The problem asks to determine how many purchasing plans the store can adopt, given the constraints on total investment and profit.

Let’s break down the information:

• There are two commodities, A and B.
• Commodity A has a unit price of 120 dollars, and commodity B has a unit price of 200 dollars.
• The store can spend no more than 8400 dollars in total on 50 pieces of the commodities.
• The profit per unit for commodity A is 20 dollars, and for commodity B is 40 dollars.
• The total profit needs to be greater than 1500 dollars.

Constraints:

1. Total units purchased: $A + B = 50$ (where $A$ is the number of commodity A purchased and $B$ is the number of commodity B purchased).

2. Total cost constraint: $120A + 200B \leq 8400$.

3. Profit constraint: $20A + 40B \geq 1500$.

We already have:

• $B = 50 - A$.

Step 1: Substitute $B$ in the equations

Substitute $B = 50 - A$ into both the total cost and profit inequalities:

1. Cost: $120A + 200(50 - A) \leq 8400$ Simplifying: $120A + 10000 - 200A \leq 8400$ $-80A \leq -1600$ $A \geq 20$

2. Profit: $20A + 40(50 - A) \geq 1500$ Simplifying: $20A + 2000 - 40A \geq 1500$ $-20A + 2000 \geq 1500$ $-20A \geq -500$ $A \leq 25$

Step 2: Determine the range for $A$

We know that:

• $A \geq 20$
• $A \leq 25$

Thus, $A$ can take values from 20 to 25, which means $A$ can be 20, 21, 22, 23, 24, or 25.

Step 3: Count the number of valid purchasing plans

For each value of $A$ (20 through 25), there corresponds a value of $B = 50 - A$, so there are 6 possible purchasing plans.

Final Answer:

The correct answer is C. 6 purchasing plans.

Would you like to go over any specific part of the solution in more detail?

Here are a few follow-up questions:

1. How do you determine the possible values for a variable in inequality constraints?
2. How would you handle a scenario where multiple variables have multiple constraints?
3. What would change if the total investment limit were raised or lowered?
4. How can the graph of these inequalities help visualize the feasible solutions?
5. How do we interpret inequalities with "greater than or equal to" conditions?

Tip: When solving linear inequalities with multiple constraints, always substitute for one variable to simplify the system of inequalities.