To solve this, let’s define a two-digit number where the tens digit is $x$ and the units digit is $y$. The number itself can then be expressed as:

$10x + y$

The problem gives two conditions:

1. The sum of the digits is one-seventh of the number: $x + y = \frac{1}{7}(10x + y)$

2. If the digits are reversed, the number decreases by 36: $10x + y - (10y + x) = 36$

Let’s work through these equations step-by-step.

Step 1: Simplify the First Condition

Starting with the first condition: $x + y = \frac{1}{7}(10x + y)$

Multiply both sides by 7 to clear the fraction: $7(x + y) = 10x + y$

Expanding and simplifying: $7x + 7y = 10x + y$ $7y - y = 10x - 7x$ $6y = 3x$ $2y = x$

Step 2: Simplify the Second Condition

Now, using the second condition: $10x + y - (10y + x) = 36$

Expanding and simplifying: $10x + y - 10y - x = 36$ $9x - 9y = 36$ $x - y = 4$

Step 3: Solve the System of Equations

Now we have the following two equations:

1. $x = 2y$
2. $x - y = 4$

Substitute $x = 2y$ into the second equation: $2y - y = 4$ $y = 4$

Substitute $y = 4$ back into $x = 2y$: $x = 2(4) = 8$

Step 4: Form the Number

Now we know $x = 8$ and $y = 4$, so the two-digit number is: $10x + y = 10(8) + 4 = 80 + 4 = 84$

Conclusion

The two-digit number is 84.

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to explore:

1. How would the solution change if the sum of the digits was one-fifth of the number?
2. What if reversing the digits decreased the number by 45 instead of 36?
3. Can you apply a similar approach to solve for a three-digit number?
4. How does the method differ if we consider a base other than 10 (like hexadecimal)?
5. What are the implications of this approach in coding problems involving number manipulations?

Tip: Always start by defining unknowns in terms of variables and form equations based on given conditions for easier solving.