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?

For a continued proportion definition, 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**”

### Continued Proportion Definition,

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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