A grocery store makes a 20-pound mixture of almonds and cashew nuts. The store charges $4 per pound for almonds and $5.50 per pound for cashews. The total value of the mixture is $92. Write a system to find how many pounds of each type of nut are in the mixture.
4 a + 5.5 c = 92
a + c = 20
4 a + 5.5 c = 92
4 a + 4 c = 80
------------------subtract
0 + 1.5 c = 12
c = 8
etc
To solve this problem, we can set up a system of equations based on the given information. Let's assume that x represents the number of pounds of almonds and y represents the number of pounds of cashew nuts in the mixture.
From the problem, we know that the total weight of the mixture is 20 pounds. Therefore, the first equation is:
x + y = 20
Next, we can determine the value of the mixture. The store charges $4 per pound for almonds and $5.50 per pound for cashews. The total value of the mixture is $92. Therefore, the second equation is:
4x + 5.50y = 92
Now we have a system of two equations:
x + y = 20
4x + 5.50y = 92
To solve this system, we can use substitution, elimination, or any other preferred method.
Let's use substitution. From the first equation, we can express x in terms of y:
x = 20 - y
Substituting this value of x into the second equation, we get:
4(20 - y) + 5.50y = 92
Simplifying and solving for y:
80 - 4y + 5.50y = 92
1.5y = 12
y = 8
Now that we have the value of y (the number of pounds of cashew nuts), we can substitute it back into the first equation to find x:
x + 8 = 20
x = 12
Therefore, there are 12 pounds of almonds and 8 pounds of cashew nuts in the mixture.
Let's assume that the weight of almonds in the mixture is represented by the variable 'a' (measured in pounds), and the weight of cashew nuts in the mixture is represented by the variable 'c' (measured in pounds).
We are given the following information:
1) The total weight of the mixture is 20 pounds: a + c = 20.
2) The cost of almonds per pound is $4, and the cost of cashews per pound is $5.50. The total value of the mixture is $92: 4a + 5.5c = 92.
To find the values of 'a' and 'c', we need to solve this system of equations.
The system can be written as:
a + c = 20,
4a + 5.5c = 92.
There are multiple methods to solve this system, but let's solve it using the substitution method.
We can rearrange the first equation to solve for 'a' in terms of 'c':
a = 20 - c.
Now, substitute the value of 'a' in the second equation:
4(20 - c) + 5.5c = 92.
Expanding and simplifying the equation:
80 - 4c + 5.5c = 92,
1.5c = 12,
c = 8.
Now, substitute the value of 'c' back into the first equation to find 'a':
a + 8 = 20,
a = 12.
Therefore, there are 12 pounds of almonds and 8 pounds of cashew nuts in the mixture.