Two or more components are combined to create a new substance. When various quantities of different components are combined to generate a combination with a mean value, the ratio of their quantities is inversely proportional to the cost variations between the constituents. We can rapidly determine the cost of a combination by using an alignment rule. This rule allows us to blend two separate parts with varying costs.

## Mixture and Alligation Formula

Alligation, as the name indicates, is the process of determining the ratio of the ingredients/things that have been blended and the price at which they are sold to generate profit or lose money.

Knowing how alligation is used to obtain the mean value of a mix when the ratio and amount of the substances blended are varied is essential to solving mixture and alligation issues.

The main formula in mixture and alligation is:

Quantity of Cheaper ∕ Quantity of Dearer = CP of dearer – Mean Price ∕ MP – CP of cheaper

When we have an initial quantity of a pure element (such as petrol), and we keep replacing a fixed part of this pure liquid with another element (such as water), we use the following formula to determine the quantity of pure element after ‘n’ replacements:

P x [1 – (R / P)]n

where P denotes the starting concentration of the pure element,

R is the quantity replaced each time, and

n denotes the total number of replacements.

## Mixture and Alligation Questions

**Question 1:** From a vessel containing 20 litres of pure milk, 5 litres are removed and replaced with water to maintain a capacity of 20 litres. This procedure is performed five times more. Calculate the proportion of pure milk remaining in the vessel following five replacements.

**Suggestion:** In this case, we must use the formula P x [1 – (R / P)]n

P = 20 litres

R = Quantity replenished = 15 litre

n = 5

Thus, the quantity of pure milk after five substitutions is 20 x [1 – (15/ 20)]5

Quantity of pure milk following 5 substitutions = 20 x ¼ x ¼ x ¼ x ¼ x ¼ = 5/256

Thus, the proportion of pure milk remaining in the vessel following five replacements is calculated as 5 / 256 x 20 = 5/5120 x 100 = 500/5120 = 0.09765625% of pure milk.

**Question 2:** Thirty litres of a milk-water combination contain 10% water; how much water should be added to increase the water content to 25% in the new mixture? Calculate the volume of water that will be added.

**Solution:** The quantity of water in the mixture = 30 x 10/100 = 3 litres

Milk in the mixture = 30 x 90/100 = 27 litres

Let the water to be added be x.

Therefore, the water to be added will be (3 + x)/ (30 + x) = 25/100

100(3 + x) = 25(30 + x)

300 + 100x = 750 + 25x

75x = 450

x = 6

We will need to add 6 litres of water to the mixture.

## Alternate Formula for Mixture and Alligation

Where c is the CP of 1 unit of the cheaper element,

d is the CP of 1 unit of the dearer element,

M is the mean price,

d – m is the quantity of the cheaper element, and

m – c is the quantity of the dearer element.

For example, a total of 25000 candidates took a test. 60% of men and 40% of women passed the exams. If the overall percentage of students that passed the test is 55%, how many females took the exam?

So, 25,000 students appeared in the exam.

25,000 x ¼ = 6,250

The number of girls that took the exam is 6,250.

### Conclusion

A mixture is made up of two or more ingredients that combine to generate a third element. The combination questions are typically based on Ratio and Proportion and do not involve using any particular formulas.

Alligation is a mathematical method that helps us rapidly compute a mixture’s price when it consists of two parts with distinct costs. Alligation is the rule that enables us to determine the ratio in which two or more elements at a particular price must be combined to get the desired price mixture. The Mean Price of a unit amount of such a combination is the cost price of that unit quantity.

Bear in mind that the cost of the more expensive component > the cost of the mixture > the cost of the less expensive ingredient.