Six year old Rosie wants to write down all counting numbers up to one thousand. When she finished writing 12, she had 15 digits.
Rosie gave up after she had correctly written 2358 digits.
What was the last number she wrote?
9 1-digit numbers
90 2-digit numbers
so, there were (2358-99)/3 3-digit numbers
But you aren't trying to count how many numbers but many many digits (so 1-12 there would be 15 digits) so how many numbers is 2358 digits?
Umm, I guess that would be
(2358-99)/3
right?
Actually we can represent this with algebra:
Note there are three stages of digits:
Stage 1 - one digit numbers (1 to 9) there are 9 digits total
Stage 2 - two digit numbers (10 to 99) there are 89 x 2 = 178 digits total
Stage 3 - We don't no how far to go so we stop here
Rosie reached 2358 which gets into stage three of three digit numbers.
We would represent our situation with an equation:
2358 = 189 + 3x
now keep terms of x on one side, and the rest on the other side
2358 - 189 = 189-189 +3x
2169 = 3x
Now simple divide both sides by 3 to get X
723 = x
To find the last number Rosie wrote, we first need to determine the pattern in the number of digits for each counting number.
Let's observe:
1 has 1 digit.
2 has 1 digit.
3 has 1 digit.
...
9 has 1 digit.
10 has 2 digits.
11 has 2 digits.
...
99 has 2 digits.
100 has 3 digits.
101 has 3 digits.
...
999 has 3 digits.
1000 has 4 digits.
From the given information, we know that after writing 12 numbers, Rosie had written 15 digits. This means that the numbers 1-12 combined have 15 digits.
We also know that when Rosie stopped writing, she had written 2358 digits in total.
Using this information, we can find how many digits Rosie wrote for numbers 13 to the last number she wrote.
The difference in the number of digits between consecutive numbers beyond 12 is as follows:
Numbers 13-99 each have 2 digits, so (99-12) x 2 = 174 digits.
Numbers 100-999 each have 3 digits, so (999-99) x 3 = 2700 digits.
We still need to add the number of digits for the last number Rosie wrote.
Now, let's calculate the number of digits for the last number Rosie wrote:
2358 - (digit count for numbers 1-12) = (digit count for numbers 13 to the last number)
2358 - 15 = 2343 digits
Since the first 174 of those digits are used for numbers 13-99, we have:
2343 - 174 = 2169 digits remaining for numbers 100 to the last number.
Each number between 100 and 999 has 3 digits, so:
2169 / 3 = 723 numbers remaining.
If we add this count to the last number of the previous range, which is 99, we get:
99 + 723 = 822
Therefore, the last number Rosie wrote is 822.
Keep in mind that Rosie might have stopped at a different point, so this answer assumes Rosie continued with the same pattern of writing numbers.