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
A mixture of acid and water contains 20 percent acid. When 10 litres of water is added to the mixture, then the percentage of acid becomes 15 percent. What is the original quantity of mixture ?
A container contains 20 L mixture in which there is 10% sulphuric acid. Find the quantity of sulphuric acid to be added in it to make the solution to contain 25% sulphuric acid.
The ratio of milk to water in a 100 litres mixture is 2 ∶ 3. 10 litres of this mixture is withdrawn and replaced with milk. This process is repeated 2 more times, What is the percentage of milk in final mixture ?
- 80% and 90% pure acid solutions are mixed to obtain 20 litres of 87% pure acid solution. Find the quantity (in litres) of 80% pure acid solution taken to form the mixture.
Some fruits are bought at a rate of 11 for Rs. 100 and an equal number at a rate of 9 for Rs. 100. If all the fruits are sold at a rate of 10 for Rs. 100, then what is the gain or loss percent in the entire transaction?
