Final answer:
The coffee distributor requires 20 pounds of $3.00 coffee, 50 pounds of $5.00 coffee, and 30 pounds of $6.00 coffee to create a 100-pound blend that costs $4.90 per pound.
Step-by-step explanation:
To solve the problem of blending coffee to achieve a specific price per pound, we can set up a system of equations based on the given information. We know the distributor wants 100 pounds of blended coffee, which should cost $4.90 per pound. The mixture will include coffees priced at $3.00, $5.00, and $6.00 per pound respectively. Let's define the variables: x for the amount of $3.00 coffee, y for the amount of $5.00 coffee, and z for the amount of $6.00 coffee.
According to the problem, we have the following constraints:
- The total weight is 100 pounds: x + y + z = 100
- The total cost of the blend per pound is $4.90: 3x + 5y + 6z = 490 (since the total cost is 100 pounds times $4.90)
- The distributor has the same amount of $5.00 coffee as the combined amounts of $3.00 and $6.00 coffees: y = x + z
Substituting the third equation into the first equation, we get 2x + 2z = 100. Dividing by 2, we simplify to x + z = 50. Using these equations, we can solve for the amounts of each type of coffee needed:
- x = 50 - z
- y = 50
- z (unknown)
Substitute x and y into the second equation: 3(50 - z) + 5(50) + 6z = 490. This simplifies to 150 - 3z + 250 + 6z = 490. Combining like terms, we get 3z = 90, which means z = 30. Then, from x = 50 - z, we get x = 20. Therefore, the distributor needs 20 pounds of $3.00 coffee, 50 pounds of $5.00 coffee, and 30 pounds of $6.00 coffee.