GMAT Math : DSQ: Calculating arithmetic mean

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #21 : Dsq: Calculating Arithmetic Mean

Given five distinct positive integers - \(\displaystyle A,B,C,D,E\) - which of them is the median?

Statement 1: \(\displaystyle A < B < E\)

Statement 2: \(\displaystyle D < B < C\)

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. 

These two orderings are both consistent with the ordering given in Statement 1:

\(\displaystyle A < B < E < C < D\) - median \(\displaystyle E\)

\(\displaystyle C < D < A < B < E\) - median \(\displaystyle A\).

Therefore, Statement 1 alone provides insufficient information to answer the question. For a similar reason, so does Statement 2. 

Assume both statements to be true. Then \(\displaystyle B\) is greater than both \(\displaystyle A\) and \(\displaystyle D\) and less than both \(\displaystyle C\) and \(\displaystyle E\). That makes \(\displaystyle B\) the middle element, and, thus, the median.

Example Question #22 : Dsq: Calculating Arithmetic Mean

Give the arithmetic mean of \(\displaystyle A\) and \(\displaystyle B\).

Statement 1: A rectangle with length \(\displaystyle A\) and width \(\displaystyle B\) has area 500.

Statement 2: A triangle with base of length \(\displaystyle A\) and height \(\displaystyle B\) has area 250.

Possible Answers:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

The area of a rectangle is the product of its length and width; the area of a triangle is half the product of its height and the length of its base. Therefore, from Statement 1, we get that 

\(\displaystyle AB = 500\).

From Statement 2, we get that

\(\displaystyle \frac{1}{2}AB = 250\), or, equivalently,

\(\displaystyle \frac{1}{2}AB \cdot 2 = 250 \cdot 2\), or

\(\displaystyle AB = 500\)

In other words, the two statements are equivalent, so one of two things happens - either statement alone is sufficient, or both together are insufficient. We show that the latter is the case:

Case 1: \(\displaystyle A = 20, B = 25\)

\(\displaystyle AB = 20 \cdot 25= 500\)

The mean of the two is \(\displaystyle \mu = \frac{A + B}{2} = \frac{20 + 25}{2} = 22.5\).

Case 2: \(\displaystyle A = 10, B = 50\)

\(\displaystyle AB = 10 \cdot 50= 500\)

The mean of the two is \(\displaystyle \mu = \frac{A + B}{2} = \frac{10 + 50}{2} = 25\).

Therefore, knowing the area of the rectangle with these dimensions is not helpful to determining their arithmetic mean. This makes Statement 1, and, equivalently, both statements together, unhelpful.

Example Question #23 : Dsq: Calculating Arithmetic Mean

Joseph's final grade is calculated from the mean of his test scores. His teacher also allows them to drop the lowest score before calculating the final grade. If Joseph received a \(\displaystyle \textup{67, 82, 73, 91, 77, and, 75}\) on his tests, what was his final grade rounded to the nearest whole number?

Possible Answers:

\(\displaystyle 82\)

\(\displaystyle 81\)

\(\displaystyle 80\)

\(\displaystyle 78\)

\(\displaystyle 79\)

Correct answer:

\(\displaystyle 80\)

Explanation:

The average or mean is found by taking all of the scores and dividing by the total number of scores. Remember, we must first find the lowest score and not include that in the calculation. Therefore, we get: 

\(\displaystyle \frac{82+73+91+77+75}{5}=79.6=80\) when rounded. 

Example Question #21 : Arithmetic Mean

You are given the data set \(\displaystyle \left \{14, 35, 44, 47, 49, 49, 50, 51, 56, N \right \}\) , where \(\displaystyle N\) is an integer not necessarily greater than 56. What is the value of \(\displaystyle N\)?

Statement 1: The mean of the data set is 44.3

Statement 2: The median of the data set is 48.5

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question. 

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Knowing the mean of the set is enough to calculate \(\displaystyle N\):

\(\displaystyle \frac{14+35+ 44+ 47+ 49+ 49+ 50+ 51+ 56+ N }{10} = 44.3\)

\(\displaystyle \frac{395+ N }{10} = 44.3\)

\(\displaystyle 395+ N= 443\)

\(\displaystyle N=48\)

Knowing the median is enough to deduce \(\displaystyle N\). Since there are ten elements - an even number - the median is the arithmetic mean of the fifth and sixth highest elements. If \(\displaystyle N\leq 47\), those two elements are 47 and 49, making the median 48. If \(\displaystyle N \geq 49\), those two elements are both 49, making the median 49. This forces \(\displaystyle N\) to be 48, making the median 48.5.

Therefore, either statement alone is sufficient to answer the question.

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