In what ratio must a mixture of 15% spirit strength be mixed with that of 25% spirit strength to get a mixture of 21% spirit strength?
2 : 3
Solving Spirit Mixture Ratio Problems using Alligation
This problem asks for the ratio in which two spirit mixtures of different strengths must be combined to achieve a desired intermediate strength. This type of problem is efficiently solved using the Alligation method, also known as the rule of mixture or alligation alternate.
Understanding the Alligation Method
The Alligation method is a simple technique used to find the ratio in which two ingredients of different prices or strengths should be mixed to get a mixture of a desired price or strength. It involves finding the difference between the individual values and the mean value.
Applying Alligation to Spirit Strengths
We have two mixtures with spirit strengths of 15% and 25%. We want to obtain a final mixture with a spirit strength of 21%. Let's represent these values in an alligation diagram format:
| Ingredient 1 (Weaker) | Mean Value | Ingredient 2 (Stronger) |
|---|---|---|
| 15% | 21% | 25% |
Now, we find the absolute difference between the mean value and each ingredient's value. These differences are then placed diagonally opposite to the respective ingredient's value.
- Difference between the stronger strength (25%) and the mean strength (21%): \(25\% - 21\% = 4\%\). This difference corresponds to the proportion of the weaker strength (15%).
- Difference between the mean strength (21%) and the weaker strength (15%): \(21\% - 15\% = 6\%\). This difference corresponds to the proportion of the stronger strength (25%).
The ratio of the quantity of the weaker mixture to the quantity of the stronger mixture is given by the inverse ratio of these differences:
| Quantity of Weaker (15%) | Quantity of Stronger (25%) |
|---|---|
| Difference corresponding to stronger (25% - 21%) | Difference corresponding to weaker (21% - 15%) |
| 4 | 6 |
So, the ratio in which the 15% spirit mixture must be mixed with the 25% spirit mixture is \(4 : 6\).
Simplifying the Ratio
The ratio \(4 : 6\) can be simplified by dividing both parts by their greatest common divisor, which is 2.
\(4 \div 2 = 2\)
\(6 \div 2 = 3\)
The simplified ratio is \(2 : 3\).
Therefore, a mixture of 15% spirit strength must be mixed with a mixture of 25% spirit strength in the ratio of 2 : 3 to get a mixture of 21% spirit strength.
Revision Table: Spirit Mixture Ratios
| Initial Strengths | Desired Strength | Difference 1 (Mean - Weaker) | Difference 2 (Stronger - Mean) | Ratio (Weaker : Stronger) |
|---|---|---|---|---|
| 15%, 25% | 21% | \(21 - 15 = 6\) | \(25 - 21 = 4\) | \(4 : 6 = 2 : 3\) |
Additional Information on Mixture Problems
Mixture and Alligation problems are common in quantitative aptitude tests. They often involve:
- Mixing liquids with different concentrations (like spirit and water, milk and water).
- Mixing articles with different prices to sell the mixture at a certain price.
- Mixing different types of tea, coffee, etc., based on cost.
The Alligation rule is a shortcut for solving problems that could also be solved using weighted averages or algebraic equations. For example, if you mix \(x\) litres of the 15% solution and \(y\) litres of the 25% solution, the total amount of spirit in the mixture is \(0.15x + 0.25y\). The total volume of the mixture is \(x+y\). The final concentration is \(\frac{0.15x + 0.25y}{x+y} = 0.21\). Solving this equation for \(x/y\) would give the same ratio.
\(0.15x + 0.25y = 0.21(x+y)\)
\(0.15x + 0.25y = 0.21x + 0.21y\)
\(0.25y - 0.21y = 0.21x - 0.15x\)
\(0.04y = 0.06x\)
\(\frac{x}{y} = \frac{0.04}{0.06} = \frac{4}{6} = \frac{2}{3}\)
So, \(x : y = 2 : 3\), which is the same ratio found using the Alligation method. The Alligation method provides a visual and faster way to arrive at this ratio.
Important Questions from To Make a Mixture from Two Mixtures
- A and B are two alloys of gold and copper prepared by mixing the metals in the ratio of 7 : 2 and 7 : 11, respectively. Equal quantities of the alloys are melted to form a third alloy, C. If the amount of copper in C is 10 kg, then what is the amount of gold in C?
Two solutions A and B having sugar concentrations of 65% and 85% respectively are required to make a mixture. In what ratio these two solutions be mixed to get a mixture of sugar concentration 73%
In what ratio should cement costing Rs. 250 per bag be mixed with cement costing Rs. 325 per bag so that the cost of the mixture is Rs. 300 per bag. (A bag of cement is 50 kg).
A money lender finds that due to the fall in the annual rate of interest from 18% to 15%, his yearly income diminishes by Rs. 750. His capital is:
- There is a 120 L mixture of vinegar and water. The ratio of vinegar to water is 7 ∶ 5. A shopkeeper mixes a certain amount of water in order to make the ratio of vinegar to water 5 ∶ 6. The new quantity of water is what percentage of the original quantity of water in the mixture?
