Problem 21 · 2009 Math Kangaroo
Stretch
Counting & Probability
careful-countingplace-value
The rooms in a hotel each have a three-digit number. The first digit gives the floor and the last two digits give the room number on that floor; for example, room 125 is on floor 1, room 25. The hotel has 5 floors (1 to 5) with 35 rooms on each floor, so floor 1 has rooms 101 to 135, and so on. How many times does the digit 2 appear in all the room numbers of the hotel?
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Answer: E — 105 times
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Hint 1 of 2
Split a room number into the floor digit and the two room digits, and count the 2s place by place.
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Hint 2 of 2
Count 2s in the hundreds, tens and units across floors 1–5 and rooms 01–35.
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Approach: count by digit place
- Rooms run 101–135, 201–235, …, 501–535: 175 rooms in all.
- Hundreds place: a 2 appears on the whole 2nd floor, 35 times.
- Tens place: rooms x20–x29 give 10 twos per floor × 5 = 50 times.
- Units place: rooms ending in 2 give 4 per floor × 5 = 20 times.
- Total = 35 + 50 + 20 = 105 times — answer E.
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