Alligation answers questions like 'in what ratio should rice at ₹40 and ₹65 per kg be mixed to sell at ₹50?'. Enter the two values and the target to get the ratio and the quantities of each.
Enter the cheaper and dearer values.
Enter the target mean and total quantity.
Read the ratio and quantities.
Ratio: Cheaper : Dearer = (d − m) : (m − c)
Alligation is a quick rule for finding the ratio in which two ingredients of different price or strength must be mixed to get a mixture of a desired mean. A shopkeeper blending two grades of rice or tea, a chemist diluting an acid, or a dairy adding water to milk all face this question.
This calculator takes the cheaper value, the dearer value, the target mean and the total quantity needed. It returns the mixing ratio and the quantity of each ingredient. It works for prices per kg, percentage concentrations, and any other quantity that averages by weight, such as marks or speeds weighted by time.
Mixture and alligation questions are common in SSC, banking and CAT-style aptitude tests, and the method is far faster than setting up equations.
1. Identify the cheaper value c, the dearer value d and the target mean m, with c < m < d.
2. Compute d − m and m − c.
3. The ratio of cheaper to dearer quantity is (d − m) : (m − c).
4. To split a total Q, cheaper quantity = Q × (d − m) ÷ (d − c).
5. Dearer quantity = Q − cheaper quantity.
6. Check: (cheaper qty × c + dearer qty × d) ÷ Q should equal m.
Suppose x units of the cheaper item and y units of the dearer item make a mixture worth m per unit. The total value is xc + yd, and it must equal (x + y)m. Rearranging, x(m − c) = y(d − m), so x : y = (d − m) : (m − c). Each ingredient's share is proportional to the distance of the other ingredient from the mean. The ingredient closer to the mean is needed in larger quantity, which matches intuition.
A weighted average of two numbers always lies between them. If the target is outside the range, no positive quantities can produce it, and the formula would give a negative ratio. If the target equals one of the values, the ratio puts all the weight on that ingredient. The calculator refuses a target outside the open range for this reason. For three or more ingredients, the answer is not unique and exam questions usually pair them up.
The rule works for any quantity that averages linearly by amount. For milk and water, treat water as 0% milk and milk as 100%. For acids, use percentage strength. When a question says a mixture is sold at a price that gives a certain profit, first find the cost price of the mixture by removing the profit, then apply alligation to cost prices. Using the selling price directly is a frequent error.
A tea merchant in Siliguri wants 30 kg of a blend costing ₹200 per kg, made from leaf that costs ₹180 per kg and a premium grade that costs ₹240 per kg.
Rule of alligation: Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper) = (240 − 200) : (200 − 180) = 40 : 20
Split the total: Cheaper = 30 × 40 ÷ 60 = 20 Dearer = 30 − 20 = 10
Answer: Mixing ratio (cheaper : dearer) 40 : 20; Quantity of cheaper item 20; Quantity of dearer item 10
Writing the ratio the wrong way round as (m − c) : (d − m) for cheaper to dearer.
Using a selling price that includes profit instead of the mixture's cost price.
Applying alligation to quantities that do not average linearly, such as successive percentages.
Choosing a target mean that is not between the two values.
Treating water as 100% instead of 0% in milk-and-water problems.
Blending grades of rice, tea, coffee or spices to hit a target price.
Diluting solutions in labs and pharmacies to a target concentration.
Solving aptitude exam questions on mixtures quickly.
Finding the split of two groups from an overall average, such as marks of boys and girls.
Does it work for milk and water?
Yes. Treat water as concentration 0% and milk as 100%.
Why must the mean be in between?
Mixing can only produce values between the two ingredients.