A grocer mixes a premium blend of coffee worth $17/kg with a coffee blend worth #7/kg. A mixture of 36 kg worth $11/kg is produced. How many kilograms of each type of coffee are included in the mixture?
amount of premium coffee --- x
amount of cheaper coffee --- 36-x
17x + 7(36-x) = 11(36)
solve for x
To solve this problem, let's denote the number of kilograms of the premium blend coffee as x and the number of kilograms of the regular blend coffee as y.
1. First, we can set up an equation based on the information given in the problem. The total weight of the mixture is 36 kg, so we have:
x + y = 36 (Equation 1)
2. The value of the mixture is $11/kg. To find the value of the mixture, we multiply the weight of each type of coffee by their respective prices and sum them up. The value of the premium blend coffee is $17/kg, so the value of the premium blend in the mixture is 17x. Similarly, the value of the regular blend coffee is $7/kg, so the value of the regular blend in the mixture is 7y. Therefore, we have:
(17x + 7y) / (x + y) = 11 (Equation 2)
3. Now, we have a system of two equations (Equation 1 and Equation 2) in two unknowns, x and y. We can use substitution or elimination to solve the system.
Let's solve using substitution:
- Solve Equation 1 for x: x = 36 - y
- Substitute x in Equation 2: (17(36 - y) + 7y) / (36 - y + y) = 11
- Simplify the expression: (612 - 17y + 7y) / 36 = 11
- Combine like terms: (612 - 10y) / 36 = 11
- Multiply both sides by 36 to eliminate the denominator: 612 - 10y = 396
- Subtract 612 from both sides: -10y = -216
- Divide both sides by -10: y = 21.6
Now, we have the number of kilograms of regular blend coffee, which is approximately 21.6 kg.
4. Substitute y = 21.6 back into Equation 1:
x + 21.6 = 36
x = 36 - 21.6
x = 14.4
So, the number of kilograms of premium blend coffee is approximately 14.4 kg.
Therefore, the grocer mixed approximately 14.4 kg of the premium blend coffee and 21.6 kg of the regular blend coffee to produce a mixture of 36 kg worth $11/kg.