Question

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

Jotham needs 70 liters of a 50% alcohol solution. He has a 30% and an 80% solution available. How many liters of the 30% and how many liters of the 80% solutions should he mix to make the 50% solution?

Short answer

The number of 30% solution is 42 litres and number of 80% solution is 28 litres.

Step-by-step solution

Step 1. Given Information  

The given data is that Jotham needs 70 liters of a 50% alcohol solution. He has a 30% and an 80% solution available.

Step 2. Explanation    

Let number of units of 30% solution be x and number of units of 80% solution be y.

The number of units of 50% solution is 70 liters.

Thus, the total units of solution can be expressed as $x+y=70--\left(1\right)$

The amount of 50% solution is $70\times 0.5=35$

And the total amount of solution can be expressed as$0.3x+0.8y=35--\left(2\right)$

Step 3. Calculation 

Multiply equation (1) with number $0.3$and write the revised equation.

$$\begin{gathered} 0.3(x+y)=0.3\left(70\right) \\ 0.3x+0.3y=21--\left(3\right) \end{gathered}$$

Solve the equation (3) and (2) by subtracting equation (3) from equation (2).

$$\begin{gathered} 0.3x+0.8y-0.3x-0.3y=35-21 \\ 0.5y=14 \\ y=28 \end{gathered}$$

Substitute the value of y in equation (1) to find the value of x.

$$\begin{gathered} x+y=70 \\ y=70-28 \\ y=42 \end{gathered}$$