**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,

**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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