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
In a mixture of liquid ,1/5 part is acid 2/5 part is alcohol and the remaining part is water. If the total quantity of the mixture is 20 litres, then how much water (in litre) does the mixture contain?
In what ratio, should rice at ₹60 per kg be mixed with rice at ₹42 per kg such that by selling the mixture at ₹56 per kg there is a gain of 12%?
A vessel contains 20 litres containing milk and water in the ratio 3 : 2. Ten litres of this milk is removed and replaced with equal amount of pure milk. If this process is repeated once again, find the final ratio of milk and water.
From a container of 50 liters pure milk, 10 liters is taken out and replaced by 10 liters of water. If this process is repeated thrice, what is the ratio of water and milk finally?
Consider the following statements about a mixture and determine which of the statements is/are correct.
1. A mixture has a variable composition.
2. In compounds, the composition of each new substance is always fixed.
3. A mixture shows the properties of the constituent substances.
