Step 1: Understanding the Question:
We are given three separate containers of different capacities containing a mixture of milk and water in different ratios.
We need to determine the final composite ratio of milk to water when all three containers are completely emptied into a single large container.
Step 2: Key Formula or Approach:
Assume convenient absolute volumes for the three containers based on their given volume ratio.
Calculate the exact quantities of milk and water independently in each container.
Sum the total milk and total water separately, and then find the final combined ratio.
Step 3: Detailed Explanation:
The volumes of the three containers are in the ratio 3 : 4 : 5.
Let the actual volumes of the containers be 300 liters, 400 liters, and 500 liters respectively, to make fraction calculations easier.
Container 1 (300 liters): The ratio of milk to water is 4 : 1. The total parts are \( 4 + 1 = 5 \).
Milk in Container 1 = \( 300 \times \frac{4}{5} = 240 \) liters.
Water in Container 1 = \( 300 \times \frac{1}{5} = 60 \) liters.
Container 2 (400 liters): The ratio of milk to water is 3 : 1. The total parts are \( 3 + 1 = 4 \).
Milk in Container 2 = \( 400 \times \frac{3}{4} = 300 \) liters.
Water in Container 2 = \( 400 \times \frac{1}{4} = 100 \) liters.
Container 3 (500 liters): The ratio of milk to water is 5 : 2. The total parts are \( 5 + 2 = 7 \).
Milk in Container 3 = \( 500 \times \frac{5}{7} = \frac{2500}{7} \) liters.
Water in Container 3 = \( 500 \times \frac{2}{7} = \frac{1000}{7} \) liters.
Total Quantities in the Fourth Container:
Total Milk = \( 240 + 300 + \frac{2500}{7} = 540 + \frac{2500}{7} \).
Finding a common denominator for total milk: \( \frac{540 \times 7 + 2500}{7} = \frac{3780 + 2500}{7} = \frac{6280}{7} \) liters.
Total Water = \( 60 + 100 + \frac{1000}{7} = 160 + \frac{1000}{7} \).
Finding a common denominator for total water: \( \frac{160 \times 7 + 1000}{7} = \frac{1120 + 1000}{7} = \frac{2120}{7} \) liters.
The final ratio of Milk to Water in the fourth container is \( \frac{\frac{6280}{7}}{\frac{2120}{7}} \).
This simplifies to \( 6280 : 2120 \).
Divide by 10 to get \( 628 : 212 \).
Dividing both sides by their greatest common divisor, 4, gives \( 157 : 53 \).
Step 4: Final Answer:
The ratio of milk and water in the fourth container is 157 : 53.