1. **State the problem:** Two children run a certain distance at 6 mph and return the same distance at 4 mph. We need to find their average speed for the entire trip. 2. **Formula for average speed when distances are equal:** $$\text{Average speed} = \frac{2 \times v_1 \times v_2}{v_1 + v_2}$$ where $v_1$ and $v_2$ are the speeds for each leg of the trip. 3. **Apply the formula:** Given $v_1 = 6$ mph and $v_2 = 4$ mph, $$\text{Average speed} = \frac{2 \times 6 \times 4}{6 + 4} = \frac{48}{10}$$ 4. **Simplify the fraction:** $$\frac{48}{10} = \frac{\cancel{48}^{\times 4 \times 12}}{\cancel{10}^{\times 2 \times 5}} = \frac{24}{5} = 4.8$$ 5. **Final answer:** The average speed for the entire trip is **4.8 mph**.