A nurse requires a 40% alcohol solution. The nurse has 20% and 70% alcohol solutions on hand. How many liters of each should the nurse use to make 25 liters of the 40% solution?
amount of the 20% stuff --- x L
amount of the 70% stuff --- 25-x
.2x + .7(25-x) = .4(25)
how about multiplying each term by 10
2x + 7(25-x) = 4(25)
2x + 175 - 7x = 100
-5x = -75
x = 15
take over
Tasha needs 75 liters of a 40% solution of alcohol. She has a 20% solution and a 50% solution available. How many liters of the 20% solution and how many liters of the 50% solution should she mix to make the 40% solution?
Joseph would like to make 12 pounds of a coffee blend for $6.25 per pound. He blends Ground Chicory at $4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use?
To solve this type of mixture problem, you can use a common algebraic approach called the "mixture equation." Let's use x to represent the amount of the 20% alcohol solution the nurse needs and y to represent the amount of the 70% alcohol solution.
Since the nurse wants to make 25 liters of the 40% alcohol solution, we know that x + y = 25 (Eq. 1), representing the total volume of the final solution.
The amount of alcohol in the mixture can also be expressed as a proportion. For the 20% solution, there will be 0.2x liters of alcohol, and for the 70% solution, there will be 0.7y liters of alcohol. So, the equation for the total amount of alcohol in the solution is 0.2x + 0.7y = 0.4 * 25 (Eq. 2), which simplifies to 0.2x + 0.7y = 10.
Now we have a system of linear equations to solve: Eq. 1 and Eq. 2.
To solve this system of equations, we can use a method called substitution:
1. Solve Eq. 1 for x:
x = 25 - y
2. Substitute this value of x into Eq. 2:
0.2(25 - y) + 0.7y = 10
3. Simplify and solve for y:
5 - 0.2y + 0.7y = 10
0.5y = 5
y = 5 / 0.5
y = 10
Now we know that the nurse needs 10 liters of the 70% alcohol solution.
4. Substitute this value of y back into Eq. 1 to find x:
x + 10 = 25
x = 25 - 10
x = 15
Therefore, the nurse needs 15 liters of the 20% alcohol solution and 10 liters of the 70% alcohol solution to make 25 liters of the 40% alcohol solution.