1600 satellites were sent up by a country for several purposes. The purposes are classified as broadcasting (B), communication (C), surveillance (S), and others (O). A satellite can serve multiple purposes; however a satellite serving either B, or C, or S does not serve O.
The following facts are known about the satellites:
1. The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2: 1:1.
2. The number of satellites serving all three of B, C, and S is 100.
3. The number of satellites exclusively serving C is the same as the number of satellites exclusively serving S. This number is 30% of the number of satellites exclusively serving B.
4. The number of satellites serving O is the same as the number of satellites serving both C and S but not B.
Question:
What is the minimum possible number of satellites serving B exclusively?
Given data:
1. Number of satellites serving B, C, and S (though maybe not exclusively) are in the ratio
2. Number of satellites serving all three of B, C, and S is 100
3. Number of satellites exclusively serving C is the same as the number of satellites exclusively serving S (= 30% of the number of satellites exclusively serving B)
4. Number of satellites serving O is the same as the number of satellites serving both C and S but not B
Since there are satellites serving a single or multiple purposes, Venn Diagram is a good method to solve.
For ease of calculation, let the number of satellites exclusively serving B = 10x.
the number of satellites exclusively serving C and S
Now, let the number of satellites serving others(O) by y.
Lastly, let the number of satellites serving B, C but not S be z.
Since the numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio
number of satellites serving B, S but not
It is given that the total number of satellites = 1600:
The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio
Solving the equations (substitute this in equation 1):
Using the boundary condition for x,
Also,
Therefore, we can say that x lies in the range 25 to 80.
The number of satellites serving B exclusively = 10x.
This will be the minimum when x is minimum.
At x (minimum value) = 25,
number of satellites serving B exclusively