Determining the Mixing Ratio for Milk and Water Mixtures

This problem involves finding the ratio in which two different mixtures of milk and water need to be combined to achieve a specific ratio in the final mixture. We can solve this using the principle of mixtures and allegation.

Analyzing the Initial Mixture Ratios

First, let's break down the composition of the milk and water in each vessel:

• Vessel 1: The ratio of milk to water is $9 : 5$.
  • Total parts = $9 + 5 = 14$.
  • The proportion (or fraction) of milk in Vessel 1 is $\frac{9}{14}$.
• Vessel 2: The ratio of milk to water is $3 : 8$.
  • Total parts = $3 + 8 = 11$.
  • The proportion of milk in Vessel 2 is $\frac{3}{11}$.
• Desired Resultant Mixture: The target ratio of milk to water is $13 : 19$.
  • Total parts = $13 + 19 = 32$.
  • The proportion of milk desired in the final mixture is $\frac{13}{32}$.

Applying the Rule of Allegation

The rule of allegation helps us find the ratio of quantities when two ingredients of known quality are mixed to produce a mixture of a desired quality. We will focus on the proportion of milk.

Let:

• $Q_1$ be the proportion of milk in the first vessel ($\frac{9}{14}$).
• $Q_2$ be the proportion of milk in the second vessel ($\frac{3}{11}$).
• $Q_m$ be the mean proportion of milk in the final mixture ($\frac{13}{32}$).

We set up the allegation as follows:

Quantity of Mixture 1            $|Q_m - Q_2|$
                                                   $Q_m$
Quantity of Mixture 2            $|Q_m - Q_1|$

The ratio in which the mixtures should be mixed is given by $|Q_m - Q_2| : |Q_m - Q_1|$.

Calculation Steps:

1. Calculate the first difference: $|Q_m - Q_1|$
   • $|\frac{13}{32} - \frac{9}{14}|$
   • Find a common denominator for 32 and 14. The least common multiple (LCM) is 224.
   • $|\frac{13 \times 7}{32 \times 7} - \frac{9 \times 16}{14 \times 16}| = |\frac{91}{224} - \frac{144}{224}|$
   • $= |-\frac{53}{224}| = \frac{53}{224}$
2. Calculate the second difference: $|Q_m - Q_2|$
   • $|\frac{13}{32} - \frac{3}{11}|$
   • Find a common denominator for 32 and 11. The LCM is 352.
   • $|\frac{13 \times 11}{32 \times 11} - \frac{3 \times 32}{11 \times 32}| = |\frac{143}{352} - \frac{96}{352}|$
   • $= |\frac{47}{352}| = \frac{47}{352}$

Determining the Final Mixing Ratio

The ratio in which Mixture 1 and Mixture 2 should be mixed is the ratio of the differences calculated:

Ratio = (Difference corresponding to Vessel 2) : (Difference corresponding to Vessel 1)

Ratio = $\frac{47}{352} : \frac{53}{224}$

Simplifying the Ratio:

To simplify this ratio, we can multiply both sides by the LCM of the denominators (352 and 224).

• Find the LCM of 352 and 224.
  • Prime factorization: $352 = 2^5 \times 11$ and $224 = 2^5 \times 7$.
  • LCM$(352, 224) = 2^5 \times 7 \times 11 = 32 \times 77 = 2464$.
• Multiply the ratio by the LCM:
  • $(\frac{47}{352} \times 2464) : (\frac{53}{224} \times 2464)$
  • Since $2464 / 352 = 7$ and $2464 / 224 = 11$:
  • $(47 \times 7) : (53 \times 11)$
  • $329 : 583$

Conclusion

Therefore, the mixture from the first vessel and the mixture from the second vessel should be mixed in the ratio $329 : 583$ to obtain the desired resultant mixture with a milk to water ratio of $13 : 19$.