Two kinds of nuts,

one cost $5.00 per kilogram
another cost $5.40 per kilogram.
How many kilograms of each type of nut should be mixed in order to have 40 kg of a mixture worth $5.25 per kilogram?

To solve this problem, let's denote the number of kilograms of the first kind of nut as x, and the number of kilograms of the second kind of nut as y.

We are given the following information:
- The cost of the first kind of nut is $5.00 per kilogram.
- The cost of the second kind of nut is $5.40 per kilogram.
- The target weight of the mixture is 40 kg.
- The desired average cost of the mixture is $5.25 per kilogram.

To find the amounts of each type of nut needed, we can set up a system of equations based on the given information.

Equation 1: The sum of the weights of both types of nuts is equal to the target weight of the mixture.
x + y = 40

Equation 2: The cost of the mixture is calculated by taking the weighted average of the costs of the two types of nuts.
(5.00x + 5.40y) / (x + y) = 5.25

Now we can use these equations to solve for x and y.

First, let's solve Equation 1 for one variable. We can choose x.
x = 40 - y

Substituting this value of x into Equation 2, we can solve for y.
(5.00(40 - y) + 5.40y) / (40 - y + y) = 5.25
(200 - 5y + 5.40y) / 40 = 5.25
(200 + 0.4y) / 40 = 5.25
200 + 0.4y = 210
0.4y = 10
y = 10 / 0.4
y = 25

Now, substitute the value of y back into Equation 1 to solve for x.
x + 25 = 40
x = 40 - 25
x = 15

Therefore, you would need 15 kilograms of the first kind of nut and 25 kilograms of the second kind of nut in order to have a mixture of 40 kg worth $5.25 per kilogram.