Question

11. A laboratory technician at Beth Israel Hospital needs to make a 10-liter batch of antisepticthat is 60% alcohol. How can she combine a batch of antiseptic that is 30% alcohol with anothervariety that is 70% alcohol to formulate the desired concentration? (10 points)

Answer 1

Assuming the mixture volumes doen'st change from one percent to another, we can calculate this using a system of equations.

Let x be de volume of 30% alcohol and y be the volume of 70% alcohol, both in liters.

If we want a 10-liter batch, the sum of the volumes needs to be 10, so er have the first equation:

`x+y=10`

Now, assuming the percentages are of volumes, in the x volume of 30% alcohol, 30% of it is alcohol, so the amount of alcohol in it is 0.30x. Similarly, the amount of alcohol in the y volume of 70% alcohol is 0.70y. At the end we have 10 liters of 60% alcohol, so the amount of alcohol is 0.60*10. the sum of the first two, 0.30x and 0.70y, needs to be the third, 0.60*10, which gives us the second equation:

`\begin{gathered} 0.30x+0.70y=0.60\cdot10 \\ 0.3x+0.7y=6 \end{gathered}`

So, the system of equations is:

`\begin{gathered} x+y=10 \\ 0.3x+0.7y=6 \end{gathered}`

To solve it, we can multiply the first equation by 0.3 and substract is from the second:

`\begin{gathered} (x+y=10)\cdot0.3 \\ 0.3x+0.3y=3 \\ \\ 0.3x+0.7y=6 \\ -(0.3x+0.3y=3) \\ 0.3x-0.3x+0.7y-0.3y=6-3 \\ 0.4y=3 \\ y=(3)/(0.4)=7.5 \end{gathered}`

Now, with this value, we can substitute it into the first equation and solve for x:

`\begin{gathered} x+y=10 \\ x+7.5=10 \\ x=10-7.5 \\ x=2.5 \end{gathered}`

Thus, y = 7.5 and x = 2.5.

This means that we cancombine 2.5 liters of the 30% alcohol with 7.5 liters of the 70% alcohol to get 10 liters of the 60% alcohol.