How many three digits numbers can be made if the tens digit is twice as big as the units digit and the sum of the three digits is less than 10?

Question

a. 10

b. 18

c. 29

d. 87

e. 245

Answer 1

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.