## Alligation and Mixtures

**Important terms:**

**1) Alligation:**Rule of alligation enables us to find the ratio in which two or more ingredients at a given price must be mixed to produce a resultant mixture of desired price.

**2) Mean Price:**It is the cost of unit quantity of a mixture.

**3) Mixture:**Mixture is formed by mixing two or more quantities. It can be expressed in the form of percentage or ratio.

- 10 % of sugar in water

- A solution of water and sugar is 12 : 20, which means water : sugar = 12 : 20

**According to the rule of alligation:**When two ingredients are mixed,

Cheaper Quantity (X) | = | (C.P. of Y) – (Mean Price) |

(Quantity of Y) | (Mean Price) – (C.P. of X) |

**If a vessel contains A liters of milk and if B litres of milk is withdrawn and replaced by water, and again if B litres of mixture is withdrawn and replaced by water and this operation is replaced n times in all, then**

(Quantity of milk left after n^{th} operation) | = | (A – B) | ^{n} | ||

(Initial quantity of milk) | A |

Quantity of milk left after n^{th} operation = A x | [A(1 – (B/A))^{n}] |

A |

^{th}operation = [A(1 – (B/A))

^{n}]

**Tricks and Tips:**

**1) Rule of alligation:**

Ratio = | (M – B) | = | (B – M) |

(A – M) | (M – A) |

**Points to Remember While Using the Rule of Alligation**

- The three values alligated should always represent the same variable and should have same units.

- alligation of 3 values of cost gives the ratio in terms of number and vice-versa.

- If two values of cost price and selling price of the mixture are given, then in such cases first calculate the cost price of the mixture and then allegate the 3 values of cost price.

- A and B represent concentration if the numerical is based on mixing of solutions.

**Question Variety**

Type : 1 Based on Rule of Alligation

**Examples:**

**Q 1.**In what ratio must wheat A at Rs. 10.50 per kg be mixed with wheat B at Rs. 12.30 per kg, so that the mixture be worth of Rs. 11 per kg?

a. 13 : 5

b. 18 : 3

c. 17 : 5

d. 11 : 5

View solution

Correct option :(a)**Convert Rs into paise, to make the calculation easy**

Ratio = | (B – M) |

(M – A) |

The required ratio = 130 : 50 = 13 : 5

**Q 2.**In what ratio must a shopkeeper mix Peas and Soybean of Rs. 16 and Rs. 25 per kg respectively, so as to obtain a mixture of Rs. 19.50 ?

a. 9 : 5

b. 7 : 5

c. 11 : 7

d. 12 : 8

View solution

Correct option: (c)**Use rule of alligation, to determine the ratio**

The required ratio of Soybean and Peas = 5.50 : 3.50 = 11 : 7