#### 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:If at least 100 of the 1600 satellites were serving O, what can be said about the number of satellites serving S? Option: 1 At most 475   Option: 2 Exactly 475   Option: 3 At least 475   Option: 4 No conclusion is possible based on the given information

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

$\mathrm{ 10 x+2 z+100 \quad \ldots(2) }$

$\mathrm{\quad \Rightarrow z+100+3 x+y=1 }$

$\mathrm{ \quad \Rightarrow 10 x+2 z+100=2(z+100+3 x+y) \\ }$

$\mathrm{\quad \Rightarrow 4 x=100+2 y }$

$\mathrm{\quad \Rightarrow 2 x=50+y }$

$\mathrm{\Rightarrow y=2 x-50 }$.......(2)

Using the boundary condition for x,

$\mathrm{\Rightarrow 2x-50\geq 0 }$

$\mathrm{\Rightarrow x\geq 25 }$

Also,

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

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

Therefore, we can say that x lies in the range 25 to 80.

Since it is given that at least 100 of the 1600 satellites were serving O.

$\mathrm{\Rightarrow 2x-50\geq 100}$

$\mathrm{\Rightarrow x\geq 75}$

Number of satellites serving S

\mathrm{ \begin{aligned} & =100+800-10 x+2 x-50+3 x \\ & =850-5 x \end{aligned} }
At x=75, the number of satellites serving S
\mathrm{ \begin{aligned} & =850-5 \times 75 \\ & =475 \end{aligned} }
At x=80, the number of satellites serving S
\mathrm{ \begin{aligned} & =850-5 \times 80 \\ & =450 \end{aligned} }

Hence, we can say that the number of satellites serving S must be from 425 to 475