A dataset consists of 10 positive integers: 4, 6, 8, 8, 10, 12, 12, 14, 16, and x. If the … — SAT

A dataset consists of 10 positive integers: $4, 6, 8, 8, 10, 12, 12, 14, 16$ , and $x$ . If the median of the dataset is equal to the mean of the dataset, and $x > 16$ , what is the value of $x$ ?

Answer: 20

Answer

The correct answer is 20. When the 10 positive integers are sorted in ascending order, the condition

x > 16

ensures that

x

is the largest value and occupies the final position:

4, 6, 8, 8, 10, 12, 12, 14, 16, x

. The median of this 10-value dataset is the average of the 5th and 6th values, which are 10 and 12. Thus, the median is

\frac{10 + 12}{2} = 11

. The mean of the dataset is the sum of the 10 values divided by 10. The sum of the 9 known values is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

, so the mean is

\frac{90 + x}{10}

. Setting the mean equal to the median gives the equation

\frac{90 + x}{10} = 11

. Multiplying both sides by 10 gives

90 + x = 110

, and subtracting 90 yields

x = 20

. This value is consistent with the condition

x > 16

Step-by-Step Solution

Sort the dataset including the variable

x

The sorted dataset of 10 integers is

4, 6, 8, 8, 10, 12, 12, 14, 16, x

Since

x > 16

, it must be the largest value in the dataset and will be positioned at the end of the sorted list.

Find the median of the sorted dataset.

The median is 11.

For an even number of values (10), the median is the average of the two middle values (the 5th and 6th values), which are 10 and 12:

(10 + 12) / 2 = 11

Find the sum of the known values and write an expression for the mean.

The mean is

\frac{90 + x}{10}

The sum of the 9 known integers is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

. Adding

x

gives a total sum of

90 + x

, and dividing by the total count of 10 gives the mean.

Equate the mean to the median and solve for

x

x = 20

Setting the mean expression equal to the median yields

\frac{90 + x}{10} = 11

. Multiplying by 10 gives

90 + x = 110

, which solves to

x = 20

Key Concept

Calculating and equating the mean and median of a dataset containing a variable.

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