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?

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 \( \boxed{200} \).