A dataset consists of 7 integers listed in ascending order: 3, 4, 4, x, 9, y, 15. If the u… — GMAT

A dataset consists of 7 integers listed in ascending order: $3, 4, 4, x, 9, y, 15$ . If the unique mode of the dataset is $4$ and the arithmetic mean of the dataset is equal to its median, what is the value of $y$ ?

Cevap: 13

Cevap

The value of

y

is 13.

Because the dataset is in ascending order, the median of the 7 terms is the 4th term,

x

. Setting the mean

\frac{35 + x + y}{7}

equal to

x

yields

y = 6x - 35

. Since

y

lies between

9

and

15

inclusive, solving

9 \le 6x - 35 \le 15

gives

x = 8

. Substituting

x = 8

into

y = 6x - 35

gives

y = 13

Adım Adım Çözüm

Determine the median of the 7-element ordered set.

The median is the 4th term,

x

For an odd number of ordered terms (

n = 7

), the median is the

\frac{7+1}{2} = 4\text{th}

term.

Set up the equation equating the arithmetic mean to the median.

\frac{35 + x + y}{7} = x

, which simplifies to

y = 6x - 35

The mean of a dataset is the sum of all elements divided by the total number of elements.

Apply the ascending order constraint to find the value of

x

Since

9 \le y \le 15

, we have

9 \le 6x - 35 \le 15

, giving

44 \le 6x \le 50

. The only integer value for

x

in this range is

8

The dataset is listed in ascending order, so the element

y

after

9

and before or equal to

15

must satisfy

9 \le y \le 15

Calculate the value of

y

y = 6(8) - 35 = 13

Substitute

x = 8

back into the simplified linear equation relating

x

and

y

Anahtar Kavram

Relating arithmetic mean and median in an ordered set using inequalities derived from ascending order constraints.

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