This problem can be solved two ways, with a formula or with reason.

Using the formula, the intersection of the Venn diagram for which classes students take is:

By using reason, it is clear that 60 + 70 is greater than 100 by 30. It is assumed that this extra 30 students come from students who were counted twice because they took both classes.

To find the intersection of a set of numbers, we need to find all numbers that are in both sets (definition of intersection). We can see that all the primes less than  that are odd are also odd numbers less than , therefore they are in both sets and in the intersection. Thus the answer:

 means the complement of  union  - this is everything in the universe that is not in  or . First we need to find :

 - we find this by knowing anything in either set is in the union. Next we need to find things in the universe that are NOT in :

Since the universe is integers less than or equal to 10, we see that the only ones that fit that description that are not in  are 

First, subtract the students that are in neither class; 75 – 30 = 45 students.

Thus, 45 students are enrolled in Chemistry, Physics, or both.  Of these 45 students, we know 40 are in Chemistry, so that leaves 5 students who are enrolled in Physics only; with 15 total students in Physics, that means 10 must be in Chemistry as well. So 10 students are in both Physics and Chemistry.

Since a given player’s scoring average can be determined by dividing the sum total of points scored by the number of games, we can determine the total points of the scoring leader by multiplying the average points per game by the total number of games. 18 x 7 = 126. Angela would have to score 1 more point than the current scoring leader.  Angela’s current total is 87 points; therefore, she must score 40 (87 + 40 = 127) over the course of the final two games to have the highest average points per game in the league.

mean = sum of all values divided by the number of values.

The equation for the mean of a group of numbers is to find the sum of all of the numbers and then divide by how many numbers are in the group. This means that if we know the mean and how many numbers are in the group, we can find the sum of those numbers.

(sum of all five numbers) / 5 = 40 --> sum of all five numbers = 200

(sum of two smallest numbers) / 2 = 25 --> sum of two smallest numbers = 50

Subtracting the sum of the two smallest numbers from the sum of all five gives us the sum of the remaining three. We can then divide by three to find the mean of those three remaining numbers.

200 – 50 = 150

150 / 3 = 50

To find the mean, you have to add all of the items and then divide by the number of items.   Finding the sum (31+30+55+14+29+20+12)= 191 and then dividing by 7 would give a result of 27.3.  [Side Note: The Median is 29 and the range is 43.]

To find your answer, you would have to set it up like a weighted average because we don’t know what he scored for the 81 games, but we can approximate how many actual points he scored by multiplying 29.5 times 81.  Thus, we would get the equation  (29.5(81)+x)/82=30.  Multiply both sides by 82 to get 2389.5+x=2460.  Thus x=70.5 and we would round up to get 71 points. 

In order to calculate the average, or arithmetic mean, of numbers in a set, we add up all of the numbers, then divide that total sum by the n (the count of numbers in the set).

Here we have 86, 90, 88, and 96. This gives us n = 4.

We add up 86 + 90 + 88 + 96 = 360. 360 / 4 = 90.