The __ proportion definition__ page introduced the topic of ratio and proportion.

Such as how  2  ratios are classed as in proportion if they are equal to each other in value overall.

As an example:

The ratios   3 : 5   and   6 : 10   are in proportion.

“

**3**  is to  

**5**“      as      ”

**6**  is to  

**10**”

__or__

“

**3**  out of  

**5**”     is the same ratios as     “

**6**  out of  

**10**“.

In addition to this, there is also another concept to learn which is called  ”continued proportion”.

### What is Continued Proportion?

Three or more quantities are considered to be in continued proportion if the ratio between successive quantities is the same.

For example, the ratio    3 : 6 : 12    is in continued proportion.

**6**  is double  

**3**,  while 

**12**  is double  

**6**.

The ratio between successive quantities is the same.

“

**3**  is to  

**6**” ,  as  “

**6**  is to  

**12**”

### General Case:

So if we have  

these quantities will be in continued proportion if    a : b  =  b : c.

Using proportion notation this looks like   =>   

**a : b :: b : c**

Or if one was writing ratios in fraction form,

**a : b** = 

**b : c**   can be written as    \bf{\frac{a}{b}} = \bf{\frac{b}{c}}.

What is particularly important to note and remember is that when two ratios are in continued proportion,

then   a × c  =  b × b.

So   a × c  =  b^{2}.

When these conditions are present,  b  is known as the “mean proportional”.

More information on the mean proportional can be seen on the fractions cross multiplication page  __ here__.

__Examples    __

*(1.1) ***4 : 8 : 16**    is in continued proportion.

**4**×

**16**=

**64**    ,    

**8**×

**8**=

**64**

*(1.2) ***2 : 4 : 6**    is NOT in continued proportion.

**2**×

**6**=

**12**    ,    

**4**×

**4**=

**16**

**12**≠

**16**

## Continued Proportion, Further Examples

*(2.1) *Find the value of  

*t*  if the ratio    

**2 :**    is in continued proportion.

*t*: 32

*Solution*   So we have   

**2 :**.

*t*::*t*: 32Now,   

**2**×

**32** = 

*t*^{2}.

=>   

**64** = 

*t*^{2}      ,      

**√64** = 

*t*=>   

** = **

*t***8**

*(2.2) *Find the value of  

**s**  if the ratio    

**3 : 9 : s**    is in continued proportion.

*Solution*   We have   

**3 : 9 :: 9 : s**.

**3**×

**s** = 

**9**

^{2}.

=>   

**3s** = 

**9**

^{2}      ,      

**3s** = 

**81**

=>   

**s** =  \bf{\frac{81}{3}}       ,      

**s** = 

**27**

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