Two vessels P and Q contain liquid A and liquid B in the ratio $4 : 3$ and $5 : 4$ respectively. In what ratio must the mixtures from vessel P and vessel Q be combined to obtain a new mixture in vessel R containing liquid A and liquid B in the ratio $11 : 8$?
$28 : 9$
Detailed Solution for Mixture Ratio Problem
This problem involves finding the ratio in which two different mixtures, P and Q, must be combined to create a new mixture R with a specific proportion of two liquids, A and B. We are given the ratios of A to B in vessels P and Q, and the desired ratio in vessel R.
Understanding the Given Ratios
- Vessel P: Contains Liquid A and Liquid B in the ratio $4 : 3$.
- The total parts in the ratio are $4 + 3 = 7$.
- Fraction of Liquid A in P = $\frac{4}{7}$.
- Fraction of Liquid B in P = $\frac{3}{7}$.
- Vessel Q: Contains Liquid A and Liquid B in the ratio $5 : 4$.
- The total parts in the ratio are $5 + 4 = 9$.
- Fraction of Liquid A in Q = $\frac{5}{9}$.
- Fraction of Liquid B in Q = $\frac{4}{9}$.
- Vessel R (Desired Mixture): Needs Liquid A and Liquid B in the ratio $11 : 8$.
- The total parts in the desired ratio are $11 + 8 = 19$.
- The desired fraction of Liquid A in R = $\frac{11}{19}$.
- The desired fraction of Liquid B in R = $\frac{8}{19}$.
Applying the Method of Alligation
The method of alligation is a quick way to solve such mixture problems. We can focus on the proportion of either Liquid A or Liquid B. Let's use the proportion of Liquid A. We need to find the ratio in which mixture P and mixture Q should be mixed.
Let the ratio in which mixtures from P and Q are combined be $x : y$.
- Fraction of Liquid A in P = $\frac{4}{7}$
- Fraction of Liquid A in Q = $\frac{5}{9}$
- Target Fraction of Liquid A in R = $\frac{11}{19}$
According to the rule of alligation, the ratio of the quantities is inversely proportional to the differences between the quantity of the component in the individual mixtures and the mean quantity (the desired quantity in the final mixture).
Calculation using Liquid A
Difference between the fraction of A in P and the desired fraction in R:
$ \text{Diff}_P = \left| \frac{4}{7} - \frac{11}{19} \right| $ $ \text{Diff}_P = \left| \frac{4 \times 19 - 11 \times 7}{7 \times 19} \right| $ $ \text{Diff}_P = \left| \frac{76 - 77}{133} \right| $ $ \text{Diff}_P = \left| \frac{-1}{133} \right| = \frac{1}{133} $Difference between the fraction of A in Q and the desired fraction in R:
$ \text{Diff}_Q = \left| \frac{5}{9} - \frac{11}{19} \right| $ $ \text{Diff}_Q = \left| \frac{5 \times 19 - 11 \times 9}{9 \times 19} \right| $ $ \text{Diff}_Q = \left| \frac{95 - 99}{171} \right| $ $ \text{Diff}_Q = \left| \frac{-4}{171} \right| = \frac{4}{171} $The ratio $x : y$ (Mixture P : Mixture Q) is given by the inverse ratio of these differences:
$ x : y = \text{Diff}_Q : \text{Diff}_P $ $ x : y = \frac{4}{171} : \frac{1}{133} $Simplifying the Ratio
To simplify the ratio $\frac{4}{171} : \frac{1}{133}$, we can cross-multiply or find a common denominator.
First, find the least common multiple (LCM) of the denominators $171$ and $133$. $171 = 9 \times 19$ $133 = 7 \times 19$ LCM$(171, 133) = 7 \times 9 \times 19 = 1197$.
Convert the fractions:
$ \frac{4}{171} = \frac{4 \times 7}{171 \times 7} = \frac{28}{1197} $ $ \frac{1}{133} = \frac{1 \times 9}{133 \times 9} = \frac{9}{1197} $So the ratio is:
$ x : y = \frac{28}{1197} : \frac{9}{1197} $This simplifies to:
$ x : y = 28 : 9 $Verification using Liquid B (Optional)
Let's verify this using the proportion of Liquid B.
- Fraction of Liquid B in P = $\frac{3}{7}$
- Fraction of Liquid B in Q = $\frac{4}{9}$
- Target Fraction of Liquid B in R = $\frac{8}{19}$
Difference between the fraction of B in P and the desired fraction in R:
$ \text{Diff}_P(B) = \left| \frac{3}{7} - \frac{8}{19} \right| = \left| \frac{3 \times 19 - 8 \times 7}{7 \times 19} \right| = \left| \frac{57 - 56}{133} \right| = \frac{1}{133} $Difference between the fraction of B in Q and the desired fraction in R:
$ \text{Diff}_Q(B) = \left| \frac{4}{9} - \frac{8}{19} \right| = \left| \frac{4 \times 19 - 8 \times 9}{9 \times 19} \right| = \left| \frac{76 - 72}{171} \right| = \frac{4}{171} $The ratio $x : y$ should be $\text{Diff}_Q(B) : \text{Diff}_P(B)$:
$ x : y = \frac{4}{171} : \frac{1}{133} $This yields the same simplified ratio $28 : 9$.
Final Conclusion
The mixtures from vessel P and vessel Q must be combined in the ratio $28 : 9$ to obtain the new mixture R containing Liquid A and Liquid B in the ratio $11 : 8$.
Important Questions from To Make a Mixture from Two Mixtures
In a mixture of 75 liters, the ratio of milk to water is 3 : 2. If the ratio is to be 1 : 2, how much of water should be added?
A mixture contains acid and alcohol in the ratio of 3 : 2. On adding 10 litres of alcohol in mixture, the ratio of acid to alcohol becomes 3 : 5. The quantity of acid (in litres) in the original mixture was:
There are two containers Xand Y. Xcontains 100 ml of milk and Ycontains 100 ml of water. 20 ml of milk from Xis transferred to Y. After mixing well, 20 ml of the mixture in Yis transferred back to X. If mdenotes the proportion of milk in Xand ndenotes the proportion of water in Y, then which one of the following is correct?
In a vessel, a mixture of milk and water is in ratio $9 : 5$, while in another vessel mixture of milk and water is in ratio $3 : 8$. In what ratio mixture of both the vessels should be mixed together so that in the resultant mixture ratio of milk and water becomes $13 : 19$?
30 litres of salt solution contains 5% salt. How many litres of water must be added so as to get a resulted solution containing 3% salt?
A. 20 litres
B. 25 litres
C. 30 litres
D. 35 litres
