90% and 97% pure acid solutions are mixed to obtain 21 litres of 95% pure acid…

Alligation is the rule for finding the ratio in which two ingredients at two different concentrations must be mixed to give a mixture at a target (mean) concentration. For a cheaper ingredient at concentration c and a dearer ingredient at concentration d mixed to give a mean m, the rule states: (quantity of cheaper) : (quantity of dearer) = (d − m) : (m − c).

1. Here c = 90% (the cheaper, lower-concentration solution), d = 97% (the dearer, higher-concentration solution), and the required mean m = 95%.

2. By the alligation rule, the ratio of (90% solution) : (97% solution) = (d − m) : (m − c) = (97 − 95) : (95 − 90) = 2 : 5.

3. The total mixture is 21 litres, and the ratio 2 : 5 has 2 + 5 = 7 parts, so 1 part = 21 ÷ 7 = 3 litres.

4. Quantity of 90% solution = 2 × 3 = 6 litres; quantity of 97% solution = 5 × 3 = 15 litres.

Cross-check by direct substitution: with x = 6 litres of the 90% solution and y = 15 litres of the 97% solution, x + y = 6 + 15 = 21 litres (matches the required total volume), and the acid content is 0.90 × 6 + 0.97 × 15 = 5.4 + 14.55 = 19.95 litres, which is exactly 95% of 21 litres (0.95 × 21 = 19.95). Both conditions hold independently, confirming the result.

So 6 litres of the 90% solution and 15 litres of the 97% solution must be mixed.