#### 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? Option: 1 $100$Option: 2 $200$Option: 3 $500$Option: 4 $250$

Given data:

1. Number of satellites serving B, C, and S (though maybe not exclusively) are in the ratio $2:1:1.\$

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.

$\Rightarrow$ the number of satellites exclusively serving C and S
$\mathrm{= 0.30\times10x }$
$\mathrm{= 3x }$

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 $2:1:1.$

$\Rightarrow$ number of satellites serving B, S but not $\mathrm{C = z.}$

It is given that the total number of satellites = 1600:

\mathrm{\begin{aligned} & \Rightarrow \quad 10 x+2 z+2 y+6 x=1600 \\ & \Rightarrow \quad 8 x+z+y=750 \ldots(1) \end{aligned}}

The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio $2:1:1$

$10x + 2z + 100 ... (2)$

$\mathrm{\begin{array}{ll} \Rightarrow & z+100+3 x+y=1 \\ \Rightarrow & 10 x+2 z+100=2(z+100+3 x+y) \\ \Rightarrow & 4 x=100+2 y \\ \Rightarrow & 2 x=50+y \\ \Rightarrow & y=2 x-50 \end{array}}$

Solving the equations (substitute this in equation 1):

\mathrm{\begin{aligned} \Rightarrow 8 x+z+2 x-50 & =750 \\ z=800-10 x & \ldots(3) \end{aligned}}

Using the boundary condition for x,

\mathrm{\begin{aligned} & \Rightarrow 2 x-50 \geq 0 \\ & \Rightarrow x \geq 25 \end{aligned}}

Also,

$\mathrm{800 -10x \geq 0}$

$\mathrm{\Rightarrow\: x\leq 80 }$

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

$\mathrm{=10\times25 }$

$\mathrm{=250 }$