Alligation or Mixture
Alligation or Mixture - Aptitude Formulas
Definitions
Alligation: Alligation is a rule that enables us to find the ratio in which two or more ingredients at different prices must be mixed to produce a mixture at a given price.
Mixture: A mixture is formed by combining two or more ingredients in a certain ratio.
Basic Formulas
Alligation Rule
If two ingredients are mixed, one costing \(C_1\) per unit and the other costing \(C_2\) per unit, to form a mixture costing \(C_m\) per unit, the ratio of quantities of the two ingredients is given by:
\[ \text{Ratio} = \frac{C_2 - C_m}{C_m - C_1} \]Mixture Formula
If two quantities are mixed, \(Q_1\) of the first ingredient and \(Q_2\) of the second ingredient, the cost of the mixture is given by:
\[ \text{Cost of Mixture} = \frac{Q_1 \times C_1 + Q_2 \times C_2}{Q_1 + Q_2} \]Properties of Mixtures
Mean Price
The mean price of a mixture is the average cost per unit of the mixture. It is calculated using the total cost of the mixture divided by the total quantity:
\[ \text{Mean Price} = \frac{\text{Total Cost of Mixture}}{\text{Total Quantity of Mixture}} \]Replacement Rule
If a certain quantity of a mixture is replaced with one of the ingredients, the new concentration can be found using the formula:
\[ \text{New Concentration} = \left( 1 - \frac{\text{Quantity Removed}}{\text{Total Quantity}} \right) \times \text{Old Concentration} + \left( \frac{\text{Quantity Added}}{\text{Total Quantity}} \right) \times \text{Concentration of New Ingredient} \]Example Problems
Example 1: Finding the Ratio of Ingredients
If ingredient A costs $10 per unit and ingredient B costs $20 per unit, and the mixture costs $15 per unit, the ratio of the quantities of A and B is:
\[ \text{Ratio} = \frac{20 - 15}{15 - 10} = \frac{5}{5} = 1:1 \]Example 2: Finding the Cost of Mixture
If 3 units of ingredient A (costing $10 per unit) and 2 units of ingredient B (costing $20 per unit) are mixed, the cost of the mixture is:
\[ \text{Cost of Mixture} = \frac{3 \times 10 + 2 \times 20}{3 + 2} = \frac{30 + 40}{5} = \frac{70}{5} = 14 \]Example 3: Replacement Rule
If 2 liters of a 10% solution is replaced with a 20% solution in a 10-liter mixture, the new concentration is:
\[ \text{New Concentration} = \left( 1 - \frac{2}{10} \right) \times 10\% + \left( \frac{2}{10} \right) \times 20\% = 0.8 \times 10\% + 0.2 \times 20\% = 8\% + 4\% = 12\% \]Alligation or Mixture
Alligation, also known as mixture, is a method used to solve problems involving combinations of different substances with varying properties.
Types of Alligation
- Alligation Medial: Used to find the average or mean value of a mixture.
- Alligation Alternate: Used to determine the ratio of quantities for a specific property.
Example 1: Finding the Average Concentration
Suppose you have 4 liters of a 10% salt solution and 6 liters of a 30% salt solution. To find the concentration of salt in the resulting mixture:
Total amount of salt:
- For the 10% solution:
4 × 0.10 = 0.4 liters - For the 30% solution:
6 × 0.30 = 1.8 liters
Total salt and total volume:
- Total salt:
0.4 + 1.8 = 2.2 liters - Total volume:
4 + 6 = 10 liters
Mean concentration:
\[ \text{Mean Concentration} = \frac{2.2 \, \text{liters}}{10 \, \text{liters}} \times 100\% = 22\% \]
Example 2: Mixing Ratios
You want to mix two solutions, one with 15% concentration and another with 25% concentration, to get 20 liters of a solution with 20% concentration. Find the required quantities:
Let \(x\) be the amount of 15% solution and \(20 - x\) be the amount of 25% solution.
Set up the equation based on concentration:
\[ 0.15x + 0.25(20 - x) = 0.20 \times 20 \] \[ 0.15x + 5 - 0.25x = 4 \] \[ -0.10x = -1 \] \[ x = 10 \]
So, you need 10 liters of 15% solution and \(20 - 10 = 10\) liters of 25% solution.
Alligation Formulas
Alligation is a technique used to solve problems involving the mixing of two or more different quantities to find a mixture's properties. Here are the basic formulas and examples.
1. Alligation Medial
Used to find the mean or average value of a mixture.
The formula to find the mean value of a mixture is:
\[ \text{Mean Value} = \frac{\sum (\text{Quantity}_i \times \text{Value}_i)}{\sum \text{Quantity}_i} \]
2. Alligation Alternate
Used to find the ratio in which two or more quantities should be mixed to achieve a specific property.
The formula for finding the ratio is:
\[ \frac{\text{Quantity of first part}}{\text{Quantity of second part}} = \frac{\text{Difference between final value and second part's value}}{\text{Difference between first part's value and final value}} \]
Examples
Example 1: Alligation Medial
Find the mean concentration of a mixture of 5 liters of 10% solution and 10 liters of 20% solution.
Total amount of solute:
\[ \text{Amount from first solution} = 5 \times 0.10 = 0.5 \text{ liters} \] \[ \text{Amount from second solution} = 10 \times 0.20 = 2 \text{ liters} \] \[ \text{Total amount of solute} = 0.5 + 2 = 2.5 \text{ liters} \] \[ \text{Total volume} = 5 + 10 = 15 \text{ liters} \] \[ \text{Mean concentration} = \frac{2.5}{15} \times 100\% = 16.67\% \]
Example 2: Alligation Alternate
Mix two solutions with concentrations of 15% and 25% to get a 20% solution. Find the ratio of the two solutions needed.
Let \( x \) be the amount of 15% solution, and \( 100 - x \) be the amount of 25% solution.
Set up the equation based on the final concentration:
\[ 0.15x + 0.25(100 - x) = 0.20 \times 100 \] \[ 15x + 25(100 - x) = 20 \times 100 \] \[ 15x + 2500 - 25x = 2000 \] \[ -10x = -500 \] \[ x = 50 \]
Thus, the ratio of 15% solution to 25% solution is \( 50:50 \) or \( 1:1 \).