. A coffee merchant has two types of coffee beans, one selling for $3 per pound and the other for $6 per pound. The beans are to be mixed to provide 100 lb of a mixture selling for $4.98 per pound. How much of each type of coffee bean should be used to form 100 lb of the mixture?

Question

. A coffee merchant has two types of coffee beans, one selling for $3 per pound and the other for $6 per pound. The beans are to be mixed to provide 100 lb of a mixture selling for $4.98 per pound. How much of each type of coffee bean should be used to form 100 lb of the mixture?

Answer 1

To solve this problem, we can use a system of equations. Let's assume that x represents the amount (in pounds) of the $3 coffee beans, and y represents the amount (in pounds) of the $6 coffee beans.

Given that the total amount of the mixture is 100 pounds, we have the equation x + y = 100.

Since the desired mixture sells for $4.98 per pound, we can also set up an equation for the total cost of the mixture. The cost of the $3 coffee beans is 3x, and the cost of the $6 coffee beans is 6y. Thus, the total cost of the mixture is 3x + 6y.

Since the mixture sells for $4.98 per pound, we can set up another equation for the average cost, which is the total cost divided by the total weight of the mixture. This gives us the equation (3x + 6y) / 100 = $4.98.

Now, we have two equations:
x + y = 100 (Equation 1)
(3x + 6y) / 100 = $4.98 (Equation 2)

To simplify Equation 2, we can multiply both sides of the equation by 100 to remove the denominator:
3x + 6y = 498 (Equation 3)

Now, we have a system of equations:
x + y = 100 (Equation 1)
3x + 6y = 498 (Equation 3)

We can solve this system of equations using substitution or elimination. Let's use the elimination method to solve for x and y:

Multiply Equation 1 by 6 to make the coefficients of y cancel out:
6x + 6y = 600 (Equation 4)

Subtract Equation 3 from Equation 4:
(6x + 6y) - (3x + 6y) = 600 - 498
3x = 102

Divide both sides of the equation by 3:
x = 34

Now, substitute the value of x into Equation 1 to solve for y:
34 + y = 100

Subtract 34 from both sides of the equation:
y = 66

Therefore, you should use 34 pounds of the $3 coffee beans and 66 pounds of the $6 coffee beans to form 100 pounds of the mixture.

Answer 2

Let the amount of $3 beans used be x pounds, let the amount of $6 beans used be 100-x pounds

then
3x + 6(100-x) = 4.98(100)

solve for x (I got x=34)