A shopkeeper packages together two types of hard candy: peppermint and cinnamon. Separately the peppermint candy sell for $1.80/kg and the cinnamon candy sells for $2.40/kg. The mixture sells for $2.16/kg. Determine the amount of each type of candy that is required to make 50kg of the mixture. Verify your answer.
-Determine a linear equation to represent the total mass of the new mixture.
-Determine a linear equation to represent the cost of the new mixture.
-Apply a process to verify the point of intersection
we have p kg of peppermint and c kg of cinnamon
p+c=50, so c = 50-p
the price of all the peppermint + the price of the cinnamon is the price of all the mixed bags.
1.80p + 2.4(50-p) = 2.16(50)
1.80p + 120 - 2.4p = 108
.60p = 12
p = 20
so,
c = 30
To determine the amount of each type of candy required to make 50kg of the mixture, we can set up a system of linear equations based on the given information.
Let's represent the amount of peppermint candy in kg as 'x' and the amount of cinnamon candy in kg as 'y'.
1. The total mass of the new mixture can be represented by the equation:
x + y = 50 --------(Equation 1)
2. The cost of the new mixture can be represented by the equation:
(1.80x + 2.40y) = 2.16(50) --------(Equation 2)
Now, let's solve this system of equations to find the values of 'x' and 'y'.
To solve the system, we'll use the method of substitution. First, let's solve Equation 1 for 'x':
x = 50 - y
Now we substitute this value of 'x' in Equation 2:
1.80(50 - y) + 2.40y = 2.16(50)
Now simplify:
90 - 1.80y + 2.40y = 108
Combine like terms:
0.60y = 18
Divide both sides by 0.60 to solve for 'y':
y = 18/0.60
y = 30
Now substitute the value of 'y' back into Equation 1 to find 'x':
x + 30 = 50
x = 50 - 30
x = 20
So, 20kg of peppermint candy and 30kg of cinnamon candy are required to make 50kg of the mixture.
To verify this answer, let's check if the cost of the mixture matches. Using Equation 2:
(1.80 * 20) + (2.40 * 30) = 2.16 * 50
36 + 72 = 108
The cost of the mixture is indeed 108, verifying our answer.
The point of intersection of the linear equations represents the solution to the system, which in this case is the amounts of peppermint and cinnamon candy required to make 50kg of the mixture.