Let's go through the problem step-by-step and form the corresponding equations.
1. Defining the Digits:
- Let [tex]\( h \)[/tex] represent the hundreds digit.
- Let [tex]\( t \)[/tex] represent the tens digit.
- Let [tex]\( u \)[/tex] represent the units digit.
2. Forming the Equations:
- Equation for the sum of the digits:
We are told that the sum of the digits is 20. This can be expressed as:
[tex]\[
h + t + u = 20
\][/tex]
- Equation for the tens digit:
We are told that the tens digit [tex]\( t \)[/tex] is three times the sum of the other two digits. This can be written as:
[tex]\[
t = 3(h + u)
\][/tex]
- Equation for the reversal condition:
The number itself can be represented as:
[tex]\[
100h + 10t + u
\][/tex]
When the digits are reversed, the number becomes:
[tex]\[
100u + 10t + h
\][/tex]
We are also told that this reversed number exceeds the original number by 99:
[tex]\[
100u + 10t + h = 100h + 10t + u + 99
\][/tex]
3. Simplifying the Reversal Condition:
- Start with the reversal equation:
[tex]\[
100u + 10t + h = 100h + 10t + u + 99
\][/tex]
- Subtract [tex]\( 10t \)[/tex] from both sides to simplify:
[tex]\[
100u + h = 100h + u + 99
\][/tex]
- Rearrange terms to isolate [tex]\( h \)[/tex] and [tex]\( u \)[/tex]:
[tex]\[
100u - u + h - 100h = 99
\][/tex]
[tex]\[
99u - 99h = 99
\][/tex]
- Divide through by 99:
[tex]\[
u - h = 1
\][/tex]
This can be written as:
[tex]\[
h + 99 = 100u - 10t + u
\][/tex]
So, we end up with the following set of equations:
1. [tex]\( h + t + u = 20 \)[/tex]
2. [tex]\( t = 3(h + u) \)[/tex]
3. [tex]\( h + 10t + 100u = 100h + 10t + u + 99 \)[/tex]
These equations represent the conditions stated in the problem.
