$
$$n=45-28=17$$
Sum of all possible values of n = 2+7+17 = 26
$$Answer\ is\ Option\ B$$
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Re: By adding n into {3, 6, 9, 10}, the average is equal to the median. What is the sum of possible values of n?

by Max@Math Revolution » Mon Mar 02, 2020 1:09 am
=>

The possible medians are 6, n, and 9 only.

Case 1: Assume n ≤ 6.
The median of n, 3, 6, 9, 10 is 6 and we have (n + 28) / 5 = 6 or n + 28 = 30.
Thus, we have n = 2.

Case 2: Assume 6 < n ≤ 9.
The median of 3, 6, n, 9, 10 is n and we have (n + 28) / 5 = n or n + 28 = 5n.
Thus, we have 4n = 28 or n = 7.

Case 3: Assume 9 < n.
The median of 3, 6, 9, n, 10 is 9 and we have (n + 28) / 5 = 9 or n + 28 = 45.
Thus, we have n = 17.

Hence, the possible values of n are 2, 7, and 17.
Then we have 2 + 7 + 17 = 26.

Therefore, B is the answer.
Answer: B
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