A nut wholesaler sells two types of mixes of cashews and peanuts. He makes a low-grade mix containing 8 pounds of peanuts and 4 pounds of cashews and high-grade mixture containing 6 pounds of peanuts and 6 pounds of cashews. Let x and y denote the numbers of low-grade and high-grade packages that the wholesaler can make from 100 pounds of peanuts and 80 pounds of cashews. Represent the possible combinations of packs of the two mixes that can be made.

Question

A nut wholesaler sells two types of mixes of cashews and peanuts. He makes a low-grade mix containing 8 pounds of peanuts and 4 pounds of cashews and high-grade mixture containing 6 pounds of peanuts and 6 pounds of cashews. Let x and y denote the numbers of low-grade and high-grade packages that the wholesaler can make from 100 pounds of peanuts and 80 pounds of cashews. Represent the possible combinations of packs of the two mixes that can be made.

Answer 1

Okay, for x pkgs of low-grade nuts, he needs 8x lbs of peanuts and 4x lbs of cashews.

For y pkgs of high-grade nuts, he needs 6x lbs of peanuts and 6x lbs of cashews.
That means that we need
8x+6y <= 100
4x+6y <= 80
Assume we use all the available nuts. That changes our inequalities to equations, with solution
x=5 and y=10
Now you can just list the combinations where x <=5 and y >= 10
or x >= 10 and y <= 5
where the totals meet the requirements. For example,
x=1 means
8+6y <= 100 so y <= 16
4+6y <= 80 so y <= 12
So, with x=1, y<=12 because there are not enough cashews to use up all the available peanuts.
And so on.

Answer 2

Or you can just plot both lines, and any lattice points (with positive integer coordinates) in the shaded region will be solutions.

https://www.wolframalpha.com/input/?i=solve+8x%2B6y+%3C%3D+100,+4x%2B6y+%3C%3D+80+over+the+positive+integers

Answer 3

thank u oobleck

Answer 4

To solve this problem, let's set up a system of equations based on the given information.

Let x be the number of low-grade packages and y be the number of high-grade packages that the wholesaler can make.

We know that the low-grade mix contains 8 pounds of peanuts and 4 pounds of cashews. Therefore, the total weight of peanuts in the low-grade packages will be 8x and the total weight of cashews will be 4x.

We also know that the high-grade mix contains 6 pounds of peanuts and 6 pounds of cashews. Therefore, the total weight of peanuts in the high-grade packages will be 6y and the total weight of cashews will be 6y.

We are given that there are 100 pounds of peanuts and 80 pounds of cashews available. So, we can set up the following equations:

8x + 6y = 100 (equation 1)
4x + 6y = 80 (equation 2)

Now, we can solve this system of equations to find the values of x and y, which represent the possible combinations of packs of the two mixes that can be made.

To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method.

Multiply equation 1 by 2:
16x + 12y = 200 (equation 3)

Subtract equation 2 from equation 3:
16x + 12y - (4x + 6y) = 200 - 80
12x + 6y = 120

Divide equation 4 by 6:
2x + y = 20 (equation 4)

Now we have a new equation (equation 4) that represents the relationship between x and y.

To find the possible combinations of packs of the two mixes, we need to consider the constraints. Since the number of packages cannot be negative, x and y must be greater than or equal to zero.

With this information, we can create a table of possible combinations:

| x | y |
|-------|-------|
| 0 | 20 |
| 1 | 18 |
| 2 | 16 |
| 3 | 14 |
| 4 | 12 |
| 5 | 10 |
| 6 | 8 |
| 7 | 6 |
| 8 | 4 |
| 9 | 2 |
| 10 | 0 |

These combinations of x and y represent the number of low-grade and high-grade packages that can be made from 100 pounds of peanuts and 80 pounds of cashews.

For example, when x = 0 and y = 20, the wholesaler can make 0 low-grade packages and 20 high-grade packages.