mary has a solution that is 60% alcohol and another that is 20% alcohol. how much of each should she use to make 100 milliliters of a solution that is 52% alcohol?
please help, I don't understand how to do this problem!
Let's say Mary used X ml of the 60% solution and Y ml of the 20% solution. You have two requirements:
X + Y = 100 (the total amount needed), and
0.6X + 0.2Y = 52 (the amount of a alcohol in the final mixture)
Solve those two equations in two unknowns. One way to do that would be to substitute 100 - X for Y in the second equation.
0.6X + 20 -0.2X = 52
0.4X = 32
etc
To solve this problem, you can set up a system of equations representing the amounts of each solution needed. Let's say Mary needs to use x milliliters of the 60% alcohol solution and y milliliters of the 20% alcohol solution.
Since she wants to end up with a 100 milliliters solution, we have the equation: x + y = 100.
We also know that she wants the resulting solution to be 52% alcohol, so we can calculate the amount of alcohol in each solution.
For the 60% alcohol solution, the amount of alcohol can be calculated as 0.6x (since 60% is equal to 0.6 in decimal form).
Similarly, for the 20% alcohol solution, the amount of alcohol can be calculated as 0.2y (since 20% is equal to 0.2 in decimal form).
The total amount of alcohol in the resulting 100 milliliters solution can be calculated as 0.52 * 100 = 52.
Putting it all together, we have the equations:
x + y = 100
0.6x + 0.2y = 52
Now we can solve this system of equations to find the values of x and y.
One way to solve this system of equations is by substitution:
1. Solve the first equation for x:
x = 100 - y
2. Substitute x in the second equation with the expression found in step 1:
0.6(100 - y) + 0.2y = 52
3. Simplify and solve for y:
60 - 0.6y + 0.2y = 52
0.6y - 0.2y = 60 - 52
0.4y = 8
y = 8 / 0.4
y = 20
4. Substitute the value of y back into the first equation to find x:
x + 20 = 100
x = 100 - 20
x = 80
Therefore, Mary should use 80 milliliters of the 60% alcohol solution and 20 milliliters of the 20% alcohol solution to make 100 milliliters of a solution that is 52% alcohol.
To solve this problem, you can use the concept of "mixture" or "alligation." Here's how you can solve it step by step:
Step 1: Start by setting up a proportion based on the percentages of alcohol in the solutions.
Let's say Mary uses x milliliters of the 60% alcohol solution and (100 - x) milliliters of the 20% alcohol solution.
The proportion can be set up as follows:
(60/100) * x + (20/100) * (100 - x) = (52/100) * 100
Step 2: Simplify the equation by converting the percentages to decimals:
0.6x + 0.2(100 - x) = 0.52 * 100
Step 3: Solve the equation for x:
0.6x + 20 - 0.2x = 52
Step 4: Combine like terms:
0.4x + 20 = 52
Step 5: Subtract 20 from both sides of the equation:
0.4x = 32
Step 6: Divide both sides by 0.4:
x = 80
Step 7: Calculate the amount of each solution to use:
Mary should use 80 milliliters of the 60% alcohol solution and (100 - 80) = 20 milliliters of the 20% alcohol solution to make 100 milliliters of a solution that is 52% alcohol.
So, to make the desired solution, Mary needs to use 80 milliliters of the 60% alcohol solution and 20 milliliters of the 20% alcohol solution.