Ned, the owner of Ned's Nut Shop, sells peanuts for $\mathrm{\$}10$ per pound and cashews for $\mathrm{\$}9$ per pound. Ned wants to create a 20 pound barrel of mixed nuts and sell it for $\mathrm{\$}9.70$ per pound. How many pounds of peanuts and cashews should Ned use?
Ned should mix pounds of peanuts and pounds of cashews to make 20 pounds of a blend that sells for $\mathrm{\$}9.70$ per pound.

Step 1 ：Ned, the owner of Ned's Nut Shop, sells peanuts for $10perpoundandcashewsfor$9 per pound. He wants to create a 20 pound barrel of mixed nuts and sell it for $9.70 per pound. We need to find out how many pounds of peanuts and cashews Ned should use.

Step 2 ：We can set up two equations based on the information given in the problem. The first equation will be based on the total weight of the nuts, and the second equation will be based on the total cost of the nuts.

Step 3 ：Let's denote the weight of peanuts as p and the weight of cashews as c. We know that $p+c=20$ (since the total weight of the mixed nuts is 20 pounds).

Step 4 ：The second equation will be based on the total cost. We know that the cost of peanuts per pound is $10andthecostofcashewsperpoundis$9. The total cost of the mixed nuts per pound is $9.70. So, we have $10p+9c=9.70\ast 20$.

Step 5 ：We can solve this system of equations to find the values of p and c.

Step 6 ：The solution to the system of equations is $c=6$ and $p=14$.

Step 7 ：Final Answer: Ned should mix $\boxed{{\displaystyle 14}}$ pounds of peanuts and $\boxed{{\displaystyle 6}}$ pounds of cashews to make 20 pounds of a blend that sells for $9.70 per pound.