Arithmetic Mean

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GMAT Quantitative › Arithmetic Mean

Questions 1 - 10

The average of 10, 25, and 70 is 10 more than the average of 15, 30, and x. What is the missing number?

$30$

$25$

$15$

$20$

$35$

Explanation

The average of 10, 25, and 70 is 35: $\frac{10+25+70}{3}=35$

So the average of 15, 30, and the unknown number is 25 or, 10 less than the average of 10, 25, and 70 (= 35)

so $\frac{15+30+x}{3}=25$

$15+30+x=75$

$45+x=75$

$x=30$

What is the mean of the following data set in terms of and ?

$\left \{a -x, a, a + x, a + 2x, 2a-x, 2a, 2a+x, 2a+2x \right \}$

$\frac{3}{2} a +\frac{1}{2} x$

$\frac{3}{2} a -\frac{1}{2} x$

$\frac{3}{2} a$

Explanation

Add the expressions and divide by the number of terms, 8.

The sum of the expressions is:

$\left (a -x \right ) + a + \left ( a + x \right ) +\left ( a +2x \right ) + \left ( 2a-x \right ) + 2a + \left ( 2a+x \right ) +\left ( 2a+2x \right )$

Divide this by 8:

$\left ( 12a + 4x \right )\div 8 =\frac{3}{2} a +\frac{1}{2} x$

What is the mean of the following data set in terms of and ?

$\left \{a -x, a, a + x, a + 2x, 2a-x, 2a, 2a+x, 2a+2x \right \}$

$\frac{3}{2} a +\frac{1}{2} x$

$\frac{3}{2} a -\frac{1}{2} x$

$\frac{3}{2} a$

Explanation

Add the expressions and divide by the number of terms, 8.

The sum of the expressions is:

$\left (a -x \right ) + a + \left ( a + x \right ) +\left ( a +2x \right ) + \left ( 2a-x \right ) + 2a + \left ( 2a+x \right ) +\left ( 2a+2x \right )$

Divide this by 8:

$\left ( 12a + 4x \right )\div 8 =\frac{3}{2} a +\frac{1}{2} x$

The average of 10, 25, and 70 is 10 more than the average of 15, 30, and x. What is the missing number?

$30$

$25$

$15$

$20$

$35$

Explanation

The average of 10, 25, and 70 is 35: $\frac{10+25+70}{3}=35$

So the average of 15, 30, and the unknown number is 25 or, 10 less than the average of 10, 25, and 70 (= 35)

so $\frac{15+30+x}{3}=25$

$15+30+x=75$

$45+x=75$

$x=30$

The average high temperature for the week is 85. The first six days of the week have high temperatures of 89, 76, 92, 90, 80, and 84, respectively. What is the high temperature on the seventh day of the week?

Explanation

$average = \frac{sum}{days}$

$85=\frac{89+76+92+90+80+84+x}{7}=\frac{511+x}{7}$

The average of the following 6 digits is 75. What is a possible value of ?

80, 78, 78, 70, 71,

Explanation

Therefore, the sum of all 6 digits must equal 450.

Subtract 377 from both sides.

Find such that the arithmetic mean of is equal to the arithmetic mean of

Explanation

The formula for the arithmetic mean is:

Mean= $\frac{x_{1}+x_{2}+...+x_{n}}{n}$

We can then write:

$\frac{n+13+27+35}{4}=\frac{20+16+41+23}{4}$

Explanation

$average = \frac{sum}{days}$

$85=\frac{89+76+92+90+80+84+x}{7}=\frac{511+x}{7}$

The average of the following 6 digits is 75. What is a possible value of ?

80, 78, 78, 70, 71,

Explanation

Therefore, the sum of all 6 digits must equal 450.

Subtract 377 from both sides.

When assigning a score for the term, a professor takes the mean of all of a student's test scores except for his or her lowest score.

On the first five exams, Donna has achieved the following scores: 76, 84, 80, 65, 91. There is one more exam in the course. Assuming that 100 is the maximum possible score, what is the range of possible final averages she can achieve (nearest tenth, if applicable)?

Minimum 79.2; maximum 86.2

Minimum 66; maximum 71.8

Minimum 66; maximum 99.2

Minimum 79.2; maximum 99.2

Minimum 80, maximum 84