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j's question at Yahoo! Answers regarding using a Venn diagram to determine the cardinality of sets

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MarkFL

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Feb 24, 2012
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Here is the question:

Very difficult precalculus quantitative methods questions, please help!?

Of 150 students registered at a local university, 76 are attending an accounting course, 49 are taking statistics, 55 in economics, 24 attending the accounting and the statistics course, 22 registered in both statistics and economics courses and 15 are taking all three courses. (A Venn diagram may prove useful)

1) how many students are taking both the statistics and the economics courses without taking the accounting course?

2) how many students are attending the statistics course but nether the accounting nor the economics courses?

3) how many students are taking the accounting course or the statistics course or the economics course?

4) how many of these 150 registered students are not taking any of these three courses?
I have posted a link there to this topic so the OP can see my work.
 
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MarkFL

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Feb 24, 2012
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Hello j,

I have drawn a Venn diagram and filled it in, which I will explain:

jvenn.jpg

I began with the fact that we are told 15 are taking all 3 courses, so I wrote 15 in the intersection of all 3 sets, representing the 3 courses.

Next, I looked at the statement that 22 registered in both statistics and economics, and 15 of those are already accounted for in the first step, so that leaves 7 to go where statistics and economics intersect but outside of accounting.

Next, I looked at the statement that 24 are attending accounting and statistics, and 15 of those are already accounted for in the first step, so that leaves 9 to go where statistics and accounting intersect but outside of economics.

No statement is made regarding the number attending accounting and economics, so I put 0 where accounting and economics intersect outside of statistics.

Next, I looked at the statement that 55 are in economics, and since 15 + 7 = 22 are already accounted for, this leaves 33 to write in economics but outside of the other two courses.

Next, I looked at the statement that 49 are in statistics, and since 15 + 9 + 7 = 31 are already accounted for, this leaves 18 to write in statistics but outside of the other two courses.

Lastly, I looked at the statement that 76 are in accounting, and since 15 + 9 = 24 are already accounted for, this leaves 52 to write in accounting but outside of the other two courses.

Now we are able to answer the questions:

1.) How many students are taking both the statistics and the economics courses without taking the accounting course?

We see this is 7.

2.) How many students are attending the statistics course but neither the accounting nor the economics courses?

We see this is 18.

3.) How many students are taking the accounting course or the statistics course or the economics course?

Adding all the numbers, we find this number is 52 + 9 + 18 + 15 + 7 + 33 = 134.

4.) How many of these 150 registered students are not taking any of these three courses?

We see this is 150 - 134 = 16.