ALLEGATION AND MIXTURE

Key Concept

1. Allegation :

This is a rule which we can use to find the ratio in which we have to mix two or more types of ingredients to produce the mixture in a desired price. 

2. Mean price :

The cost price of the mixture per given unit is called mean price. 

3. Rule to find the ratio for producing mixtures :

To produce the mixture, the ratio in which the cheaper and dearer to be mixed is

=  (d - m) : (m - c) 

4. Say, a bucket has "x" units of liquid. From which "y" units of liquid taken out and replaced by water. After repeating this process "n" times,

Practice Problems

Problem 1 :

What is the ratio in which the two types of wheat mixed where the price of the first type is $9.30 per kg and the second type is $10.80 per kg so the mixture is having worth $10 per kg ? 

Solution :

From the given information, we have 

Cost price of the cheaper (c)  =  $9.30

Cost price of the dearer (d)  =  $10.80 

Cost price of the mixture (m)  =  $10

 Rule to find the ratio for producing mixture is

=  (d - m) : (m - c) 

So, we have

(d - m) : (m - c)  =  (10.8 - 10) : (10 - 9.3)

(d - m) : (m - c)  =  0.8 : 0.7

(d - m) : (m - c)  =  8 : 7 

So, the ratio in which the first kind and second kind to be mixed is 8 : 7.

Problem 2 :

Find the ratio in which, water to be mixed with milk to gain 20% by selling the mixture at cost price.

Solution :

Let us consider the water as cheaper and milk as dearer.

Let the cost price of 1 ltr of pure milk be $1 and water be $0.  

Now we take some quantity of milk (less than 1 ltr), add some water and make it to be 1 ltr milk-water mixture. 

Let "x" be the cost price in 1 ltr of the mixture.  

Because the gain is 20%, selling price of 1 ltr of the mixture is

=  x + 20% of x 

=  x +  (20x / 100)

=  (100x / 100)  +  (20x / 100)

=  (100x + 20x) / 100

=  120x / 100

=  6x / 5

Given : The mixture is sold at the cost price of pure milk.

That is, 

S.P of 1 ltr of mixture  =  C.P of 1 ltr of pure milk

6x / 5  =  1

Multiply  both sides by 5 / 6.

x  =  5 / 6

So, the cost price of 1 ltr of the mixture is $5/6.

Now, we can consider the following points.

Cheaper : Cost price of 1 ltr water (c)  =  $0

Dearer : Cost price of 1 ltr pure milk (d)  =  $1

Mean price : Cost price of 1 ltr mixture (m)  =  $5/6

Rule to find the ratio for producing mixture is 

=  (d - m) : (m - c)

Then, we have

(d - m) : (m - c)  =  (1 - 5/6) : (5/6 - 0)

(d - m) : (m - c)  =  1/6 : 5/6

(d - m) : (m - c)  =  1 : 5

So, water and milk  have to be mixed in the ratio to gain 20% is 1 : 5.

Problem 3 :

The milk and water in two vessels A and B are in the ratio 4:3 and 2:3 respectively. In what ratio, the liquids in vessels A and B be mixed to obtain a new mixture in vessel C consisting half milk and half water ?

Solution :

Let the cost price of 1 liter pure milk be $1.

Amount of milk in 1 liter of mixture in A  =  4/7 liter 

Amount of milk in 1 liter of mixture in B  =  2/5 liter 

Amount of milk in 1 liter of mixture in C  =  1/2 liter 

Let us assume the cost price of the liquid mixture in A as cheaper, B as dearer and C as mean price. 

C.P of 1 liter mixture in A (c)  =  $4/7 

C.P of 1 liter mixture in B (d)  =  $2/5 

C.P of 1 liter mixture in C (m)  =  $1/2 

Rule to find the ratio for producing mixture is 

=  (d - m) : (m - c)

So, we have

(d - m) : (m - c)  =  (2/5 - 1/2)  :  (1/2 - 4/7)

(d - m) : (m - c)  =  (4 - 5) / 10  :  (7 - 8) / 14

(d - m) : (m - c)  =  (-1) / 10  :  (-1) / 14

(d - m) : (m - c)  =  1 / 10  :  1 / 14

L.C.M of (10, 14) is 70.

So, multiply both the terms of the ratio (1/10 : 1/14) by 70.

(d - m) : (m - c)  =  70 / 10  :  70 / 14

(d - m) : (m - c)  =  7 : 5

So, the ratio in which the liquids in vessels A and B be mixed is 7 : 5.

Problem 4 :

A dealer mixes tea costing $6.92 per kg with tea costing $7.77 per kg and sells the mixture at $8.80 per kg and earns a profit of 17.5 % on his sale price. In what ratio does he mix them ?   

Solution :

Given : The mixture is sold at $8.80 per kg and 17.5% profit earned on the sale price. 

Profit  =  17.5% of 8.80

Profit  =  0.175 ⋅ 8.80

Profit  =  1.54

Then the cost price of of the mixture is 

=  Selling price - Profit

=  8.80 - 1.54

=  7.26

Let the cost price of tea costing $6.92 per kg as cheaper and $7.77 as dearer. 

Then the mean price is $7.26. 

More clearly, 

Cheaper (c)  =  6.92

Dearer (d)  =  7.77

Mean price (m)  =  7.26

Rule to find the ratio for producing mixture is 

=  (d - m) : (m - c)

So, we have

(d - m) : (m - c)  =  (7.77 - 7.26)  :  (7.26 - 6.92)

(d - m) : (m - c)  =  0.51 :  0.34

(d - m) : (m - c)  =  51  :  34

(d - m) : (m - c)  =  3 : 2

So, the required ratio is 3 : 2. 

Problem 5 :

A container has 10 liters of milk. 1 liter of liquid is taken out and replaced by water. If this processed is repeated thrice, find the amount of pure milk in the container. 

Solution :

The formula to find the amount of pure milk is, 

=  x ⋅ (1 - y/x)ⁿ

Here, 

x  =  10 ltrs

y  =  1 ltr

n  =  3

Then, we have 

=  10 ⋅ (1 - 1/10)³ 

=  10 ⋅ (10/10 - 1/10)³ 

=  10 ⋅ (9 / 10)³

=  10 ⋅ (0.9)³

=  10 ⋅ 0.729

=  7.29

So, the amount of pure milk in the container will be 7.29 liters. 

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