**DILR – Logical Reasoning – Set**

**The following set contains four questions related to Logical Reasoning. Choose the best answer to each question.**

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.

**1)** What best can be said about the number of satellites serving C?

a) Must be between 450 and 725

b) Must be at least 100

c) Cannot be more than 800

d) Must be between 400 and 800

**2)** What is the minimum possible number of satellites serving B exclusively?

a) 500

b) 100

c) 200

d) 250

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

a) No conclusion is possible based on the given information

b) Exactly 475

c) At most 475

d) At least 475

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

a) All 1600 satellites serve B or C or S

b) The number of satellites serving B is more than 1000

c) The number of satellites serving C cannot be uniquely determined

d) The number of satellites serving B exclusively is exactly 250

**Solution: **

It is given that the satellites serving either B, C or S do not serve O.

From point (1), let the number of satellites serving B, C and S be 2K, K, K respectively.

Let the number of satellites exclusively serving B be x.

From point (3), the number of satellites exclusively serving C and exclusively serving S will each be 0.3x

From point (4), the number of satellites serving O is same as the number of satellites serving only C and S. Let that number be y.

Since the number of satellites serving C is same as the number of satellites serving S,

we can say that (number of satellites serving only B and C) + 0.3x + 100 + y = (number of satellites serving only B and S) + 0.3x + 100 + y

Let the number of satellites serving only B and C = the number of satellites serving only B and S = Z

Therefore, the venn diagram will be as follows:

Given that there are a total of 1600 satellites So

x + z + 0.3x + z + 100 + y + 0.3x + y = 1600

1.6x + 2y + 2z = 1500—————(1)

Also K = 0.3x + z + y +100

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

Or 2(0.3x + z + y + 100)= x + 2z + 100

0.4x = 2y + 100

x = 5y + 250 ——————-(2)

Substituting (2) in (1), we will get

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

10y + 2z = 1100

z= 550 – 5y ———— (3)

**1)** The number of satellites serving C = z + 0.3x + 100 + y = (550 – 5y) + 0.3(5y + 250) + 100 + y = 725 – 2.5y This number will be maximum when y is minimum. Minimum value of y is 0. So the maximum number of satellites serving C will be 725.

From equation 3), z = 550 – 5y, Since the number of satellites cannot be negative

z ≥ 0

or 550 – 5y ≥ 0

y ≤ 110.

Maximum value of y is 110.

When y = 110, the number of satellites serving C will be 725 – 2.5 × 110 = 450.

This will be the minimum number of satellites serving C. The number of

satellites serving C must be between 450 and 725.

Answer: a) Must be between 450 and 725

**2)** From equation 2) , the number of satellites serving B exclusively is x = 5y + 250

This is minimum when y is minimum. Minimum value of y = 0. The minimum number of satellites serving B exclusively = 5 × 0 + 250 = 250.

**Answer: d) 250**

**3)** Given, at least 100 satellites serve O.

We can say in this case that y ≥ 100.

So Number of satellites serving S = 0.3x + z +100 + y. = 725 – 2.5y

This is minimum when y is maximum, i.e. 110, (from eq 3)

Minimum number of satellites serving = 725 – 2.5 × 100 = 450.

This is maximum when y is minimum, i.e., 100 in this case.

Maximum number of satellites serving = 725 – 2.5 × 100 = 475

Therefore, the number of satellites serving S is at most 475

Answer: **c) at most 475**

**4)** The number of satellites serving at least two of B, C or S = number of satellites serving exactly two of B, C or S + Number of satellites serving all the three

= z + z + y + 100 = 2(550 – 5y) + y + 100 = 1200 – 9y.

Given that this is equal to 1200 so

1200 – 9y = 1200

Or y = 0

If y = 0, x = 5y + 250 = 250 z = 550 – 5y = 550

No. of satellites serving C = k = z + 0.3x + 100 + y = 550 + 0.3 x 250 + 100 + y = 725

No. of satellites serving B = 2k = 2 x 725 = 1450.

From the given options, we can say that the option “the number of satellites serving C cannot be uniquely determined” must be FALSE.

Answer: c) The number of satellites serving C cannot be uniquely determine

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