A nut store normally sells cashews for​ $4.00 per pound and peanuts for​ $1.50 per pound.  But at the end of the month the peanuts had not sold​ well, so, in order to sell 

4040 

pounds of​ peanuts, the manager decided to mix the 

4040 

pounds of peanuts with some cashews and sell the mixture for 

$ 2.00$2.00 

per pound.  How many pounds of cashews should be mixed with the peanuts to ensure no change in the​ revenue?

To solve this problem, we need to determine how many pounds of cashews should be mixed with 40 pounds of peanuts so that the total revenue from selling the mixture at $2.00 per pound equals the revenue that would have been earned by selling the peanuts and cashews separately.

### Step 1: Define Variables
- Let $ x $ be the number of pounds of cashews to be mixed with the 40 pounds of peanuts.

### Step 2: Set Up Revenue Equations
1. **Revenue from selling peanuts and cashews separately:**
   - The revenue from 40 pounds of peanuts sold at $1.50 per pound:
     \[
     \text{Revenue from peanuts} = 40 \times 1.50 = 60 \text{ dollars}
     \]
   - The revenue from $ x $ pounds of cashews sold at $4.00 per pound:
     \[
     \text{Revenue from cashews} = 4.00x
     \]
   - Total revenue from selling them separately:
     \[
     \text{Total Revenue} = 60 + 4.00x
     \]

2. **Revenue from selling the mixture:**
   - The total weight of the mixture is $ 40 + x $ pounds.
   - The mixture is sold at $2.00 per pound:
     \[
     \text{Revenue from mixture} = 2.00 \times (40 + x) = 80 + 2.00x
     \]

### Step 3: Set Up the Equation for No Change in Revenue
The revenue from selling the mixture should equal the revenue from selling the peanuts and cashews separately:
\[
60 + 4.00x = 80 + 2.00x
\]

### Step 4: Solve the Equation
1. **Isolate $ x $:**
   \[
   4.00x - 2.00x = 80 - 60
   \]
   \[
   2.00x = 20
   \]
   \[
   x = 10
   \]

### Conclusion:
The manager should mix 10 pounds of cashews with the 40 pounds of peanuts to ensure no change in the revenue.

Would you like more details, or do you have any questions?

Here are five related questions:
1. What would be the selling price per pound if the manager wanted to maintain the same revenue with 20 pounds of cashews?
2. How would the required amount of cashews change if the selling price of the mixture was $2.50 per pound?
3. What would the mixture price need to be if the manager only had 5 pounds of cashews to mix?
4. How would the solution change if the cashews were priced at $5.00 per pound?
5. What happens to the revenue if the peanuts are sold at $2.00 per pound without mixing?

**Tip:** In problems involving mixtures, always ensure that the total revenue before and after mixing remains consistent to maintain profitability.

Explore how to maximize revenue by mixing cashews and peanuts. Solve the problem of determining the amount of cashews needed to mix with peanuts to maintain revenue. Detailed step-by-step solution provided. Suitable for students and enthusiasts interested in algebra and profit maximization.

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