Q. 365 - Intermediate Algebra

Question

In the following exercises,translate to a system of equations and solve.

A scientist needs 70 liters of a 40% solution of alcohol.He has a 30% and a 60%  solution available.How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?

Step-by-Step Solution

Answer

The number of liters of 40% solution should mix to make the 40% solution is about $46\frac{2}{3}$.

Step 1.Given information

Let x represents the number of liters of 30% solution.

Let y represents the number of liters of 60% solution.

Suppose a scientist needs 70 liters of a 40% alcohol solution.

He has 30% and 60% in his lab.

That means,$x+y=70$

The trail mix is $30\%x+60\%y=0.40\left(70\right)$

$$\begin{gathered} 0.30x+0.60y=0.40\left(70\right) \\ 0.30x+0.60y=28 \end{gathered}$$

Consider the system of linear equations

$$\begin{gathered} x+y=70 \\ 0.30x+0.40y=28 \end{gathered}$$

Step 2.Substitute x = 70 - y in the second equation 0 . 30 x + 0 . 60 y = 28 .

From first equation,

$$\begin{gathered} x+y=70 \\ x=70-y \end{gathered}$$

$$\begin{gathered} 0.30(70-y)+0.60y=28 \\ 21-0.30y+0.60y=28 \\ 0.30y=28-21 \\ 0.30y=7 \\ y=\frac{7}{0.30} \\ y=23\frac{1}{3} \end{gathered}$$

Step 3.Substitute y = 23 1 3 into the equation x + y = 70 and solve for x

$$\begin{gathered} x+y=70 \\ x+23\frac{1}{3}=70 \\ x=70-23\frac{1}{3} \\ x=46\frac{2}{3} \end{gathered}$$

Therefore,The number of liters of 40% solution should mix to make the 40% solution is about 

$46\frac{2}{3}$.