A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?
More than 320 litres but less than 1280 litres of the 2% boric acid solution will have to be added.
Explanation
This problem involves setting up linear inequalities based on the concentration constraints. Following the pattern from Example 13 in the textbook, we let x be the unknown quantity of solution to be added, then set up two inequalities representing the lower and upper bounds of the desired concentration range. The solution involves solving both inequalities simultaneously to find the valid range for x.
Solution Steps
Step 1: Let x litres of 2% boric acid solution be added. Total mixture = (x + 640) litres.
Step 2: Set up the inequality for more than 4% concentration: 8% of 640 + 2% of x > 4% of (x + 640)
Step 3: This gives: 0.08(640) + 0.02x > 0.04(x + 640), which simplifies to: 51.2 + 0.02x > 0.04x + 25.6
Step 4: Solving: 25.6 > 0.02x, therefore x < 1280
Step 5: Set up the inequality for less than 6% concentration: 8% of 640 + 2% of x < 6% of (x + 640)
Step 6: This gives: 0.08(640) + 0.02x < 0.06(x + 640), which simplifies to: 51.2 + 0.02x < 0.06x + 38.4
Step 7: Solving: 12.8 < 0.04x, therefore x > 320
Step 8: Combining both inequalities: 320 < x < 1280. Thus, the 2% solution should be more than 320 litres but less than 1280 litres.