Given a budget of [tex]$20, let's determine how to allocate this budget among the three goods A, B, and C in order to maximize the total utility.
### Step-by-Step Solution
1. Determine Quantity of Good A:
- The price of good A is $[/tex]10.
- Let’s decide to buy 2 units of good A.
- Cost of 2 units of good A: [tex]\( 2 \times 10 = \$20 \)[/tex].
- The total expenditure on good A is [tex]$20, consuming the entire budget.
- This leaves us with a remaining budget of \( \$[/tex]20 - \[tex]$20 = \$[/tex]0 \), which means no money is left to spend on goods B or C.
2. Evaluate Good B:
- The price of good B is [tex]$2.
- Since there is no remaining budget after buying good A, we can't purchase any units of good B.
- Therefore, the quantity of good B is 0.
3. Evaluate Good C:
- The price of good C is $[/tex]6.
- Again, with no remaining budget after the purchase of good A, we can't purchase any units of good C.
- Therefore, the quantity of good C is 0.
### Summary
- We should buy 2 units of good A.
- We cannot buy any units of good B or good C due to budget constraints.
Thus, the optimal purchase given the budget of $20 is:
- 2 of good A,
- 0 of good B,
- 0 of good C.
Therefore, the quantities of goods purchased are:
- [tex]\(\boxed{2}\)[/tex] of good A,
- [tex]\(\boxed{0}\)[/tex] of good B,
- [tex]\(\boxed{0}\)[/tex] of good C.
