There are two categories of data here: students who study math, and students who study science. So we can using a Venn diagram that looks like the following:

We know that 58 students study both math and science. So we can put this number in the center section, where the two circles overlap:

We also know that 12 students are studying math but not science. These students belong in the left circle but not the right circle:

Lastly, we have also been told that 23 students are not studying either of these subjects. This number should be in the rectangle that represents the whole set, but not in either of the circles:

We can use the fact that there are 100 students in the data set in total to find how many study science.

The total number of students will be equal to the number who study science at all, plus the number who do not study science.

The number who do not study science will be equal to the number who study math but not science, which is 12, plus the number who do not study either, which is 23.

If we let the number of students who study science be x, the we can put all of this together to get x + 23 + 12 = 100.

Solving this gives x = 65, and so there are 65 students in total who study science.

Looking at the Venn diagram we have created, the students who are studying science will be those in the circle on the right. This includes both the 58 students in the overlap who study both math and science, as well as the unknown number who study science but not math.

So we could also have solved this by first finding the number of students who study science but not math, and then adding that amount to 58.

We should find 401 in the table and look up and across to identify the corresponding headings.

The number 401 lies in the row for people who chose meat and in the column for people who did not add spice. This means that there were 401 people who chose meat chili and did not add spice.

We can find this total in a variety of ways:

• Add the four numbers in middle
• Add the total numbers for each type of chili (the total column on the right)
• Add the total numbers for each type of spice level (the total row along the bottom)

Let's add the numbers in the total colum on the right:

\text{Meat}+\text{Vegetarian}=732+168=900

So there were 900 people who ate chili at Chili Fest.

We can check this by adding up the four numbers in the middle, which represent each specific combination of meat and spice, meat and no spice, vegetarian and spice, and vegetarian and no spice:

331+401+125+43=900

We can first fill in the given information, and then use the sums of rows and columns to find missing values.

Filling in what we already know, we have:

Based on this information, we can now calculate:

• The total number that flowered: 1000-136=864
• The number that did not flower and were planted in peat soil: 136-30=106

With these newly calculated values, we can now find the remaining unknown values:

• Total number planted in peat soil: 394+106=500
• Then the total number planted in sandy soil must be: 1000-500=500
• The number that flowered and were planted in sandy soil: 864-394=470

We can always check by adding across each row and down each column to ensure that the totals add up.

Keshawn seems to have a misconception around how the frequencies relate to one another. He has stated that \text{sandy and not flowered}+\text{peat and flowered}=\text{Total} while it should be\text{sandy and flowered}+\text{sandy and not flowered}+\text{peat and flowered}+\text{peat and not flowered}=\\ \text{Total}

We need to consider all possible cases, not just opposite cases.

He should have said that because there were a total of 1000 plants observed and 30 plants that were planted in sandy soil did not flower, this means that 1000-30=970 were planted in peat soil or were planted in sandy soil and did flower.