A distributor wants to make 1000 pounds of trail mix that is 8% raisins. He will use a mix that is 5% raisins and a mix that is 20% raisins. How many pounds of each should he use?
The Correct Answer and Explanation is:
To solve the problem, we can use a system of equations. Let’s define the variables:
- Let x = pounds of the 5% raisin mix
- Let y = pounds of the 20% raisin mix
We know two things:
- The total weight must be 1000 pounds: x+y=1000(Equation 1)x + y = 1000 \quad \text{(Equation 1)}x+y=1000(Equation 1)
- The final mix should be 8% raisins. The 5% mix contributes 0.05x pounds of raisins, and the 20% mix contributes 0.20y pounds. The total amount of raisins should be 8% of 1000 pounds, or 80 pounds: 0.05x+0.20y=80(Equation 2)0.05x + 0.20y = 80 \quad \text{(Equation 2)}0.05x+0.20y=80(Equation 2)
Solving the system:
From Equation 1:y=1000−xy = 1000 – xy=1000−x
Substitute into Equation 2:0.05x+0.20(1000−x)=800.05x + 0.20(1000 – x) = 800.05x+0.20(1000−x)=800.05x+200−0.20x=800.05x + 200 – 0.20x = 800.05x+200−0.20x=80−0.15x+200=80-0.15x + 200 = 80−0.15x+200=80−0.15x=80−200=−120-0.15x = 80 – 200 = -120−0.15x=80−200=−120x=−120−0.15=800x = \frac{-120}{-0.15} = 800x=−0.15−120=800
Now find y:y=1000−x=1000−800=200y = 1000 – x = 1000 – 800 = 200y=1000−x=1000−800=200
Final Answer:
- 800 pounds of the 5% raisin mix
- 200 pounds of the 20% raisin mix
Explanation
This type of problem is a classic mixture problem that can be solved using algebra. The distributor needs 1000 pounds of a trail mix that contains 8% raisins. He has two types of mixes available: one with 5% raisins and another with 20% raisins. The goal is to determine how much of each mix he should combine to get exactly 1000 pounds of an 8% raisin mix.
We start by assigning variables: let x be the amount of the 5% mix, and y be the amount of the 20% mix. Since the total amount must be 1000 pounds, we write the first equation:
x + y = 1000.
Next, we consider the percentage of raisins in each mix. The 5% mix contributes 0.05x pounds of raisins, and the 20% mix contributes 0.20y pounds. Since we want the final 1000-pound mix to be 8% raisins, we want a total of 80 pounds of raisins (because 8% of 1000 is 80). This gives us the second equation:
0.05x + 0.20y = 80.
By substituting one equation into the other, we eliminate a variable and solve for x. After finding x, we substitute back to find y. The solution gives us 800 pounds of the 5% mix and 200 pounds of the 20% mix. When combined, these produce a 1000-pound mix with exactly 8% raisins, satisfying both conditions of the problem.
