Calculating Combined Milk and Water Ratio

The problem asks for the final ratio of milk and water when three mixtures from bottles of equal capacity are combined into a bigger bottle. The three bottles have different milk and water ratios.

Let's assume the capacity of each of the three bottles is $V$ units.

The given ratios of Milk : Water for the three bottles are:

• Bottle 1: 2 : 1
• Bottle 2: 3 : 1
• Bottle 3: 2 : 3

We need to find the amount of milk and water in each bottle based on its ratio and the equal capacity $V$.

Bottle 1:

Ratio = 2 : 1. Total parts = $2 + 1 = 3$.

Amount of Milk in Bottle 1 = $\frac{2}{3} \times V$

Amount of Water in Bottle 1 = $\frac{1}{3} \times V$

Bottle 2:

Ratio = 3 : 1. Total parts = $3 + 1 = 4$.

Amount of Milk in Bottle 2 = $\frac{3}{4} \times V$

Amount of Water in Bottle 2 = $\frac{1}{4} \times V$

Bottle 3:

Ratio = 2 : 3. Total parts = $2 + 3 = 5$.

Amount of Milk in Bottle 3 = $\frac{2}{5} \times V$

Amount of Water in Bottle 3 = $\frac{3}{5} \times V$

When these three bottles are emptied into a bigger bottle, the total amount of milk in the bigger bottle will be the sum of the milk from each bottle, and the total amount of water will be the sum of the water from each bottle.

Total amount of Milk in the bigger bottle:

Total Milk = Milk in Bottle 1 + Milk in Bottle 2 + Milk in Bottle 3

Total Milk = $\frac{2}{3}V + \frac{3}{4}V + \frac{2}{5}V$

To sum these fractions, we find the least common multiple (LCM) of the denominators 3, 4, and 5. The LCM of 3, 4, and 5 is 60.

Total Milk = $\left(\frac{2}{3} + \frac{3}{4} + \frac{2}{5}\right)V$

Total Milk = $\left(\frac{2 \times 20}{3 \times 20} + \frac{3 \times 15}{4 \times 15} + \frac{2 \times 12}{5 \times 12}\right)V$

Total Milk = $\left(\frac{40}{60} + \frac{45}{60} + \frac{24}{60}\right)V$

Total Milk = $\frac{40 + 45 + 24}{60}V = \frac{109}{60}V$

Total amount of Water in the bigger bottle:

Total Water = Water in Bottle 1 + Water in Bottle 2 + Water in Bottle 3

Total Water = $\frac{1}{3}V + \frac{1}{4}V + \frac{3}{5}V$

Again, using the LCM of 60:

Total Water = $\left(\frac{1}{3} + \frac{1}{4} + \frac{3}{5}\right)V$

Total Water = $\left(\frac{1 \times 20}{3 \times 20} + \frac{1 \times 15}{4 \times 15} + \frac{3 \times 12}{5 \times 12}\right)V$

Total Water = $\left(\frac{20}{60} + \frac{15}{60} + \frac{36}{60}\right)V$

Total Water = $\frac{20 + 15 + 36}{60}V = \frac{71}{60}V$

The ratio of milk and water in the bigger bottle is the ratio of Total Milk to Total Water.

Ratio (Milk : Water) = Total Milk : Total Water

Ratio = $\frac{109}{60}V : \frac{71}{60}V$

Since $V$ is a common factor and the denominators are the same, they cancel out.

Ratio = 109 : 71

Thus, the ratio of milk and water in the bigger bottle is 109 : 71.