Max A. answered • 10/14/19

Professional Engineer with a Strong Tutoring/Academic Background

Let's start by defining our three digit number as "xyz", with x being the hundreds place, y being the tens place, and z being the ones place. It's good to start this problem by defining our domain for each digit. Well, we know they must all be single digit integers (0-9). Additionally, we know x cannot be 0 because it would result in a two digit number, not three (for example 061). So our domain is as follows:

Domain x = [1, 2, 3, 4, 5, 6, 7, 8, 9]

Domain y = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Domain z = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

The tens digit is twice as big as the ones digit. This can be written in an equation as follows:

y = 2z (eq. 1)

The sum of the three digits is less than 10. This can be written in an equation as follows:

x + y + z < 10 (eq. 2)

Plug in our expression for y from eq. 1 into eq. 2:

x + 2z + z < 10

x + 3z < 10

Let's look at the above inequality and compare it to our domain for z. We can see right away that z cannot be greater than or equal to 4. If we plug z=4 into the above equation, we are already greater than 10. Additionally, if z = 3, the only way the inequality is true is if x = 0, which we have already determined it cannot be. Therefore, our new domain for z is [0, 1, 2]. If it is any higher, our inequality will be false.

Let's start by looking at the possible three digit numbers when z = 2:

If z = 2, then y = 2z = 4, and 3z = 6. This means that x could be 1, 2, or 3, and the inequality x + 3z < 10 would be true. The resulting 3 digit numbers are listed below:

142, 242, 342

Next, we can do the same for z = 1:

If z = 1, then y = 2z = 2, and 3z = 3. This means that x could be 1, 2, 3, 4, 5, or 6. The resulting 3 digit numbers are listed below:

121, 221, 321, 421, 521, 621

Finally, if z = 0, then y = 2z = 0, and 3z = 0. This means that x could be 1, 2, 3, 4, 5, 6, 7, 8, or 9. The resulting 3 digit numbers are listed below:

100, 200, 300, 400, 500, 600, 700, 800, 900

Simply add up the total number of three digit numbers from above and we get 3 + 6 + 9 = **18**.