Question

Gabe Amodeo, a nuclear physicist needs 60 liters of a 40% acid solution. He currently has a 20% solution and a 50% solution. How many liters of each class does he need to make the needed 60 liters of 40% acid solution?

Answer 1

The correct answer is:

He needs 20 liters of the 20% solution and 40 liters of the 50% solution.

Step-by-step explanation:

Let x represent the amount of the 20% solution and y represent the amount of the 50% solution.

The total amount of acid in the 20% solution would then be 0.2x; the total amount of acid in the 50% solution would be 0.5y.

We know that together, these make 60 liters of a 40% solution; this gives us the equation

0.2x+0.5y = 0.4(60)

Simplifying, we get

0.2x+0.5y = 2.4

We also know that the amounts of the 20% solution and 50% solution together give us 60 liters; this gives us the equation

x+y = 60

We now have a system of equations:

`\left \{ {{0.2x+0.5y=24} \atop {x+y=60}} \right.`

We will use elimination to solve this. First we will make the coefficients of y equal; to do this, we will multiply the top equation by 2:

`\left \{ {{2(0.2x+0.5y=24)} \atop {x+y=60}} \right. </p><p>\\</p><p>\\\left \{ {{0.4x+y=48} \atop {x+y=60}} \right.`

We will now cancel the y variables by subtracting the bottom equation:

`\left \{ {{0.4x+y=48} \atop {-(x+y=60)}} \right. </p><p>\\</p><p>\\-0.6x = -12`

Divide both sides by -0.6:

`(-0.6x)/(-0.6)=(-12)/(-0.6)</p><p>\\</p><p>\\x=20`

Substituting this into the second equation, we have

x+y = 60

20+y = 60

Subtract 20 from each side:

20+y-20 = 60-20

y = 40

Answer 2

First, let x be the amount of 20% acid solution in liters. With this, 60 - x is the amount of 50% solution. The acid balance before and after mixing becomes,
(0.2)(x) + (0.5)(60 - x) = (0.40)(60)
The value of x is 20. Therefore, Gabe needs 20 liters of 20% acid solution and 40 liters of 50% acid solution.