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answers to sample questions

question 1

Alex owns a bookstore, and his weekly revenue from it are the following: 
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
$260
$230
$260
$250
$365
$450
$320
Find the mean revenue, median, and mode.
To answer this question, first of all, think which among the three M’s will be the easiest to answer. If you recall the definitions of mean, median and mode on the previous page, you will note that the mean and median would involve some arithmetic because you are looking for the average revenue, the “central” value, and the number that appears most frequently, respectively. So, to find the three numbers:

  • Find the mode first since it only involves finding the most frequent number. In this case, the value is $260 (since it was the revenue made for Monday and Wednesday).

  • Find the median next since it will involve rearranging the numbers in an ascending order. In this case, it is an odd set of numbers.
$230  $250  $260  $260  $320  $365  $450
Since the number of values in this set is odd, find the number found in the middle of the entire set. In this case, the value is, again, at $260 — same as the mode.
The question then becomes, “will the average (mean) be the same as the median and mode above?” We’ll find out!

  • Finally, find the mean of the set. Add all the numbers in the set and divide them by seven. Thus:
230 + 250 + 260 + 260 + 320 + 365 + 450
————————————————-
7

=
2135
——
7
Divide 2135 by 7, and you will get $305. So Alex’s average revenue for that particular week is $305.

question 2

Ms. Roberts owns a bakery, and she loves making bread. She samples pastry weights before baking, and her observations are the following (in grams):
175   190   165   140   185   180
Find the mean, median, and mode.
Again, this question is similar to earlier. You need to decide which of the three M’s is easiest to find. The easiest to find is the mode, in which there is none since all the numbers are different. No common number exists, so just write “none” when you answer the question.

Now with the mode done, the next thing to find–and you get to choose–would either be the median first before the mean, or vice versa. In this case, I would start with the median because it is simply rearranging the numbers in ascending order. In this case:

140   165   175   180   185   190
With six numbers in a set, you might wonder what the median is because from the definition, the median is the “central” number in a given set. In the earlier problem, it was easy to find the median because it is located exactly in the middle of the set. However, in this problem, the central number could not be determined. What to do to find the median in this case?

First, split the numbers in sets of three by drawing a line in between three numbers:

140   165   175   ||   180   185   190
Then indicate the numbers immediately on the left and right of the line (indicated in red); in this case, the numbers are 175 and 180. Add the two  together, then divide the sum by two to find the median. In this case:
   175
+ 180
———
   355

==>


355
——–
2

=


The median weight is
177.5 grams.

Finally, find the mean of the set. In this case:

140 + 165 + 175 + 180 + 185 + 190
—————————————–
6

=


1035
——
6
Divide 1035 by 6, and you will get 172.5 grams as the average (mean) weight of the breads.

question 3

Peter wants to maintain his good grades at school. His dream is to make his Grade Point Average (GPA) at least 3.50. His GPA averages from the previous five semesters have been the following:
3.75  3.43  3.50  3.35  3.28
a) What GPA does he need this semester so that he can attain his dream GPA?

b) Find the median and the mode, including your answer from letter A.

In this question, it is a two-pronged question wherein you are going to perform two independent, but inter-related tasks:

  • Finding the missing GPA, as asked in question A; and
  • Finding out the median and the mode, if there is one.

To perform part A of the problem, you will need to add up the GPAs first:

3.75 + 3.43 + 3.50 + 3.35 + 3.28   =   17.31
Next, multiply the GPA he wants to attain by six. Why? Because, as how I set up the problem, I asked “what GPA does he need this semester to attain his dream GPA [of 3.50]?” In this case, you will need to multiply 3.50 by six, such that:
3.50 x 6   =   21
Finally, subtract 21 from 17.31 to find the needed GPA:
21 – 17.31   =   3.69
Peter, then, will need to work very hard to attain a GPA of 3.69 to attain his dream GPA of 3.50.

However, the problem is not yet over. Part B wants you to find the median and the mode of the set, including your answer from part A, which is 3.69. In this case, you will need to find the mode first since it only lists the GPAs in no particular order, and you’ll be pleasantly surprised that there is no mode. Finally, find the median of the set:

3.28   3.35   3.43   3.50   3.69   3.75
Similar to question 2 above, you have an even-numbered set wherein you will not immediately find the median of the set. What to do?
3.28   3.35   3.43   ||   3.50   3.69   3.75
Once you find the two numbers that are next to the dividing line, add them up and divide it by two. Your median should appear as 3.465.
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