Question

Three friends decided to celebrate a birthday together. They agreed that the total number of guests would be 40. Mary invited 17 friends. The number of guests invited only by Anna is 3 more than the number of guests invited only by Dora. Since they also have mutual friends, the following happened: All three invited 4 mutual friends. Mary and Anna invited 5 mutual friends, Anna and Dora 6, and Mary and Dora 7 mutual friends. How many guests did Anna invite, and how many did Dora?​

Answer 1

Answer:

you try to use Venn diagram to solve the question

Step-by-step explanation:

of course it will be easy when you will use Venn diagram

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Answer 2

Given:

No. of guests at party = 40

No. of guests invited by Mary = 17

No. of guests invited by Anna only = 3 + No. of guests invited by Dora only

No. of mutual friends = 4

No. of mutual friends invited by Mary and Anna = 5

No. of mutual friends invited by Dora and Anna = 6

No. of mutual friends invited by Mary and Dora = 7

To find:

No. of guests Anna invited.

No. of guests Dora invited.

Solution:

Let Mary be represented as M, Anna as A and Dora as D.

n(M∪A∪D) = 40, n(M) = 17, n(A) = 3 + n(D), n(M∩A∩D) = 4, n(M∩A) = 5, n(M∩D) = 7, n(A∩D) = 6, n(A) = ?, n(D) = ?

n(M∪A∪D) = n(M) + n(A) + n(D) - n(M∩A) - n(A∩D) - n(M∩D) + n(M∩A∩D)

$40=17+3+n(D)+n(D)-5-7-6+4$

$40=2n(D)+6$

$40-6=2n(D)$

$34=2n(D)$

$n(D)=\frac{34}{2}=17$

∴ No. of friends invited by Dora only is 17.

$n(A)=3+n(D)$

$n(A)=3+17=20$

∴ No. of friends invited by Anna only is 20.

Anna invited 20 guests and Dora invited 17 guests for their birthday party.