Mixture and Alligation: Formulas, Methods and Tricks
Mixture and Alligation deals with combining substances having different prices, quantities, concentrations or qualities. The topic uses weighted averages to find the mean value of a mixture and the alligation rule to find the required ratio of its components. Common applications include price mixtures, concentration problems, dilution and repeated replacement.
What is Mixture and Alligation?
For components with values a and b mixed in quantities x and y, the mean value is (ax + by) / (x + y). If the lower value is L, the higher value is H and the required mean is M, then the alligation ratio is lower-value quantity : higher-value quantity = (H - M) : (M - L), provided L < M < H.
Mixture and Alligation Formula & Tricks
Important Formulas
Multiply each component's quantity by its value, add the products, and divide by the total quantity.
The cross-differences give the required quantities. The mean value must lie between the two component values.
This form applies when a new component is added to an existing mixture to obtain a specified mean.
Here V is the initial volume, x is the volume removed and replaced each time, and n is the number of operations.
Use the same concentration unit for all components, such as percent or decimal form.
Quick Tricks
For two components, subtract the required mean from the higher value and the lower value from the required mean. Place these differences opposite their corresponding components.
The required mean must lie between the two component values. If it equals one component value, the other component quantity is zero; if it lies outside the range, a positive mixture ratio is not possible.
Convert quantities, prices and concentration values into compatible units before using a formula. The final ratio is unchanged if both quantities use the same unit.
If x/V of a mixture is removed each time, the fraction left after n replacements is (1 − x/V)ⁿ.
Mixture and Alligation Concepts
Weighted Average in Mixtures
For quantities q₁ and q₂ having values v₁ and v₂, mean value = (q₁v₁ + q₂v₂)/(q₁ + q₂). The same method applies to prices, concentrations, speeds represented as rates, and other measurable values.
Alligation Rule for Two Components
If L and H are the lower and higher values and M is the required mean, then the quantity ratio is L:H = (H − M):(M − L). The larger difference is placed against the component whose value is farther from the mean, so that the weighted average becomes M.
Finding Quantity or Price from a Known Mixture
Set total value before mixing equal to the sum of component values: q₁v₁ + q₂v₂ = (q₁ + q₂)M. If one quantity is unknown, substitute the known values and solve the linear equation.
Concentration and Dilution
Solute amount = concentration × volume. Therefore, final concentration = total solute amount / total volume. Adding pure solvent reduces concentration, while adding pure solute increases it.
Repeated Replacement of a Mixture
If V is the initial volume and x is removed each time, the fraction retained after one operation is (V − x)/V. After n operations, the original quantity remaining is V[(V − x)/V]ⁿ.
Mixture and Alligation Video Lessons
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Alligation Rule: Cross Method
Learn how to apply the rule of alligation using the cross method to compare mixture values and determine the required ratio of ingredients.
Practice Mixture and Alligation Questions
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Mixture and Alligation Quick Quiz
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Mixture and Alligation Revision Points
Use these formulas and checks for quick revision.
- Mean value = total value ÷ total quantity.
- For two components, quantity ratio = (higher value − mean):(mean − lower value).
- The required mean must lie between the two component values for a positive two-component mixture.
- In concentration questions, calculate solute quantity before calculating final concentration.
- For repeated replacement, remaining original quantity = V(1 − x/V)ⁿ.
- Use the same units for all quantities and values before applying a mixture formula.
- In a mixture with positive quantities, the mean lies strictly between the component values unless one component quantity is zero.
Mixture and Alligation FAQs
What is the mixture alligation formula?
Quantity of lower-value component : quantity of higher-value component = (Higher value − Mean value) : (Mean value − Lower value).
What ratio of ₹25 and ₹40 items gives a mixture worth ₹31 per kg?
Ratio = (40 − 31):(31 − 25) = 9:6 = 3:2. Thus, ₹25 items and ₹40 items are mixed in the ratio 3:2.
How do you calculate the mean price of a mixture?
Mean price = total cost ÷ total quantity. For 2 kg at ₹30 and 3 kg at ₹50, it is (2 × 30 + 3 × 50)/5 = ₹42 per kg.
Can alligation be used when the required mean is outside the component values?
No. A positive mixture of two components cannot have a mean below the lower value or above the higher value.
What is the formula for repeated replacement?
Remaining original substance = V(1 − x/V)ⁿ, where V is the initial volume, x is the volume replaced each time and n is the number of replacements.
How is the concentration of two solutions calculated after mixing?
Final concentration = (C₁V₁ + C₂V₂)/(V₁ + V₂). For 2 L of 10% and 3 L of 30%, it is (20 + 90)/5 = 22%.