Sat math Essentials

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N U M B E R S A N D O P E R AT I O N S R E V I E W
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The 
mode
of a set of numbers is the number that occurs most frequently.
Example
For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs three times. The other
numbers occur only once or twice.
Practice Question
If the mode of a set of three numbers is 17, which of the following must be true?
I.
The average is greater than 17.
II.
The average is odd.
III.
The median is 17.
a.
none
b.
I only
c.
III only
d.
I and III
e.
I, II, and III
Answer
c.
If the mode of a set of three numbers is 17, the set is {
x
, 17, 17}. Using that information, we can evalu-
ate the three statements:
Statement I:
The average is greater than 17
.
If
x
is less than 17, then the average of the set will be less than 17. For example, if
x
2, then we can find the
average:
2 
17 
17 
36
36 
3 
12
Therefore, the average would be 12, which is not greater than 17, so number I isn’t necessarily true. Statement
I is FALSE.
Statement II:
The average is odd
.
Because we don’t know the third number of the set, we don’t know that the average must be even. As we just
learned, if the third number is 2, the average is 12, which is even, so statement II ISN’T NECESSARILY TRUE.
Statement III:
The median is 17
.
We know that the median is 17 because the median is the middle value of the three numbers in the set. If
X
>
17, the median is 17 because the numbers would be ordered:
X
, 17, 17. If
X
< 17, the median is still 17 because
the numbers would be ordered: 17, 17,
X
. Statement III is TRUE.
Answer: Only statement III is NECESSARILY TRUE.