James prints all of the numbers from 1 to 885 . How many times does he print the digit 7

Question

GPT-4o mini · Accepted Answer

To find how many times the digit '7' appears in the numbers from 1 to 885, we can break it down by analyzing each digit position: the hundreds, tens, and units.

Step 1: Count how many times '7' appears in each digit position.

1. Units place:

We look at the numbers from 1 to 885 and check how many times '7' appears in the units place:

The units digit is '7' for the numbers: 7, 17, 27, ..., 877.
This forms an arithmetic sequence where the first term $a = 7$ and the common difference $d = 10$ .
The last term must be less than or equal to 885 (which is 877).

To find how many terms (n) are in this sequence, we can use the formula for the n-th term of an arithmetic sequence: $a_{n} = a + (n - 1) \cdot d$ Setting $a_{n} \leq 885$ : $877 = 7 + (n - 1) \cdot 10$ $870 = (n - 1) \cdot 10$ $n - 1 = 87 \Rightarrow n = 88$ Thus, '7' appears 88 times in the units place.

2. Tens place:

Next, we check how many times '7' appears in the tens place:

The relevant numbers are those in the range 70-79, 170-179, 270-279, 370-379, 470-479, 570-579, 670-679, 770-779, 870-879.
Each complete set from 70-79, 170-179, ..., 770-779 has 10 occurrences of '7' in the tens place (as the units digit varies from 0 to 9).

We can count these:

70-79: 10 occurrences
170-179: 10 occurrences
270-279: 10 occurrences
370-379: 10 occurrences
470-479: 10 occurrences
570-579: 10 occurrences
670-679: 10 occurrences
770-779: 10 occurrences
870-879: 10 occurrences

Total for the tens place: $10 (for each of the 8 groups) = 10 \cdot 8 = 80$ Thus, '7' appears 80 times in the tens place.

3. Hundreds place:

Lastly, we see how many times '7' appears in the hundreds place:

From 1 to 885, '7' only appears in the hundreds place in the range 700 to 799.
All numbers from 700 to 799 include '7' in the hundreds place.

Counting these:

There are $799 - 700 + 1 = 100$ occurrences of '7' in the hundreds place.

Final Count:

Now we can combine these counts:

Occurrences in the units place: 88
Occurrences in the tens place: 80
Occurrences in the hundreds place: 100

Adding these together gives: $88 + 80 + 100 = 268$

Thus, the digit '7' is printed 268 times when James prints all the numbers from 1 to 885.