Counting the Number of Times 8 Appears in Numbers from 1 to 300
Qwen, a virtual assistant created by Alibaba Cloud, has analyzed the frequency of the digit 8 in numbers counting from 1 to 300. This task involves examining different ranges and iterating through each range to identify the occurrences of the digit. Let's break down the process in detail.
Counting 8 in Each Range
From 1 to 99
8 in the Tens Place
The numbers from 80 to 89 include 8 in the tens place. This range covers 10 numbers:
80, 81, 82, 83, 84, 85, 86, 87, 88, 89
8 in the Units Place
The numbers are 8, 18, 28, 38, 48, 58, 68, 78, 88, 98. These also include 10 numbers:
8, 18, 28, 38, 48, 58, 68, 78, 88, 98
Note that 88 is counted twice, so we subtract one instance:
Total from 1 to 99 10 (tens) 10 (units) - 1 double count 19
From 100 to 199
8 in the Hundreds Place
No numbers in this range contain the digit 8:
Total from 100 to 199 0
8 in the Tens Place
The numbers from 180 to 189 include 8 in the tens place:
180, 181, 182, 183, 184, 185, 186, 187, 188, 189
Total from 100 to 199 10
8 in the Units Place
The numbers are 108, 118, 128, 138, 148, 158, 168, 178, 188, 198. These also include 10 numbers:
108, 118, 128, 138, 148, 158, 168, 178, 188, 198
Note that 188 is counted twice, so we subtract one instance:
Total from 100 to 199 10 (tens) 10 (units) - 1 double count 19
From 200 to 299
8 in the Hundreds Place
No numbers in this range contain the digit 8:
Total from 200 to 299 0
8 in the Tens Place
The numbers from 280 to 289 include 8 in the tens place:
280, 281, 282, 283, 284, 285, 286, 287, 288, 289
Total from 200 to 299 10
8 in the Units Place
The numbers are 208, 218, 228, 238, 248, 258, 268, 278, 288, 298. These also include 10 numbers:
208, 218, 228, 238, 248, 258, 268, 278, 288, 298
Note that 288 is counted twice, so we subtract one instance:
Total from 200 to 299 10 (tens) 10 (units) - 1 double count 19
From 300
No 8 in 300
300 does not contain the digit 8:
Total from 300 0
Total Calculation
Now, we can sum up the counts from each range:
Total from 1 to 99: 19 Total from 100 to 199: 19 Total from 200 to 299: 19 Total from 300: 0Total 19 19 19 0 57
Therefore, there are a total of 57 numbers from 1 to 300 that contain the digit 8.
Alternative Method
To further simplify the task of counting 8, let's consider each number range step-by-step:
1 to 100
Counting from 1 to 10, 8 appears only once (8 itself).
11 to 20: 8 appears once (18) 21 to 30: 8 appears once (28) 31 to 40: 8 appears once (38) 41 to 50: 8 appears once (48) 51 to 60: 8 appears once (58) 61 to 70: 8 appears once (68) 71 to 80: 8 appears once (78) 81 to 90: 8 appears twice (88, 81-89) 91 to 100: 8 appears once (98)Total 8's from 1 to 100 10 1 11
Thus, the total number of times the digit 8 appears from 1 to 300 is 57.