On the bright side, mrs gargantua thinks a slighty altered version of the same medicine might very well clear up her gooey scalp. So now she is mixing a liquid containing 12% reactor fuel with a liquid containing 32% reactor fuel to create a 4 ounce liquid containing 20% reactor fuel. How many ounces of the 12% liquid must she use? round your answer to the nearest one decimal place (tenths)

Answer 1

Let \( x \) be the amount (in ounces) of the 12% reactor fuel solution that Mrs. Gargantua uses. Since the total volume of the final mixture is 4 ounces, the amount of the 32% solution used will be \( 4 - x \) ounces.

We can set up an equation based on the concentration of reactor fuel in the mixtures. We know:

1. The amount of reactor fuel from the 12% solution is \( 0.12x \).
2. The amount of reactor fuel from the 32% solution is \( 0.32(4 - x) \).
3. The total amount of reactor fuel in the final mixture is \( 0.20 \times 4 = 0.8 \).

Now, we can write the equation:

\[ 0.12x + 0.32(4 - x) = 0.8 \]

Expanding the equation:

\[ 0.12x + 1.28 - 0.32x = 0.8 \]

Combining like terms:

\[ -0.2x + 1.28 = 0.8 \]

Subtracting 1.28 from both sides gives:

\[ -0.2x = 0.8 - 1.28 \] \[ -0.2x = -0.48 \]

Dividing by -0.2:

\[ x = \frac{-0.48}{-0.2} = 2.4 \]

Thus, Mrs. Gargantua must use 2.4 ounces of the 12% liquid. Rounding to the nearest one decimal place, the answer is:

\[ \boxed{2.4} \]