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CAT 2019 Slot 1 — DILR Question 16

Mixed PracticeEasy
Passage / Data

Answer the following question based on the information given below.

​​​​​​​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.

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?

Answer & solution

Correct answer: 5

  • 5

  • B

    3

  • C

    6

  • D

    4

Solution

Considering the given options, the maximum number of different types can be 6.

Assume that there are 6 items.

Now number of items of same type as the one in box 45 = 1 + 31 + 43 = 75

So number of remaining items = 25

1 + 2 + 4 + 8 + 16 = 31. If there are 5 types of items, the minimum number of items of 5 types = 31.

31 + 75 >100

So, there cannot be 6 types of items.

Now consider that there are 5 types of items.

Now number of items of same type as the one in box 45 = 1 + 31 + 43 = 75

So number of remaining items = 25

Now, 25 = (1 + 2 + 4 + 17) or (1 + 3 + 6 + 16) ,… etc

So there can be 5 types of different items.

Hence, option (a).

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CAT 2019 Slot 1 DILR Q16: You ask for the type of item in box 45. Instead of being given a direct answer, you are told that there are 31 — Solution | TheCATExam