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Let's work through this problem step by step:

### Part a: Maximize Utility Given Income and Prices
- You are given $9 of income.
- The price of good X is $2, and the price of good Y is $1.

Your goal is to choose the quantities of goods X and Y to maximize your utility. To do this, you will compare the **marginal utility per dollar** for both goods and select the combination that gives you the highest total utility.

- **Marginal Utility per Dollar (MU/$):**
  - For X: $ \frac{MU_X}{P_X} $ = $ \frac{MU_X}{2} $
  - For Y: $ \frac{MU_Y}{P_Y} $ = $ \frac{MU_Y}{1} = MU_Y $

#### Step 1: Calculate Marginal Utility per Dollar for each unit of X and Y.
| Units | MUx | MUx per $ | MUy | MUy per $ |
|-------|-----|-----------|-----|-----------|
| 1     | 10  | 5         | 8   | 8         |
| 2     | 8   | 4         | 7   | 7         |
| 3     | 6   | 3         | 6   | 6         |
| 4     | 4   | 2         | 5   | 5         |
| 5     | 3   | 1.5       | 4   | 4         |
| 6     | 2   | 1         | 3   | 3         |

#### Step 2: Maximize Utility
- You want to buy the goods with the highest marginal utility per dollar first, while not exceeding your $9 budget.

1. Start with Y (1 unit) since $ MU_Y = 8 $ per $1.
   - Remaining budget: $8.
   
2. Next, Y (2 units), $ MU_Y = 7 $.
   - Remaining budget: $7.
   
3. Next, X (1 unit), $ \frac{MU_X}{2} = 5 $.
   - Remaining budget: $5.
   
4. Next, Y (3 units), $ MU_Y = 6 $.
   - Remaining budget: $4.
   
5. Next, X (2 units), $ \frac{MU_X}{2} = 4 $.
   - Remaining budget: $2.
   
6. Finally, Y (4 units), $ MU_Y = 5 $.
   - Remaining budget: $1.

Now, we have exhausted $9, and the optimal choice is:
- **2 units of X and 5 units of Y.**

### Part b: Total Utility Realized
To calculate the total utility, sum the marginal utilities of the chosen quantities:
- X (2 units): MU = 10 + 8 = 18
- Y (5 units): MU = 8 + 7 + 6 + 5 + 4 = 30

Thus, the total utility realized is:
- **Total Utility = 18 (from X) + 30 (from Y) = 48 utils.**

Would you like more details or have any other questions?

Here are 5 related questions to expand on this:
1. What would happen if your income increased to $12?
2. How would the allocation change if the price of Y doubled to $2?
3. What is the marginal utility per dollar for the third unit of X?
4. How does diminishing marginal utility affect your purchasing decisions?
5. Could you reallocate your budget and get the same utility with fewer units?

**Tip:** Always consider the marginal utility per dollar when allocating resources to maximize utility efficiently.

You are choosing between two goods, X and Y, and your marginal utility from each is shown in the following table. If your income is $9 and the prices of X and Y are $2 and $1, respectively, what quantities of each will you purchase to maximize utility?

Explore how to maximize utility with a budget of $9 using the goods X and Y with different prices and marginal utilities. This example teaches key economic principles including the utility maximization rule and the budget constraint.

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