The modern grocery has cashews that sell for $4.00 a pound and peanuts that sell for 2.50 a pound. How much of each must Albert the grocer mixed to get 60 pounds of mixture that he can sell for three dollars per pound? Express the problem as a system of linear equations and solve using the method of your choice
c+p = 60
4.00c + 2.50p = 3.00*60
Now just solve for c and p
well, since c+p=60,
p = 60-c
4.00c + 2.50(60-c) = 3.00*60
That is, add up the values of the separate nuts, and it must equal the value of the final mixture.
4.00c + 150 - 2.5c = 180
1.5c = 30
c = 20
So, 20 lbs cashews and 40 lbs peanuts
Note that since the final cost of $3.00 is 2/3 of the way from $4.00 to $2.50, the amount of peanuts is 2/3 of the total.
How do you solve this
To solve this problem, we can set up a system of linear equations based on the given information.
Let's assume Albert mixes x pounds of cashews and y pounds of peanuts to get a total of 60 pounds of mixture.
1) The first equation represents the total weight of the mixture:
x + y = 60
2) The second equation represents the total cost of the mixture:
($4.00 * x) + ($2.50 * y) = $3.00 * 60
Now we have a system of linear equations:
x + y = 60
4x + 2.5y = 180
There are several methods to solve this system of equations. Let's use the method of substitution in this case:
1) Solve the first equation for x:
x = 60 - y
2) Substitute x in the second equation:
4(60 - y) + 2.5y = 180
Expanding and simplifying:
240 - 4y + 2.5y = 180
-1.5y = -60
y = -60 / -1.5
y = 40
3) Substitute the value of y back into the first equation to find x:
x + 40 = 60
x = 60 - 40
x = 20
Therefore, Albert needs to mix 20 pounds of cashews and 40 pounds of peanuts to get 60 pounds of mixture that he can sell for three dollars per pound.