How many 2-digit numbers, with tens digit less than units digit, have the sum of digits equal to 3 times the difference?
To calculate the number of 2-digit numbers that meet the given criteria, we need to break the problem down into steps:
Step 1: Determine the range of possible tens digits.
Since we need a 2-digit number, the tens digit cannot be zero. Therefore, the possible values for the tens digit range from 1 to 9.
Step 2: Determine the range of possible units digits.
Since the tens digit must be less than the units digit, the possible values for the units digit depend on the chosen tens digit. For example, if the tens digit is 1, the units digit can be any number from 2 to 9. If the tens digit is 2, the units digit can be any number from 3 to 9.
Step 3: Find the 2-digit numbers that meet the given conditions.
For each possible tens digit, we need to count the number of corresponding units digits that satisfy the condition of the sum of digits being equal to 3 times the difference. For example, if the tens digit is 1, we need to find the number of units digits that satisfy: (1 + units digit) = 3 × (units digit - 1).
Step 4: Calculate the total number of 2-digit numbers.
Finally, we sum up the number of 2-digit numbers that meet the condition for each valid tens digit.
Let's go through the steps and calculate the answer.
Step 1: The possible values for the tens digit range from 1 to 9.
Step 2: For each tens digit, we need to determine the range of possible units digits.
- If the tens digit is 1, the units digit can be any number from 2 to 9.
- If the tens digit is 2, the units digit can be any number from 3 to 9.
- If the tens digit is 3, the units digit can be any number from 4 to 9.
- And so on, until the tens digit is 9, and the units digit can be 10 (which is not a 2-digit number).
Step 3: Count the units digits that satisfy the condition for each tens digit.
For each possible tens digit, we need to count the number of units digits that satisfy the condition: (tens digit + units digit) = 3 × (units digit - tens digit).
- If the tens digit is 1, we have the equation: (1 + units digit) = 3 × (units digit - 1)
Simplifying this equation, we get: 1 + units digit = 3 × units digit - 3
Rearranging the terms, we get: 4 = 2 × units digit
The only possible units digit that satisfies this equation is 2.
- If the tens digit is 2, we have the equation: (2 + units digit) = 3 × (units digit - 2)
Simplifying this equation, we get: 2 + units digit = 3 × units digit - 6
Rearranging the terms, we get: 8 = 2 × units digit
The only possible units digit that satisfies this equation is 4.
- If the tens digit is 3, we have the equation: (3 + units digit) = 3 × (units digit - 3)
Simplifying this equation, we get: 3 + units digit = 3 × units digit - 9
Rearranging the terms, we get: 6 = 2 × units digit - 3
Adding 3 to both sides, we get: 9 = 2 × units digit
The only possible units digit that satisfies this equation is 9.
Step 4: Calculate the total number of 2-digit numbers.
Now that we have counted the units digits that satisfy the condition for each tens digit, we can sum up these counts:
For the tens digit 1, there is 1 corresponding units digit (2).
For the tens digit 2, there is 1 corresponding units digit (4).
For the tens digit 3, there is 1 corresponding units digit (9).
....
Continuing this pattern up to the tens digit 9, we can see that there is 1 corresponding units digit (9).
Adding up these counts, we get 1 + 1 + 1 + ... + 1 = 9.
Therefore, there are a total of 9 two-digit numbers that have the sum of digits equal to 3 times the difference with the tens digit less than the units digit.