Allegation and mixture is a very common topic asked in the competitive exams. The topic contains the sums related to ratio and proportion and profit and loss. Today we will be discussing on mean price. The mean price questions are common on this topic.

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## Alligation Concept

When different quantities of different costs are mixed together to produce a mixture of a mean cost also known as average cost, then the ratio of the quantities is inversely proportional to the difference in their cost from the mean cost.

### The Rule of Alligation

The quantity of cheaper/ Quantity of dearer = Price of dearer – Mean price/ Mean price – Price of cheaper

Some of the points you need to remember while solving mean price in alligation and mixture:

1. The cost prices are considered in alligation and mixture.

2. Usually, the cost price of water is considered to be zero.

3. If it is given that the vessel contains ‘x’ liters of milk and if ‘y’ liters is to be withdrawn and replaced by water, and if ‘y’ liters of the mixture is withdrawn and replaced by water, and the operation is repeated for ‘n’ number of times, then

Milk left in vessel after nth operation / initial quantity of milk in vessel = (x – y)^{n} / x^{n}

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### Solved Example

1. A shopkeeper has 50 kg of rice, out of that he sells some at 8 % gain and the rest at 18 % gain. He gains 14 % in all. Find the amount of rice sold at 8 % profit.

A. 16 kg

B. 18 kg

C. 20 kg

D. 22 kg

Ans: Here before solving the question, we will divide the data given to us and based on it we will put the values into the equation.

Lower gain = 8% (cheaper)

Higher gain = 18% (dearer)

Overall gain = 14% (mean)

Now, the given question can be solved by two methods. We will get the answer by both these methods. One is formula based and the other is figure based. Both the methods are simple to remember.

Method 1:

Formula based

(Quantity of cheaper)/(Quantity of dearer) = (18 – 14)/(14 – 8) = 4/6 = 2/3.

In this method, you have just put the values into the above equation given to you. If not then you will be requiring the second method to solve the question.

Method 2:

In this method, you will be using find the ratio of the two values and use the cross-multiplication method. Further, in this method, the overall gain should be written in the middle of the figure. And the two values 18 and 8 on the upper side. The 14 in the middle is deducted from the two upper values. And the values obtained should be on the downward side. The figure will be as follows:

Thus, we have found 6: 4 as the required ratio. But 6 : 4 = 3 : 2. So, the rice was sold at 8 % profit was 20 kg. Thus, the correct answer is C.

### Examples of Gain and Loss

1. A shopkeeper has 50 kg of sugar, part of which he sells at 8% gain and the rest at 18%. He gains 14% on the whole process. Find the quantity of sugar sold at 18% profit.

A. 25 kg

B. 30 kg

C. 35 kg

D. 40 kg

As we have mentioned above in the important points always calculate the profit on cost price. Thus, in this question, we will start by assuming that the cost price of sugar per kg is Rs. x. Therefore, the price of the sugar sold at 8 % gain is 1.08x per kg. And the price of sugar sold at 18% gain is 1.18x per kg. Also, the mean price at which the sugar is sold is Rs. 1.14x per kg.

Therefore, putting values in the given equation the ratio of quantities will be,

(1.18x – 1.14x)/(1.14x – 1.08x) = 0.04x/0.06x = 2/3.

So, the quantity sold at 18% gain = 3/5(50) = 30 kg. Thus, the correct answer is B.

## Practice Questions

1. A businessman has 50 kg of rice, a part of which he sells at 10% profit and the rest at 5% loss. He gains 7% on the whole. Find the quantity of rice sold at 5% loss.

A. 10 kg

B. 15 kg

C. 22 kg

D. 25 kg

The correct answer is A.

2. 2 kg of sugar has the cost price of Rs. 4 should be mixed with how many kgs of sugar worth Rs. 5 kg to be sold at 5 Rs/kg gaining 100/9% profit?

A. 2 kg

B. 2.5 kg

C. 3 kg

D. 4 kg

The correct answer is A.

3. 30 kgs of rice costing Rs. 12/kg is mixed with some kgs of rice costing Rs. 18 to get the mixture costing Rs. 13.5. Find the quantity of rice costing Rs. 18.

A. 20 kg

B. 10 kg

C. 12 kg

D. 15 kg

The correct answer is B.

4. 100 kgs of wheat are sold in two parts. One at 5% profit and other at 15% profit. If there is an overall profit of 12%, find the quantity sold at 15% profit.

A. 30 kg

B. 40 kg

C. 70 kg

D. 80 kg

The correct answer is C.

5. A shopkeeper mixes 50 kg of salt worth of Rs. 60 per kg with 40 kg of another variety of slat to get a mixture costing Rs. 70 per kg. What is the price of the costlier variety?

A. Rs. 80/kg

B. Rs. 82.5/kg

C. Rs. 85/kg

D. Rs. 90/kg

The correct answer is B.

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