Let's denote the two numbers as $x$ and $y$.

Given:

1. The sum of the two numbers is 45: $x + y = 45$
2. One of the numbers is two-thirds of the other. Assuming $x$ is two-thirds of $y$: $x = \frac{2}{3}y$

Now, we can substitute the expression for $x$ from the second equation into the first equation:

$\frac{2}{3}y + y = 45$

To simplify, combine like terms:

$\frac{2}{3}y + \frac{3}{3}y = 45$

$\frac{5}{3}y = 45$

Now, solve for $y$ by multiplying both sides by $\frac{3}{5}$:

$y = 45 \times \frac{3}{5} = 27$

With $y = 27$, we can find $x$ using the relationship $x = \frac{2}{3}y$:

$x = \frac{2}{3} \times 27 = 18$

So, the two numbers are $18$ and $27$.

Summary:

• The two numbers are 18 and 27.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What if the sum of the two numbers was 60 instead of 45?
2. How would the solution change if one number was three-quarters of the other?
3. Can you solve the problem using a different method, such as substitution or elimination?
4. What if the problem involved three numbers with a similar relationship?
5. How does this problem relate to ratio and proportion in algebra?

Tip: When dealing with word problems, always translate the words into equations and solve step-by-step.