Mathematics, 20.12.2020 14:00 seaotter7140

A group of 62 students were surveyed, and it was found that each of the students surveyed
like at least one of the following three fruits:
apples, bananas, and guavas.
34 liked apples, 30 liked bananas, 33 liked
guavas, 11 like apples and bananas, 15 liked
bananas and guavas, 17 like apples and
guavas, 5 liked three fruits. (20 pts)
• How many students liked apples, but not
bananas and guavas?
. How many students liked bananas only?
• How many students liked apples or bananas
but not guavas?
• How many students liked bananas or guavas
or apples?
• How many students liked like apples and
guavas but not bananas?

Answers: 2

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Answers

students liked apples, but not bananas and guavas at the same time. ( students liked apples and liked neither bananas nor guavas.)

students liked bananas only.

students liked apples or bananas but not guavas.

students liked bananas or guavas or apples, (That is the number of students that liked one or more of the three fruits.)

students liked apples and guavas but not bananas.

Step-by-step explanation:

Start by drawing a Venn Diagram. (Refer to the first attachment.)

The outermost rectangle denotes the set of all

students that were surveyed.The top-left circle denotes the set of surveyed students that liked apples.The top-right circle denotes the set of surveyed students that liked bananas.The other circle denotes the set of surveyed students that liked guavas.

Using information from the question, fill in the number of students in each section.

Start with intersection of all three circles denotes the set of surveyed students that liked all three fruits.

The question states that there are students in this set. Besides, this set isn't a superset of any other set. Therefore, write the number $5\!$ in the corresponding place without doing any calculation.

Continue with the intersection of students that liked two fruits only. For example, the question states that surveyed students liked apple and bananas. However, that $11\!$ students also include the surveyed students that liked all three fruits (apple, banana, and guava.) Therefore, only 11 - 5 = 6 surveyed students liked apple and banana only (but not guava.)

Similarly:

surveyed students liked bananas and guavas only. 17 - 5 = 12

surveyed students liked apples and guavas only.

The question states that surveyed students liked guavas. However, among that $33\!$ students:

of them also liked banana but not apples (bananas and guavas only.)

of them also liked apples but not bananas (apples and guavas only.)

of them also liked apples and bananas (all three fruits.)

Therefore, only 33 - 10 - 12- 5 =6 surveyed students liked guava but neither apple nor banana (guava only.)

Similarly:

of the surveyed students liked apples but neither guava nor banana (apple only.) 30 - 6 - 5 - 10 = 9

of the surveyed students liked bananas but neither apple nor guava (bananas only.)

Among the surveyed students, 62 - (11 + 6 + 9 + 12 + 5 + 10 + 6) = 3 of them liked none of the three fruits.

Refer to the second diagram attached for the Venn Diagram with the corresponding numbers.

Students that liked apples but not bananas and guavas at the same time include:

Students that liked apples only.Students that liked apples and bananas, but not guavas (apple and banana only.)Students that liked apple and guavas, but not bananas (apple and guava only.)

These three subsets include 11 + 6 + 12 = 29 surveyed students.

It was previously found that students liked bananas only.

Students that liked apples or bananas but not guavas include:

Students that liked apples only.Students that liked bananas only.Students that liked apples and bananas, but not guavas (apple and banana only.)

That include 11 + 6 + 9 = 26 surveyed students.

Among these surveyed students, 11 + 6 + 9 + 12 + 5 + 10 + 6 = 59 of them liked at least one of the three fruits.

It was previously found that among these surveyed students, of them liked apples and guavas only. That gives the number of students that liked apples and guavas, but not bananas.