In three vessels, each of 25 litres capacity, mixture of milk and water is filled. The ratio of milk and water are `3:1,2:3,4:3` in the respective vessels. If all the three vessels are emptied into a single large vessel, then that will be the ratio of water to milk in the resultant mixture?

Options

A. `179:241`
B. `197:214`
C. `219:117`
D. `179:234`

Answer

To solve the problem, we need to find the ratio of water to milk when the mixtures from three vessels are combined. Let's break it down step by step.

### Step 1: Determine the amount of milk and water in each vessel.

**Vessel 1:**
- Ratio of milk to water = 3:1
- Total capacity = 25 litres
- Total parts = 3 + 1 = 4 parts
- Milk = (3/4) * 25 = 18.75 litres
- Water = (1/4) * 25 = 6.25 litres

**Vessel 2:**
- Ratio of milk to water = 2:3
- Total capacity = 25 litres
- Total parts = 2 + 3 = 5 parts
- Milk = (2/5) * 25 = 10 litres
- Water = (3/5) * 25 = 15 litres

**Vessel 3:**
- Ratio of milk to water = 4:3
- Total capacity = 25 litres
- Total parts = 4 + 3 = 7 parts
- Milk = (4/7) * 25 = 14.29 litres (approximately)
- Water = (3/7) * 25 = 10.71 litres (approximately)

### Step 2: Calculate the total amount of milk and water when all vessels are combined.

**Total Milk:**
- From Vessel 1: 18.75 litres
- From Vessel 2: 10 litres
- From Vessel 3: 14.29 litres

Total Milk = 18.75 + 10 + 14.29 = 43.04 litres (approximately)

**Total Water:**
- From Vessel 1: 6.25 litres
- From Vessel 2: 15 litres
- From Vessel 3: 10.71 litres

Total Water = 6.25 + 15 + 10.71 = 32.96 litres (approximately)

### Step 3: Find the ratio of water to milk.

Now we need to find the ratio of water to milk:
- Water : Milk = Total Water : Total Milk
- Water : Milk = 32.96 : 43.04

To simplify the ratio, we can divide both sides by the total amount of milk:
- Water : Milk = 32.96 / 43.04 : 43.04 / 43.04
- Water : Milk = 0.766 : 1

To express this as a ratio of whole numbers, we can multiply both sides by 100:
- Water : Milk = 76.6 : 100

This can be approximated to:
- Water : Milk = 77 : 100 (after rounding)

### Step 4: Final Ratio of Water to Milk

To express the ratio of water to milk in the simplest form, we can also express it as:
- Water : Milk = 77 : 100

Thus, the final answer is that the ratio of water to milk in the resultant mixture is approximately **77:100**.