Thread: How many in the intersection of two sets

1. How many in the intersection of two sets

I'm seeking confirmation that the correct answer to this question is $0$, which is one of the choices in the multiple-choice version of the question, and why.

Q: A dance studio has $35$ students. A poll shows that $12$ study tap and $19$ study ballet. What is the minimum number of students who are studying both tap and ballet?

I'm not clear if the question's original wording (which I don't have handy) meant that $n$ students studying tap means " $12$ students who study only tap" or " $12$ students who are studying at least tap and possibly also ballet" (and the same ambiguous wording for ballet).

But note that $12+19$ is less than $35$ total students. So are there $4$ other students who might have been taking both ( $4$ was not answer choice).

Also, $12$ students was not an answer choice - as in there are $12$ students taking tap and all are also taking ballet, which has $5$ more students, not taking tap.

Please explain the question and answer? Thanks

2. Re: How many in the intersection of two sets

Originally Posted by mathDad
I'm seeking confirmation.
that the correct answer to this question is $0$, which is one of the choices in the multiple-choice version of the question, and why.
Q: A dance studio has $35$ students. A poll shows that $12$ study tap and $19$ study ballet. What is the minimum number of students who are studying both tap and ballet?
I'm not clear if the question's original wording (which I don't have handy) meant that $n$ students studying tap means " $12$ students who study only tap" or " $12$ students who are studying at least tap and possibly also ballet" (and the same ambiguous wording for ballet).
But note that $12+19$ is less than $35$ total students. So are there $4$ other students who might have been taking both ( $4$ was not answer choice).
Also, $12$ students was not an answer choice - as in there are $12$ students taking tap and all are also taking ballet, which has $5$ more students, not taking tap.
Unless you can post the exact wording of the original, it is pointless to even try to help.
As posted, the numbers are inconstant, unless there are others qualifiers left out
For example: if 19 take only ballet and 12 take only tap, then this can be answered.

Try to find the exact wording.

3. Re: How many in the intersection of two sets

Originally Posted by mathDad
Q: A dance studio has $35$ students. A poll shows that $12$ study tap and $19$ study ballet. What is the minimum number of students who are studying both tap and ballet?
I see four possibilities: the numbers can refer to students that study (1) those disciplines and possibly something else or (2) those disciplines only, and (a) there are other disciplines besides tap and ballet or (b) there are no other disciplines.

(1a) The answer is 0 under the following scenario: 12 people study only tap. 19 people study only ballet and 4 people study salsa.

(1b) This variant is impossible.

(2a) The answer is 0 under the same scenario as in (1a).

(2b) The answer is 4.

The question as it is phrased strongly suggests (1a).

4. Re: How many in the intersection of two sets

Originally Posted by Plato
Unless you can post the exact wording of the original, it is pointless to even try to help.
As posted, the numbers are inconstant, unless there are others qualifiers left out
For example: if 19 take only ballet and 12 take only tap, then this can be answered.

Try to find the exact wording.
The original question's wording is ambiguous. That was my point. And it seems that it must be a typo or something.
But since you asked I went and looked for the exact wording:

In a dance school with 35 students, a poll shows that 12 are studying tap dance and 19 are studying ballet. What is the minimum number of students in the school who are studying both tap dance and ballet?
A. 0
B. 7
C. 9
D. 12
E. 31

5. Re: How many in the intersection of two sets

Originally Posted by emakarov
I see four possibilities: the numbers can refer to students that study (1) those disciplines and possibly something else or (2) those disciplines only, and (a) there are other disciplines besides tap and ballet or (b) there are no other disciplines.

(1a) The answer is 0 under the following scenario: 12 people study only tap. 19 people study only ballet and 4 people study salsa.

(1b) This variant is impossible.

(2a) The answer is 0 under the same scenario as in (1a).

(2b) The answer is 4.

The question as it is phrased strongly suggests (1a).
You defined scenario (1a) to mean that those numbers represent people who are studying that discipline and possibly others. So for (1a) 12 ppl study tap and ballet, 19 ppl study ballet (7 of them study only ballet), and 4 ppl study salsa. Answer 12 (which is a choice - contrary to what I said in the OP).

Scenario (2b) seems impossible: tap and ballet are the only disciplines and there are not even 35 students in those two classes.

6. Re: How many in the intersection of two sets

Originally Posted by mathDad
The original question's wording is ambiguous. That was my point. And it seems that it must be a typo or something.
But since you asked I went and looked for the exact wording:

In a dance school with 35 students, a poll shows that 12 are studying tap dance and 19 are studying ballet. What is the minimum number of students in the school who are studying both tap dance and ballet?
A. 0
B. 7
C. 9
D. 12
E. 31
The only slightly ambiguous thing I see about this question is whether there are other disciplines studied in the school. There is no ambiguity with respect to studying tap or ballet exclusively. If I say that I am studying computer science, it means just that. It does not mean that I am not studying anything else besides CS. Similarly, if it is said that 12 people are studying tap dance, it means that they are studying tap dance for sure, but possibly other disciplines as well.

Concerning other disciplines besides tap and ballet in the school, in the absence of any statement to the contrary it is natural to allow this possibility. Also, as post #3 says and as explained below, variant (1b) is impossible and is not one of the answer options. Therefore, the most natural interpretation is (1a).

Originally Posted by emakarov
I see four possibilities: the numbers can refer to students that study (1) those disciplines and possibly something else or (2) those disciplines only, and (a) there are other disciplines besides tap and ballet or (b) there are no other disciplines.

(1a) The answer is 0 under the following scenario: 12 people study only tap, 19 people study only ballet and 4 people study salsa.

(1b) This variant is impossible.

(2a) The answer is 0 under the same scenario as in (1a).

(2b) The answer is 4.

The question as it is phrased strongly suggests (1a).
Originally Posted by mathDad
You defined scenario (1a) to mean that those numbers represent people who are studying that discipline and possibly others. So for (1a) 12 ppl study tap and ballet, 19 ppl study ballet (7 of them study only ballet), and 4 ppl study salsa.
No, the phrase "possibly others" admits many scenarios. The 12 people who study tap dance don't have to study ballet as well: they may study nothing else, some of them may study tap dance and ballet, some of them may study tap dance and ballet while others tap dance and salsa, etc. However, the minimum number of students who study both tap dance and ballet is zero and is achieved, in particular, by the scenario in the quote above.

Originally Posted by mathDad
Scenario (2b) seems impossible: tap and ballet are the only disciplines and there are not even 35 students in those two classes.
Under interpretation (2b), 12 people study only tap and 19 people study only ballet. What do the rest 4 study? Since there are no other disciplines besides tap and ballet, they must study both. These 4 and not included into groups of 12 and 19 who study one dance only: this is consistent with interpretation (2). On the other hand, (1b) is indeed impossible because we cannot say that the remaining 4 people study both dances: otherwise, they would be included into the groups of tap and ballet students according to (1). However, this is just thinking of what might theoretically have been. Any interpretation except (1a) is extremely unlikely.