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Let's solve each of the problems step-by-step.

### Problem 1:
Pascal walks $\frac{1}{4}$ mile for every $\frac{2}{3}$ mile Lucien runs.

The question asks how many miles Pascal walks for every 1 mile Lucien runs.

We need to set up a proportion:
\[
\frac{1}{4} \text{ mile Pascal} = \frac{2}{3} \text{ mile Lucien}
\]
We want to find how many miles Pascal walks for every 1 mile Lucien runs. So, we need to find:
\[
\frac{1}{4} \div \frac{2}{3} = \frac{1}{4} \times \frac{3}{2} = \frac{3}{8}
\]
Thus, Pascal walks $\frac{3}{8}$ mile for every 1 mile Lucien runs.

### Problem 2:
**Part A**: The recipe uses 5 cups of strawberries for every $\frac{1}{4}$ cup of sugar. The constant of proportionality is the ratio of strawberries to sugar.

To find the constant of proportionality, we divide the number of cups of strawberries by the number of cups of sugar:
\[
\frac{5 \text{ cups strawberries}}{\frac{1}{4} \text{ cup sugar}} = 5 \times 4 = 20
\]
So, the constant of proportionality is 20, which means for every 1 cup of sugar, 20 cups of strawberries are used. The correct answer for Part A is **D: 20**.

**Part B**: The cost of 2 cups of strawberries is $6. We need to find an equation that shows the relationship between the cost $c$ and the number of cups of strawberries $s$.

The cost per cup of strawberries is:
\[
\frac{6}{2} = 3 \text{ dollars per cup}
\]
So, the equation relating cost $c$ and the number of cups $s$ is:
\[
c = 3s
\]

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Do you want further details or have any questions? 

Here are 5 related questions to expand your understanding:
1. How do you solve proportions involving fractions in different scenarios?
2. What other examples can we create where constant of proportionality applies?
3. How can we check if a ratio is directly proportional?
4. What happens if the number of cups of sugar changes in the shortcake recipe?
5. How can you graph the cost equation $c = 3s$?

**Tip**: Always simplify fractions and ratios to their simplest form to avoid errors in proportionality problems!

1. For every 1/4 mile Pascal walks, Lucien runs 2/3 mile. How many miles does Pascal walk for every 1 mile Lucien runs? 2. The number of cups of strawberries in a shortcake recipe is proportional to the number of cups of sugar in the recipe. The recipe uses 5 cups of strawberries per 1/4 cup of sugar. What is the constant of proportionality for the relationship of cups of strawberries to cups of sugar? The cost of 2 cups of strawberries is $6. Write an equation to show the relationship between the cost, c, and the cups of strawberries, s.

Solve two problems involving proportions and unit rates. Learn how far Pascal walks for every mile Lucien runs, and how to find the constant of proportionality between cups of strawberries and sugar. A detailed look into the cost equation for strawberries in a recipe.

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