A set consists of 5 numbers. What is the arithmetic mean of the 5 numbers? (1) The sum of … — GMAT

A set consists of $5$ numbers. What is the arithmetic mean of the $5$ numbers?

(1) The sum of the $5$ numbers is $45$ .
(2) The median of the $5$ numbers is $9$ and the range of the $5$ numbers is $8$ .

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The mean of a dataset of

n

items is defined as the sum of all elements divided by

n

. Since

n = 5

, knowing that the sum of the

5

numbers is

45

immediately gives the mean as

45 / 5 = 9

. Thus, the first statement provides sufficient information on its own. Conversely, knowing only the median and range allows for multiple distinct sets of numbers with different total sums and arithmetic means, making the second statement insufficient.

Adım Adım Çözüm

Rephrase the question stem

The arithmetic mean of

5

numbers is equal to

\frac{\text{Sum of numbers}}{5}

. Thus, knowing the sum of the

5

numbers is necessary and sufficient to find the mean.

By definition,

\text{Mean} = \frac{\text{Sum}}{n}

where

n = 5

Evaluate Statement (1)

Statement (1) gives

\text{Sum} = 45

. Therefore,

\text{Mean} = \frac{45}{5} = 9

. This gives a unique value.

Statement (1) alone provides all required information.

Evaluate Statement (2)

Consider two possible sets with median

9

and range

8

: Set X =

\{5, 9, 9, 9, 13\}

(Sum =

45

, Mean =

9

) and Set Y =

\{5, 5, 9, 13, 13\}

(Sum =

45

, Mean =

9

), but also Set Z =

\{5, 6, 9, 10, 13\}

(Sum =

43

, Mean =

8.6

). Since the mean is not uniquely determined, Statement (2) is not sufficient.

Median and range leave the remaining elements and the overall sum indeterminate.

Anahtar Kavram

Arithmetic Mean Definition in Data Sufficiency

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