{3, 5, 9, 13, y} If the average (arithmetic mean) and the median of the set of numbers shown above are equal, then what

BTGmoderatorDC wrote: ↑

Sat Nov 14, 2020 6:01 pm

{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


OA D

Solution:

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,pending 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.

Answer: D