Grades 5-6 Video Solutions 2024-2024_5-6

Video Transcription

Question 10. The rooms in a hotel are numbered in ascending order, starting from 1. No number is omitted. Kangaroo counted all the digits in the room numbers and found the digit 2  14 times and the digit 5 3 times. What is the largest possible number of rooms in the hotel? First, let's list out all the numbers with 2s in them. We use up our 14th 2 on the room  numbered 32. This means we cannot have the next room with a 2, which will be room number 42. Now, let's list out all the numbers with a 5 in them. We use up our 3rd 5 on room  number 25, which means that we cannot have the next room with a 5, which is room 35. Since we cannot skip over a room number and we cannot have rooms 35 and 42, the highest  room number we can have must be 34, which means the largest possible number of rooms in the hotel is 34.

Video Summary

In the given problem, the hotel rooms are numbered sequentially from 1 with no numbers skipped. Kangaroo counted the digit '2' appearing 14 times and '5' 3 times in these numbers. The 14th occurrence of the digit '2' happens at room number 32. The 3rd occurrence of the digit '5' happens at room number 25. Thus, rooms like 35 (which contains a '5') and 42 (which contains a '2') are not possible. Therefore, the highest possible room number is limited to 34, as higher numbers involve additional '2's or '5's, hence the largest number of rooms is 34.

Keywords

hotel rooms

sequential numbering

digit occurrence

room limit

number restriction