Milk and Water Ratio Problem

This problem involves mixing two different solutions of milk and water and finding the ratio of milk and water in the final mixture. We are given the ratios in two vessels and the ratio in which the contents of these vessels are mixed.

Analyzing Vessel Contents

Let's break down the contents of each vessel:

• Vessel 1: Milk and Water are in the ratio 7 ∶ 8.
  • Total parts = 7 (Milk) + 8 (Water) = 15 parts.
  • Proportion of Milk in Vessel 1 = \(\frac{7}{15}\).
  • Proportion of Water in Vessel 1 = \(\frac{8}{15}\).
• Vessel 2: Milk and Water are in the ratio 13 ∶ 5.
  • Total parts = 13 (Milk) + 5 (Water) = 18 parts.
  • Proportion of Milk in Vessel 2 = \(\frac{13}{18}\).
  • Proportion of Water in Vessel 2 = \(\frac{5}{18}\).

Mixing the Vessels in Ratio 1:1

The two vessels are mixed in the ratio 1 ∶ 1. This means an equal quantity is taken from Vessel 1 and Vessel 2. Let's assume we take 'Q' quantity from Vessel 1 and 'Q' quantity from Vessel 2.

In the new mixture, the total amount of milk will be the sum of the milk from Vessel 1 and the milk from Vessel 2. Similarly, the total amount of water will be the sum of the water from Vessel 1 and the water from Vessel 2.

• Amount of Milk from Vessel 1 = \(\frac{7}{15} \times Q\)
• Amount of Water from Vessel 1 = \(\frac{8}{15} \times Q\)
• Amount of Milk from Vessel 2 = \(\frac{13}{18} \times Q\)
• Amount of Water from Vessel 2 = \(\frac{5}{18} \times Q\)

Calculating the Total Milk and Water

Total Milk in the new mixture = (Milk from Vessel 1) + (Milk from Vessel 2)

Total Milk = \(\frac{7}{15} Q + \frac{13}{18} Q\)

To add these fractions, we find a common denominator for 15 and 18. The least common multiple (LCM) of 15 and 18 is 90.

Total Milk = \(\frac{7 \times 6}{15 \times 6} Q + \frac{13 \times 5}{18 \times 5} Q = \frac{42}{90} Q + \frac{65}{90} Q = \frac{42 + 65}{90} Q = \frac{107}{90} Q\)

Total Water in the new mixture = (Water from Vessel 1) + (Water from Vessel 2)

Total Water = \(\frac{8}{15} Q + \frac{5}{18} Q\)

Using the same common denominator (90):

Total Water = \(\frac{8 \times 6}{15 \times 6} Q + \frac{5 \times 5}{18 \times 5} Q = \frac{48}{90} Q + \frac{25}{90} Q = \frac{48 + 25}{90} Q = \frac{73}{90} Q\)

Finding the Final Ratio

The ratio of Milk to Water in the new mixture is:

(Total Milk) ∶ (Total Water)

\(\frac{107}{90} Q : \frac{73}{90} Q\)

Since 'Q' is a common factor on both sides of the ratio, it cancels out.

The ratio is \(\frac{107}{90} : \frac{73}{90}\). To get whole numbers, we multiply both parts of the ratio by 90.

\(\frac{107}{90} \times 90 : \frac{73}{90} \times 90\)

107 ∶ 73

So, the ratio of milk and water in the new mixture is 107 ∶ 73.

Summary of Calculations

Mixing ratio is 1 ∶ 1.

Final Milk Proportion = \(\frac{7}{15} + \frac{13}{18} = \frac{107}{90}\)

Final Water Proportion = \(\frac{8}{15} + \frac{5}{18} = \frac{73}{90}\)

Final Ratio = \(\frac{107}{90} : \frac{73}{90} = 107 : 73\)