1. **State the problem:** Jack mixes two sugar solutions, Brand A with 25% sugar and Brand B with 50% sugar, to get a mixture with 40% sugar. We need to find the mass ratio of solution A to solution B. 2. **Formula and explanation:** Let the mass of solution A be $x$ and solution B be $y$. The total sugar in the mixture is the sum of sugar from A and B. The sugar content equation is: $$0.25x + 0.50y = 0.40(x + y)$$ 3. **Solve the equation:** $$0.25x + 0.50y = 0.40x + 0.40y$$ Rearranging terms: $$0.25x - 0.40x = 0.40y - 0.50y$$ $$-0.15x = -0.10y$$ Divide both sides by $-0.05$: $$3x = 2y$$ 4. **Find the ratio:** $$\frac{x}{y} = \frac{2}{3}$$ 5. **Interpretation:** The mass ratio of solution A to solution B is $2 : 3$. **Final answer:** The mass ratio of solution A to solution B is **2 : 3**.