How many 9s are there between 1 and 200? please answer
The Correct Answer and Explanation is:
To find how many times the digit 9 appears between 1 and 200, consider how often it shows up in each digit position—units and tens—for numbers from 1 to 199 (since 200 does not contain a 9).
Step-by-Step Breakdown
1. Units Place (1–199)
The units digit goes from 0 to 9 repeatedly. In every group of 10 consecutive numbers, the digit 9 appears once in the units place:
- Examples:
- 9, 19, 29, …, 189, 199
Count how many full sets of 10 there are from 1 to 199:
19910=19 full groups\frac{199}{10} = 19 \text{ full groups}10199=19 full groups
Each group contributes one 9 in the units place.
→ 19 nines in the units place
2. Tens Place (1–199)
Now examine the tens digit. It becomes 9 for numbers from 90 to 99 and again from 190 to 199. That’s 10 numbers each time:
- 90–99: tens digit is 9
- 190–199: tens digit is 9
Each of those two ranges includes 10 numbers where the tens digit is 9.
→ 10 (from 90–99) + 10 (from 190–199) = 20 nines in the tens place
Total Count of the Digit 9
- Units place: 19
- Tens place: 20
Final Total: 19 + 20 = 39
Explanation
Counting how often a specific digit appears within a range requires separating the digit positions. For numbers between 1 and 199, both the units and tens digits need examination.
Begin with the units place. From 1 to 199, every 10-number interval features one number ending in 9. For example, the first few are 9, 19, 29, and so on. Each group of 10 contributes one such number. Since there are 19 full groups of 10 before reaching 199, this results in 19 occurrences of the digit 9 in the units position.
Next, consider the tens place. The tens digit turns into 9 for numbers from 90 to 99 and from 190 to 199. Each of these segments contains 10 consecutive numbers. In both segments, the tens digit stays constant at 9, adding another 20 occurrences of the digit 9.
Adding the two contributions—19 from the units place and 20 from the tens place—gives a total of 39.
This approach ensures that each instance of the digit 9 is accounted for exactly once. It avoids overcounting and breaks the problem into manageable parts. Only numbers within the range 1 to 199 are considered, excluding 200 since it has no 9.
Understanding patterns in digit positions, especially in repetitive number systems like base 10, simplifies digit frequency questions. The total number of times the digit 9 appears between 1 and 200 is 39.
