Question

In a group of 54 people, each likes music or dance.If the ratio of people who like music only and dance only is 5:4 and the number who like both is 18,find the number of people who like dance only by using venn-diagram
it is from set chapter ​

Answer 1

10

Let D denotes dancing, P denotes painting and S denotes singing.

∴n(D∪P∪S)=265

n(S)=200,n(D)=110,n(P)=55,n(S∩D)=60,n(S∩P)=30 and n(D∩P∩S)=10

∵n(D∪P∪S)=n(D)+n(P)+n(S)−n(D∩P)−n(P∩S)−n(S∩D)+n(D∩P∩S)

∴265=110+55+200−n(D∩P)−30−60+10

⇒265=285−n(D∩P)

⇒n(D∩P)=20

∴ Only person who like dancing and painting =n(D∩P)−n(D∩P∩S)

=20−10=10

Answer 2

Question for you:

Tell me if my answer is correct or not.

Step-by-step explanation:

let the music be M and dance be D;

Then,

n(M U D) = 54

n(M n D) = 18

let the ratio of people who like music only be 5x and dance only be 7x;

Then,

n(M) = 5x

n(D) = 4x

now,

       n(M U D) = {n(M) + n(D)} - n(M n D)

    or, 54 = (5x + 4x) - 18

    or, 54 + 18 = 9x

    or, 72/9 = x

    or, x = 8

  Therefore, x = 8

Then,

        n(D) = 4x

                = 4 × 8

                = 32

again,

         n₀(D) = 32 - 18

                  = 14

Therefore the number of people who like dance only is 14.

(Thank You!! :-)