Sat Math Hard Practice Quiz Worksheet With Answer Key Page 18
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18SAT Math Hard Practice Quiz Answers
Data, Statistics, and Probability
3. E
(Estimated Difficulty Level: 4)
Using the definition of average gives:
1. A
(Estimated Difficulty Level: 4)
a + b + c
= 100
3
The first nine terms of the sequence are:
so that a + b + c = 300. Since a, b, and c are all positive,
n, n + 5, n + 10, n + 15, . . . , n + 40.
the smallest possible value for any of the numbers is 1.
The largest possible value of one of the three numbers
(You should probably write all nine terms out to avoid
then occurs when the other two numbers are both 1. In
mistakes.) Adding these terms up gives: 9n + 180. The
this case, the numbers are 1, 1, and 298, so that the
average is the sum (9n + 180) divided by the number of
largest possible value is 298. Answer E can not be a
terms (9). The average is then: (9n + 180)/9 = n + 20.
possible value, so it is the correct answer.
4. E
(Estimated Difficulty Level: 5)
2. B
(Estimated Difficulty Level: 5)
Plug in real numbers for set M to make this problem
concrete. For example, if M is the set of consecutive
The average of a set of numbers is the sum of the num-
integers from 1 to 5, then the median and average are
bers divided by the number of numbers:
both 3. If M is the set of consecutive integers from 1
to 4, then the median and average are both 2.5. From
sum
average =
.
these examples, we can see that the number of numbers
N
in set M needs to odd, otherwise the median is not an
We can solve this equation for the sum:
integer. Choice II must be true.
Also, if the number of numbers in a set of consecutive
sum = average × N.
integers is odd, then when the first number is odd, the
last number is odd. Or, when the first number is even,
Here, since there are 7 numbers and the average is 12,
the last number is even. This is because the difference
the sum of the numbers is 7 × 12 = 84. The sum of the
of the largest number and the smallest number will be
new set of numbers is 7 × 15 = 105. Now, suppose that
even when the number of numbers is odd. Choice III
the seven numbers are a, b, c, d, e, f , and g, and that
must then be true, since the sum of two odd numbers
g gets replaced with the number 6. Then, we have:
or two even numbers is an even number.
a + b + c + d + e + f + g = 84,
At this point, the only answer with choices II and III is
answer E, so that must be the correct answer. Why is
and
choice I also correct? The average of a set of consecutive
a + b + c + d + e + f + 6 = 105.
integers is equal to the average of the first and the last
The second equation says that a + b + c + d + e + f = 99.
integers in the set. The average of two integers that are
Substituting into the first equation gives 99 + g = 84 so
both odd or both even is the integer halfway between
that g =
15.
the two, which is also the median of the set. Whew!
pg. 18
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