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.

Q.1: What best can be said about the number of satellites serving C?

1. Must be between 450 and 725
2. Cannot be more than 800
3. Must be between 400 and 800
4. Must be at least 100

Q.2: What is the minimum possible number of satellites serving B exclusively?

1. 100
2. 200
3. 500
4. 250

Q.3: If at least 100 of the 1600 satellites were serving O, what can be said about the number of satellites serving S?

1. At most 475
2. Exactly 475
3. At least 475
4. No conclusion is possible based on the given information

Q.4: If the number of satellites serving at least two among B, C, and S is 1200, which of the following MUST be FALSE?

1. The number of satellites serving C cannot be uniquely determined
2. The number of satellites serving B is more than 1000
3. All 1600 satellites serve B or C or S
4. The number of satellites serving B exclusively is exactly 250

1. 1
2. 4
3. 3
4. 1

### Explanation

Let the number of satellites serving B, C and S be 2x, x, and x respectively.

Let the number of satellites exclusively serving B be t. So, the number of satellites exclusively serving C and exclusively serving S will each be 0.3t.

The number of satellites serving O is same as the number of satellites serving only C and S. Let this number be y.

Number of satellites serving C = Number of satellites serving S

⇒ Number of satellites serving only B and C + 0.3t + 100 + y = Number of satellites serving only B and S + 0.3t + 100 + y

So, Number of satellites serving only B and C = Number of satellites serving only B and S = z (assume)

There are a total of 1600 satellites. So,

t + z + 0.3t + z + 100 + y + 0.3t + y = 1600

1.6t + 2y + 2z = 1500 ... (i)

Also, x = 0.3t + z + y +100

Satellites serving B = 2x = t + 2z + 100

⇒ 2(0.3t + z + y +100) = t + 2z + 100

0.4t = 2y + 100

t = 5y + 250

Substituting in (i)

1.6(5y + 250) + 2y + 2z = 1500

10y + 2z = 1100

z = 550 – 5y