CAT 2019 – Slot 1 – Logical Reasoning and Data Interpretation – A new game show on TV has 100 boxes numbered 1, 2, ...

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A new game show on TV has 100 boxes numbered 1, 2, . . . , 100 in a row, each containing a mystery prize. The prizes are items of different types, a, b, c, . . . , in decreasing order of value. The most expensive item is of type a, a diamond ring, and there is exactly one of these. You are told that the number of items at least doubles as you move to the next type. For example, there would be at least twice as many items of type b as of type a, at least twice as many items of type c as of type b and so on. There is no particular order in which the prizes are placed in the boxes.

Q. 1: What is the minimum possible number of different types of prizes?

Q. 2: What is the maximum possible number of different types of prizes?

Q. 3: Which of the following is not possible?
1. There are exactly 45 items of type c.
2. There are exactly 30 items of type b.
3. There are exactly 75 items of type e.
4. There are exactly 60 items of type d.

Q. 4: You ask for the type of item in box 45. Instead of being given a direct answer, you are told that there are 31 items of the same type as box 45 in boxes 1 to 44 and 43 items of the same type as box 45 in boxes 46 to 100.

What is the maximum possible number of different types of items?

1. 5
2. 4
3. 3
4. 6

1. 2
2. 6
3. Option: 1
4. Option: 1

Explanation:

a. Minimum possible number is when there is 1 of type a and 99 of type b which is in accordance to the condition.

b. Maximum possible number is when 1 of a, 2 of b, 4 of c, 8 of d, 16 of e, 32 of f,

Now left boxes woud be 100 – (1+2+…32) = 37

Now if one more type is to be added then we need at least 64 which is not available thus maximum possible is 6.

c. Lets try to prove the given options possible using easy numbers.

op1:never possible

op2: 1,30,69 is possible

op3: 1,2,4,18.75 is possible

op4: 1,9,30,60 is possible.

d. There have to be then at least 31 + `1 + 43 = 75 gifts of same type,

Thus maximum possible number of boxes = 5 when all types are lest 1,2,4,18,75 

 

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