What quantity (in mL) of a 45% acid solution of a mono-protic strong acid must be mixed with a 20% solution of the same acid to produce 800 mL of a 29.875% acid solution?

We have two solutions of the same mono-protic strong acid:

• One solution is $$45\%$$ acid.
• The other solution is $$20\%$$ acid.

Let us assume that we take $$x$$ mL of the $$45\%$$ solution. Because the final mixture must have a total volume of $$800$$ mL, the remaining volume, coming from the $$20\%$$ solution, will be $$800 - x$$ mL.

The mass (or volume, since densities cancel for percentage calculations) of pure acid present in each portion is obtained by multiplying the volume by the percentage (expressed as a decimal). Hence:

• Pure acid from the $$45\%$$ portion = $$0.45 \times x$$ mL.
• Pure acid from the $$20\%$$ portion = $$0.20 \times (800 - x)$$ mL.

The required final mixture is $$800$$ mL of a $$29.875\%$$ acid solution. Therefore, the total pure acid in the final mixture must be

$$0.29875 \times 800$$ mL.

Now we equate the total pure acid contributed by both initial solutions to the pure acid required in the final mixture:

$$0.45x + 0.20(800 - x) = 0.29875 \times 800.$$

We next simplify the left-hand side by distributing the $$0.20$$:

$$0.45x + 0.20 \times 800 - 0.20x = 0.29875 \times 800.$$

Since $$0.20 \times 800 = 160$$, this becomes

$$0.45x + 160 - 0.20x = 0.29875 \times 800.$$

Combine the like terms $$0.45x - 0.20x$$ to obtain $$0.25x$$:

$$0.25x + 160 = 0.29875 \times 800.$$

Now compute the right-hand side. We note that

$$0.29875 \times 100 = 29.875,$$

so multiplying by $$800 = 8 \times 100$$ gives

$$0.29875 \times 800 = 29.875 \times 8 = 239.$$

Substituting this value, the equation becomes

$$0.25x + 160 = 239.$$

Subtract $$160$$ from both sides:

$$0.25x = 239 - 160 = 79.$$

Finally, divide by $$0.25$$ to solve for $$x$$:

$$x = \frac{79}{0.25} = 316.$$

Thus, we must mix $$316$$ mL of the $$45\%$$ acid solution with $$800 - 316 = 484$$ mL of the $$20\%$$ solution to obtain $$800$$ mL of a $$29.875\%$$ acid solution.

Hence, the correct answer is Option A.