Two vessels A and B contain spirit and water in the ratio 2 : 7 and 5 : 4…

Concept: When two mixtures are combined in a stated ratio, the amount of each ingredient (spirit or water) in the resulting mixture equals the sum of the contributions from each source mixture. A source's contribution to an ingredient is found by multiplying the quantity taken from that source by that ingredient's fraction within the source's own ratio.

Application:

1. Vessel A has spirit : water = 2 : 7 (total parts = 9), so the spirit fraction = 2/9 and the water fraction = 7/9.

2. Vessel B has spirit : water = 5 : 4 (total parts = 9), so the spirit fraction = 5/9 and the water fraction = 4/9.

3. The two vessels are mixed in the ratio 1 : 2, so let x litres be taken from vessel A and 2x litres from vessel B.

4. Spirit from vessel A = x × 2/9 = 2x/9; water from vessel A = x × 7/9 = 7x/9.

5. Spirit from vessel B = 2x × 5/9 = 10x/9; water from vessel B = 2x × 4/9 = 8x/9.

6. Total spirit = 2x/9 + 10x/9 = 12x/9; total water = 7x/9 + 8x/9 = 15x/9.

7. Ratio of spirit to water = 12x/9 : 15x/9 = 12 : 15 = 4 : 5, dividing both terms by 3.

Cross-check: The total mixture taken is x + 2x = 3x. Adding the spirit and water totals gives 12x/9 + 15x/9 = 27x/9 = 3x, which matches the total quantity taken — confirming the split is consistent.

∴ The ratio of spirit and water in the new mixture is 4 : 5.