Three vessels contain equal mixtures of milk and water in the ration 6:1, 5:2 and 3:1 respectively. If all the solutions are mixed together, the ratio of milk to water in the final mixture will beOptions :64:6565:6419:6565:19

Sure, let's solve this step by step.

1. First, let's find the ratio of milk to water in each vessel:

   • Vessel 1: The ratio is 6:1, which means there are 6 parts milk and 1 part water. So, the total parts in the vessel are 6+1 = 7 parts. The fraction of milk in the vessel is 6/7 and the fraction of water is 1/7.

   • Vessel 2: The ratio is 5:2, which means there are 5 parts milk and 2 parts water. So, the total parts in the vessel are 5+2 = 7 parts. The fraction of milk in the vessel is 5/7 and the fraction of water is 2/7.

   • Vessel 3: The ratio is 3:1, which means there are 3 parts milk and 1 part water. So, the total parts in the vessel are 3+1 = 4 parts. The fraction of milk in the vessel is 3/4 and the fraction of water is 1/4.

2. Now, let's assume that the total volume of the mixture from all the vessels is 1 unit. This means that the volume of the mixture from each vessel is 1/3 unit.

3. The volume of milk from each vessel is the volume of the mixture from that vessel multiplied by the fraction of milk in that vessel. Similarly, the volume of water from each vessel is the volume of the mixture from that vessel multiplied by the fraction of water in that vessel.

   • Vessel 1: Volume of milk = 1/3 * 6/7 = 2/7 unit, Volume of water = 1/3 * 1/7 = 1/21 unit
   • Vessel 2: Volume of milk = 1/3 * 5/7 = 5/21 unit, Volume of water = 1/3 * 2/7 = 2/21 unit
   • Vessel 3: Volume of milk = 1/3 * 3/4 = 1/4 unit, Volume of water = 1/3 * 1/4 = 1/12 unit

4. The total volume of milk in the final mixture is the sum of the volumes of milk from all the vessels. Similarly, the total volume of water in the final mixture is the sum of the volumes of water from all the vessels.

   • Total volume of milk = 2/7 + 5/21 + 1/4 = 64/84 unit
   • Total volume of water = 1/21 + 2/21 + 1/12 = 19/84 unit

5. Therefore, the ratio of milk to water in the final mixture is 64:19. So, the correct option is 65:19.

To solve this problem, we need to calculate the total amount of milk and water in each vessel and then add them together.

1. In the first vessel, the ratio of milk to water is 6:1. This means that for every 7 parts of the mixture, 6 parts are milk and 1 part is water. If we assume that the total mixture in the vessel is 7 units, then there are 6 units of milk and 1 unit of water.

2. In the second vessel, the ratio is 5:2. This means that for every 7 parts of the mixture, 5 parts are milk and 2 parts are water. If we again assume that the total mixture in the vessel is 7 units, then there are 5 units of milk and 2 units of water.

3. In the third vessel, the ratio is 3:1. This means that for every 4 parts of the mixture, 3 parts are milk and 1 part is water. If we assume that the total mixture in the vessel is 4 units, then there are 3 units of milk and 1 unit of water.

Now, we add up the total amount of milk and water from all the vessels:

Total milk = 6 units (from the first vessel) + 5 units (from the second vessel) + 3 units (from the third vessel) = 14 units

Total water = 1 unit (from the first vessel) + 2 units (from the second vessel) + 1 unit (from the third vessel) = 4 units

So, the ratio of milk to water in the final mixture is 14:4, which simplifies to 7:2.

None of the options provided match this ratio. There might be a mistake in the question or the options provided.