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, I order to sell 60 pounds of peanuts, the manager decided to mix the 60 pounds of peanuts with some cashews and sell the mixture for $2.50 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 can set up an equation using the concept of weighted averages, ensuring that the revenue generated by selling the mixture remains the same as if the peanuts and cashews had been sold separately.

### Step-by-Step Explanation:

Let $ x $ represent the number of pounds of cashews to be mixed with the 60 pounds of peanuts.

1. **Price per pound of peanuts**: $1.50
2. **Price per pound of cashews**: $4.00
3. **Price per pound of the mixture**: $2.50
4. **Amount of peanuts**: 60 pounds
5. **Amount of cashews**: $ x $ pounds

The total revenue from selling the peanuts and cashews separately would be the sum of the revenue from the peanuts and the cashews:
\[
\text{Revenue from peanuts} = 60 \times 1.50 = 90 \text{ dollars}
\]
\[
\text{Revenue from cashews} = x \times 4.00 = 4x \text{ dollars}
\]
Thus, the total revenue from selling them separately is:
\[
90 + 4x \text{ dollars}
\]

Now, for the mixture, the total weight of the mixture is $ 60 + x $ pounds, and it is sold at $2.50 per pound. Therefore, the revenue from selling the mixture is:
\[
(60 + x) \times 2.50 = 150 + 2.5x \text{ dollars}
\]

Since the revenue must remain the same, we can set the two revenue expressions equal to each other:
\[
90 + 4x = 150 + 2.5x
\]

### Solving the Equation:

1. Subtract $ 2.5x $ from both sides:
   \[
   90 + 4x - 2.5x = 150
   \]
   \[
   90 + 1.5x = 150
   \]

2. Subtract 90 from both sides:
   \[
   1.5x = 60
   \]

3. Divide both sides by 1.5:
   \[
   x = \frac{60}{1.5} = 40
   \]

### Conclusion:
The manager should mix **40 pounds of cashews** with the 60 pounds of peanuts to ensure no change in the revenue.

Would you like further details or explanations?

Here are five related questions you might consider:
1. How would the result change if the price per pound of the mixture were $3.00 instead of $2.50?
2. What if the peanuts were sold at $2.00 per pound and cashews at $5.00 per pound?
3. How much revenue would be lost if the manager sold the mixture for $2.00 per pound?
4. What is the percentage of cashews in the final mixture?
5. How many pounds of the mixture would be sold if the price were reduced by 10%?

**Tip:** In mixture problems, weighted averages help balance the costs across different components, maintaining consistency in revenue or quality.

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 60 pounds of peanuts, the manager decided to mix the 60 pounds of peanuts with some cashews and sell the mixture for $2.50 per pound. How many pounds of cashews should be mixed with the peanuts to ensure no change in the revenue?

Learn how to calculate the amount of cashews needed to mix with 60 pounds of peanuts to ensure no change in revenue when sold at $2.50 per pound. This problem involves weighted averages and solving linear equations, suitable for Grades 9-10 students.

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