A local library recorded the number of books checked out by 9 patrons on a certain day. Th… — SAT

A local library recorded the number of books checked out by 9 patrons on a certain day. The numbers of books checked out were:

2, 3, 3, 5, 6, 7, 8, 9, 12

If a 10th patron who checked out $x$ books is included, the mean of the numbers of books checked out by the 10 patrons is equal to the median of the numbers of books checked out by the 10 patrons. If $x$ is a positive integer, what is the value of $x$ ?

Cevap: 10

Cevap

To find the value of

x

, we first find the sum of the 9 given numbers:

2 + 3 + 3 + 5 + 6 + 7 + 8 + 9 + 12 = 55

. When a 10th number

x

is added, the mean of the 10 numbers is

\frac{55 + x}{10}

. Since

x

is a positive integer, we analyze the median of the 10 numbers depending on the value of

x

. If

x \ge 8

, the sorted list of the 10 numbers is

2, 3, 3, 5, 6, 7, 8, 9, 12, x

(with the positions of

9

12

, and

x

potentially rearranged depending on how large

x

is). In this case, the 5th and 6th numbers in the sorted list are

6

and

7

. Thus, the median is

\frac{6 + 7}{2} = 6.5

. Setting the mean equal to the median gives

\frac{55 + x}{10} = 6.5

, which simplifies to

55 + x = 65

, or

x = 10

. Since

10

is indeed a positive integer and is

\ge 8

10

is the correct answer.

Adım Adım Çözüm

Find the sum of the original 9 data points.

Sum = 55

This is needed to write the formula for the mean of the 10 data points: Mean = (55 + x)/10.

Determine the median of the 10 data points when x is a large positive integer.

Median = 6.5

For x >= 8, the ordered dataset is 2, 3, 3, 5, 6, 7, 8, 9, 12, x. The median of a 10-element dataset is the average of the 5th and 6th elements (6 and 7), which is 6.5.

Equate the mean and median expressions and solve for x.

x = 10

Setting (55 + x)/10 = 6.5 gives 55 + x = 65, which results in x = 10. Since 10 is a positive integer and is greater than or equal to 8, it is the correct value.

Anahtar Kavram

Calculating and comparing the mean and median of a dataset after adding a new value.

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