let

x = the hundreds digit

y = the tens digit

z = the units

the sum of the digits in a thee digit whole number is 12:

x + y + z = 12

the tens digit is 10 less than the sum of the units and hundreds digit:

y = (x + z) - 10

72 is added to the number but the hundreds digit remains unchanged.

we have the 2 following sub-cases:

First case: z < 8

then the new number after adding 72 has:

x = the hundred digits

y + 7 = the tens digits

z + 2 = the units

the tens digit of the sum would thrice the hundreds digit of the sum:

y + 7 = 3x

the units digit of the sum would be one less than the tens digit of the original number:

z + 2 = y - 1

by solving the system of equations:

y = (x + z) - 10

y + 7 = 3x

z + 2 = y - 1

we find:

x = 13

y = 32

z = 29

click here to see the step by step solution of the system of equations:

but the digits also need to satisfy the condition x, y and z <= 9 as well as x + y + z = 12 and 13 + 32 + 29 > 12 thus this is not the solution.

Second case: z >= 8

For z = 8:

then the new number after adding 72 has:

x = the hundred digits

y + 7 + 1 = y + 8 the tens digits

z + 2 = 8 + 2 = 10 and 0 is the units

the tens digit of the sum would thrice the hundreds digit of the sum:

y + 8 = 3x

the units digit of the sum would be one less than the tens digit of the original number:

0 = y - 1 => y = 1

y + 8 = 3x

3x = 9

x = 3

we have the number 318

the sum of the digits is: 3 + 1 + 8 = 12

lets check if y = (x + z) - 10 that is 1 = (3 + 8) - 10

thus we have the solution, the number is 318.

For z = 9

then the new number after adding 72 has:

x = the hundred digits

y + 7 + 1 = y + 8 the tens digits

z + 2 = 9 + 2 = 11 and 1 is the units

the tens digit of the sum would thrice the hundreds digit of the sum:

y + 8 = 3x

the units digit of the sum would be one less than the tens digit of the original number:

1 = y - 1

y = 2

from y + 8 = 3x we have 3x = 10 but since x must be integer follows this is not the solution.

We have considered all the possible cases.

The solution of the problem is the number 318.