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:

- The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2:1:1.
- The number of satellites serving all three of B, C, and S is 100.
- 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.
- 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?**

- Must be between 450 and 725
- Cannot be more than 800
- Must be between 400 and 800
- Must be at least 100

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

- 100
- 200
- 500
- 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?**

- At most 475
- Exactly 475
- At least 475
- 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?**

- The number of satellites serving C cannot be uniquely determined
- The number of satellites serving B is more than 1000
- All 1600 satellites serve B or C or S
- The number of satellites serving B exclusively is exactly 250

### Answers

- 1
- 4
- 3
- 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