__Ratio and proportion__

__What is Ratio?__

A ratio is a relationship between two numbers by division of the same kind. The ration of a to b is written as a : b = a / b In ratio a : b , we can say that a as the **first term** or **antecedent** and b, the**second term** or **consequent**.

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__What is Proportion?__

The idea of proportions is that two ratio are equal.

If a: b = c: d, we write a: b: : c: d,

** Example:** 3 / 15 = 1 / 5

**a**and

**d**called

**extremes,**where as

**b**and

**c**called

**mean terms**

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__Example:__

If P : Q : R = 2 : 3 : 4 , Then P / Q : Q / R : R / P = ?

__Answer:__

P : Q : R = 2 : 3 : 4 . Let P = 2k, Q = 3k, R = 4k. Then,

= =, = = and = =.

SO, : : = : : = 8 : 9 : 24 .

__Example:__

If a : b = 3 : 7 and b : c = 5 : 9, find a : b : c .

__Answer:__** ** A : B = 3 : 7, B : C = 5 : 9

= ( 5 x 7 / 5 ) : ( 9 x 7 / 5 ) = 7 : 63 / 5 .

= A : B : C = 3 : 7 : 63 / 5 = 15 : 35 : 63 .

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__Example:__

If 75 is divided into three parts proportional to 3, 5, 8, 9 then the smallest part is :

__Answer:__

ratio is = 3 : 5 : 8 : 9, sum of ratio terms is = 25.

So the smallest part is = ( 75 x 3 / 25 ) = 9 .

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__Example:__

The Salary of Three friend A, B, C are divided into ratio 5 : 6 : 8.If the increment has given of 10%, 20%, 25%, Find the new ratio of three friend salary ?

__Answer:__

**Step 1:** We assume ration as 5x, 6x, 8x

now the increment of new salary is A = 110 / 100, B = 120 / 100, C = 125 / 100.

**Step 2:** A,s new salary is 110 X 5x / 100 = 55x / 10.

B,s new salary is 120 X 6x / 100 = 36x / 5.

C,s new salary is 125 X 8x / 100 = 10.

**Step 3:**New ratio is 55x /10: 36x / 5 : 10.

__Example:__

Find the mean proportional between given two number that is 64 and 49.

__Answer:__

The mean proportion of two numbers is

Root of 64 and 49 is √8 x √ 7 = 8 x 7 = 56.

**So, the mean proportional is 56.**

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** **

__Example:__

Which of the following ratio is the greatest ?

7 : 15 , 15 : 23 , 17 : 25 , 21 : 29

__Answer:__

7 / 15 = 0.466

15 / 23 = 0.652

17 / 25 = 0.68

21 / 29 = 0.724

**So, 0.724 is greatest and therefore, 21 : 29 **

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__Example:__

What number has to be added to each term of 3 : 5 to make the ratio 5 : 6 :

__Answer :__

Let the number to be added x , Then

3 + x / 5 + x = 5 / 6

6 ( 3 + x ) = 5 ( 5 + x )

x = ( 25 – 18 ) = 7

**So , the number to be added is 7 .**

__Example:__

A money bag contains 50 p, 25 p, and 10 p coins in the ratio 5 : 9 : 4, and the total amounting to Rs.206.

Find the individual number of coins of each type.

__Answer:____ __

**Step 1****:** Let the number of 50 p,25 p, and 10 p coins be 5x, 9x, 4x respectively.

Then, 5x / 2 + 9x / 4 + 4x / 10 = 206

= 50x + 45x + 8x = 4120= 103x = 4120

= x = 40.

**Step 2:** Number of 50 p coins is **( 5 x 40 = 200 ),**

Number of 25 p coins is**( 9 x 40 = 360 ),**

Number of 10 p coins **( 4 x 40 = 160 ),**

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__Example:__

In a bottle mixture of 80 liters and the ratio of milk and water is 3 : 2. If this mixture ratio is to be 2 : 3. What the quantity of water to be further added?

__Answer:__

**Step 1:** Quantity of Milk ( 80 x 3 / 5 ) = 48 liters, So Quantity of water in it ( 80 – 48 ) = 32 liters.

**Step 2:** New Ratio required 2: 3, Let x water to be added, Then Milk: Water is = 48: (32+x)

=48 / (32 + x).

Step 3: Now 48 / (32 + x) = 2: 3

48 / (32 + x) = 2 / 3

2x = 144 – 64

x = 80/2

=40 liters.

__Example:__

The ratio of copper and zinc is 9 : 3 in an alloy . If quantity of zinc is 24 kg. The quantity of copper is would be?

__Answer:__

ratio of copper and zinc is 9 : 3

Let the required quantity of copper be x kg.

So, 9: 3: 24: x

9x = 24 x 3

x = 72 / 9

x = 8 kg

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__Example:____ __In a liquid mixture of 60 liters, the ratio of milk and water is 2:1. If this ratio is to be 1: 2, then the quantity of water to be further added is:

__Answer:__

So Quantity of milk = (60 x 2 / 3) = 40 liters.

Water in it = (60 – 40) = 20 liters.

New ratio required = 1: 2.

Let quantity of water to be added further be x liters

milk: water = 40 / (20 + x)

Now,

40 / ( 20 + x ) = 1 / 2

20 + x = 80

x = 60 litres

So, 60 litres of water to be added further.

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__Example:__

20% alcohol present in a 15 litres mixture and rest of water . If 3 litres of water be mixed with it , the percentage of alcohol in the new mixture would be ?

__Answer:__

20% present in a 15 litres mixture

So, 20 x 15 / 100 = 3 litres .

Water in it = (15 – 3) 12 liters.

New quantity of mixture = (15 + 3) = 18 liters.

Percentage of alcohol in new mix is = (3 x 100 / 18) % = 50 / 3.

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