1. **State the problem:** We have a ratio of flour to water in an art paste and a table showing some values. We need to find the missing values in the table and specifically answer: How many cups of water do you need for 1/2 cup of flour? 2. **Identify the given ratio:** From the top row, cups of flour = 1 and cups of water = \frac{1}{2}. So the ratio of flour to water is: $$\text{flour} : \text{water} = 1 : \frac{1}{2}$$ 3. **Express the ratio as a fraction:** $$\frac{\text{water}}{\text{flour}} = \frac{\frac{1}{2}}{1} = \frac{1}{2}$$ This means for every 1 cup of flour, we need \frac{1}{2} cup of water. 4. **Find water for 1/2 cup flour:** Let water needed be $w$ cups. $$\frac{w}{\frac{1}{2}} = \frac{1}{2}$$ Multiply both sides by \frac{1}{2}: $$w = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$ So, for 1/2 cup of flour, you need \frac{1}{4} cup of water. 5. **Complete the table:** - For 4 cups of flour, water needed is: $$w = 4 \times \frac{1}{2} = 2$$ - For 3 cups of water, flour needed is: Let flour be $f$: $$\frac{3}{f} = \frac{1}{2} \Rightarrow 3 = \frac{1}{2} f$$ Multiply both sides by 2: $$2 \times 3 = f \Rightarrow f = 6$$ - For 1/2 cup of water, flour needed is: $$\frac{\frac{1}{2}}{f} = \frac{1}{2} \Rightarrow \frac{1}{2} = \frac{1}{2} f$$ Multiply both sides by 2: $$1 = f$$ So flour = 1 cup. **Final completed table:** | Cups of Flour | Cups of Water | |---------------|--------------| | 1 | \frac{1}{2} | | 4 | 2 | | 6 | 3 | | 1 | \frac{1}{2} | | \frac{1}{2} | \frac{1}{4} | **Answer:** You need \frac{1}{4} cup of water for 1/2 cup of flour.