Solve the problem - How many three digit numbers do not contain a five? Show evidence of solving the problem in a systematic way.

try figuring

how many do contain a five?

Then subtract that from the number of 3-digit values.

So we only have 8 digits to choose from for our leading digit, then 9 digits for each of the remaining spots

number of three digit numbers with no 5
= 8x9x9
= 648

To solve this problem systematically, we need to consider all possible three-digit numbers and count the ones that do not contain the digit five.

Step 1: Determine the range of three-digit numbers.
The smallest three-digit number is 100, and the largest is 999. So, our range is from 100 to 999 inclusive.

Step 2: Break down the number into hundreds, tens, and ones place.
Let's consider each place value separately.

Step 3: Count the possibilities for each place value.
For the hundreds place:
The hundreds place can be any digit from 1 to 9 (excluding 0), giving us 9 options.

For the tens place:
The tens place can be any digit from 0 to 9 (including 0) except for 5, as the number should not contain a five. So, we have 9 options here as well.

For the ones place:
Similarly, the ones place can also be any digit from 0 to 9 (including 0) except for 5. Again, we have 9 options here.

Step 4: Multiply the possibilities together to get the total count.
Since the choices for each place value are independent of each other, we can multiply the number of options for each place together to obtain the total count.

Total count = (Number of options for hundreds place) * (Number of options for tens place) * (Number of options for ones place)
= 9 * 9 * 9
= 729

Therefore, there are 729 three-digit numbers that do not contain the digit five.

Evidence of systematic solving:
By systematically breaking down the problem and considering each digit's place value separately, we determined all the possible options for each place and multiplied them together to find the total count. This approach ensures that no solution is missed and provides a logical and organized way to arrive at the answer.