Let's solve this problem step by step.

Let the two numbers be $x$ and $y$.

1. We are given that the sum of the two numbers is 180: $x + y = 180$

2. We are also told that one number is $\frac{1}{3}$ of the other. Without loss of generality, let's assume $x$ is $\frac{1}{3}$ of $y$: $x = \frac{1}{3}y$

3. Substitute $x = \frac{1}{3}y$ into the first equation: $\frac{1}{3}y + y = 180$

4. Combine like terms: $\frac{1}{3}y + \frac{3}{3}y = \frac{4}{3}y = 180$

5. To solve for $y$, multiply both sides by 3 to get rid of the fraction: $4y = 540$

6. Now, divide both sides by 4: $y = 135$

7. Substitute $y = 135$ back into the equation $x = \frac{1}{3}y$ to find $x$: $x = \frac{1}{3} \times 135 = 45$

So, the two numbers are $45$ and $135$.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. What would the numbers be if the sum was 240 instead of 180?
2. How would the solution change if one number was 1/4th of the other?
3. What is the relationship between the numbers if the sum is negative?
4. How can you express the smaller number as a fraction of the larger number for different sums?
5. What if both numbers were integers and one was 2/3rd of the other?

Tip: When solving problems with fractions, always consider multiplying through by the denominator to simplify the equation.