(PLEASE HELP) Combinations Question, A survey of 100 people who volunteered information about their reading habits showed that...

Question

A survey of 100 people who volunteered information about their reading habits showed that,

1. 75 read newspapers daily.
2. 35 read books at least once a week.
3. 45 read magazines regularly.
4. 25 read both newspapers and books.
5. 15 read both books and magazines.
6. 10 read newspapers, books, and magazines.

a) Construct a Venn diagram to determine the maximum number of people in the survey who read both newspapers and magazines.

I had already made the diagram but I didn't know how to get the number between magazines and newspapers, can someone please explain and show the steps to getting the answer? I'm very confused!! Thanks.

I had already sketched the Venn diagram, but I didn't understand how to find the number between magazines and newspapers.

b) Explain why you cannot determine exactly how many of the people surveyed read both newspapers and magazines.

Answer 1

Hi,

Below is a lengthy explanation of a possible solution. I you follow the mathematical reasoning for why the answer can’t be found.

My opinion:

You could put 25 and 5 for the remaining 30 M readers. But, since a direct relationship isn’t given you wouldn’t be able to give a concrete answer and clear up the ambiguity.

Work I did to find the answers:

The first step to solving this problem is the following: you write the corresponding numbers of each section to double check that it matches the given information.

For example, I noticed that your numbers for books and magazines show 15. However, we know that the end result is 25.

Initially, one would think that the “15” in the other circle is saying that 25 people read books, newspapers and...magazines. But, we have to consider the additional conditions that are expressed.

.................................................................................

A survey of 100 people who volunteered information about their reading habits showed that,

1. 75 read newspapers daily.
2. 35 read books at least once a week.
3. 45 read magazines regularly.
4. 25 read both newspapers and books.
5. 15 read both books and magazines.
6. 10 read newspapers, books, and magazines.

So we should separate each group into sections based on what we do know.

B and M =15

N, B and M = 10

N and B = 25

N and M??

The circle for N, B and M is right.

(100-10 leaves 90 readers that don’t do this).

15 read books and magazines; we know, however, that 10 are already locked in for N, B and, M.

(The circle with five is correct).

*Also, 85 readers don’t affect this number.

25 read N and B-the circle with 15 is fine: ten overlap from the previous condition of N, B and M.

This would leave 75 readers that don’t affect the results of N and B.

All circles for B meet the conditions.

To check the numbers for B and M we must find the remaining numbers for M:

45 magazine readers-15 M and B readers = 30 remaining, magazine readers initially.

But, if you add all the numbers of readers for M you’ll see that the total number for M could be the following:

45 regular readers + 15 readers of B and M + 10 for N, B and M = 70

..................................................................................

N = 75

-25 = 50 remaining N

50-x = 45 (M)

-45

5-x = 0

5= x

Answer 2

Congratulations on getting the Venn Diagram!

Call the missing number x.

Then M is 30 - x and N is 50 - x

Add all of the numbers and set equal to 100.

Solve for x