Three containers A, B and C are having mixtures of milk and water in the ratio…

When several mixtures with different milk:water ratios are combined in a given capacity ratio, the combined ratio is found using the weighted-average (alligation) method for multiple mixtures: convert each container's ratio into a fraction of milk and a fraction of water, multiply each fraction by that container's capacity to get the actual milk and water quantities it contributes, then add all the milk quantities and all the water quantities separately. The final milk:water ratio is simply (total milk) : (total water).

1. Let the capacities of A, B and C (in the ratio 5:4:5) be 5k, 4k and 5k litres respectively.

2. Container A has milk:water = 1:6, i.e. 7 total parts, so its milk fraction is 1/7 and water fraction is 6/7. Milk from A = 5k x 1/7 = 5k/7; Water from A = 5k x 6/7 = 30k/7.

3. Container B has milk:water = 3:4, i.e. 7 total parts, so its milk fraction is 3/7 and water fraction is 4/7. Milk from B = 4k x 3/7 = 12k/7; Water from B = 4k x 4/7 = 16k/7.

4. Container C has milk:water = 5:6, i.e. 11 total parts, so its milk fraction is 5/11 and water fraction is 6/11. Milk from C = 5k x 5/11 = 25k/11; Water from C = 5k x 6/11 = 30k/11.

5. Total milk = 5k/7 + 12k/7 + 25k/11 = 17k/7 + 25k/11. Taking the LCM of 7 and 11 (which is 77): = 187k/77 + 175k/77 = 362k/77.

6. Total water = 30k/7 + 16k/7 + 30k/11 = 46k/7 + 30k/11 = 506k/77 + 210k/77 = 716k/77.

7. Ratio of milk to water = 362k/77 : 716k/77 = 362 : 716. Dividing both terms by 2 gives 181 : 358.

181 is a prime number and 358 = 2 x 179, so the two share no common factor beyond the one already removed - 181/358 is fully reduced, confirming it as the final answer.