Question

A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be added to it so that acid content in the resulting mixture will be more than 15% but less than 18%.

Answer 1

Hint: To compare the volume of acid to be added in the 600 litres of a 12% solution of acid we will assume the volume x and then compare the volume of 30% acid solution and 12% acid solution with the total solution.

Complete answer:
- In the given question, we have to find the volume of the solution which contains 30% acid and when mixed with the 600 litres solution then it has an acidity less than 18% and more than 15%.
- As we know that the percentage of acid in a 600-litre solution is 12%. So, we will assume the volume of the solution as 'x' in which the percentage of acid is 30%.
- Now, we have to add this x L solution in the 600 litres so that the net acidity of the solution should be less than 18% and more than 15%.
- So, the total mixture will be (x + 600).
- So, we can write the expression for the given data as:
30% x + 12% of 600 > 15% of (x + 600)
\[\dfrac{30\text{x}}{100}\text{ + }\dfrac{12}{100}\text{ }\times \text{ 600 }\ge \text{ }\dfrac{15}{100}\text{ }\times \text{ (x + 600)}\]
-Now, by dividing by 100 from the left side and right side we will get,
\[30\text{x + 12 }\times \text{ 600 }\ge \text{ 15 }\times \text{ (x + 600)}\]
\[\text{30x + 7200 }\ge \text{ 15x + 9000}\]
\[\begin{align}
  & 15\text{x }\ge \text{ 1800} \\
 & \text{x }\ge \text{ 120} \\
\end{align}\]
- So, the volume should greater than 120 litres and to keep acidity less than 18% the value of volume will be:
\[30\text{x + 12 }\times \text{ 600 }\le \text{ 18 }\times \text{ (x + 600)}\]
\[\text{30x + 7200 }\le \text{ 18x + 10800}\]
\[\begin{align}
  & 12\text{x }\le \text{ 3600} \\
 & \text{x }\le \text{ 300} \\
\end{align}\]
- So, the volume should be less than 300 litres.

Therefore, the volume of solution should be less than 300 and more than 120 to keep the acidity less than 18% and more than 15%.

Note:
The acidity of the solution can be measured by using the ph scale or litmus paper. According to the ph scale, as we move from 1 to 6 in the ph scale, the acidity decreases and basicity increases.