Calculating number of satellites $1600$ satellites were sent up by a country.The purposes are classified as B,C,S and O. A satellite can serve multiple purposes however a satellite serving either B, or C, or S doesn't serve O. 
The following facts about satellites are known:- 
1)the ratio of satellites serving B:C:S(though maybe not exclusively)$=2:1:1$. 
2) the number of satellites serving all three of B C and S is $100$. 
3) the number of satellites exclusively serving S=number of satellites exclusively serving C$=0.3$ times the number of satellites serving B.
4)the number of satellites serving O=number of satellites serving both C and S but not B.
Then what is the minimum possible number of satellites serving B exclusively? Options are $a)100 b)200 c)250 d)500$
$$\text{Attempt}$$ 
Let a=satellites serving B b=satellites serving C c=no of satellites serving S d=satellites serving O e=no of satellite serving O,B f=O,C g=O,S h=O,B,C i=O,B,S j=O,C,S k=O,B,C,S(It would help if someone tells the code for an array.)
 We know $k=100 ,b=c=0.3a,d=j,\text{a+e+h+i+j+k}=2\text{b+f+h+j+k}=2\text{c+g+i+j+k}$ and $\sum_a^k x=1600$.  Doing some manipulations I found that $\text{number of satellites serving B}=\frac{1700-(a+e)}{2}$ . But I dont think this is of any help.Dont know how to begin with the problem.
 A: If I am not wrong, the smallest solution for B is this configuration:
Satellites serving O -> 0
Satellites serving B -> 250
Satellites serving S -> 75
Satellites serving C -> 75
Satellites serving CS -> 0
Satellites serving BC -> 550
Satellites serving BS -> 550
Satellites serving BCS -> 100

Edit: Explanation
We have 7 types of satellites attending to their purpose: Q (purpose O), B, C, S, BC, BS, CS, BCS. We know:
0) Q + B + C + S + BC + BS + CS + BCS = 1600
1) B + BC + BS + BCS = 2(C + BC + CS + BCS) = 2(S + BS + CS + BCS)
2) BCS = 100
3) S = C = 0.3B
4) Q = CS

Implications
By 3) S=C, and taking the last equality of 1) we have BC=BS

Now we use the first equality of 1)
B + BC + BS + BCS = 2(C + BC + CS + BCS)

and we replace there what we know:
B + BS + BS + 100 = 2(0.3B + BS + Q + 100) =>
B + 2BS + 100 = 0.6B + 2BS + 2Q + 200 =>
0.4B = 2Q + 100 =>
B = 5Q + 250

Now the minimum B is obtained by assuming Q=0, and the rest follows as consequence
A: I'd start with the following diagram:

The statements you have been given can be used to create equations:
1)the ratio of satellites serving B:C:S(though maybe not exclusively)=2:1:1.
$b+x+100+y=2(x+c+z+100)$
$s+y+100+z=x+c+z+100$
2) the number of satellites serving all three of B C and S is 100.
Placed already in centre of diagram.
3) the number of satellites exclusively serving S=number of satellites exclusively serving C=0.3 times the number of satellites serving B.
$s=c$
$s=0.3(b+x+100+y)$
4)the number of satellites serving O=number of satellites serving both C and S but not B. 
$o=z$
There is also an equation that uses the total number:
$o+b+c+s+x+y+z+100=1600$
Your equations can be simplified to give this system:
$b+x+100+y=2x+2c+2z+200 \Rightarrow b-2c-x+y-2z=100$
$-c+s-x+y=0$
$s=c$
$-3b-3x-3y+10s=300$
$o=z$
$o+b+c+s+x+y+z=1500$
Can you go from there?
