1. **State the problem:** We are given the harmonic mean (HM) and geometric mean (GM) of two numbers and need to find the two numbers. 2. **Given:** - Harmonic mean $HM = \frac{24}{5}$ - Geometric mean $GM = 6$ 3. **Recall formulas:** - Harmonic mean of two numbers $x$ and $y$ is given by: $$HM = \frac{2xy}{x+y}$$ - Geometric mean is: $$GM = \sqrt{xy}$$ 4. **Use the GM to find the product:** $$\sqrt{xy} = 6 \implies xy = 6^2 = 36$$ 5. **Use the HM to find the sum:** $$\frac{2xy}{x+y} = \frac{24}{5} \implies \frac{2 \times 36}{x+y} = \frac{24}{5}$$ 6. **Solve for $x+y$:** $$\frac{72}{x+y} = \frac{24}{5} \implies 72 \times 5 = 24 (x+y) \implies 360 = 24 (x+y)$$ $$x+y = \frac{360}{24} = 15$$ 7. **Now we have:** $$x+y = 15$$ $$xy = 36$$ 8. **Find the numbers by solving the quadratic:** $$t^2 - (x+y)t + xy = 0 \implies t^2 - 15t + 36 = 0$$ 9. **Calculate the discriminant:** $$\Delta = 15^2 - 4 \times 36 = 225 - 144 = 81$$ 10. **Find roots:** $$t = \frac{15 \pm \sqrt{81}}{2} = \frac{15 \pm 9}{2}$$ 11. **Two numbers:** $$t_1 = \frac{15 + 9}{2} = 12$$ $$t_2 = \frac{15 - 9}{2} = 3$$ **Answer:** The two numbers are $12$ and $3$.