Read the problem. Drawing a figure similar to the one in Example 5.4.30 may help you visualize the situation. Then creating a table, similar to the table in that same example, can help you to organize the information. Anatole must mix some of the $10\%$ solution and some of the $35\%$ solution together to create $250$mL of the $25\%$ solution. Name what we are looking for. The two unknowns are the amount of each type of solution being mixed together. Let $x$ represent the amount of $10\%$ solution (in mL) and let $y$ represent the amount of $35\%$ solution (in mL). One quantity given to us in the problem is 250 mL, which is the amount of the $25\%$ solution that needs to be created. These three amounts, as well as the concentrations for each type of solution, can be entered into the table and then the amount of hydrochloric acid in each type of solution can be calculated by multiplying the number of milliters of solution times the concentration (in decimal form). Translate into a system of equations. Since $250$mL is the total volume of the $25\%$ solution, we must have for one of the equations in the system: $$(x\,\text{mL})+(y\,\text{mL})=250\,\text{mL}$$ Or without units: $$x+y=250$$ To build the second equation, we have to think about the amount (in mL) of hydrochloric acid being mixed. If we have $x$ milliliters of solution that is $10\%$ hydrochloric acid, then $0.10x$ is the actual amount (in mL) of hydrochloric acid in that solution. Similarly, $0.35y$ is the amount of hydrochloric acid in the $35\%$ solution. The final solution is 250 mL of solution that is $25\%$ hydrochloric acid, so it has $0.25(250)=62.5$mL of hydrochloric acid. All of this together means: $$0.10(x\,\text{mL})+0.35(y\,\text{mL})=62.5\,\text{mL}$$ Or without units: $$0.10x+0.35y=62.5$$ So our system is: $$\left\{ \begin{alignedat}{3} x\amp+{}\amp y\amp=250 \\ 0.10x\amp+{}\amp0.35y\amp=62.5 \end{alignedat} \right.$$ Solve the system of equations. To solve this system, we’ll solve for $x$ in the first equation: $$\begin{aligned} x+y\amp=250\\ x\amp=250-y \end{aligned}$$ And then substitute $x$ in the second equation with $250-y\text{:}$ $$\begin{aligned} 0.10x+0.35y\amp=62.5\\ 0.10(\substitute{250-y})+0.35y\amp=62.5\\ 25-0.10y+0.35y\amp=62.5\\ 25+0.25y\amp=62.5\\ 0.25y\amp=37.5\\ \divideunder{0.25y}{0.25}\amp=\divideunder{37.5}{0.25}\\ y\amp=150 \end{aligned}$$ As a last step, we will determine $x$ using the equation where we had isolated $x\text{:}$ $$\begin{aligned} x\amp=250-y\\ x\amp=250-\substitute{150}\\ x\amp=100 \end{aligned}$$ Answer the question. In summary, Anatole needs to combine 100 mL of $10\%$ hydrochloric acid solution with 150 mL of $35\%$ hydrochloric acid solution to create 250 mL of solution that is $25\%$ hydrochloric acid by volume. The check is left to the reader.