Question

A laboratory technician needs to make a 54 -liter batch of a 20 % acid solution. How can the laboratory technician combine a batch of an acid solution that is pure acid with another that is 10 % to get the desired concentration?

Answer 1

According to the information given in the exercise:

- The laboratory technician needs to make a 54 ​-liter batch of a 20 ​% acid solution.

- The combination has a batch of an acid solution that is pure acid with another solution that is 10​% acid solution.

Let be "p" the amount of acid solution that is pure acid (in liters) and "t" the amount of solution that is 10% acid.

Using the information given, you can set up the following:

`\begin{cases}p+t=54 \\ p+0.1t=(0.2)(54)\end{cases}`

Notice that when you simplify the second equation:

`\begin{cases}p+t=54 \\ p+0.1t=10.8\end{cases}`

As you can see, the second equation represents the addition of the amount of 100% acid solution and the amount of 10% acid solution, which will result in a 20% acid solution.

Remember that a percent can be written as a Decimal Number by dividing it by 100:

`\begin{gathered} (100)/(100)=1 \\ \\ (10)/(100)=0.1 \\ \\ (20)/(100)=0.2 \end{gathered}`

Those are the coefficients used in the second equation.

To solve the system of equations, you can follow these steps:

1. Multiply the second equation by -1:

`\begin{cases}p+t=54 \\ -p-0.1t=-10.8\end{cases}`

2. Add the equations.

3. Solve for "t".

Then:

`\begin{gathered} \begin{cases}p+t=54 \\ -p-0.1t=-10.8\end{cases} \\ ------------ \\ 0+0.9t=43.2 \\ \\ t=(43.2)/(0.9) \\ \\ t=48 \end{gathered}`

4. Substitute the value of "t" into any original equation:

`\begin{gathered} p+t=54 \\ p+(48)=54 \end{gathered}`

5. Solve for "p":

`\begin{gathered} p=54-48 \\ p=6 \end{gathered}`

Therefore, you can determine that the answer is:

The laboratory technician can combine 6 liters of pure acid solution and 48 liters of 10% acid solution to get the desired​ concentration.