Question

Find out how many litres of water will have to be added to $ 1125 $ litres of the $ 45\% $ solution of acid so that the resulting mixture will contain more than $ 25\% $ but less than $ 30\% $ acid content?

Answer 1

Hint: This question is related to linear inequalities. Any two real numbers or algebraic expressions when related by ‘ $ < $ ’, ‘ $ > $ ’, ‘ $ \leqslant $ ’ or ‘ $ \geqslant $ ’ form a linear inequality. Here, in this question we will solve both the inequalities to get our required answer. We have to be extra careful while using the inequality signs.

Complete step by step solution:
Let $ x $ litres of water is required to be added.
Then, total mixture = $ (x + 1125) $ litres
It is evident that the amount of acid contained in the resulting mixture is $ 45\% $ of $ 1125 $ litres and as a result the mixture will contain more than $ 25\% $ but less than $ 30\% $ acid content.
 $ \therefore 30\% \;of\;\left( {1125 + x} \right) > 45\% \,of\,1125 $
And, $ 25\% \;of\;\left( {1125 + x} \right) < 45\% \,of\,1125 $
So, $ 30\% \;of\;\left( {1125 + x} \right) > 45\% \,of\,1125 $
 $
   \Rightarrow \dfrac{{30}}{{100}}\left( {1125 + x} \right) > \dfrac{{45}}{{100}} \times 1125 \\
   \Rightarrow 30\left( {1125 + x} \right) > 45 \times 1125 \\
   \Rightarrow 30 \times 1125 + 30x > 45 \times 1125 \\
   \Rightarrow 30x > 45 \times 1125 - 30 \times 1125 \\
   \Rightarrow 30x > \left( {45 - 30} \right) \times 1125 \\
   \Rightarrow x > \dfrac{{15 \times 1125}}{{30}} = 562.5 \;
  $
Whereas, $ 25\% \;of\;\left( {1125 + x} \right) < 45\% \,of\,1125 $
 $
   \Rightarrow \dfrac{{25}}{{100}}\left( {1125 + x} \right) < \dfrac{{45}}{{100}} \times 1125 \\
   \Rightarrow 25\left( {1125 + x} \right) < 45 \times 1125 \\
   \Rightarrow 25 \times 1125 + 25x < 45 \times 1125 \\
   \Rightarrow 25x < 45 \times 1125 - 25 \times 1125 \\
   \Rightarrow 25x < \left( {45 - 25} \right) \times 1125 \\
   \Rightarrow x < \dfrac{{20 \times 1125}}{{25}} = 900 \;
  $
Therefore, we can say that $ 562.5 < x < 900 $
Hence, the required number of litres of water that is to be added will have to be more than $ 562.5 $ but less than $ 900 $ .

So, the correct answer is “ $ 562.5 < x < 900 $ ”.

Note: The given question was an easy application-based question. The common mistake which students can make is wrong usage of the inequality signs. While solving the question, the students should ensure that the inequality symbol should not change by mistake. There are certain rules of inequality which need to be followed while solving inequality problems. Some of them are:
Only equal numbers should be added or subtracted from both the sides of inequality.
Both the sides of inequality can only be multiplied and divided with the same positive number.