We are given two conditions for the numbers:

1. The difference of two numbers is 12. $x - y = 12$ where $x$ is the greater number and $y$ is the lesser number.

2. Two-fifths of the greater number is six more than one-third of the lesser number. $\frac{2}{5}x = \frac{1}{3}y + 6$

Now, we have two equations:

1. $x - y = 12$
2. $\frac{2}{5}x = \frac{1}{3}y + 6$

Solving the system of equations:

From equation (1), we can express $x$ in terms of $y$: $x = y + 12$

Substitute this into equation (2): $\frac{2}{5}(y + 12) = \frac{1}{3}y + 6$

Now, expand and simplify: $\frac{2}{5}y + \frac{24}{5} = \frac{1}{3}y + 6$

Multiply through by 15 to eliminate the denominators: $15\left(\frac{2}{5}y + \frac{24}{5}\right) = 15\left(\frac{1}{3}y + 6\right)$ This gives: $6y + 72 = 5y + 90$

Simplifying further: $6y - 5y = 90 - 72$ $y = 18$

Now substitute $y = 18$ into $x = y + 12$: $x = 18 + 12 = 30$

Final Answer:

The two numbers are 30 and 18.

Would you like further details or explanations?

Here are 5 related questions:

1. How can we solve the same system of equations graphically?
2. What happens if the difference between the numbers changes?
3. Can we solve for $x$ and $y$ using substitution or elimination in all systems of equations?
4. How does multiplying both sides of the equation by the least common denominator help?
5. Can these equations be solved using matrix methods?

Tip: Always check your solution by substituting values back into the original equations!