Question

There are 61 students in the travel club. They discovered that 21 members have visited country A. 29 have visited country B. 25 have been to country C, 8 have visited countries A and B, 14 have been only to country C, and 11 have been only to country A. Some club members have not been to any of the three foreign countries, and curiously, an equal number have been to all three countries.a. How many students have been to all three countries?b. How many students have been only to country B?

Answer 1

Given:

Total No. of students in the travel club = 61

Students visited Country A = 21

Students visited Country B = 29

Students visited Country C = 25

Students visited Country A and B = 8

Students visited Country C only = 14

Students visited Country A only = 11

No. of members who have not been to any country = No. of members who have been to all 3 countries ( *let's call the two of them x since they are equal)

Required:

a. No. of students whho have been to all 3 countries

b. No. of studens who have been to Country B only

Solution:

Total Number of Students = A + B + C - AB - AC -BC + ABC + NONE

61 = 21 + 29 + 25 - 8 - AC - BC + X + X

61 = 67 - AC - BC + 2X

AC + BC = 67 - 61 + 2X

AC + BC = 6 + 2X

Only in Country A = A - AB - AC + ABC

11 = 21 - 8 - AC + X

11 = 13 - AC + X

AC = 13 -11 + X

AC = 2 + X

Only in Country C = C - AC - BC + X ( but, AC = 2 + X )

14 = 25 - ( 2 + X ) - BC + X

14 = 25 - 2 - X - BC + X

BC = 25 - 2 -14 -X + X

BC = 9

Only in Country B = B - AB - BC + ABC

Only in Country B = 29 - 8 - 9 + X

Only in Country B = 12+ X

Also, we know that AC + BC = 6 + 2X ( but, AC = 2 + X and BC = 9 )

Thus,

AC + BC = 6 + 2X

( 2 + X ) + 9 = 6 + 2X

2 + X + 9 = 6 + 2X

11 + X = 6 + 2X

11 - 6 = 2X - X

5 = X

If X = 5 then,

Only in Country B = 12+ X = 12 + 5 = 17

and AC = 2 + X = 2 + 5 = 7

Answer:

a. No. of students whho have been to all 3 countries = 5

b. No. of studens who have been to Country B only = 17