A nut store normally sells cashews for $4.00 a pound and peanuts for $1.50 per pound. In order to sell 50 pounds of peanuts, the manager decides to mix 50 pounds of peanuts with some cashews and sell the mixture for $3.50 per pound. How many pounds of cashews should be mixed with the peanuts to ensure no change in profit?

Answer 1

Let \( x \) be the number of pounds of cashews mixed with 50 pounds of peanuts.

1. Calculate the total weight of the mixture: \[ \text{Total weight} = 50 + x \text{ pounds} \]

2. Calculate the selling price of the mixture: The manager plans to sell the mixture at $3.50 per pound. Therefore, the total revenue from selling the mixture is: \[ \text{Revenue} = 3.50 \times (50 + x) \]

3. Calculate the cost of the peanuts and cashews:

   • The cost for 50 pounds of peanuts at $1.50 per pound: \[ \text{Cost of peanuts} = 50 \times 1.50 = 75 \]
   • The cost for \( x \) pounds of cashews at $4.00 per pound: \[ \text{Cost of cashews} = 4.00 \times x \]

4. Total cost of the mixture: \[ \text{Total cost} = 75 + 4.00x \]

5. To ensure there is no change in profit: We set the revenue equal to the total cost: \[ 3.50(50 + x) = 75 + 4.00x \]

6. Distributing the left side: \[ 175 + 3.50x = 75 + 4.00x \]

7. Rearranging the equation to find \( x \): Subtract \( 3.50x \) from both sides: \[ 175 = 75 + 0.50x \] Subtract 75 from both sides: \[ 100 = 0.50x \] Divide by 0.50: \[ x = 200 \]

Thus, the number of pounds of cashews that should be mixed with the peanuts to ensure no change in profit is \( \boxed{200} \).