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, 2017

Algebra Difficulty 1.6 Junior Find the answer Canada

The mean (average) of the four integers $78,83,82,$ and $x$ is 80. Which one of the following statements is true?

$\text{}$ x is 2 greater than the mean}
$\text{}$ x is 1 less than the mean}
$\text{}$ x is 2 less than the mean}
$\text{}$ x is 3 less than the mean}
$\text{}$ x ${ is equal to the mean}$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution 1

Since 78 is 2 less than 80 and 82 is 2 greater than 80, the mean of 78 and 82 is 80.

Since the mean of all four integers is 80, then the mean of 83 and $x$ must also equal 80.

The integer 83 is 3 greater than 80, and so $x$ must be 3 less than 80.

That is, $x=80-3=77$ .

(We may check that the mean of $78,83,82,$ and $77$ is indeed 80.)

Solution 2

Since the mean of the four integers is 80, then the sum of the four integers is $4\times80=320$ .

Since the sum of 78, 83 and 82 is 243, then $x=320-243=77$ .

Therefore, $x$ is 77 which is 3 less than the mean 80.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.