A copier at the high school has a counter which tallies how many copies have been made on the machine. The counter, however, has a glitch: it always skips the digit 4. For example, the counter will jump from 23 copies to 25 copies, or from 39 copies to 50 copies. If the counter currently reads 2005, how many copies has the machine actually made?
for every thousand, no 400's ... down to 1805
for every 100, no 40's ... down to 1625
for every 10, no 4's ... down to 1462
thx
To find out how many copies the machine has actually made, we can examine the pattern of how the counter skips the digit 4.
First, we notice that the counter does not skip the digit 4 until it reaches a multiple of 10 that ends in 4 (for example, 14, 24, 34, etc.). After it skips 4, it resumes normal counting until it reaches the next multiple of 10 that ends in 4, where it skips 4 again.
Since the counter currently reads 2005, we know that it has not yet encountered a multiple of 10 that ends in 4 after 2000. Therefore, it has not skipped the digit 4 again beyond 2000.
To find out how many copies the machine has made, we can subtract the number of multiples of 10 that end in 4 that the machine has encountered so far from the counter reading.
To determine the number of multiples of 10 that end in 4 that are less than or equal to 2000, we divide 2000 by 10 (to get the number of multiples of 10) and then divide that result by 10 (to get the number of multiples that end in 4). So, 2000 divided by 10 is 200, and 200 divided by 10 is 20. Therefore, there are 20 multiples of 10 that end in 4 that are less than or equal to 2000.
Finally, we subtract the number of multiples of 10 that end in 4 (20) from the counter reading (2005):
2005 - 20 = 1985
Therefore, the machine has actually made 1985 copies.