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.

I totally forget how to do this kind of problem. Please help if you can! Thank you so much! :)

let the amount of the 60% solution be x ml

then the amount of the 20% solution must be (100-x) ml
then isn't .6x + .2(100-x) = .52(100) ?

solve for x

Good

To solve this problem, we can use a simple algebraic equation.

Let's assume Mary should use x milliliters of the 60% alcohol solution and (100 - x) milliliters of the 20% alcohol solution to obtain 100 milliliters of the 52% alcohol solution.

Now let's set up the equation based on the amount of alcohol:

Amount of alcohol in the 60% solution + amount of alcohol in the 20% solution = amount of alcohol in the final 52% solution

0.60x + 0.20(100 - x) = 0.52(100)

Now we can solve the equation to find the value of x:

0.60x + 20 - 0.20x = 52

0.40x = 32

x = 32 / 0.40

x = 80

So, 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.

To solve this kind of problem, we can set up a system of equations using the concept of mixtures. Let's call the amount of the 60% alcohol solution that Mary needs to use as x milliliters, and the amount of the 20% alcohol solution as y milliliters.

We know that the total volume of the resulting solution is 100 milliliters. Therefore, we can write our first equation:

x + y = 100

Next, we need to consider the amount of alcohol in the resulting solution. Since the 60% alcohol solution contains 60% alcohol, we can say that 0.60x is the amount of alcohol from that solution. Similarly, the 20% alcohol solution contains 20% alcohol, so that gives us 0.20y. Adding these two amounts together will give us the total amount of alcohol in the resulting solution. We then divide this total by the total volume of the solution (100 mL) to get the average alcohol concentration of 52%. Using this information, we can write our second equation:

(0.60x + 0.20y) / 100 = 0.52

Now, we have a system of two equations that we can solve simultaneously to determine the values of x and y. There are multiple methods to solve a system of equations, such as substitution or elimination. Let's use the elimination method here:

Multiply both sides of the second equation by 100 to eliminate the denominator:

0.60x + 0.20y = 0.52 * 100
0.60x + 0.20y = 52

Now, we can multiply the first equation by -0.20 and add it to the second equation to eliminate y:

-0.20(x + y) = -0.20 * 100
-0.20x - 0.20y = -20

0.60x - 0.20x = 52 - 20
0.40x = 32

Divide both sides of the equation by 0.40:

x = 32 / 0.40
x = 80

Now, substitute the value of x back into the first equation to solve for y:

80 + y = 100
y = 100 - 80
y = 20

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.