Bunuel wrote:
A safari is held twice in a day either in the morning or in the evening. A person is allowed to participate in the safari in both the timings. A total of 108 people participate in the safari. The number of men and women in the safari are in the ratio 5 : 4. The total number of men who participate in the safari in the morning is 50 percent of the total number of men who participate in the safari. If every person participated in the safari at least once, what is the number of people who participate in the safari both in the morning and in the evening?
(1) The number of men who participate in the safari only in the morning is 80 percent of the women who participate in the safari only in the morning. Also, the number of men who participate in the safari only in the evening is double the number of women who participate in the safari only in the evening.
(2) The number of women who participate in the safari only in the evening is 60 percent of the girls who participate in the safari only in the morning.
I struggled a lot with this question, is there a simpler way to solve it? It took me a really long time.
Let:
Men:
- X = only in the morning
- Y = only in the evening
- XY = both (note that it's not a multiplication, just a convenient notation)
- M = X + Y + XY
Women:
- A = only in the morning
- B = only in the evening
- W = A + B + AB (same here)
We want AB + XYInfo from stimulus:
4 M = 5 W -> 1 ~unit~ = \(\frac{108}{9}\) = 12
Therefore M = 5 * 12 = 60, W = 48
Men that participate in the morning = X + XY = 30
(1) 5 X = 4 A
Y = 2 B
We have far more variables than equations (it's easy to plug in some numbers and get viable results)
(2) 5 B = 3 A
The same
(1) + (2):
To combine 5 X = 4 A and 5 B = 3 A, we multiply the first by 3 and the second by 4. Y = 2 B can than be inserted by mathing the B = Y/2
15 X = 12 A = 20 B = 10 Y (3 equations)
Combine with:
X + XY = 30 (1 eq)
AB+XY = 108 - (A+B+X+Y) (1 eq)
Variables: X, Y, XY, A, B, AB -> 6
We need one more equation. Maybe taking into account that te solution needs to comprise only of integers:
Solution 1: X = Y = A = B = 0
Solution 2: X = 4
Therefore, E.