The vendor of a coffee cart mixes coffee beans that cost $8 per pound with coffee beans that cost $4 per pound. How many pounds of each sould be used to make a 60-pound blend that sells for $5.25 per pound?
amount of $8 coffee --- x
amount of $ 4 coffee -- 60-x
8x + 4(60-x) = 5.25(60)
8x + 240 - 4x = 315
4x = 75
x = 18.75 lbs
To solve this problem, we can use a system of equations to represent the given information.
Let's assume that x pounds of coffee beans that cost $8 per pound are used, and y pounds of coffee beans that cost $4 per pound are used.
Since we are trying to find the number of pounds for each type of coffee bean, we can set up the following system of equations:
Equation 1: x + y = 60 (since the total weight of the blend is 60 pounds)
Equation 2: 8x + 4y = 5.25(60) (since the cost of the blend is $5.25 per pound)
To solve this system of equations, we can use the method of substitution or elimination.
Method 1: Substitution
From Equation 1, we can solve for x in terms of y: x = 60 - y.
Substituting x in Equation 2, we get: 8(60 - y) + 4y = 5.25(60).
Expanding the equation gives us: 480 - 8y + 4y = 315.
Combining like terms, we have: -4y = -165.
Dividing both sides of the equation by -4, we get: y = 41.25.
Substituting this value of y back into Equation 1, we get: x + 41.25 = 60.
Simplifying the equation gives us: x = 18.75.
So, 18.75 pounds of coffee beans that cost $8 per pound should be used, and 41.25 pounds of coffee beans that cost $4 per pound should be used to make the 60-pound blend.
Method 2: Elimination
We can multiply Equation 1 by 4 to eliminate y: 4(x + y) = 4(60), giving us 4x + 4y = 240.
Subtracting this equation from Equation 2, we get: (8x + 4y) - (4x + 4y) = 5.25(60) - 240.
Simplifying the equation gives us: 4x = 75.
Dividing both sides of the equation by 4, we get: x = 18.75.
Substituting this value of x back into Equation 1, we get: 18.75 + y = 60.
Simplifying the equation gives us: y = 41.25.
So, 18.75 pounds of coffee beans that cost $8 per pound should be used, and 41.25 pounds of coffee beans that cost $4 per pound should be used to make the 60-pound blend.