# GMAT Math : Calculating arithmetic mean

## Example Questions

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### Example Question #51 : Arithmetic Mean

Give the arithmetic mean of  and .

Statement 1:

Statement 2:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide 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.

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.

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

Explanation:

The arithmetic mean of  and  is equal to .

Statement 1 alone provides insufficient information to find the mean. For example:

If

then

and

But if

then

and

Statement 2 alone provides sufficient information:

.

### Example Question #52 : Arithmetic Mean

Give the arithmetic mean of  and .

Statement 1: A trapezoid with bases of length  and  and height 20 has area 1,000.

Statement 2: If  and  are, respectively, the first and second terms of a geometric sequence with positive terms, the third term is 810.

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.

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 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does 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.

Explanation:

The arithmetic mean of two numbers is half their sum.

Assume Statement 1 alone. The area of a trapezoid can be calculated from its height  and the lengths of its bases  and  as follows:

Setting , and , this becomes

Therefore, the arithmetic mean of  and  is, by definition, 50.

Assume Statement 2 alone. A geometric sequence is one in which the ratio of each term to the previous term is constant - a common ratio. Consider these two sequences:

Both sequences have 810 as their third term. Both are geometric; the common ratio of the first sequence is 9, and that of the second is 10. The arithmetic means of the first two terms, however, are different. In the first sequence it is

In the second sequence, it is

Therefore, Statement 2 alone gives insufficient information.

### Example Question #53 : Arithmetic Mean

Some balls are placed in a large box, which include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1".

A carnival wants to set up a game by which a player can pay to draw a ball and win an amount of money equal to the number marked on the ball drawn. What should the carnival charge per play in order to make the expected value $1 per game in the carnival's favor? Possible Answers: Correct answer: Explanation: The total number of balls in the box will be . Since , it follows that the number of balls is . The frequencies out of 55 of each outcome from 1 to 10, in order, is as follows: Their respective probabilities are their frequencies divided by 55: The expected value of the payment that the church will have to make per game can be calculated by multiplying the frequency of each outcome by the respective payment: ...and then adding these products. The carnival should expect to pay a prize of$4 per draw, so to expect an average profit of $1 per game, it should charge$5 per play.

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