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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. What is the minimum possible number of satellites serving B exclusively?
  • a)
    250
  • b)
    100
  • c)
    500
  • d)
    200
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
1600 satellites were sent up by a country for several purposes. The p...
It is given that a satellite serving either B, or C, or S does not serve O. So we can say that it's basically 3 satellites broadcasting (B), communication (C), surveillance (S) which can have intersections. Those satellites which are not part of any category are placed in others. We can draw the Venn diagram as follows.
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.
Let '10x' be the number of satellites exclusively serving B. Then, the number of satellites exclusively serving C and S = 0.30*10x = 3x
Let 'y' be the number of satellites serving others(O).
Let 'z' be the number of satellites serving B, C but not S. Since the numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2:1:1. Therefore, we can say that the number of satellites serving B, S but not C = z.
It is given that 
⇒10x + 2z + 2y + 6x = 1600 
⇒8x + z + y = 750 ... (1) 
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
⇒Z + 100 + 3x + y = 1
⇒10x + 2z +100 = 2(z + 100 + 3x + y)
⇒4x = 100 + 2y
⇒2x = 50 + y
⇒Y = 2x - 50 …(2)
We can substitute this in equation (1) 
⇒8x + z + 2x - 50 = 750
⇒z = 800 - 10x ... (3)
Let us define boundary condition for x, 
⇒2x - 50 ≥ 0 
⇒x ≥ 25 
Also, 800 - 10x ≥ 0 
⇒x ≤ 80 
Therefore, we can say that x ∈[25, 80].
The number of satellites serving B exclusively = 10x. This will be minimum when 'x' is minimum.
At x = 25, The number of satellites serving B exclusively = 10*25 =250. Hence, option A is the correct answer.
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Most Upvoted Answer
1600 satellites were sent up by a country for several purposes. The p...
Given Information:
- 1600 satellites were sent up by a country.
- The purposes of the satellites are classified as broadcasting (B), communication (C), surveillance (S), and others (O).
- A satellite can serve multiple purposes.
- A satellite serving either B, or C, or S does not serve O.
- The number 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.

Objective:
To find the minimum possible number of satellites serving B exclusively.

Solution:
Let's break down the given information and solve step by step:

Step 1: Determine the number of satellites serving B, C, and S
Let the number of satellites serving B, C, and S be 2x, x, and x respectively.
Therefore, the total number of satellites serving B, C, and S (though may be not exclusively) is 2x + x + x = 4x.

Step 2: Determine the number of satellites exclusively serving B, C, and S
Let the number of satellites exclusively serving B, C, and S be a, b, and c respectively.
From the given information:
- a + b + c = 100 (as 100 satellites serve all three B, C, and S)
- b = c (the number of satellites exclusively serving C is the same as the number of satellites exclusively serving S)
- b = 0.3a (the number of satellites exclusively serving C is 30% of the number of satellites exclusively serving B)
Using these equations, we can solve for a, b, and c.
Substituting b = c and b = 0.3a in the equation a + b + c = 100:
a + 0.3a + 0.3a = 100
1.6a = 100
a = 62.5 (approx)
Therefore, b = c = 62.5

Step 3: Determine the number of satellites serving O
Let the number of satellites serving O be d.
From the given information, d = b + c - a (the number of satellites serving O is the same as the number of satellites serving both C and S but not B)
Substituting the values of a, b, and c:
d = 62.5 + 62.5 - 62.5 = 62.5

Step 4: Determine the number of satellites serving B exclusively
The number of satellites serving B exclusively can be found using the equation:
Number of satellites serving B exclusively = Number of satellites serving B - Number of satellites serving B, C, and S + Number of satellites serving all three B, C, and S
Number of satellites serving B exclusively = 2x - (4x - 100) + 100
Number of satellites serving B exclusively =
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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. What is the minimum possible number of satellites serving B exclusively?a) 250b) 100c) 500d) 200Correct answer is option 'A'. Can you explain this answer?
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