Unit 4: Reasoning with Ratios Independent Practice 5 1. Model each of the following with either expressions, equations or tables that can be used to answer the question. Show your full process. Then, state a conclusion. (a) How many times would you have to use the \( 2 / 3 \) of a cup measuring cup to add \( 3 / 4 \) of a cup of flour to a recipe? (b) How many times would you have to use the \( 3 / 4 \) of a cup measuring cup to add \( 2 / 3 \) of a cup of four to a recipe? (c) How many \( 1 / 3 \) cup servings of yogurt are found in a small yogurt cup that holds 1 Y cups of yogurt?

Let's solve each part of the problem step by step. ### Part (a) **Question:** How many times would you have to use the \( \frac{2}{3} \) of a cup measuring cup to add \( \frac{3}{4} \) of a cup of flour to a recipe? **Modeling the Problem:** To find out how many times we need to use the \( \frac{2}{3} \) cup to reach \( \frac{3}{4} \) cup, we can set up the equation: \[ x \cdot \frac{2}{3} = \frac{3}{4} \] where \( x \) is the number of times we need to use the \( \frac{2}{3} \) cup. **Solving the Equation:** To solve for \( x \), we can rearrange the equation: \[ x = \frac{\frac{3}{4}}{\frac{2}{3}} \] This can be simplified by multiplying by the reciprocal: \[ x = \frac{3}{4} \cdot \frac{3}{2} = \frac{9}{8} \] **Conclusion for Part (a):** You would need to use the \( \frac{2}{3} \) cup approximately \( 1.125 \) times to add \( \frac{3}{4} \) of a cup of flour to the recipe. --- ### Part (b) **Question:** How many times would you have to use the \( \frac{3}{4} \) of a cup measuring cup to add \( \frac{2}{3} \) of a cup of flour to a recipe? **Modeling the Problem:** We can set up a similar equation: \[ y \cdot \frac{3}{4} = \frac{2}{3} \] where \( y \) is the number of times we need to use the \( \frac{3}{4} \) cup. **Solving the Equation:** Rearranging gives us: \[ y = \frac{\frac{2}{3}}{\frac{3}{4}} \] This can be simplified by multiplying by the reciprocal: \[ y = \frac{2}{3} \cdot \frac{4}{3} = \frac{8}{9} \] **Conclusion for Part (b):** You would need to use the \( \frac{3}{4} \) cup approximately \( 0.89 \) times to add \( \frac{2}{3} \) of a cup of flour to the recipe. --- ### Part (c) **Question:** How many \( \frac{1}{3} \) cup servings of yogurt are found in a small yogurt cup that holds \( 1Y \) cups of yogurt? **Modeling the Problem:** To find the number of \( \frac{1}{3} \) cup servings in \( 1Y \) cups, we can set up the equation: \[ z \cdot \frac{1}{3} = 1Y \] where \( z \) is the number of \( \frac{1}{3} \) cup servings. **Solving the Equation:** Rearranging gives us: \[ z = \frac{1Y}{\frac{1}{3}} = 1Y \cdot 3 = 3Y \] **Conclusion for Part (c):** You would have \( 3Y \) servings of \( \frac{1}{3} \) cup yogurt in a small yogurt cup that holds \( 1Y \) cups of yogurt. --- In summary: - For part (a), you need to use the \( \frac{2}{3} \) cup approximately \( 1.125 \) times. - For part (b), you need to use the \( \frac{3}{4} \) cup approximately \( 0.89 \) times. - For part (c), you have \( 3Y \) servings of yogurt in a \( 1Y \) cup.