Question

How many litres of water must be added to \[8\] litres of a \[40\% \] acid solution to obtain a \[10\% \] acid solution?

Answer 1

Hint: In order to solve this question, the first thing we will assume is that the density of the solution remains constant. After that we will let \[x\] litres of water be required to be added. Then the total mixture will be equal to \[\left( {x + 8} \right)\] litres.After that we will form a linear equation by using the given conditions and solve it to get the required answer.

Complete step by step answer:
Let \[x\] litres of water are required to be added. Then, the total mixture will be equal to \[\left( {x + 8} \right)\] litres. Now according to the question, it is evident that the amount of acid contained in the resulting mixture is \[40\% \] of \[8\] litres and as a result the mixture contains \[10\% \] acid content.

Therefore, we have
\[10\% {\text{ }}of{\text{ }}(x + 8) = 40\% {\text{ }}of{\text{ }}8{\text{ }} - - - \left( 1 \right)\]
Here, \[40\% \] concentration means,
\[40\% = \dfrac{{40L{\text{ }}acid}}{{100L{\text{ }}solution}}\]
So, the volume of acid in \[8L\] of solution is
Volume of acid \[ = 8L{\text{ }}solution \times \dfrac{{40L{\text{ }}acid}}{{100L{\text{ }}solution}}\]

On solving, we get
Volume of acid \[ = 3.2L{\text{ }}acid\]
\[ \Rightarrow 40\% {\text{ }}of{\text{ }}8{\text{ }}litres = 3.2L{\text{ }}acid\]
On substituting in equation \[\left( 1 \right)\] we have
\[10\% {\text{ }}of{\text{ }}(x + 8) = 3.2L{\text{ }}acid\]
\[ \Rightarrow \dfrac{{10}}{{100}}\left( {x + 8} \right) = 3.2\]

On multiplying by \[100\] both sides, we get
\[ \Rightarrow 10\left( {x + 8} \right) = 3.2 \times 100\]
\[ \Rightarrow 10x + 80 = 320\]
On subtracting \[80\] both sides, we get
\[ \Rightarrow 10x = 240\]
On dividing by \[10\] both sides, we get
\[ \therefore x = 24\]

Hence, the required number of litres of water that is to be added will be \[24{\text{ }}litres\].

Note: The given question was an easy application-based question. The common mistake which students can make is while making the linear equation. So, be careful while making the equation. Read the question properly and then accordingly arrange the terms and make an equation.
Also, you can check your answer by substituting the value of \[x\] in \[\dfrac{{10}}{{100}}\left( {x + 8} \right) = 3.2\]
So, we get
\[\dfrac{{10}}{{100}}\left( {24 + 8} \right) = 3.2\]
\[ \Rightarrow \dfrac{{10}}{{100}}\left( {32} \right) = 3.2\]
\[ \Rightarrow \dfrac{{32}}{{10}} = 3.2\]
\[ \therefore 3.2 = 3.2\]
Hence, we get the right answer.