Three bottles of equal capacity contain mixtures of milk and water in ratio 2 : 5 , 3 : 4 and 4 : 5 respectively . These three bottles are emptied into a large bottle. What will be the ratio of milk and water respectively in the large bottle ?

Options

A. 73 : 106
B. 73 : 116
C. 73 : 113
D. 73 : 189

Answer

To solve the problem step by step, we will calculate the amount of milk and water in each of the three bottles and then find the total amounts to determine the final ratio of milk to water in the large bottle.

### Step 1: Understand the Ratios
We have three bottles with the following ratios of milk to water:
- Bottle 1: 2:5
- Bottle 2: 3:4
- Bottle 3: 4:5

### Step 2: Calculate the Total Parts in Each Ratio
For each bottle, we need to find the total parts in the ratio:
- Bottle 1: \(2 + 5 = 7\) parts
- Bottle 2: \(3 + 4 = 7\) parts
- Bottle 3: \(4 + 5 = 9\) parts

### Step 3: Find a Common Capacity
Since all bottles have equal capacity, we can assume each bottle has a capacity of 63 liters (the least common multiple of the total parts).

### Step 4: Calculate Milk and Water in Each Bottle
Now, we will calculate the amount of milk and water in each bottle based on the assumed capacity of 63 liters.

**For Bottle 1:**
- Milk: \(\frac{2}{7} \times 63 = 18\) liters
- Water: \(\frac{5}{7} \times 63 = 45\) liters

**For Bottle 2:**
- Milk: \(\frac{3}{7} \times 63 = 27\) liters
- Water: \(\frac{4}{7} \times 63 = 36\) liters

**For Bottle 3:**
- Milk: \(\frac{4}{9} \times 63 = 28\) liters
- Water: \(\frac{5}{9} \times 63 = 35\) liters

### Step 5: Calculate Total Milk and Water
Now, we will sum the amounts of milk and water from all three bottles.

**Total Milk:**
- From Bottle 1: 18 liters
- From Bottle 2: 27 liters
- From Bottle 3: 28 liters
- Total Milk = \(18 + 27 + 28 = 73\) liters

**Total Water:**
- From Bottle 1: 45 liters
- From Bottle 2: 36 liters
- From Bottle 3: 35 liters
- Total Water = \(45 + 36 + 35 = 116\) liters

### Step 6: Find the Ratio of Milk to Water
The final ratio of milk to water in the large bottle is:
\[ \text{Ratio of Milk to Water} = 73 : 116 \]

### Step 7: Simplify the Ratio
To simplify the ratio, we can divide both sides by their greatest common divisor (GCD). The GCD of 73 and 116 is 1 (since 73 is a prime number), so the ratio remains:
\[ \text{Final Ratio} = 73 : 116 \]

### Conclusion
Thus, the ratio of milk to water in the large bottle is \(73 : 116\).