Let's start by listing these numbers in an organized way. We may be able to see a pattern that will help us find the answer quickly. (If we don't find a pattern them we can just list them all and count.)
Numbers do not start with a zero (that counts) so we will start with the two-digit numbers that start with 1:
How many Two-digit numbers
1 10
All the other two-digit numbers that start with 1 will have a units/ones digit that is not less than the tens digit. Now we'll add the numbers that start with 2's and 3's:
How many Two-digit numbers
1 10
2 20 21
3 30 31 32
At this point we may see a pattern that helps. We can see that the first row has 1 number, the 2nd row has two numbers, the 3rd row has 3 numbers. So our last row, the 9th, will have nine numbers. You may have learned a formula for the sum of consecutive Integers: n*(n+1)/2. Since we will have 9 numbers to add this would be: 9*(9+1)/2 = 9*10/2 = 45.
If you don't know about this formula then picture this:- The numbers are forming a right triangle as we go down the list.
- Imagine another triangle just like the one our list makes.
- Imagine this 2nd triangle flipped upside down so that the longest row (the numbers in the nineties) is on the top.
- Now imagine the two triangles next to each other so that the "10" of the first triangle is in front of the 9 nineties numbers from the flipped triangle; the "20 21" is in front of the 8 eighties numbers fron the flipped triangle; and all the way down the the 9 nineties numbers from the first triangle in front of the "10" of the flipped triangle.
- The combined shape of the two triangles should be a rectangle with 9 rows and 10 numbers in each row. This means there are 9*10 or ninety numbers in this rectangle.
- But this rectangle has two copies of every number, one from each triangle. So the count of the numbers in just one triangle is going to be 1/2 of the total for the rectangle: 90/2 = 45.
Note how we just figured out the n*(n+1)/2 formula by using the two triangles and a rectangle like this!
Or you could just add up how many numbers there are in each row:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9