#### 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 best can be said about the number of satellites serving C? Option: 1 Must be between 450 and 725   Option: 2 Cannot be more than 800   Option: 3 Must be between 400 and 800   Option: 4 Must be at least 100

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
$=0.30\times10\: \mathrm{x}$

$=3\: \mathrm{x}$

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 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{ \begin{aligned} 10 x+2 z+100 \quad \ldots(2) \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow z+100+3 x+y=1 \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow 10 x+2 z+100=2(z+100+3 x+y) \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow 4 x=100+2 y \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow 2 x=50+y \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow y=2 x-50 \end{aligned} } ..........(2)
Solving the equations (substitute this in equation 1):
\mathrm{ \begin{aligned} \Rightarrow 8 x+z+2 x-50 =750 \\ \end{aligned}}

\mathrm{ \begin{aligned} 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{ \begin{aligned} & 800-10 x \geq 0 \\ & \Rightarrow x \leq 80 \end{aligned} }

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

Number of satellites serving C

$\mathrm{= 800 - 10x + 100 + 3x + 2x - 50 }$

$\mathrm{= 850 -5x }$

At x = 25, The number of satellites serving C

$\mathrm{= 850 -5x }$

$\mathrm{= 850 - 5 \times 25 }$

$\mathrm{= 725 }$

At $\mathrm{x = 80, }$ The number of satellites serving C

$\mathrm{= 850 -5x }$

$\mathrm{= 850 -5 \times 80 }$

$\mathrm{= 450 }$

Hence, we can say that the number of satellites serving C must be between 450 and 725.