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# If the average (arithmetic mean) and the median of the set of numbers

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Senior Manager
Status: Gathering chakra
Joined: 05 Feb 2018
Posts: 433
If the average (arithmetic mean) and the median of the set of numbers  [#permalink]

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06 Feb 2019, 19:39
1
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Difficulty:

15% (low)

Question Stats:

80% (01:11) correct 20% (00:58) wrong based on 20 sessions

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{3, 5, 9, 13, y}

If the average (arithmetic mean) and the median of the set of numbers shown above are equal, then what is the value of y ?

A) 7
B) 8
C) 10
D) 15
E) 17

1) Find avg to find median, Avg = (30+y)/5 or 6+(5/y)
2) From this we know that y has to be divisible by 5, so C or D.
3) Pick C, if y=10 then the Avg is 40/5 = 8. Looking at the set, if y=10 then 9 is the median, not 8
4) Check D, if y=15 then the Avg is 45/5 = 9. Now the avg = med, thus D.

Official explanation:
This problem is a great opportunity to Plug In the Answers. Start with (C) and substitute 10 for y in the problem. The average of the numbers {3, 5, 9, 13, 10} is 8, but the median of those numbers is 9. Eliminate (C). The value of y needs to be greater, so try (D), 15. The average of the numbers {3, 5, 9, 13, 15} is 9, and the median of those numbers is also 9. We’re done. The correct answer is (D).
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Re: If the average (arithmetic mean) and the median of the set of numbers  [#permalink]

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06 Feb 2019, 23:36
energetics wrote:
{3, 5, 9, 13, y}

If the average (arithmetic mean) and the median of the set of numbers shown above are equal, then what is the value of y ?

A) 7
B) 8
C) 10
D) 15
E) 17

average (arithmetic mean) and the median of the set of numbers shown above are equal

(30 + y)/ 5 = y

From the given options plug in values for y now

$$\frac{{30 + 15}}{5}$$ = 9

9 = 9

D
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Re: If the average (arithmetic mean) and the median of the set of numbers  [#permalink]

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07 Feb 2019, 03:06
energetics wrote:
{3, 5, 9, 13, y}

If the average (arithmetic mean) and the median of the set of numbers shown above are equal, then what is the value of y ?

A) 7
B) 8
C) 10
D) 15
E) 17

1) Find avg to find median, Avg = (30+y)/5 or 6+(5/y)
2) From this we know that y has to be divisible by 5, so C or D.
3) Pick C, if y=10 then the Avg is 40/5 = 8. Looking at the set, if y=10 then 9 is the median, not 8
4) Check D, if y=15 then the Avg is 45/5 = 9. Now the avg = med, thus D.

Official explanation:
This problem is a great opportunity to Plug In the Answers. Start with (C) and substitute 10 for y in the problem. The average of the numbers {3, 5, 9, 13, 10} is 8, but the median of those numbers is 9. Eliminate (C). The value of y needs to be greater, so try (D), 15. The average of the numbers {3, 5, 9, 13, 15} is 9, and the median of those numbers is also 9. We’re done. The correct answer is (D).

If placed in increasing order, we have 3 possibilities for median
- y is less than 5 and median is 5
- y is between 5 and 9 and y is the median
- y is more than 9 and hence median is 9

There is no option with y less than 5 so ignore case 1.
If y is 7 or 8 median will be 7 or 8. But in neither case, can mean be 7 or 8 (check using the deviation method discussed in the link given below. It takes just a few secs)
If y is 10, median is 9 but it cannot again be the mean.
If y is 15, median is 9 and mean will be 9 too.

For more on mean using deviation method, check: https://www.veritasprep.com/blog/2012/0 ... eviations/
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Re: If the average (arithmetic mean) and the median of the set of numbers  [#permalink]

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10 Feb 2019, 08:46
energetics wrote:
{3, 5, 9, 13, y}

If the average (arithmetic mean) and the median of the set of numbers shown above are equal, then what is the value of y ?

A) 7
B) 8
C) 10
D) 15
E) 17

Looking at the numbers, we see that 5 is 4 less than 9, and 13 is 4 more than 9. We also see that 3 is 6 less than 9, so if y is 6 more than 9, then 9 would be both the mean and the median. In that case, y would be 9 + 6 = 15. Since 15 is one of the choices, it is the correct answer.

Alternate Solution:

We observe that depending on the value of y, the median is either 5 (if y is less than or equal to 5), y (if y is between 5 and 9) or 9 (if y is greater than or equal to y).

If the median is 5, then, since the mean is also 5, the sum of the five numbers must be 5 x 5 = 25. We see that this is not possible since the sum of the numbers without y is already 30 and there are no negative numbers in the answer choices.

If the median is y, then the sum of the five numbers must be 5 x y = 5y. Then, we get 3 + 5 + 9 + 13 + y = 5y; which simplifies to 30 + y = 5y. Then, 4y = 30 and y = 7.5. We see that this is not among the answer choices.

Finally, if the median is 9, then the sum of the five numbers must be 5 x 9 = 45. Then, we get 3 + 5 + 9 + 13 + y = 45; which simplifies to 30 + y = 45. This yields y = 15.

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Re: If the average (arithmetic mean) and the median of the set of numbers   [#permalink] 10 Feb 2019, 08:46
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