How many three-digit numbers exist such that the ratio of the hundreds digit and the tens digit is the same as the ratio of the tens digit and the units digit?

To solve this problem, we need to understand that the ratio of the hundreds digit to the tens digit is the same as the ratio of the tens digit to the units digit. This means that the three digits of the number are in a geometric progression.

Step 1: Identify the possible digits The digits can be any number from 0 to 9. However, since we are looking for a three-digit number, the hundreds digit cannot be 0. So, the hundreds digit can be any number from 1 to 9, the tens digit and the units digit can be any number from 0 to 9.

Step 2: Identify the possible ratios Since the digits are in a geometric progression, the ratio can be any number from 1 to 9. However, since the tens digit and the units digit can be 0, the ratio can also be 0.

Step 3: Count the possible numbers For each possible hundreds digit (1 to 9), there are 10 possible tens digits (0 to 9). For each possible tens digit, there are as many possible units digits as the tens digit (because the ratio is the same). So, for each hundreds digit, there are 10 * 10 = 100 possible three-digit numbers.

However, we have to subtract the numbers where the tens digit and the units digit are 0, because these are not valid (the ratio would be undefined). There are 9 such numbers (one for each possible hundreds digit).

So, the total number of three-digit numbers is 9 * 100 - 9 = 891.

Therefore, there are 891 three-digit numbers such that the ratio of the hundreds digit and the tens digit is the same as the ratio of the tens digit and the units digit.