A cashew and raisin mixture costs $2 per pound. You buy a 4.5,pound mixture of cashews and raisins. If cashews cost $2.50 per pound and raising cost $1.25 per pound, how many pounds of cashews and how many pounds of raisins make up the mixture?

amount of cashews --- x lbs

amount of raisins ---- 4.5 - x lbs

2.5x + 1.25(4.5-x) = 2(4.5)

To solve this problem, we can set up a system of equations. Let's use x to represent the number of pounds of cashews and y to represent the number of pounds of raisins in the mixture.

1) The cost of the mixture is given as $2 per pound, so the total cost of the 4.5-pound mixture is:
4.5 * $2 = $9

2) The cost of the cashews is given as $2.50 per pound, so the total cost of the cashews in the mixture is:
$2.50x

3) The cost of the raisins is given as $1.25 per pound, so the total cost of the raisins in the mixture is:
$1.25y

4) Since the total cost of the mixture is $9, we can write the equation:
$2.50x + $1.25y = $9

Now, we need to relate the weights of cashews and raisins in the mixture. We know that the mixture is 4.5 pounds, so we can write another equation:
x + y = 4.5

Now we have a system of two equations:
$2.50x + $1.25y = $9 ----(1)
x + y = 4.5 ----(2)

To solve this system, let's multiply equation (2) by 2.50 to make the coefficients of x match:
2.50 * (x + y) = 2.50 * 4.5
2.50x + 2.50y = 11.25

Now we have the system:
$2.50x + $1.25y = $9 ----(1)
2.50x + 2.50y = 11.25 ----(3)

Now, we can subtract equation (1) from equation (3) to eliminate the x-term:
(2.50x + 2.50y) - ($2.50x + $1.25y) = 11.25 - $9
(2.50x - 2.50x) + (2.50y - $1.25y) = $2.25
(0) + (1.25y) = $2.25
1.25y = $2.25

Divide both sides by 1.25 to isolate y:
y = $2.25 / 1.25
y = 1.8

Therefore, there are 1.8 pounds of raisins in the mixture.

Now, let's substitute the value of y back into equation (2) to find x:
x + 1.8 = 4.5
x = 4.5 - 1.8
x = 2.7

Therefore, there are 2.7 pounds of cashews in the mixture.

In conclusion, there are 2.7 pounds of cashews and 1.8 pounds of raisins in the 4.5-pound mixture.