SOLUTION: How many 2 digit numbers are there, such that the units digit is strictly smaller than the tens digit?

Algebra ->  Customizable Word Problem Solvers  -> Numbers -> SOLUTION: How many 2 digit numbers are there, such that the units digit is strictly smaller than the tens digit?       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 706129: How many 2 digit numbers are there, such that the units digit is strictly smaller than the tens digit?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
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:
  1. The numbers are forming a right triangle as we go down the list.
  2. Imagine another triangle just like the one our list makes.
  3. Imagine this 2nd triangle flipped upside down so that the longest row (the numbers in the nineties) is on the top.
  4. 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.
  5. 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.
  6. 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