Mixtures and Alligation: Method with Worked Examples

The mixture problems below cover one situation: combining two mixtures or two priced goods into one blend. Two tools solve them all, the balance method and the alligation cross, under four stated assumptions: quantities are positive, mixing is uniform so every part of the finished blend has the same composition, amounts and volumes add exactly, and all strengths in one problem use the same unit, such as percent or rupees per kilogram. Single-solution questions, such as plain dilution of one solution, are outside this article's scope.

Two methods for mixture problems

The balance method works for any number of ingredients: the total amount of a component before mixing equals the total after mixing. The alligation cross is a shortcut for exactly two ingredients whose mean strength lies strictly between theirs.

strength

the amount of component per unit of ingredient, such as percent of salt or price per kilogram

balance equation

(strength 1 x quantity 1) + (strength 2 x quantity 2) = mean strength x total quantity

alligation cross

for strengths a (higher) and b (lower) with mean m between them, the quantity at strength a : the quantity at strength b = (m - b) : (a - m)

mean strength

the strength of the finished mixture; with positive quantities it is a weighted average, so it lies between the two ingredient strengths

Worked examples

Example 1: two salt mixtures (balance method)

A blend is made from 10 kg of a mixture that is 30% salt by mass and 20 kg of a mixture that is 60% salt by mass. How much salt does the blend contain, and what is its concentration by mass?

1. Salt from the first mixture: 30% of 10 = 3 kg.
2. Salt from the second mixture: 60% of 20 = 12 kg.
3. Total salt: 3 + 12 = 15 kg, in 10 + 20 = 30 kg of blend.
4. Concentration: 15/30 = 0.5 = 50% by mass.

The blend contains 15 kg of salt and is 50% salt by mass.

Example 2: rice at two prices (alligation cross)

A grocer mixes rice at Rs 40 per kg with rice at Rs 60 per kg so the mixture has a mean price of Rs 48 per kg. In what ratio are the two kinds mixed?

1. The mean 48 lies strictly between 40 and 60, so the cross applies.
2. Share of the Rs 40 rice: 60 - 48 = 12. Share of the Rs 60 rice: 48 - 40 = 8.
3. Ratio: 12 : 8 = 3 : 2, cheaper kind to dearer kind.
4. Check: 3 kg at Rs 40 = Rs 120 and 2 kg at Rs 60 = Rs 120, so Rs 240 for 5 kg = Rs 48 per kg.

The kinds are mixed in the ratio 3 : 2.

Example 3: mixing two solutions to a target strength

How many litres of a 20% alcohol solution (alcohol by volume) must be mixed with 9 L of a 50% alcohol solution (alcohol by volume) to make a 30% solution by volume? Assume ideal mixing, so the volumes add exactly.

1. The mean 30 lies strictly between 20 and 50, so the cross applies.
2. Shares: the 20% solution takes 50 - 30 = 20; the 50% solution takes 30 - 20 = 10. Ratio 20 : 10 = 2 : 1.
3. With 9 L of the 50% solution, the 20% solution needs 2 x 9 = 18 L.
4. Check: alcohol = 20% of 18 + 50% of 9 = 3.6 + 4.5 = 8.1 L in 18 + 9 = 27 L, and 8.1/27 = 0.3 = 30%.

18 L of the 20% solution are needed.

Example 4: blending tea at two prices

A trader blends 4 kg of tea at Rs 240 per kg with 6 kg of tea at Rs 340 per kg. Find the mean price of the blend.

1. Cost of the first tea: 4 x 240 = Rs 960. Cost of the second: 6 x 340 = Rs 2,040.
2. Total cost: 960 + 2,040 = Rs 3,000, for 4 + 6 = 10 kg.
3. Mean price: 3,000/10 = Rs 300 per kg.
4. Check: 300 lies between 240 and 340; with more of the dearer tea, it sits above the midpoint 290, as expected.

The blend's mean price is Rs 300 per kg.

Summary

Balance method

component amounts add across the mixing; works for any number of ingredients

Alligation cross

exactly two ingredients; the higher-strength share is (mean - lower) and the lower-strength share is (higher - mean)

Weighted mean

mean strength = (total component amount or total cost) / total quantity; with positive quantities it lies between the two strengths

Cross-check

recompute the blend's component or cost from the answer and confirm the stated mean

Frequently asked questions

When does the alligation cross work?

Only for mixing exactly two ingredients to a mean strength strictly between the two, with uniform mixing and additive amounts. For three or more ingredients, use the balance equation instead.

Does the cross give ingredient quantities or component amounts?

Ingredient quantities. The differences (higher - mean) and (mean - lower) are the shares of the two ingredients, assigned crosswise: the lower-strength ingredient takes the (higher - mean) share.

Can one cross mix percentages with prices?

No. All three numbers in one cross must use the same unit, such as percent or rupees per kilogram. The method itself works for any strength measured per unit of ingredient.

What if there are three or more ingredients?

The alligation cross handles exactly two ingredients at a time. For three or more, write the balance equation: the sum of strength x quantity over all ingredients equals the mean strength x total quantity.

Can the mean strength lie outside the two ingredient strengths?

No. With positive quantities the mean is a weighted average of the two strengths, so it sits between them; if both strengths are equal, the mean is that same strength.

Sources and scope

Method checked against Wikipedia, Alligation, Purplemath, Mixture Word Problems, OpenStax, Elementary Algebra 2e, section 3.3, and GeeksforGeeks, Mixture and Alligation, all retrieved 15 November 2026. This article teaches arithmetic under stated assumptions only and makes no claim about any syllabus, board or class. Real blending can involve volume contraction or impurities; the examples are the stated model, not measurements.

Related reading on this site: how to solve maths word problems, the Pair of Linear Equations practice page (balance equations), and maths tuition at EZ Tuitions.