(40) A recipe for pie dough calls for a ratio of butter to flour of \( \frac{2}{3}: 1 \frac{3}{4} \). If you plan to use 7 cups of flour, how much butter should you use to keep the same ratio?

To determine the amount of butter needed to maintain the same ratio when using 7 cups of flour, follow these steps: ### Given: - **Ratio of Butter to Flour:** \( \frac{2}{3} : 1\frac{3}{4} \) - **Flour to be used:** 7 cups ### Step 1: Convert Mixed Numbers to Improper Fractions Convert \(1\frac{3}{4}\) to an improper fraction: \[ 1\frac{3}{4} = \frac{7}{4} \] So, the ratio becomes: \[ \frac{2}{3} : \frac{7}{4} \] ### Step 2: Set Up the Proportion Let \( b \) be the amount of butter needed for 7 cups of flour. The proportion based on the given ratio is: \[ \frac{\frac{2}{3}}{\frac{7}{4}} = \frac{b}{7} \] ### Step 3: Solve for \( b \) Simplify the left side of the equation: \[ \frac{2}{3} \div \frac{7}{4} = \frac{2}{3} \times \frac{4}{7} = \frac{8}{21} \] Now, set up the equation: \[ \frac{8}{21} = \frac{b}{7} \] Multiply both sides by 7 to solve for \( b \): \[ b = 7 \times \frac{8}{21} = \frac{56}{21} = \frac{8}{3} = 2\frac{2}{3} \] ### **Answer:** You should use \( \boxed{2\,\dfrac{2}{3}} \) cups of butter.