Understanding the Problem
The goal is to determine the mixing ratio of two solutions containing milk and water to achieve a final mixture with equal proportions of milk and water (1:1 ratio).
- Initial Mixture 1 has milk and water in the ratio 3:5.
- Initial Mixture 2 has milk and water in the ratio 6:1.
- The desired final mixture has milk and water in the ratio 1:1.
Calculating Milk Proportions
Determine the fraction of milk in each mixture:
- Mixture 1: Total parts = $3 + 5 = 8$. Fraction of milk = $\frac{3}{8}$.
- Mixture 2: Total parts = $6 + 1 = 7$. Fraction of milk = $\frac{6}{7}$.
- Desired Final Mixture: Total parts = $1 + 1 = 2$. Fraction of milk = $\frac{1}{2}$.
Applying the Alligation Method
The rule of alligation helps find the ratio in which two ingredients must be mixed to obtain a desired mean value.
Let the quantities of Mixture 1 and Mixture 2 be $Q_1$ and $Q_2$, respectively. The ratio $\frac{Q_1}{Q_2}$ is found by:
$ \frac{Q_1}{Q_2} = \frac{|\text{Proportion of Milk in Mixture 2} - \text{Desired Proportion}|}{|\text{Proportion of Milk in Mixture 1} - \text{Desired Proportion}|} $Substitute the fractions:
$ \frac{Q_1}{Q_2} = \frac{|\frac{6}{7} - \frac{1}{2}|}{|\frac{3}{8} - \frac{1}{2}|} $Calculate the differences:
- Difference 1 (Mixture 2 vs Final): $|\frac{6}{7} - \frac{1}{2}| = |\frac{12}{14} - \frac{7}{14}| = \frac{5}{14}$
- Difference 2 (Mixture 1 vs Final): $|\frac{3}{8} - \frac{1}{2}| = |\frac{3}{8} - \frac{4}{8}| = |-\frac{1}{8}| = \frac{1}{8}$
Calculate the final ratio:
$ \frac{Q_1}{Q_2} = \frac{\frac{5}{14}}{\frac{1}{8}} $To simplify, multiply the numerator fraction by the reciprocal of the denominator fraction:
$ \frac{Q_1}{Q_2} = \frac{5}{14} \times \frac{8}{1} = \frac{40}{14} $Reduce the ratio to its simplest form by dividing both numbers by their greatest common divisor (2):
$ \frac{Q_1}{Q_2} = \frac{20}{7} $Therefore, the contents of the two glasses should be mixed in the ratio 20:7.
Important Questions from Mixture Problems
If the ratio of alcohol and water in a mixture of 85 litres is 11 ∶ 6. How much water should be added to make the ratio 5 ∶ 3?
Two bottles A and B contain diluted acid. In bottle A, the amount of water is double the amount of acid while in bottle B, the amount of acid is 3 times that of water. How much mixture(in litres) should be taken from each bottle A and B respectively in order to prepare 5 liters diluted acid containing an equal amount of acid and water?
A solution of milk and water contains milk and water in the ratio of 3 : 2. Another solution of milk and water contains milk and water in the ratio of 2 : 1. Forty litres of the first solution is mixed with 30 litre of the second solution. The ratio of milk and water in the resultant solution is:
A 70 litre mixture has liquids A and B in the ratio 5 ∶ 9. How many litres of liquid A must be added so that the ratio becomes 2 ∶ 3?
In a mixture of 60 litres, the ratio of milk and water is 2 : 1 respectively. How much more water must be added to make its ratio 1 : 2 respectively?
