Q. 146 - Intermediate Algebra

Question

A scientist needs 65 liters of a 15% alcohol solution. She has available a 25% and a 12% solution. How many liters of the 25% and how many liters of the 12% solutions should she mix to make the 15% solution?

Step-by-Step Solution

Answer

 liters of the 25% solution =15
 liters of the 12% solution=50

Step 1. Given Information

Total liters required= 65 liters.

65 liters of a 15% solution of an alcohol solution is required.

We have 25%  and 12%  solutions available.

Step 2. Formation of two linear Equations

Let number of liters of  25%  solutions required= $x$

Let number of liters of  12%  solutions requires =$y$

As we want  65 liters  in total so,

$x+y=65$

The trial mix is 

$$\begin{gathered} \left(0.25\right)x+\left(0.12\right)y=\left(0.15\right)\times 65 \\ \left(0.25\right)x+\left(0.12\right)y=9.75 \end{gathered}$$

Step 3. Solving the equations

From the first equation,

$x=65-y$

Now we will substitute this value in the second equation.

$$\begin{gathered} \left(0.25\right)\left(65-y\right)+\left(0.12\right)y=9.75 \\ 16.25-\left(0.25\right)y+\left(0.12\right)y=9.75 \\ -\left(0.13\right)y=-6.5 \\ 0.13y=6.5 \\ 13y=650 \\ y=50 \end{gathered}$$

No. of  liters of the 12% solution =50

Now put this value in the equation 

$$\begin{gathered} x=65-y \\ x=65-50 \\ x=15 \end{gathered}$$

No. of  liters of the 25% solutions =15

Step 4. Check the solution

Substitute $15$ for $x$ and $50$ for $y$ in the first equation formed.

$\begin{array}{rcl}x+y& =& 65\\ 15+50& =& 65\\ 65& =& 65\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}0.25x+0.12y& =& 9.75\\ 0.25·15+0.12·50& =& 9.75\\ 3.75+6& =& 9.75\\ 9.75& =& 9.75\end{array}$

This is also a true statement.

So the point satisfies both the equations.