Let's break down Jack's dessert budget situation: --- ### **1. Variables** - Let \( P \) be the number of popcorn servings Jack buys per week. - Let \( J \) be the number of Jelly servings Jack buys per week. --- ### **2. Prices** - **Popcorn:** $1.00 per serving - **Jelly:** $0.50 per serving --- ### **3. Budget Constraint** Jack has **$10 per week** to spend. So, the total amount spent on popcorn and jelly should be less than or equal to $10: \[ 1.00P + 0.50J \leq 10 \] The **budget line** (when Jack spends exactly $10) is: \[ 1.00P + 0.50J = 10 \] --- ### **4. Rearranging the Equation** Let's solve for \( J \) in terms of \( P \): \[ 1.00P + 0.50J = 10 \] \[ 0.50J = 10 - 1.00P \] \[ J = \frac{10 - 1.00P}{0.50} \] \[ J = 20 - 2P \] --- ### **5. Intercepts** - **If Jack buys only Jelly (\( P = 0 \)):** \[ J = 20 - 2(0) = 20 \] So, he can buy **20 Jelly**. - **If Jack buys only Popcorn (\( J = 0 \)):** \[ 1.00P = 10 \implies P = 10 \] So, he can buy **10 Popcorn**. --- ### **6. **Graph** - X-axis: Popcorn (\( P \)), from 0 to 10 - Y-axis: Jelly (\( J \)), from 0 to 20 - Connect points (0, 20) and (10, 0). **Budget Line Equation:** \( J = 20 - 2P \) --- #### **Rough Sketch** ``` Jelly (J) | 20| *---------------------- | | / | | / 10| | / | | / | | / | | / | | / +---------------------------- Popcorn (P) 0 5 10 ``` - The line starts at (0, 20) and ends at (10, 0). --- ### **Summary** Jack’s **budget line** is: \[ 1.00P + 0.50J = 10 \] or equivalently, \[ J = 20 - 2P \] with intercepts at \( (0, 20) \) and \( (10, 0) \). **He can afford any combination of Popcorn and Jelly on or below this line.**

**1. Introduction:** In consumer choice theory, a budget line (or budget constraint) illustrates all the combinations of two goods that a consumer can purchase given their income and the prices of those goods. It represents the maximum quantities of each good that can be bought without exceeding the consumer’s budget. Understanding the budget line helps in analyzing how consumers allocate their limited income among different goods to maximize their satisfaction. Key concepts include: - **Budget constraint:** The total amount of money available to spend on goods. - **Prices of goods:** The cost per unit of each good. - **Quantities of goods:** The number of units of each good purchased. - **Intercepts:** The maximum quantity of one good that can be bought if all income is spent on that good alone. In this problem, Jack has a fixed weekly budget of $10 to spend on popcorn and Jelly. The prices of these goods are given, and the goal is to formulate the budget line equation and then graph it to understand the possible combinations Jack can afford. **Explanation:** By defining the budget constraint and understanding the prices and income, one can determine the feasible combinations of popcorn and Jelly Jack can buy. The intercepts indicate the maximum quantity of each good that Jack can purchase when spending all his money on only one good, which is foundational for plotting the budget line. --- **2. Presentation of Relevant Formulas and Representation of Given Data:** **Formulas:** - **Budget constraint equation:** \[ \text{Price of good 1} \times \text{Quantity of good 1} + \text{Price of good 2} \times \text{Quantity of good 2} = \text{Total income} \] Mathematically: \[ P_1 \times Q_1 + P_2 \times Q_2 = I \] - **Rearranged form (to express one good in terms of the other):** \[ Q_2 = \frac{I - P_1 Q_1}{P_2} \] **Given Data:** | Variable | Description | Value | |------------|--------------|---------| | \( I \) | Total weekly income | \$10 | | \( P \) | Price of popcorn | \$1.00 per serving | | \( J \) | Price of Jelly | \$0.50 per serving | --- **Explanation:** The primary formula relates the total expenditure on both goods to the total income. It enables the derivation of the budget line by substituting the known prices and income. The systematic representation of data ensures clarity and accuracy in subsequent calculations. --- **3. A Detailed Step-by-Step Solution:** **Step 1:** Write the budget constraint equation: \[ 1.00 P + 0.50 J = 10 \] **Explanation:** This equation states that the total amount spent on popcorn (\( P \)) and Jelly (\( J \)) cannot exceed Jack’s weekly budget of \$10. --- **Step 2:** Solve for \( J \) in terms of \( P \): \[ 0.50 J = 10 - 1.00 P \] \[ J = \frac{10 - 1.00 P}{0.50} \] \[ J = 20 - 2 P \] **Explanation:** Rearranging the equation allows expressing Jelly quantity as a function of popcorn quantity, facilitating graphing and understanding the trade-offs. --- **Step 3:** Determine the intercepts: - **When \( P = 0 \):** \[ J = 20 - 2 \times 0 = 20 \] Jack can buy **20 units of Jelly** if he spends all \$10 on Jelly only. - **When \( J = 0 \):** \[ 0 = 20 - 2 P \Rightarrow 2 P = 20 \Rightarrow P = 10 \] Jack can buy **10 units of popcorn** if he spends all \$10 on popcorn only. **Explanation:** Intercepts help visualize the maximum possible quantities of each good achievable with the entire budget, serving as the endpoints of the budget line. --- **Step 4:** Graph the budget line: - Plot the point (0, 20) for maximum Jelly. - Plot the point (10, 0) for maximum popcorn. - Draw a straight line connecting these points. **Explanation:** The line represents all combinations of popcorn and Jelly that Jack can afford with his \$10 weekly budget. Any point on or below this line is affordable; points above are not. --- **Conclusion:** The budget line equation for Jack's weekly purchases is: \[ J = 20 - 2 P \] with intercepts at (0, 20) and (10, 0). This line graphically demonstrates all possible combinations of popcorn and Jelly that Jack can purchase given his \$10 budget, considering the prices of each dessert. It provides a visual tool to analyze Jack's consumption choices and trade-offs between the two goods. --- **End of Solution**