1. The problem states that a cleaning spray contains $\frac{1}{2}$ cup of vinegar for every 4 teaspoons of lemon oil. We want to find how many cups of vinegar are needed for 2 teaspoons of lemon oil. 2. The ratio given is $\frac{1}{2}$ cup vinegar to 4 teaspoons lemon oil, or $\frac{1/2}{4}$ cups per teaspoon. 3. To find the equivalent amount of vinegar for 2 teaspoons, we set up a proportion: $$\frac{1/2}{4} = \frac{x}{2}$$ where $x$ is the cups of vinegar for 2 teaspoons. 4. Cross-multiply: $$4x = (1/2) \times 2$$ 5. Simplify the right side: $$4x = 1$$ 6. Divide both sides by 4 to solve for $x$: $$x = \frac{1}{4}$$ 7. So, $\frac{1}{4}$ cup of vinegar is needed for 2 teaspoons of lemon oil. 8. To find the rate in cups of vinegar per teaspoon of lemon oil, divide the vinegar amount by the lemon oil amount: $$\text{Rate} = \frac{1/2}{4} = \frac{1/2}{4} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$$ 9. Therefore, the rate is $\frac{1}{8}$ cups of vinegar per teaspoon of lemon oil. Final answers: - $\frac{1}{4}$ cup of vinegar for 2 teaspoons of lemon oil. - Rate: $\frac{1}{8}$ cups vinegar per teaspoon lemon oil.