Question

A pharmacist needs to strengthen a $10\%$ alcohol solution to one of $20\%$ alcohol. How much pure alcohol should be added to 400 ml of $10\%$ solution?

Answer 1

Hint: We have to strengthen the alcohol solution that means we have to add some amount of alcohol in the solution. Let us assume the amount of alcohol to be added in the $10\%$ alcohol solution be ‘x’ ml.

Complete step-by-step answer:
First have to understand the term ‘solution’ and its properties. So, a solution is a homogeneous mixture of two or more substances. Homogeneous mixture means a mixture in which the solute particle is evenly present in the solvent. Here, in this question alcohol is a solute particle and water is solvent. The $\%$ age strength of a solution is defined as the amount of solute present in per 100 parts of the solution. For example: $10\%$ alcohol solution means 10 ml of alcohol is present in 100 ml of solution.
Now, come to the question. We have assumed that the amount of alcohol to be added is ‘x’ ml. Now, $10\%$ alcohol solution contains 10 ml alcohol in 100 ml solution.
Therefore by unitary method, 400 ml solution contains:
\[\dfrac{10}{100}\times 400=40\text{ mL}\] of alcohol.
When we add ‘x’ ml of pure alcohol to this alcohol solution then the total amount of solution becomes $(400 + x)$ ml and the amount of alcohol in the solution becomes $(40 + x)$ ml.
Now the resultant strength of the solution is $20\%$. Therefore,
\[\begin{align}
  & \dfrac{40+x}{400+x}\times 100=20 \\
 & \dfrac{40+x}{400+x}=\dfrac{20}{100} \\
 & \dfrac{40+x}{400+x}=\dfrac{1}{5} \\
\end{align}\]
By cross multiplication, we get,
\[\begin{align}
  & 5\times \left( 40+x \right)=400+x \\
 & 200+5x=400+x \\
 & 5x-x=400-200 \\
 & 4x=200 \\
 & x=\dfrac{200}{4} \\
 & x=50. \\
\end{align}\]
Hence, 50 ml pure alcohol should be added.

Note: Always note the total volume of solution and whenever there is addition of some solute or solvent, we must add the quantity to the total volume of solution. Here, when we added $x\text{ mL}$ pure alcohol to the $400\text{ mL}$ solution then, the total volume of solution formed is $(400+x)\text{ mL}$.